considerations for and implementations of deliberative and

Considerations for and Implementations of Deliberative andReactive Motion Planning Strategies for the Novel Actuated

Rimless Spoke Wheel Robot IMPASS in the Two-DimensionalSagittal Plane

Shawn C. Kimmel

Thesis submitted to the Faculty of theVirginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Sciencein

Mechanical Engineering

Dr. Dennis W. Hong, ChairDr. Alfred L. WicksDr. Robert L. West

May 30, 2008Blacksburg, Virginia

Keywords: Spoke, Wheel, Deliberative, Reactive, Robot, IMPASS, Autonomous, RoMeLa,Rimless, Mobility, Motion, PlanningCopyright 2008, Shawn C. Kimmel

Considerations for and Implementations of Deliberative and ReactiveMotion Planning Strategies for the Novel Actuated Rimless Spoke Wheel

Robot IMPASS in the Two-Dimensional Sagittal Plane

Shawn C. Kimmel

(ABSTRACT)

IMPASS is a novel spoke-wheel robot invented by researchers at the Robotics and Mecha-nisms Lab (RoMeLa) at Virginia Tech. The robot is driven by a rimless spoke wheel whichcan alter the length of any given spoke in the hub. This form of novel locomotion com-bines the efficiency of a wheeled robot and the mobility of a legged robot, arriving at a verypractical mobility platform. A highly mobile robot such as IMPASS could prove very valu-able in applications where the terrain is complex and dangerous, such as search and rescue,reconnaissance, or anti-terror response. A prototype has been constructed that effectivelydemonstrates the actuated spoke wheel concept using two wheels containing six spokes each.

Manually controlling the motion of two wheels and twelve spokes would be a daunting taskfor any operator. Due to this inherent complexity, automated motion control is a necessityfor the IMPASS platform. The work presented here will discuss two different approaches tothe motion planning problem for the two-dimensional sagittal plane. The first approach isdeliberative in nature and depends on fairly accurate terrain sensing. The motion planningfirst decides on a set of contact points based on obstacle configurations and a Lagrangianinterpolation of the terrain. A lower level motion planning component then executes themovements that guide the spoke ends to the contact points. The second motion planningapproach is reactive in nature. Proprioceptive and tactile sensors are used to determine therobot’s pose and immediate surroundings. These sensors directly affect the motion profile ofthe robot. The reactive approach follows much simpler logic, which theoretically will makeit more robust.

Motion planning strategies were tested in simulation and on the IMPASS prototype. Bothstrategies proved to be well suited for different applications. The deliberative control wasvery successful in a structured environment, whereas the reactive control was able to cross awider variety of terrain. The results from the testing also provided some insight into variablesintroduced by the hardware. Future improvements to the motion planning control includeaccounting for these variables in the hardware and eventually developing three-dimensionalmotion planning algorithms based on the lessons learned from the two-dimension case.

The author would like to thank the National Science Foundation (NSF) for their support ofthis project under grant No. IIS- 0535012.

Dedication

I would like to dedicate this work to my mother, who is the definition of strength andperseverance.

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Acknowledgments

The last two years have been a most enjoyable scholarly journey which would have beenimpossible without the friendship and guidance of my advisor Dr. Dennis Hong. He is avery perceptive and patient mentor, from whom I have learned much. Dr. Hong has createdan excellent research environment which reflects his love and dedication to the field of roboticsand mechanisms. To the other professors on my committee, thank you for your support andguidance through my Masters experience. Dr. Wicks, thank you for the invaluable stories,and barbecues and pizza when hours got long. Dr. West, thank you for the enlighteningconversations. I would also like to take this time to thank a professor not on my committee,for without his encouragement I would not be where I am: Dr. Reinholtz. Your passionfor knowledge and life is contagious. My gratitude goes out to Jesse Hurdus, an unofficialmember of my committee who I was able to bounce ideas at on a daily basis.

Within the student community, I would like to acknowledge first and foremost the seniordesign projects who worked tirelessly to create prototypes of the IMPASS robot. Withoutthem, this work would not be possible. In particular, I would like to acknowledge Eric Russelland Blake Jeans for their contributions to the IMPASS software in the Robot Driver, HighClimbing, and Reactive Motion Planning components. It has been a pleasure and a privilegeto be a part of the senior design teams. Thank you Sean Eggar for providing assistance intesting and building of the robots. Finally, I would like to thank my fellow graduate studentsin the Robotics and Mechanisms Lab (RoMeLa) at Virginia Tech. Especially, I would like tothank the lab members that worked on IMPASS with me: Ping Ren and Ya Wang. Thankyou both for the discussions and brainstorming sessions. In addition to these two, I wouldlike to thank the other individuals in RoMeLa who has helped me in one capacity or another:Brad, Eric, Derek, Gabe, Ivette, Joe, John, Karl, and Robert; the best of luck to you in yourresearch/ careers.

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Contents

1 Introduction 1

2 Background 4

2.1 Leg-Wheel Hybrids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1.1 Non-Integrated Leg-Wheel Hybrids . . . . . . . . . . . . . . . . . . . 5

2.1.2 Integrated Leg-Wheel Hybrids . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 IMPASS’s Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.4 Autonomous Robotic Control . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Platform Development 15

3.1 Spoke Actuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

3.2 Hub Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3 Hub Drive Train . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Body Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

4 Step Transitions 26

4.1 One-Point Contact Step Transitions . . . . . . . . . . . . . . . . . . . . . . . 27

4.1.1 Constant Angular Velocity of Hub . . . . . . . . . . . . . . . . . . . . 29

4.1.2 Constant Switching Angle of Spoke . . . . . . . . . . . . . . . . . . . 33

4.2 Two-Point Contact Step Transitions . . . . . . . . . . . . . . . . . . . . . . . 50

4.2.1 Experimenting with Two-Point Contact Transitions . . . . . . . . . . 52

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5 Motion Planning 53

5.1 Deliberative Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.1.1 Representation of the Environment and the Robot . . . . . . . . . . . 55

5.1.2 Defining Motion of the Robot . . . . . . . . . . . . . . . . . . . . . . 57

5.1.3 Initial Contact Point Selection (ICPS) . . . . . . . . . . . . . . . . . 58

5.1.4 Step Execution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

5.1.5 Experimenting with Deliberative ICPS Motion Planning . . . . . . . 75

5.2 Reactive Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6 Conclusion 80

6.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

Bibliography 84

A Deliberative Motion Planning Simulations 87

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List of Figures

1.1 Potential uses for an actuated spoke wheel robot. . . . . . . . . . . . . . . . 2

1.2 A prototype of the IMPASS actuated spoke wheel platform has been developedand used for motion planning research. . . . . . . . . . . . . . . . . . . . . . 3

1.3 Development of motion planning algorithms was conducted in a custom builtsimulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Non-integrated leg-wheel hybrid robots. . . . . . . . . . . . . . . . . . . . . . 5

2.2 High mobility planetary explorers that use non-integrated leg-wheel mechanisms. 6

2.3 Integrated leg-wheel hybrid robots. . . . . . . . . . . . . . . . . . . . . . . . 7

2.4 The integrated leg-wheel hybrid, Scout, uses an umbrella mechanism with onedegree of freedom to change the wheel diameter [1]. . . . . . . . . . . . . . . 8

2.5 Stop motion pictures of RHex climbing a set of stairs using “Ned’s ImprovedAscending Algorithm” (NIAA). Each picture depicts the state of the robotupon completion of one of the four stages. Shaded dots below each pictureindicate legs in contact with ground [2]. . . . . . . . . . . . . . . . . . . . . . 9

2.6 Mobility analysis of a single actuated spoke wheel describing degrees of free-dom for different configurations [3]. . . . . . . . . . . . . . . . . . . . . . . . 11

2.7 Geometry and coordinate system for a planar single actuated spoke wheel [3]. 12

2.8 Frame of reference for the kinematic model of the actuated spoke wheel [4]. . 12

2.9 Shakey was the first robot to preform autonomous operations by acting onsensor input [5]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

3.1 The IMPASS prototype climbing a set of stairs with ease. . . . . . . . . . . . 16

3.2 Potential mechanisms for spoke actuation. . . . . . . . . . . . . . . . . . . . 17

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3.3 The first iteration of the chain belt and sprocket design for the spokes ofIMPASS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

3.4 The final iteration of the chain belt and sprocket design for the spokes ofIMPASS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5 The hub driveshaft is connected to the hub by a plate pictured on the left.The three spoke motors attach to this hub plate. All the wires for the hubrun through the center of the shaft. . . . . . . . . . . . . . . . . . . . . . . . 19

3.6 Hub plates transmit torque through a set of teeth that mesh with the adjacenthub plate. The plates are held in compression by six screws that run throughhollow shafts around the edges of the hubs. . . . . . . . . . . . . . . . . . . . 21

3.7 The drive mechanism for the spokes is a belt and sprocket. Tension in thebelt is maintained with two coupled idler sprockets. . . . . . . . . . . . . . . 21

3.8 The hub drivetrains share their axis of rotation, but can rotate independently.A 90◦ degree gear box allows the motors to be positioned further back, helpingwith placement of the CG. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.9 The maximum torque experienced by the hub motors occurs when the robot isclimbing with a spoke that is extended to full length and oriented horizontally.The free-body diagram for this case is shown. . . . . . . . . . . . . . . . . . 23

3.10 This picture shows the new carbon fiber body for IMPASS side by side withthe old aluminum frame body. The new body saves on weight and is moreprotective of the interior components. . . . . . . . . . . . . . . . . . . . . . . 24

4.1 The two-point contact gait shown rotating clockwise through a step. Thisgait constrains the position of the hub center to a circle for which the radiusis the step length divided by

√3. The bar in the center of the robot’s spokes

is the center of the circular path, (xc, zc). [6]. . . . . . . . . . . . . . . . . . . 27

4.2 IMPASS in a two-point contact configuration, showing the velocity vector ~vand contact point vector ~dg. . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 The simulator was used to test algorithms before implementing them on therobot. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.4 The transition case for θ = 30◦. The front spokes touch the ground beforeθ = 0◦ because of compliance in the spokes and backlash in the hub gear train. 31

4.5 IMPASS’s switching angle can be described relative to the x-axis or the linenormal to the ground link of the terrain. These switching angles are referredto as θ and θ2 respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

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4.6 When in the two-point contact configuration, IMPASS’s legs make a trianglewith the ground link. The boundary conditions can be determined by the sumof the angles equal 180◦ and the value of β. . . . . . . . . . . . . . . . . . . . 34

4.7 This graph plots the spoke lengths IMPASS against each other as θ2 is changedin a 16 inch step. ~rA is the bottom axis and ~rB is at the left. The rectangularbox represents the robots physical limitations. For this 16 inch step, IMPASShas a continuous range in the solution space [7]. . . . . . . . . . . . . . . . . 35

4.8 Shown is the spoke lengths for IMPASS as θ2 is changed in a 20 inch step.| ~rBz| peaks first, followed by | ~rAz|The maximum and minimum spoke lengthsare shown as horizontal lines. The minimum θ2 is found at the vertical linewhere both spoke lengths are within the bounds. . . . . . . . . . . . . . . . . 36

4.9 There are infinite configurations in which IMPASS can transition with theone-point contact gait. This shows various configurations that are particularlyuseful from a motion planning perspective. . . . . . . . . . . . . . . . . . . . 37

4.10 These graphs plot the spoke lengths of IMPASS, ~rA and ~rB, for a given steplength. ~rA is the bottom axis and ~rB is at the left. The curve is createdby varying θ2 from −30◦ to 90◦, which rotates the robot through the fulltwo-point contact case. The dashed box represents the physical limitation forspoke lengths [7]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

4.11 IMPASS descending an obstacle with θ = 60◦. . . . . . . . . . . . . . . . . . 39

4.12 The left graph shows | ~rAz| and | ~rBz| with respect to θ for equal spoke lengths.The line that peaks first is | ~rBz|. The filled in region represents the area wherethe Descending Potential (DP)> 0 (i.e. | ~rBz| > | ~rBz|). The right graph showsthe DP, which reaches its maximum value at θ = 2∗π

3= 120◦. . . . . . . . . . 40

4.13 For any combination of spoke lengths, the greatest DP will always be achievedwith IMPASS at the ledge of an obstale and a vertical ground link. The circleshows the foot path as the robot rotates through all possible θ. . . . . . . . . 41

4.14 The left graph shows | ~rAz| at minimum spoke length and | ~rBz| at maximumspoke length with respect to θ. The filled in region represents the area wherethe Descending Potential (DP)> 0 (i.e. | ~rBz| > | ~rBz|). The right graph showsthe DP, which reaches its maximum value at θ = 2∗π

3= 120◦. . . . . . . . . . 42

4.15 The transition case for θ = 60◦. The front spokes touch the ground beforeθ = 0◦ because of compliance in the spokes and backlash in the hub gear train. 43

4.16 This figure shows the Non-Adjacent Ascending Transition configuration cho-sen for climbing large obstacles. θ is set equal to zero such that the back spokeis vertical, minimizing the effect of compliance in the spokes. . . . . . . . . . 46

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4.17 IMPASS climbing a 12 inch obstacle with the Adjacent Ascending Transition,θ = 0◦. The front spokes touch the ground before θ = 0◦ because of compliancein the spokes and backlash in the hub gear train. . . . . . . . . . . . . . . . 47

4.18 IMPASS climbing a 18 inch obstacle with the Non-Adjacent Ascending Tran-sition, θ = 0◦. The compliance of the spokes is noticeable in Figure 4.18(d) [8]. 48

4.19 The robot traversing moderate obstacles in the simulation environment withthe Default Transition, θ = 30◦. . . . . . . . . . . . . . . . . . . . . . . . . . 50

4.20 This graph plots the length of the back spoke against the length of the frontspoke during a two-point contact step. A dashed box has been included whichshows the maximum and minimum spoke lengths. This particular step lengthof 17 inches enters and leaves the solution space six distinct times. . . . . . . 51

4.21 The arcs seen in this figure shows the possible position of the hub center forthree consecutive steps. The points at which the arcs intersect is where thetransition must occur. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.22 IMPASS walking in the two-point contact gait. The transitions are determinedby the contact points. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

5.1 The software interfaces for controlling IMPASS. . . . . . . . . . . . . . . . . 53

5.2 The architecture for the Initial Contact Point Selection (ICPS) implementa-tion of the deliberative approach follows the sense-plan-act sequence. TheICPS method has three basic steps within the contact point planner: CriticalContact Region (CCR) identification and refinement, Critical Contact Point(CCP) determination, and Intermediary Contact Point (ICP) determination. 55

5.3 When navigating non-traversable segments, IMPASS must step in certain crit-ical contact regions (CCR) before and after the obstacle. These regions aregeometrically determined from the dimensions of the robot. . . . . . . . . . . 60

5.4 When navigating non-traversable segments, IMPASS must step in certain crit-ical contact regions (CCR) before and after the obstacle. These regions aregeometrically determined from the dimensions of the robot. . . . . . . . . . . 60

5.5 The critical contact regions (CCR) for obstacle i can include other non-traversable segments. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

5.6 The working contact regions (WCR) for obstacle i can include other non-traversable segments. The WCR may be significantly reduced from the criticalcontact region (CCR). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

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5.7 When challenged by certain obstacles, IMPASS will have multiple configura-tions available with which to traverse the obstacle. Some of these configura-tions will be more intelligent than others. . . . . . . . . . . . . . . . . . . . . 65

5.8 Working Contact Ranges (WCR) for multiple obstacles can be combined toreduce the number of steps required. This can reduce the WCR for futureobstacles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.9 The first step in the ICP selection algorithm is to estimate the number ofsteps that will need to be taken. This is done by calculating an average stepdistance dgavg based on the average height of the robot between hCCPi

andhCCPi+1

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

5.10 To find the roots for the Lagrange polynomial, n − 1 evenly spaced contactpoints are placed between CCPi and CCPi+1. These must be vertically ad-justed by ∆zk based on the difference in slope between the hub center line(z = m1x + z1) and the contact point line (z = m2x + z2) . . . . . . . . . . . 69

5.11 The vertically adjusted contact points (x′k, z′k) are used as the roots for a

Lagrange polynomial. The function is translated by hCCPiin the positive z

direction to be used as a hub center path. . . . . . . . . . . . . . . . . . . . 69

5.12 The first iteration of contact points for the Lagrange polynomial function,L(x), will most likely not converge on the end contact point, CCPi+1. Herethe first iteration exceeds the physical limitations of the robot. Accordingto algorithm 5.1.3, δz will be negative for every successive iteration until asolution is reached. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.13 IMPASS is shown here reaching a solution for the ICP of a given terrain andbounding CCP. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.14 It is possible to choose a hub center trajectory that is physically impossible.This will cause clipping, shown by the hashed line. . . . . . . . . . . . . . . . 73

5.15 It is possible to choose a hub center trajectory that is physically impossible.This will cause clipping, shown by the hashed line. . . . . . . . . . . . . . . . 74

5.16 A program is used to calculate IMPASS’s footpaths. These paths are used todetect future collisions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5.17 IMPASS traversing obstacles using the Deliberative ICPS Motion Planner. Aperfect terrain model was fed into the software to eliminate extraneous variables. 76

5.18 IMPASS traversing a dynamic terrain (a beanbag) using the reactive algorithmbased on inclinometer readings for the body position. . . . . . . . . . . . . . 79

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A.1 IMPASS climbing a set of three six inch steps and down an eleven inch obsta-cle. The Default Transition is used throughout most of the simulation withthe exception of the large negative obstacle. Here the Descending Transitionis used. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

A.2 IMPASS climbing a set of terrain features in simulation. CCR and CCP areshown as dots on the surface of the terrain. . . . . . . . . . . . . . . . . . . . 89

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List of Tables

5.1 Terrain Classifier Message. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Experimental Motion Profile Message. . . . . . . . . . . . . . . . . . . . . . 57

5.3 Improved Motion Profile Message. . . . . . . . . . . . . . . . . . . . . . . . . 58

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Chapter 1

Introduction

In the last decade, there has been a significant increase in the use of robots in real worldapplications. This usage will only increase in years to come, most likely at an acceleratedrate. As the demand for robotic solutions expands, so will demands on the physical abilities ofrobots. Currently one of the biggest weaknesses in robotic technology is mobility. Wheeledrobots tend to be efficient and capable of high speeds, but are often limited to relativelysmooth terrain. Legged robots on the other hand are better equipped to deal with irregularterrain. Unfortunately, legged robots are inherently more complex, often resulting in slowand inefficient operation.

Developing a highly mobile platform that is practical for real world applications has provendifficult. To achieve both mobility and speed, robots are being developed that include boththe leg and wheel concept into a single mechanism. These mechanisms tend to be simplerthan legged platforms, but far more mobile than wheeled platforms. Varying degrees ofleg-wheel integration have been experimented with. Some of these robots exhibit manydegrees of freedom while others are very simple. Existing leg-wheel hybrid platforms willbe discussed, as well as certain control strategies for these robots that are relevant to theplatform proposed in this paper.

The platform proposed in this paper is IMPASS (Intelligent Mobility Platform with ActiveSpoke System), a highly mobile leg-wheel hybrid. This robotic platform is based around arimless spoke wheel with individually actuated spokes. The spokes have the ability to increaseor decrease their length during operation. This results in a robot capable of traversing avariety of terrains, including obstacles up to 2

√3 times the nominal walking height. An

illustration of some potential capabilities of the actuated spoke wheel platform are shownin Figure 1.1. The robot would be able to maintain a consistent height over unstructuredterrains. Small obstacles and ditches could be easily stepped over, while larger ones climbed.Hazards such as water or vegetation could be kept below the undercarriage of the robot.

A prototype has been developed at RoMeLa (Robotics and Mechanisms Lab) of Virginia

1

Shawn C. Kimmel Chapter 1. Introduction 2

Figure 1.1: Potential uses for an actuated spoke wheel robot.

Tech, shown in Figure 1.2. This robot has two actuated spoke wheels and a body with a tail.Each wheel has six spokes radiating from the hub, which are actually composed of threelonger spokes that run through the middle of the hub. As a mechanical design problem,IMPASS offers many challenges due to the inherent complexity of individually actuatedspokes. Packaging the drive mechanisms for multiple spokes with all associated electronicsinside a reasonably sized hub required several design iterations. The lessons learned fromthis design will be discussed in the platform development chapter.

Manually controlling a robot with as many degrees of freedom as IMPASS would be adifficult task requiring a very skilled operator. This paper discusses approaches to automatedmotion control in the two-dimensional sagittal plane. The motion planning solutions fallinto two categories: deliberative and reactive. Deliberative control involves sensing theenvironment, building a world model from sensor data, and calculating an optimal pathwithin that world for the robot to travel. Reactive control has no planning component perse. It is a biologically inspired approach that focuses on sensor-actuator couplings and simplebehavior profiles. Both of the approaches have characteristics that make them attractive forIMPASS. Deliberative control tends to produce more a efficient and intelligent response.Reactive control results in less polished response, but because of its simplicity it tends to bemore robust in unstructured environments.

The algorithms presented in this paper were tested using a custom simulation package andthe IMPASS prototype. The simulator, shown in Figure 1.3, provided a means of checkinggeometry constraints and terrain collisions. Experiments on the robot allowed researchers tocomprehend the physical interactions of the robot with its environment. Results from theseexperiments indicated that the theoretical motion planning algorithms tuned in simulationwere initially inadequate. The main differences between reality and simulation were a productof the dynamic properties of the hardware. In particular the compliance of the spokes, geartrain backlash, and rolling contact point of the feet affected motion of the robot. Motionalgorithms were revised to incorporate many of these unforseen interactions, resulting in a

Shawn C. Kimmel Chapter 1. Introduction 3

Figure 1.2: A prototype of the IMPASS actuated spoke wheel platform has been developedand used for motion planning research.

Figure 1.3: Development of motion planning algorithms was conducted in a custom builtsimulator.

much more capable motion planner. With the revised motion planning algorithms, IMPASSis able to climb obstacles more than 1.5 times its nominal height and descend obstaclesgreater than the nominal walking height.

The IMPASS platform is a valuable addition to the world of mobile robotics. Potentialapplications for this robot range from search and rescue to anti-terror response to planetaryexploration. Future work that would greatly benefit IMPASS include a more complete per-ception suite, three-dimensional kinematic analysis, and eventually three-dimensional motionplanning algorithms.

Chapter 2

Background

While the implementation of the hybrid leg-wheel on IMPASS is novel, the concept of com-bining legs and wheels has been around for decades. The goal of this hybrid architecture isto mesh the advantages from both legged and wheeled mechanisms without inheriting thelimitations. The wheel is a very efficient and proven form of locomotion, but it lacks theability to traverse a variety of terrain. Legs on the other hand provide a great deal of mo-bility, but at the cost of efficiency. Legs possess greater degrees of freedom than wheels andthey are able to touch the ground at discrete contact points. Combining legs and wheels inrobotic locomotion is an excellent way to broaden the range of practical applications. Thispaper provides a summary of existing robots that fit into the leg-wheel hybrid classification,and their range of applications.

The motion planning for leg-wheel hybrid robots is a area of much research, since there aremany different ways to mediate leg and wheel functionalities. Traversing obstacles is themost heavily researched area within motion planning, since mobility is the main advantageof the leg-wheel hybrid concept. IMPASS has more degrees of freedom than most leg-wheelhybrids giving it more flexibility, but also more complexity. Motion planning for variousleg-wheel hybrids will be discussed, focusing on obstacle traversal.

Autonomous control of any robot in unstructured terrain provides many challenges. First ofall, accurate localization and terrain sensing can be difficult to achieve. Secondly, unforseencircumstances may arise requiring robust programming, for example loose terrain can shiftresulting in the inability to execute a planned motion. Intelligent traversal of unstructuredterrain can take the shape of deliberative planned maneuvers, reactive sensor-action cou-plings, or any combination of the two. Presented is an overview of various control methodsfor robots in unstructured terrains.

4

Shawn C. Kimmel Chapter 2. Background 5

(a) Chariot uses a mechanicallyseparated leg-wheel mechanismand is designed for exploration inforestry applications [9].

(b) Workpartner is a service robotbuilt on a leg-wheel hybrid plat-form [10].

(c) PAW uses four wheeled ap-pendages to traverse terrain byrolling, walking, or galloping [11].

Figure 2.1: Non-integrated leg-wheel hybrid robots.

2.1 Leg-Wheel Hybrids

The concept of combining legs and wheels for robot locomotion is not new. Many differ-ent configurations have been developed, some with much success. Applications of leg-wheelhybrids include planetary exploration, reconnaissance, search and rescue, and anti-terror re-sponse. The common thread among these applications is the need to operate in unstructuredterrain that is too dangerous, dirty, or dull for humans to operate in.

There are two basic classifications of leg-wheel hybrids that will be discussed in this section:non-integrated leg-wheels and integrated leg-wheels. Some robots use a combination of in-dividually distinguishable legs and wheels to accomplish their locomotion, hereafter referredto as non-integrated hybrids. Other leg-wheel hybrids actually integrate the legs into thewheels structure to enable reconfiguring of the wheel. The non-integrated hybrids tend tobe able to support more weight and move quicker across relatively flat terrain, whereas theintegrated hybrids tend to be better at traversing rougher terrains with larger obstacles.

2.1.1 Non-Integrated Leg-Wheel Hybrids

There are many examples of non-integrated leg-wheel robots. The Chariot robot developedin Tohoku University in Japan has mechanically separated wheels and legs, shown in Fig-ure 2.1(a) [9]. Chariot’s locomotion can take the form of rolling, walking, or a hybrid ofthe two. Uses for this robot include exploration of uncharted terrain, especially for forestryapplications.

PAW, shown in Figure 2.1(c), is another non-integrated leg-wheel hybrid. This robot hasfour appendages with a drivable wheel fixed to the end of each appendage [12]. Locomotionfor this robot can take the form of rolling, bouncing, or galloping [11, 13]. A similar robot,the WorkPartner, was developed as a service robot for humans [10]. Shown in Figure 2.1(b),


(a) JPL’s Sample Return Rover (SRR) useswheeled appendages for crossing unstructuredterrain [15].

(b) JPL’s Athlete uses wheeled appendages forcrossing unstructured terrain [16].

Figure 2.2: High mobility planetary explorers that use non-integrated leg-wheel mechanisms.

the WorkPartner robot has four appendages with drivable wheels attached to the ends.However, this robot can use each of the appendages very similar to legs when the wheels arestopped, achieving a walking gait [14]. JPL’s Sample Return Rover (SRR), Figure 2.2(a),and Athlete, Figure 2.2(b), are robots that are capable of traversing rough terrain and havebeen designed for designed for planetary exploration [15].

Non-integrated leg-wheel hybrids demonstrate good efficiency while still providing a greatdeal of mobility. However, these designs must usually transition between rolling and walk-ing stages. This transition is effectively alternating between continuous and discontinuousground contact. To achieve a more fluid gait across rough terrain with equal or greatermobility, the legs can be integrated into the physical structure of a wheel.

2.1.2 Integrated Leg-Wheel Hybrids

Integrated leg-wheel hybrids can take many forms. One important distinction is whetherthe wheel has a rim or not. There are an abundance of rimless leg-wheel hybrids, also calledrimless spoke wheels, but not so many that retain the rim of the wheel. One robot developedat Delaware University successfully showed a single degree of freedom leg-wheel hybrid witha rim, shown in Figure 2.3(a) [17, 18]. The wheel inscribes a polyhedral that can expandwith a single degree of freedom to change the wheel diameter.

The advantage of a rimless leg-wheel is the ability to traverse the terrain at discrete contactpoints without the inefficiencies associated with the majority of legged robots. Discretecontact points prove to be effective for dealing with discontinuities in the terrain. One robotthat has achieved excellent speed and mobility with rimless spoke wheels is Whegs seenin Figure 2.3(b). Developed at Case Western University, Whegs uses three pairs of wheels


(a) The Expanding Wheel robotuses an inscribed polyhedral mech-anism with one degree of freedomto change the wheel diameter [18].

(b) Whegs uses an alternatingtripedal gait to cross terrainquickly. The robot can traverse ob-stacles of up to 1.5 times the spokelength [19].

(c) RHex is a highly mobile leg-wheel hybrid with the capability ofclimbing 0.2 meter steps [20]

Figure 2.3: Integrated leg-wheel hybrid robots.

with each wheel having three rigid spokes. It walks using a biologically inspired alternatingtripedal gait, similar to that of many insects including cockroaches and ladybugs. Whenclimbing obstacles the front two axles passively align using a compliant axel to help pull theup the body. The robot is able to climb obstacles of up to 1.5 times the spoke length andjumping version of the robot can jump up to 2.5 times the spoke length [21]. This platformis capable of high speeds, but has a relatively high impact gait.

Another rimless spoke wheel robot similar to Whegs is a robot named RHex. Shown inFigure 2.3(c), RHex has three pairs of rimless spoke wheels with only one spoke per wheel.The semi-circular spoke shape has been optimized for traversing discontinuous terrain fea-tures [22]. The round shape of the foot allows the hip joint to remain close to the groundcontact point, which minimizes motor torque requirements. The compliance of the spokes isa natural shock absorber and transfers energy from step to step. RHex can climb a varietyof step sizes, reportedly up to 0.2 meters in height at a rate of 1.55 s/step.

Both of the previously discussed robots, RHex and Whegs, only actuate the rotation of theleg-wheel. Adding degrees of freedom to the legs can add greater mobility and capability.Scout is a robot with a single degree of freedom spoke actuation mechanism. Scout has anumbrella like mechanism that changes the diameter of the wheel. This robot is shown inFigure 2.4 [1].

The above mentioned robots all have similarities to IMPASS in that they are integratedleg-wheel hybrids. What makes IMPASS unique is that it can independently actuate eachspoke, with the capability of allowing any spoke to reach twice its nominal length. This addsa great deal of mobility allowing IMPASS to climb obstacles 2

√3 times the nominal walking

height- a larger step than any of the previously mentioned robots. It is a more complicateddesign, but as will be shown in this paper it is possible to create a robust mechanical platform


Figure 2.4: The integrated leg-wheel hybrid, Scout, uses an umbrella mechanism with onedegree of freedom to change the wheel diameter [1].

and motion planning algorithms that demonstrate the IMPASS concept.

2.2 Motion Planning

Leg-wheel hybrid robots have been developed to improve upon the mobility of wheeled robots.By having the capability to traverse the ground with discontinuous contact points, the robotsare able to overcome otherwise non-traversable segments of the terrain. Crossing complexterrain requires special consideration for the motion of the robot. Each robot is unique inits hybridization of legs and wheels. Therefore, the motion planning that best controls aleg-wheel hybrid robot tends to be unique. The motion planning for several robots that aresimilar to IMPASS will be discussed, as well as the lessons that can be applied to motionplanning algorithms for IMPASS.

Perhaps the most similar in shape to IMPASS of all the leg-wheel hybrids is the six spokedrimless wheel discussed in [23,24]. The six spoked rimless wheel has been proposed as a modelfor passive-dynamic biped gaits. When two six spoked hubs attached to the same axel rotate30 degrees out of phase such that they alternate contact points, there are significant parallelswith human walking. Work done with these rimless wheels has shown steady state passiverolling of the wheel down a slope. The wheel reaches an average speed and a steady stateperiodic motion when descending any slope [24]. The steady state is achieved when thekinetic energy gained from falling equals the energy lost in collision of a spoke. This modelcould be helpful in modeling IMPASS down a ramp. However, not much work has been donewith obstacle traversal for these rimless spoke wheels.

The period motion achieved by the unactuated rimless spoke wheel while descending a slope


Figure 2.5: Stop motion pictures of RHex climbing a set of stairs using “Ned’s ImprovedAscending Algorithm” (NIAA). Each picture depicts the state of the robot upon completionof one of the four stages. Shaded dots below each picture indicate legs in contact withground [2].

is something that IMPASS could also achieve if none of the spokes change length and thetail slid without friction behind the hubs. However, IMPASS has the advantage of beingable to actuate the spokes. This added freedom can be used to minimize the collision lossesdiscussed in [23] and create a smoother gait. Collision losses are reduced by extending thenext contact spoke in advance to touch the ground before the robot gains excess verticalmomentum. Smoother walking is accomplished by having the contacting spoke alter itslength during a step, achieving a more consistent vertical velocity of the robot. The smoothdecent for IMPASS will actually be closer to the descent of a wheel.

The Whegs robot is a similar rimless spoke wheel with only three spokes per hub. Theproposed gait for this robot is similar to the six spoked rimless wheel, in that the hubsalternate contacting the ground. The Whegs robot prototype with six wheels walks with analternating tripedal gait. This gait works efficiently and effectively in terrain without largeobstacles. However when the robot encounters a larger obstacle that it must climb onto,both front hubs must share the lifting burden on the front legs. To accomplish this, theWhegs robot was designed with passive-compliant axels that allow the front legs to alignunder heavy loads. The robot can then lift itself onto the obstacle and return to the regularalternating tripedal gait. This passive alignment of the hubs is an effective way to balance


operation in different settings with minimal complexity.

RHex is a very robust mobility platform that uses six rimless spoke wheels to traverseunstructured terrain. This platform has been shown to cross unobstructed terrain relativelyquickly. RHex also has the ability to climb successive steps of up to 0.2 meters, whichis relatively close to the nominal walking height. Climbing with a six hub robot requiresdelicate coordination of the spokes. In his thesis, Ned Moore describes an algorithm forclimbing steps of various dimensions [2]. The three pairs of rimless spoke wheels operate ina back to front wave gait, with at least two pairs of hubs always in contact with the ground.The ascending algorithm is broken down into four phases with each phase being defined bya duration and a final position for each leg. Figure 2.5 shows the legs in contact with theground at each phase of the climb. This approach would be useful for an IMPASS designwith multiple pairs of hubs.

The spoke wheeled robots already discussed have dealt with rotation control. The controls inthese robots is mostly focused on the position of contact points and pose of the legs. BecauseIMPASS can independently actuate its spokes, the motion planning problem becomes morecomplex. The length of the spokes can be changed, which alters the current and futureposition of the body. One way of approaching motion planning for IMPASS is to focus on theposition of the body, more specifically the center of gravity (CG). When focusing on the CG,the problem becomes how to resolve a path. There are a plethora of optimization techniquesfor cost based optimization. The Path Planning chapter will discuss some optimizationtechniques that can be applied to IMPASS.

2.3 IMPASS’s Kinematics

The kinematic analysis for IMPASS’s novel spoke wheel mechanism are described previouslyin [3,4] by Laney. For the purposes of this paper, it will be assumed that the spokes for bothhubs are aligned and synchronized. We will also assume that there is no slip at the contactpoints, allowing them to be modeled as revolute joints. For the given hardware design of sixspokes separated by 60◦, the robot can contact the ground with one, two, or three spokes.Each configuration has different mobility characteristics. The mobility of each case can bedetermined using Grubler’s equation, given by

M = 3(n− 1)− 2f1 − f2 (2.1)

where M is the number of relative degrees of freedom of the hub with respect to the ground,n is the number of links, f1 is the number of 1 degree of freedom (DOF) joints, and f2 isthe number of 2 DOF joints. From this equation, it can be determined that the one-pointcontact case has two DOF. This can be visualized by the hub rotating about the contactpoint while the spoke changes length. The two-point contact case has a single DOF, the


Figure 2.6: Mobility analysis of a single actuated spoke wheel describing degrees of freedomfor different configurations [3].

motion of which will be discussed in a later section. Finally, the three-point contact case haszero DOF. The mobility analysis for IMPASS is summed up in Figure 2.6. A more completekinematic analysis which takes into account asynchronous spoke movement for the two hubsis presented in [6], where the robot is treated as a mechanism with variable topology (MVT).

The geometry and coordinate system must be defined. Figure 2.8 depicts the variables andcoordinates with which IMPASS is defined. β is the angle that separates the spokes, whichfor the current prototype is fixed at 60 degrees. The six spokes are actually three fixed lengthspokes, all measuring l, that travel through the center of the hub. Therefore, by definingthe length of three spokes we know the length of all the spokes. The convention for definingthe spoke lengths dictates that the back contact point spoke length from A to O is rA. Thelengths for the two spokes counter clockwise from this contact spoke are referred to as rB

and rC , in that order. The lengths of any remaining spokes can be calculated as l minus theeffective length of the spoke located 180 degrees around the hub.

The coordinate system for IMPASS is described in Figure 2.8. It first consists of a Newtonianframe attached to the terrain, referred to as the N -frame. This frame is denoted by theinertially fixed X and Z axis in Figure 2.8. There is also a path frame, P -frame, rotatedabout angle φ to align axis p1 in the direction the robot is heading. For the scope of thispaper, the P -frame will always be aligned with the N -frame. The I -frame is an intermediateframe rotated about p2 (the contact point connecting line) by angle θ. The robot revolvesabout the plane defined by this frame. The W -frame is fixed to the wheels, and is createdby rotating by the angle ψ about the I1 axis. ψ is non-zero when the left and right spokesare of unequal length (rAL 6= rAR). Finally, the body frame, is fixed to the chassis of therobot and is rotated from the w2 axis by -θ. These frames of reference will be used to discussthe robot in the following chapters.


Figure 2.7: Geometry and coordinate system for a planar single actuated spoke wheel [3].

Figure 2.8: Frame of reference for the kinematic model of the actuated spoke wheel [4].


2.4 Autonomous Robotic Control

Control of autonomous robots has come a long way in the last four decades. The first con-trol approach was the Deliberative Paradigm, in which the robot follows a sense-plan-actsequence. Shakey, developed at Stanford Research Institute, was the first robot to exhibitthe Deliberative Paradigm [25]. Figure 2.9 shows Shakey in a typical testing environment. Arobot takes in sensor data to build a world map and/or locate itself in that map. Decisionsare then made based on the world map combined with a-priori knowledge of the environ-ment. Finally, the robot executes the plan based on its actuation scheme. The DeliberativeParadigm has the advantage of being able to use a large amount of information to makedecisions with.

The Deliberative Paradigm has its limitations. It works well with a closed-world assumptionwhere there are no unpredictable situations, and when large amounts of computing powerare available. However, the closed-world assumption is not very practical for real worldapplications, and until recently computing power has come at a premium.

In the search for a more robust solution, Rodney Brooks kicked off a fundamental shiftin artificial intelligence (AI) theory in 1986. This shift was towards biologically inspiredsolutions. After all, some of the most simple creatures on earth are capable of preformingsurprisingly complex tasks. At the most basic level, animals operate with reflexes thatare essentially sense-act couplings. The Reactive Paradigm was born out of the biologicallyinspired sense-act coupling. This paradigm dictates that a change in sensor state can directlyelicit a change in state of the robot.

At a higher level, a change in the robot’s state may be described as a behavior. For example,when a centipede is poked, it enters a defensive behavior in which it curls up into a spiral.So the question is begged: “How are behaviors selected?” In 1986, Brooks introduced theSubsumption Architecture in which multiple layers of behaviors are chosen from by a systemof inhibition and suppression [26]. Brooks noticed that with careful tuning of the behaviornetwork, emergent behaviors would arise giving the impression of a higher level of intelligence[27]. However, the Subsumption Architecture has limitations in its modularity. The lowerlayers function based on the activity of the higher layers, making it difficult to change eitherone independently.

Another approach to behavior selection is Ron Arkin’s potential fields method [28, 29]. Inthis approach, each behavior will simultaneously produce a vector that describes the desiredmotion for that behavior. For example, “obstacle avoidance” would produce a vector normalto any obstacle surfaces and “goal achievement” would direct the robot towards the goal.The vectors from all these behaviors are summed, resulting in the final vector to be carriedout by the robot. Problems with potential fields include trap situations where the robot getsstuck, inability to pass between closely spaced obstacles, and oscillations down hallways orbetween obstacles [30].


Figure 2.9: Shakey was the first robot to preform autonomous operations by acting on sensorinput [5].

Both Deliberative and Reactive control paradigms have their limitations. To cope withthese limitations, hybrid architectures exist that can combine the two control methods. Oneexample is the Three Layer Architecture that has a controller, sequencer, and deliberator [31].The deliberator takes care of intensive computations and path planning. The controllerexecutes the commands from the deliberator as defined by the sequencer. The sequencerselects a transfer function (essentially a behavior) to apply information from the deliberatorto the controller. The hybridization of control paradigms opens the doors for more advancedand robust operation of autonomous agents.

Deliberative, behavioral, and hybrid control strategies can all be practical depending on thesituation. Due to IMPASS’s wide breadth of applications, one overarching approach maynot be appropriate. This dilemma will be discussed in this paper.

Chapter 3

Platform Development

IMPASS’s actuated spoke wheel concept has been realized in a prototype built in theRobotics and Mechanisms Lab (RoMeLa) at Virginia Tech. Design iterations began in2004, mainly focusing on number of hubs, number of spokes, and spoke actuation method.After these basic elements were finalized, the design focus shifted to packaging of the hubsand body design. The current prototype can be seen in Figure 3.1.

The prototype has two actuated spoke wheels that rotate about the same axis, but areconnected to independent drive trains to allow differential turning. A single pair of hubswas chosen as opposed to multiple pairs because multiple pairs introduce more complexityto the robot and do not significantly increase mobility. In order to be able to turn the hubs,a tail is required to provide a reaction force that opposes the natural tendency for the bodyto rotate freely.

Each hub has six spokes, which are actually 3 long spokes that travel through the center ofthe hub. The spokes are fiberglass to give them compliance. The compliance helps carrymomentum from one step to the next, and doubles as a shock absorber. The spokes aredriven by a chain and sprocket through the center of the hub. Since the spokes cannotintersect, they are located in parallel planes that move successively further away from thebody with each new spoke. Each spoke is equipped with an independent Portescap 17Nservo motor controlled using an optical quadrature encoder, two magnetic reed switches,and an Allmotion EZSV10 motor controller. All of these components are packaged in thehub, measuring four inches in diameter and four inches in thickness.

The hub is actuated by a hollow drive shaft connected to a Maxon RE30 servo motor througha 90 degree gear box. The hollow drive shaft carries the wires from the hub into a slip ringin the center of the body. The Maxon motors are controlled using feedback from a MaxonHEDL 5540 optical quadrature encoder, a linear cam limit switch, and a Solutions CubedMOTM motor controller. Computing is done by a PC104 with a Pentium 4 processor runningLabVIEW RealTime or tethering to an external computer. Power is provided by a stack of

15

Shawn C. Kimmel Chapter 3. Platform Development 16

Figure 3.1: The IMPASS prototype climbing a set of stairs with ease.

6 Lithium-Ion batteries.

Placement of the components within the body of the robot is particularly important. Theplacement of the center of gravity (CG) is crucial when traversing obstacles. The currentprototype is actually being outfitted with a moving CG that helps with ascending anddescending large obstacles. The design for the moving CG involves shifting the weight ofthe batteries and computer, which are located on a cart that can drive along the length ofIMPASS’s tail.

Each system in IMPASS has been carefully designed for robust execution of this novel loco-motion in rugged environments. The sections below will go into detail of how the mechanismswere developed and what lessons were learned in the process.

3.1 Spoke Actuation

The spoke actuation of IMPASS is what truly makes it unique. At the same time, this uniquedegree of freedom offers significant design challenges. The final design must be compact, yetinclude several independently actuated spokes. At the same time the spokes need to becompliant, which makes it difficult to connect the drivetrain to the spokes.


(a) Spool and Wire Actuation (b) 4-bar Linkage and Power Screw (c) Rack and Pinion

Figure 3.2: Potential mechanisms for spoke actuation.

When considering different methods for linearly actuating a spoke, there were three majordesign criteria. The mechanism needed to be light, able to lift a significant amount ofweight, and able to move relatively quickly. An additional design goal was to make thespokes compliant, to help carry momentum while walking. Designing for compliant spokeswas especially tough, since the spokes have to be part of the actuating mechanism in orderto have maximum travel. As the spokes deformed, it could cause binding or loss of tension inthe mechanism. For the size and power requirements of the robot, electric actuation makesmore sense than hydraulic, pneumatic, or smart material actuation.

There are three main mechanisms that operate electric actuators: screw drives, gear drives,and belt drives. Screw drives are composed of a screw and a nut, where one is rotated relativeto the other via an electric motor. A common example of a screw drive is a manually raisedcar jack. Screw linear actuators are capable of handling large loads, but are slow moving.In the rest position, motor would essentially be at rest because of the contact friction ofthe screw. Two implementations of the screw drive were investigated. The first was a fourbar linkage, shown in Figure 3.2(b). The spoke would be made of two separate linkagesthat would connect in the middle with spacing rods. By driving the base links relative toeach other the spoke would change its length. Problems with this design are its complexityand in- and out-of-plane bending. A second implementation considered comes in the formof a spooled wire rather than a nut. This design, shown in Figure 3.2(a), would use a rod,preferably threaded, to spool a wire onto. The wire would be flexible, allowing compliance inthe spokes. However, when the spokes bend, tension would be lost causing the wire to driftacross the spool or bind. With tensioning members in the feet, perhaps this design would bepossible but that would add undesired weight to the feet. In the end, it would be difficultto ensure that the wire would never drift. Screw drive mechanisms would be appropriate fora robot carrying a heavy payload where speed was not much of an issue. However, a robotthat can move quickly across terrain with a robust drivetrain is more versatile.

Gear driven linear actuators are a speedier option than the screw drive. A good exampleof a gear driven actuator is the rack and pinion used to steer a car. The rack and pinion


Figure 3.3: The first iteration of the chain belt and sprocket design for the spokes of IMPASS.

mechanism is pictured in Figure 3.2(c). There are a couple of downsides to a gear drive forthis application. One is that they tend to be heavier because the materials must be highstrength to withstand the forces in the gear teeth. High strength materials are usually quiterigid, which would make it difficult to design a compliant spoke. With a rigid material, theshock from taking a step would jar the mechanism shortening its lifespan. One material wasfound that may be able to work in this application, Duracon. However this material wasstill relatively stiff.

A belt drive is attractive because it is light weight, flexible, and can move quickly. A flatbelt could potentially slip with the loads experienced by IMPASS. A chain belt solves thisproblem by introducing physical interference between the sprocket and the belt to preventslip. Figure 3.3 shows the first iteration of the belt and sprocket design. This design uses achain belt that runs along the length of the spokes and departs at the hub center to wraparound the drive sprocket. Using a flexible drive belt makes it difficult to maintain tension,especially with a compliant spoke. This can be solved in a number of ways. The methodchosen for IMPASS was to use two idler sprockets next to the drive sprocket, and pulleys toguide the spoke through the hub center.

The final design includes a chain belt and sprocket to actuate the spokes, where the beltis actually part of the spoke. Figure 3.4 shows the final design, which stacks three chainand sprocket mechanisms in parallel. Tension is maintained on the chain by two coupledidler sprockets on either side of the drive sprocket. The coupling of the idler sprocketsprevents any relative rotation, which ensures the belt will always be in tension between thetwo sprockets. The spokes are driven by Portescap 17N-78-210E1 motors mounted to theinside of the hub, as shown in Figure 3.5. The drive mechanisms for the three spokes are


(a) Cut-away isometric view (b) Cut-away side view

Figure 3.4: The final iteration of the chain belt and sprocket design for the spokes of IMPASS.

Figure 3.5: The hub driveshaft is connected to the hub by a plate pictured on the left. Thethree spoke motors attach to this hub plate. All the wires for the hub run through the centerof the shaft.

stacked in parallel. Locating the motors on the inside of the hub is ideal because of spaceconstraints, but makes it challenging to provide torque to outside spokes. To do this, thedrive shafts for the outer spokes must pass through the plane of the inside spokes. To savespace, the hub plates that separate adjacent drive mechanism are shared. Therefore, eachplate must have holes that support the drive shafts, idler shafts, and pulley shafts for thesurrounding drive mechanisms. Figure 3.4(b) shows the pattern of holes that was carefullycrafted to allow passage and support of these shafts. Hub plates were made in house usinga CNC machine to ensure precision placement of the holes. Fitting the spoke drive trainsinside of the hub was just one of the challenges faced in designing the IMPASS spoke wheel.The next section discusses other challenges to the hub design.


3.2 Hub Design

Once the design for three spoke drivetrains was determined, there were several other con-siderations that had to go into the design of the hub. Weight of the hub was an issue, somaterial had to be chosen such that it was light but could handle the stress. Because thematerial needed to be light, transmission of torque through the hub was a design challenge.In addition to material and torque transmission considerations, packaging all the compo-nents in a small footprint also posed a challenge. The list of components includes all of themechanical components already mentioned and electronics for controlling the spokes.

The first ever IMPASS hub was built of aluminum plating with aluminum spokes. TheCAD model of this design is shown is Figure 3.3. It was easily strong enough to handle thestresses of operation, but weighed over ten pounds. Providing the torque to drive this largeand heavy hub would require a large amount of power, and possibly a generator on board forlong missions. To reduce power consumption, the weight of the robot needed to be reduced.The latest version of the hub was made from nylon, which is light, rigid, and machines fairlywell.

Holding the plates in compression are six screws that run through hollow threaded nylonshafts built into the hub plates. The problem with these shafts is that they could not takemuch torque. To remedy this problem, the plates had teeth built into them that meshedwith the adjacent plate. These teeth are shown in Figure 3.6. The teeth double as a layerof protection for the mechanical components, shielding them from outside objects.

In addition to the mechanical components, the hub also must contain the electronics neces-sary to drive the spokes. The current design includes a motor controller, encoder and twolimit switches for each spoke. The motor controllers, Allmotion model EZSV10, were chosenfor their small footprint. The encoders, US Digital model E4P, can be seen on the outsideof the hub in Figure 3.7, and are connected to the drive shafts, which are extended to goall the way through the hub. Reed limit switches are attached to the rim of the hub, andmagnets have been added to each foot. Limit switches are used to initialize and calibratethe position of the spokes and prevent any software failures from driving the feet into thehub.

The original design called for compliant spokes. Since the degree of compliance was notknown, the spokes were designed to be interchangeable. To accomplish this, the belt istensioned and held in place by the feet. When the feet are removed, the spokes can be easilyremoved and replace with another material. The main material that has been tested with isfiberglass.

The current prototype houses the drive trains, motor controllers, encoders, and limit switchesfor three spokes (effectively six spokes), all in a 4 inch diameter, 4 inch deep cylinder. Thissatisfied the goal of a compact hub with multiple independently actuated spokes.


Figure 3.6: Hub plates transmit torque through a set of teeth that mesh with the adjacenthub plate. The plates are held in compression by six screws that run through hollow shaftsaround the edges of the hubs.

Figure 3.7: The drive mechanism for the spokes is a belt and sprocket. Tension in the beltis maintained with two coupled idler sprockets.


Figure 3.8: The hub drivetrains share their axis of rotation, but can rotate independently. A90◦ degree gear box allows the motors to be positioned further back, helping with placementof the CG.

3.3 Hub Drive Train

Located in the body of IMPASS is the hub drive train, which provides the torque to propelthe robot from step to step. Not much torque is needed for the average step, however climbingup large steps can require a significant amount of torque. To provide this high torque loadwhile maintaining accurate position control, we use a Maxon RE30 motor with a MaxonHEDL 5540 encoder. The motors are controlled by a Solutions Cubed motor controller,model MOTM2, which provides position and velocity control. A 230:1 gear reduction wasrequired on this motor for IMPASS to be able to climb its maximum theoretical obstacleheight of 38.1 inches (calculated based on 22 inch maximum spoke length). This gearingratio is obtained by a 90◦ gearbox and a gear head on the Maxon motor. Using the 90◦

gearbox also allows the motors to be placed more towards the rear of the robot, which helpskeep the CG more towards the rear. The next section will discuss CG placement in moredetail.

One significant problem encountered in the design of the hub drivetrain was passing thebundle of wires required for spoke operation. Normally a wheel does not need wires tooperate. Since the wheels rotate relative to the body, the wires would bind up if they couldnot rotate relative to the electronics inside the body. A slip ring was used to allow the hubwires to freely rotate. To get the wires from the slip ring in the body to the hub, wires had tobe passed through the driveshaft or along a notch in the drive shaft. The current prototypehas had much success with passing the wires through a hollow driveshaft, which is mountedto the back plate of the hub. The first iteration of the driveshaft mounted to the centerof the hub plate and required disassembly of the hub to remove. Mounting and removingthe hub to the driveshaft was made easier by including a large plate with the driveshaftthat mounted to the outside rim of the hub. A connector was placed on the hub side of thedriveshaft that was small enough to slide through the hollow in the shaft. This makes for


Figure 3.9: The maximum torque experienced by the hub motors occurs when the robot isclimbing with a spoke that is extended to full length and oriented horizontally. The free-bodydiagram for this case is shown.

easy disassembly of the drivetrain. The wires come out of the body side of the driveshaftinto the slip ring, and finally to the computer. The computer, along with other essentialcomponents within the body are covered in the next section.

3.4 Body Design

The robot is capable of being controlled by a PC-104 board running LabVIEW RealTimeor by an external computer via a tether. Power for computing and motors can also comevia two routes. Built-in Lithium-ion batteries provide a local source of power. A stack of 12batteries that displaces about 56 cubic inches is capable of running the robot for around 2hours, depending on the terrain. The power can also be provided via a tether.

Location of the electronic components was carefully considered, because the center of gravity(CG) of the robot is important. There is no single perfect location for the CG. Due to theversatile nature of IMPASS’s terrain traversal, it is sometimes desired to have the CG atthe back of the support polygon and other times at the front. During the initiation of aclimb or decent, the desired position of the CG is at the back of the tail. For the climbingcase, the torque on the hub motor is less when the CG is located further back. This can beseen by analyzing the free-body diagram in Figure 3.9 which shows IMPASS in the climbingconfiguration. The reaction force on the front spoke, Fspoke is described by the equation

Fspoke =mgA

2(A + B + C)(3.1)


Figure 3.10: This picture shows the new carbon fiber body for IMPASS side by side withthe old aluminum frame body. The new body saves on weight and is more protective of theinterior components.

where m is the mass of the robot, g is the acceleration of gravity, and A, B, and C are thelengths of the segments shown in Figure 3.9. The torque on the hub motor, Thub, is given by

Thub = FspokeL (3.2)

where L is the length of the extended spoke. By making the value of A smaller, we canreduce the torque necessary for climbing the obstacle. Therefore, a CG located towards theback of the tail is advantageous in this configuration. However, once the hub has climbedupon the obstacle and torque is again at normal levels, the robot must pull the tail up overthe obstacle. Here a CG located in the tail makes it difficult to complete the ascent of theobstacle.

For descending obstacles, the ideal CG location is different than the ascending case. At theinitiation of the step, it is more desirable to have keep the weight back in the tail for stabilityreasons. When stepping down, the support polygon for IMPASS shrinks significantly as thehub center passes over the feet contact points. The size of an obstacle that can be down-climbed is directly affected by the location of the CG. Once the hub has descended and itis time to bring the tail down, the best location for the CG is up at the front of the robot.This minimizes the impact of the tail on the ground. The product of the robot’s weight andthe moment arm between the foot contact point and the CG directly determines the amountof torque that acts on the body as soon as the tail leaves the edge of the obstacle.

As can be seen, the appropriate location of the CG depends on the situation. To deal withthis conflict, the CG can either be centrally located or be relocatable. In the current body,the CG is located centrally which allows for climbing of obstacles up to approximately halfthe of maximum theoretical obstacle height. However, a new body is under construction thatwill mount the computer and Lithium-ion batteries on a track. By moving these components


along the track, the CG can be repositioned. The shell of the new body design is shown inFigure 3.10. Another important feature of the robot is the shape of the tail. When followingbehind the body over obstacles, the tail must not get stuck. Additionally, when coming downobstacles the tail should lower as smoothly as possible. An attractive shape for the tail is acurve that is convex with respect to the ground. This design encourages continuous contactof the tail with the ground and helps prevent catching on sharp terrain features. The frontedge of the tail’s convex curve provides an angle of contact with obstacles that promotes lessforce in the direction normal to the front face of the obstacle. This results in less frictionwith the terrain features. The back side of the curve provides a smooth transition whendown-climbing obstacles.

The IMPASS prototype demonstrates that the proposed method of locomotion is mechani-cally feasible. Using a belt and sprocket drive train integrated into the spoke, it is possible toget a large range of linear spoke movement at high speeds without tensioning problems. Thehub is compact in size and effectively contains six actuated spokes and associate electronics.The hub drivetrain provides adequate torque to climb the maximum theoretical height of2√

3 times the nominal walking height. Location of the CG is central, as to moderate theadvantages associated with the different stages of climbing and descending obstacles. A taildesign has been conceived and implemented which has characteristics that promote constantcontact with the ground. Here the hardware has been presented for a highly mobile roboticplatform. In the next chapter, we will investigate the best ways for the robot to take stepsacross a variety of terrains.

Chapter 4

Step Transitions

Unlike many other mobile robots, IMPASS has the flexibility to choose the way it transitionsfrom one step to another. This capability stems from the ability to change the length of thespokes, and therefore create different geometries. This flexibility provides IMPASS with theability to consider many factors including efficiency, smoothness of motion, terrain clearance,or stability. There are technically infinite ways to leave a current step and enter the next,but certain configurations are more attractive. Transitions discussed in this chapter focuson considerations for continuous motion profiles, orientation of the spokes, and capability totraverse obstacles.

For the purposes of this paper, obstacles are defined as a segment of terrain which is non-traversable to IMPASS. Positive obstacles are ones that exhibit a positive vertical displace-ment in the direction that the robot is moving. Negative obstacles exhibit a negative verticaldisplacement in the direction that the robot is moving. This definition will be expanded inthe Motion Planning chapter, but for the discussions in this chapter the current definitionwill suffice.

The stability of the robot is determined by two factors, the center of gravity (CG) locationand the support polygon. The CG of the robot can be described by (xCG, zCG). The supportpolygon is the outline that is formed by the the contact points. For the one-point contactgait, the support polygon will be formed by three points, the tail and one spoke from eachwheel. For the two-point contact gait, the support polygon includes five points which includethe tail and two spokes from each wheel.

This paper considers static stability, which is achieved when the vertical projection of thesupport polygon includes the CG. For two dimensional sagittal plane analysis, the supportpolygon is one dimensional. As long as the horizontal component of the CG, xCG, is locatedbetween the front-most and rear-most contact points, the robot is considered stable.

The one-point and two-point contact gaits each have unique considerations that define theirstep transitions. Therefore, both gaits will be discussed in their own section. The step

26

Shawn C. Kimmel Chapter 4. Step Transitions 27

(a) (b) (c)

Figure 4.1: The two-point contact gait shown rotating clockwise through a step. This gaitconstrains the position of the hub center to a circle for which the radius is the step lengthdivided by

√3. The bar in the center of the robot’s spokes is the center of the circular path,

(xc, zc). [6].

transitions presented here will be used to help refine the control space for path planningalgorithms.

4.1 One-Point Contact Step Transitions

The one-point contact gait for IMPASS exhibits excellent mobility. This gait can be used toposition the hub anywhere in the two dimensional sagittal plane within the robot’s physicallimitations. The tradeoff for this flexibility is stability. With only three points contactingground (one on the tail, and one spoke from each hub), the robot would enter an unstablecase if one contact point were to temporarily loose contact with the ground. Additionally,the support polygon for the robot is reduced in this gait, making the placement of the CGmore important. This section will look at step transitions that can compensate for stabilityconcerns, as well as maximize the benefits of the one-point contact gait.

Any stable one-point contact transition requires that two feet contact the ground from eachhub. Therefore, the robot must obey the kinematics of the two-point contact case at theinstant of the transition. The two-point contact case has one DOF, which constrains theposition of the hub center to a range of motion that forms a circle, shown in Figure 4.1.This circle is described by a center point (xc, zc) with radius rc based on the contact points(x1, z1) and (x2, z2). The properties of the circle can be calculated by the equations

xc = x1 +|~dg|√

3cos(30◦ + αg) (4.1)


Figure 4.2: IMPASS in a two-point contact configuration, showing the velocity vector ~v andcontact point vector ~dg.

zc = z1 +|~dg|√

3sin(30◦ + αg) (4.2)

rc =|~dg|√

3(4.3)

where (x1, z1) is the back spoke contact point, ~dg is the vector that connects the back spoke

contact point to the front spoke contact point, and αg is the pitch of vector ~dg. The two

values ~dg and αg are shown in Figure 4.2 and defined by the equations

~dg =√

(x2 − x1)2 + (z2 − z1)2 (4.4)

αg = arctan(z2 − z1

x2 − x1

) (4.5)

Various configurations for IMPASS in the two-point contact gait are shown in Figure 4.1.During a one-point contact transition, the robot must be in a state that is consistent withthe geometry of the two-point contact gait. This section discusses various approaches to one-point contact transitions based on the geometry of the robot within the two-point contactcircle.


4.1.1 Constant Angular Velocity of Hub

One of the most important considerations in motion planning is providing a continuousmotion profile to the actuators. Motors cannot make discrete changes, so it is desirable tocommand a continuous motion function. For IMPASS, this applies mainly to the contactingspokes and the hub rotation. With respect to one-point contact step transitions, the spokesare not much of a consideration since they are either coming into contact with the groundor leaving the ground. Therefore they are only constrained on one side of the transition.However, the rotation of the hub must be considered since it describes motion of bothprevious and future steps simultaneously. This section discusses solving for transitions suchthat the angular velocity for the hub is constant.

At the instant before transition, the hub’s angular velocity, ~ωA is described by

~ωA =| ~vA|SinρA

| ~rA| (4.6)

where ~rA is the vector describing the rear spoke, ~vA is the hub velocity, and ρA is the anglebetween the spoke and velocity vectors. These variables are shown in Figure 4.2. At theinstant after transition, the hub’s angular velocity is a function of the new contact spoke,~rB, the velocity for that spoke, ~vB, and the relative angle ρB. Setting the angular velocitiesfor the front and back spokes equal results in the equation

| ~vA|SinρA

| ~rA| =| ~vB|SinρB

| ~rB| (4.7)

In a previous paper, Laney discusses the constant ω case for purely horizontal velocity ona horizontal terrain. His paper showed that the switching angle that gives a constant ω isalways 30 degrees [3], where the switching angle is defined as θ shown in Figure 2.8.

In reality the terrain IMPASS will be traversing will not be horizontal. Also there will bea vertical component to the velocity as the robot adjusts to the terrain. The results fromLaney’s paper can be generalized to give the following property

If ~dg‖~v, Then constant ω occurs at θ = 30◦ − α (4.8)

Vectors ~dg and ~v can be chosen in motion planning such that they are parallel. Using thisproperty provides the path planner with a simple correlation between the velocity and contactpoint vectors. It also makes for an easy calculation to find the switching angle.However, thistransition method lacks flexibility. To have access to a wider range of transitions, the velocityand contact point vectors must be decoupled, i.e. they need not be parallel.

The spoke and velocity vectors still must be chosen such that ωA is equivalent to ωB. To


prevent any significant jerk on the robot, an instantaneous change in the velocity vector willnot be allowed. Therefore, ~vA and ~vB are set equal, simplifying the problem. Additionally,the vectors ~rA and ~rB are kinematically linked by the two-point contact arrangement. Thislinkage provides us with the relation that ρA is 60◦ greater than ρB. Here a new angle ρv−d

is introduced in Figure 4.2 which describes the difference in slope between ~dg and ~v. AnglesρA and ρB can be written in terms of ρv−d and the switching angle θ shown by

ρA = ρv−d + θ + 90◦ (4.9)

ρB = ρv−d + θ + 30◦ (4.10)

Using the assumptions above, we arrive at a reduced form of Equation 4.7 in the form

0 =| ~rA|| ~rB| −

Sin(ρv−d + θ + 90◦)Sin(ρv−d + θ + 30◦)

(4.11)

It is immediately apparent that the magnitude of the velocity vector is not important, onlythe orientation. Equation 4.11 allows motion planning to choose the velocity vector andcontact points independently of each other. The switching angle is still a factor in thedecision of what values to choose for ~dg and ~v because the position of the robot depends on

the switching angle. In fact, there is an interdependence between θ, ~dg, and ~v that motionplanning must consider for any transition.

The constant ω approach to solving one-point contact transitions is attractive because itprovides a second-order continuous motion function for the hub motors. Additionally, bysetting ~vA and ~vB equal, we are preventing the robot from experiencing any significant shockduring transition. This method is beneficial for the motors, but has shortcomings in thetypes of terrain it can handle. The next section will discuss transitions that have moreadvanced mobility capabilities.

Experiments with Constant Angular Velocity Transitions

The Constant Angular Velocity Transition for the one-point contact gait was tuned initiallyin a custom built simulator. The simulation environment was created using LabVIEW withthe IMAQ toolkit. Motion messages are received by the simulator component and usedto create a visual representation of the robot and the terrain as shown in Figure 4.3. Thissoftware made it possible to detect any flaws in the motion planning algorithms when appliedto an ideal robot and terrain.

Of course, ideal conditions rarely exist in the real world. To truly validate an algorithm,it must be tested on hardware. The software architecture was designed such that a motion


(a) Starting at θ = 30◦ (b) Middle of step (c) Transition at θ = 30◦

Figure 4.3: The simulator was used to test algorithms before implementing them on therobot.

(a) Starting at θ = 0◦ (b) Front feet first touch theground, θ < 30◦

(c) Transition complete, θ = 30◦

Figure 4.4: The transition case for θ = 30◦. The front spokes touch the ground before θ = 0◦

because of compliance in the spokes and backlash in the hub gear train.

command was communicated using a standard protocol. This made the the jump fromsimulation to hardware easy. Since the motion planner is independent of the hardware, a“robot driver” software component had to be written as the go between for motion commandsand the actuators.

The simulation testing of the motion algorithms showed proper execution of the constantangular velocity transition. However, when the algorithm was implemented on the robotthere were distinct differences. These differences were due to certain physical and mechanicalproperties of the robot.

The Constant Angular Velocity Transition was tested on horizontal ground. The tests areshown in Figure 4.4. The front spoke contacts the ground prematurely, seen in Figure 4.4(b).As a result, the robot violated the no-slip condition in order to finally reach the correct tran-sition geometry in Figure 4.4(c). The discrepancy between the analytical and experimentalresults was caused by certain properties of the hardware.

Deviation from the theoretical findings due to hardware was expected to some degree. Therewere two major reasons that the robot contacted the ground early. The first reason is thatthere is noticeable backlash in the hub gear train, which is magnified by the fact it is attachedto a long spoke. Angular position of the hub is measured upstream of the gear train, rightat the motor. This makes accurate positioning for the hub itself impossible. When the hubcenter passes over the contact spokes, the gears switch to the other face of their teeth makingthe robot pitch forward 5◦ − 10◦.

Initialization and calibration for position of the hub can be done on either side of the gear


train backlash. For transitions, it is best to initialize position such that the hub is rotatedas far forward as possible relative to the body during initialization. This causes the positionof the hub to be inaccurate when leaving a transition, but this is less critical than enteringa transition.

The second factor driving the early ground contact is the compliance of the spokes. Thespoke does not radiate in a straight line from the hub when under load. Therefore therotation of the hub relative to the body is not indicative of the the true rotation of the robotabout the contact point, as it has been modeled in the software. As the load on the spokeshifts, the degree of bending changes. This variable deflection would need to be taken intoaccount to achieve an accurate value of hub rotation about the contact point.

The two factors just discussed cause an over-rotation of the hub. The result of this over-rotation is that the front spoke contacts the ground before it is supposed to. We then findIMPASS in a two-point contact configuration with the spokes not in their final geometry.To reach the final geometry, one or more of the spokes must slip because the two-pointcontact kinematics could not be followed. In experiments, we found that both front andback spokes could slip depending on the step. Determining which spoke would slip requiresan understanding of the friction forces.

In a two-point contact stance with the tail contacting the ground, we have a staticallyindeterminate system. The normal forces at the contact points depend on the stiffness ofthe members. The tail is very stiff, while the stiffness in the spokes varies significantly withtheir orientation and length. Depending on where the front spoke touches the ground withrespect to the desired contact point, the stiffness can significantly vary. With the currenttransition, the spoke is contacting the ground before the desired contact point, making itshorter and more vertical than planned; therefore it is more stiff. Because of the increasedstiffness in the front spoke for the experimental transition, there is more friction to preventslip. The back spoke would then be more likely to slip.

If either of the two spokes were to slip, it is preferred that the front spoke slips. The backcontact point is the reference for future steps. If this contact point is moved the robot willloose accurate localization, potentially causing a foot to be misplaced on a critical terrainfeature. There are two basic solutions that can prevent the over-rotation of the hub resultingin slip of the back contact point. One is to build a model to predict actual hub positionbased on mechanical and physical properties of the robot. The second method is to plan thegait such that the front contact point will slip instead of the back contact point.

If a better position estimate for the hub was available, the motion planning software wouldbe able to touch the front spoke to the ground at the ideal point. To get an accurate valuefor the rotation about the contact point, a model of the gear train backlash and spokecompliance would need to be built. Such a model is not included in the scope of this paper,but would be an attractive option for future research.

A fairly simple software solution was devised to prevent slip of the back spoke by always


Figure 4.5: IMPASS’s switching angle can be described relative to the x-axis or the linenormal to the ground link of the terrain. These switching angles are referred to as θ and θ2

respectively.

slipping the front spoke. By extending the front spoke past its desired transition length inadvance of the transition, it will prematurely contact the ground beyond the desired contactpoint and prevent the over-rotation of the hub. Adding additional support spokes helpsincrease accuracy of position readings by reversing the gear train backlash and reducingcompliance in the back spoke. The difference between this spoke pre-extension and thecurrent algorithm is that the front spoke is always longer and at a shallower angle than theback spoke. The back spoke does not slip because there is more normal force acting on it.While this solution violates the no-slip criteria, it does produce a practical solution to themechanical problems encountered with this transition.

4.1.2 Constant Switching Angle of Spoke

While a constant motion profile is beneficial, it is not necessary. There are other factors thatare just as important to the robot which can be accomplished via other configurations. Onepossible way to define a step transition is by the angle that the back contact spoke makeswith the z-axis of the inertially fixed N -frame, θ. Another, very similar way is to choose aswitching angle relative to the ground link. The switching angle based on the ground linkwill be called θ2. Both switching angles are shown in Figure 4.5. θ2 is related to θ by theequation


(a) Minimum physical switch-ing angle

(b) Isosceles triangle (c) Maximum physical switch-ing angle

Figure 4.6: When in the two-point contact configuration, IMPASS’s legs make a trianglewith the ground link. The boundary conditions can be determined by the sum of the anglesequal 180◦ and the value of β.

θ2 = θ − α (4.12)

Both methods, using a constant θ or θ2, will be discussed in this chapter.

First we will investigate the boundary conditions for the switching angle. The boundariesfor θ2 are explicitly defined as

(−90 + β)◦ < θ2 < 90◦ (4.13)

whereas the boundary conditions for θ depend on the ground link angle α. To define theboundary for θ we simply use Equation 4.12 and the boundaries for θ2 given in Equation 4.13.

Equation 4.12 was derived from simple geometry. A triangle is formed in the two-pointcontact case. This triangle includes the two spokes in contact with the ground and theground link, as seen in Figure 4.9. The interior angles of the triangle must sum to 180◦, sosubtracting β from this sum gives us the total range of motion for the spokes. For the robotproposed in this paper, β is equal to 60◦ so the maximum theoretical range is 120◦.

The theoretical range assumes that the spokes can be any length. In reality, the spokeshave physical limitations on their lengths, specifically 3.5 inches minimum and 21.5 inchesmaximum for the IMPASS prototype. With these limitations, the full 120◦ range of motioncannot be realized. This is explained in Figures 4.6(a) and 4.6(c). A graphical representationof the leg length constraint is shown in Figure 4.7. The curve plots the back spoke length,| ~rA|, versus the front spoke length, | ~rA|, for a fixed step distance, |~dg|, of 16 inches across thefull range of theoretical switching angles, −30◦ < θ2 < 90◦. The dashed rectangle shows thephysical limitations of the spoke lengths. If the step distance were to increase much morewhat is shown in this figure, the curve would no longer have a single continuous segmentwithin the box. Without a continuous curve, the robot would not be able to exist in allconfigurations at that step length. Using this analysis tool, it has been determined thatthe maximum step distance that allows the robot to rotate continuously from the minimum


0 5 10 15 20 25

0

5

10

15

20

25

rA

r B

Figure 4.7: This graph plots the spoke lengths IMPASS against each other as θ2 is changedin a 16 inch step. ~rA is the bottom axis and ~rB is at the left. The rectangular box representsthe robots physical limitations. For this 16 inch step, IMPASS has a continuous range inthe solution space [7].

angle to the maximum angle is 16.45 inches.

If the step distance is increased past 16.45 inches, then portions of the curve will still bewithin the solution space. These extreme conditions are not very useful for the two-pointcontact gait, but can still be used for one-point contact transitions. First we must findthe physical boundaries for the switching angle. The minimum θ2 is achieved when | ~rA| isat a minimum and | ~rB| is at a maximum, which is the top left corner of the dashed box.The maximum θ2 is found in the opposite configuration in the bottom right corner of thedashed box. The stepping distance for the minimum and maximum angles can be foundusing the analysis tool in Figure 4.7. Both extremes occur at a step length of 20 inches.The minimum and maximum angles can be solved for using this step distance. Figure 4.8illustrates the back and front spoke lengths as a function of θ2. Horizontal lines show thephysical limitations of the spoke lengths. A vertical line is used to show the minimum anglethatthe robot can switch at, which is is θ2 = 8.7◦. The maximum switching angle will occurat the same rotation from the horizontal x-axis, but on the opposite side of the robot. Theactual range of switching angles for IMPASS is therefore −21.30◦ < θ2 < 81.3◦.

Now that the boundaries for switching angles have been defined, we can look at the ad-vantages offered by various configurations. The switching angles have been separated intocategories based on their performance characteristics. These categories are shown in the listbelow.


Figure 4.8: Shown is the spoke lengths for IMPASS as θ2 is changed in a 20 inch step.| ~rBz| peaks first, followed by | ~rAz|The maximum and minimum spoke lengths are shown ashorizontal lines. The minimum θ2 is found at the vertical line where both spoke lengths arewithin the bounds.

A. Equivalent Spoke Length Transition, θ2 = 30◦

B. Descending Transition, θ = 60◦

C. Ascending Transition, θ = 0◦ for Adjacent and Non-Adjacent Spokes

D. Default Transition, θ = 30◦

Each of these transition configurations will be discussed individually. Testing was conductedon the robot for each type of transition. There were some interactions with the physical andmechanical properties of the robot that resulted in unexpected phenomenon. Some of thesewere easy to fix, while others would require some more advanced modeling. These problemsand solutions will be discussed for each transition.

A: Equivalent Spoke Length Transition, θ2 = 30◦

The most versatile switching angle for IMPASS is θ2 = 30◦. In this configuration IMPASS iscapable of making the largest and smallest steps across flat terrain. This is graphically shownin Figure 4.10. The point on the curve that corresponds to θ2 = 30◦ is always located on theline of unity slope that goes through (0, 0). As long as the maximum and minimum spokelengths are universal for all spokes on the hub (i.e. the solution space is a square), the lineof unity slope will go through the bottom left and top right corners of the constraint region.The Figure 4.10 shows that maximum and minimum step length occur in these corners.


(a) Equivalent Spoke Length (b) Descending (c) Adjacent Spoke Ascending

(d) Non-Adjacent Spoke Ascend-ing

(e) Default

Figure 4.9: There are infinite configurations in which IMPASS can transition with the one-point contact gait. This shows various configurations that are particularly useful from amotion planning perspective.


0 5 10 15 20 25

0

5

10

15

20

25

rA

r B

(a) Minimum Step Distance

0 5 10 15 20 25

0

5

10

15

20

25

rA

r B(b) Maximum Step Distance

Figure 4.10: These graphs plot the spoke lengths of IMPASS, ~rA and ~rB, for a given steplength. ~rA is the bottom axis and ~rB is at the left. The curve is created by varying θ2 from−30◦ to 90◦, which rotates the robot through the full two-point contact case. The dashedbox represents the physical limitation for spoke lengths [7].

With this switching angle, IMPASS is exactly halfway between the minimum and maximumswitching angle. The stability is dependent on the ground link angle α, and therefore willvary from step to step. As α becomes more negative or the step distance |~dg| becomeslarger, this transition becomes less stable. Additionally the stability depends on the terrainprevious to the obstacle. For example, the robot would be much less stable if the front ofthe robot had just descended a negative obstacle and the tail was still on the obstacle. Anin depth stability analysis will not be covered in this paper, but would be a good area forfuture research.

As discussed the the Constant Angular Velocity transition section, it is easy to choose avelocity vector parallel to the ground link. If these two vectors are chosen to be parallel,the Equivalent Spoke Length Transition will have a constant ω. It becomes apparent thatthere are cases of overlap between the transition classifications. These dual identity caseswill inherit the benefits of both of the transition classifications that describe it.

The strength of the Equivalent Spoke Length transition lies it its flexibility to reach a widerange of step lengths. This can be useful in situations when large areas of terrain are con-sidered non-traversable, forcing contact points to be spaced far apart. Also, when IMPASSencounters sudden terrain changes that require path planning adaptations within a limitedspace, the Equivalent Spoke Length transition allows for small steps to make those adapta-tions. It may seem logical that the Equivalent Spoke Length Transition would be good forobstacle traversal, but future sections will show that other transitions exist that are better


Figure 4.11: IMPASS descending an obstacle with θ = 60◦.

suited for obstacles. However, before other transitions are discussed, the simulation andexperimentation for this transition will be presented.

Experiments with Equivalent Spoke Length Transition Simulation and experimentson IMPASS using the Equivalent Spoke Length Transition were conducted on horizontalterrain. These tests had the same findings described in the experimenting with ConstantAngular Velocity Transitions. The simulator showed proper execution of the steps. However,the hub gear train backlash and spoke compliance caused problems with accurately measuringthe hub rotation. This resulted in premature contact of the front spoke with the ground. Asa result of the premature contact, the back spoke contact point had issues with slip.

Slip is a problem for an path planning that depends on accurate positioning of the feet.This is the case for deliberative motion planning. However, the reactive approach does notnecessarily require accurate foot positioning.

Ascending obstacles did not have any problems with back spoke slip, since the back spokewas always more stiff and located closer to the CG. However, descending obstacles did proveto have problems with back spoke slipping. The software solution mentioned in the ConstantAngular Velocity Transition section worked to prevent back spoke slipping.

B: Descending Transition, θ = 60◦

The switching angle for a step determines the orientation of the contact spokes during thetransition. When θ equals 30◦, the back and front spoke have the same vertical displacementper unit length. As the switching angle is increased greater than 30◦, the forward spokeis able to achieve a greater vertical displacement than the back spoke. This concept ispresented in Figure 4.11, which shows IMPASS with θ = 60 with equal lengths for the frontand back spokes. The front spoke is clearly able to achieve a greater vertical displacementin the N -frame.


0.5 1.0 1.5 2.0 2.5 3.0Θ

-20

-10

10

20

rÓz, inches

0.5 1.0 1.5 2.0 2.5 3.0Θ

-5

5

10

15

20

DP, inches

Figure 4.12: The left graph shows | ~rAz| and | ~rBz| with respect to θ for equal spoke lengths.The line that peaks first is | ~rBz|. The filled in region represents the area where the DescendingPotential (DP)> 0 (i.e. | ~rBz| > | ~rBz|). The right graph shows the DP, which reaches itsmaximum value at θ = 2∗π

3= 120◦.

The aforementioned bias in vertical displacement facilitates easier transitions to terrain thatis below the height of the current contact point, i.e. negative obstacles. Here, a metric isintroduced called ’descending potential’ or DP, which describes the ability of the robot totraverse negative terrain features. This metric is calculated by the equation

DP (θ) = | ~rBz| − | ~rAz| = | ~rB| · Cos[π

3− θ

]− | ~rA| · Cos[θ] (4.14)

The first term gives the vertical rise of the front spoke, and the second term gives the riseof the back spoke. According to this metric the switching angle with the greatest DP isachieved when both spokes are at maximum length and the ground link is vertical, such thatθ = 120◦ and α = −90◦. The DP metric for this configuration is shown in Figure 4.12.

Implementation of this maximum DP case would lead to a precarious configuration. Withboth spokes at maximum length and θ = 120◦, the hub center will be in front of the contactpoints by

√3

2times the spoke length. Placement of the CG will need to be extremely far back

for a stable stance. To be realistic, this configuration can only exist with a moving centerof gravity, which is possible but not necessary. In addition to CG issues, this configurationdoes not contact the bottom of the back foot. The back foot is essentially used as a hookto hold onto the top of the obstacle. The safety of the robot becomes a concern here. Ifthe back foot slips before the front foot can contact, the robot could roll off the obstaclepotentially damaging components or falling onto its back or side becoming immobilized.

Without a moving CG or a guaranteed foot hold for the back foot, the maximum DP con-figuration is not practical. Since neither of these issues have been solved for the currentprototype, a different configuration had to be used. In the interest of making a solution thatis appropriate for any robot with a centrally located CG, the exact location of the CG is notused. θ is constrained to less than 90◦ so that the contact spoke will have normal contactwith the ground. The back spoke length is fixed at its minimum length. This way the


Figure 4.13: For any combination of spoke lengths, the greatest DP will always be achievedwith IMPASS at the ledge of an obstale and a vertical ground link. The circle shows thefoot path as the robot rotates through all possible θ.

furthest that the hub center can overhang the obstacle is the minimum spoke length. Thefront spoke is fixed at its maximum value to provide the largest difference in | ~rB| and | ~rA|that is possible. This will maximize the value of the DP. Applying these constraints freezesIMPASS in an position in the two-point contact case that can be described as θ2 = −21.3◦.

With the back spoke at the minimum length and the front spoke at the maximum length, therobot is capable of a step length of 20 inches. Therefore, the robot is capable of descending20 inches. This is shown in Figure 4.14. The first graph shows the vertical component ofeach spoke with respect to θ. The second graph shows the actual DP metric, peaking at 20inches with θ = 68.3◦. Again, the resulting orientation of the robot is with a vertical contactpoint vector. This brings to light the relationship that for any combination of spoke lengths,the orientation with the greatest DP will always be when α = −90◦. Logically this can bedetermined by picturing a circle around the back contact point with radius equal to the stepdistance. The lowest point on the circle will always be located straight down.

Using an α of −90◦ gives the greatest DP, but it is not very practical. A vertical contact pointvector requires the back spoke to be precariously perched at the edge. This configurationalso requires the back spoke to be stuck in the bottom corner of the obstacle, which can onlyhappen with a perfectly vertical obstacle. To address these issues, a factor of safety can beadded that places the back spoke away from the edge and the front spoke forward.

With these factors of safety, θ = 60◦ becomes an attractive switching angle. Setting θ to 60◦

allows the front spoke to be vertical, which is desirable with compliant spokes. The spokewill flex minimally in the vertical orientation, providing the most reliable spoke length. Thisswitching angle suffers a DP loss of only 0.25 inches compared to other angles, while gaining


0.5 1.0 1.5 2.0 2.5 3.0Θ

-10

-5

5

10

15

20

rÓx, inches

0.5 1.0 1.5 2.0 2.5 3.0Θ

-5

5

10

15

20

DP, inches

Figure 4.14: The left graph shows | ~rAz| at minimum spoke length and | ~rBz| at maximumspoke length with respect to θ. The filled in region represents the area where the DescendingPotential (DP)> 0 (i.e. | ~rBz| > | ~rBz|). The right graph shows the DP, which reaches itsmaximum value at θ = 2∗π

3= 120◦.

a comfortable horizontal distance of 2.5 inches between the contact points. The horizontaldistance can be split into 1.25 inches of clearance from the obstacle for both contact points.

If the robot needs to descend a height of less than 19.75 inches, it would be better to increasethe rear spoke length than change the angle. This will give the hub more ground clearanceand decrease the change in angular velocity between the two steps. Maintaining the 60◦

switching angle provides consistency and keeps the front spoke vertical, which are bothbeneficial.

Experiments with Descending Transition Testing the Descending Transition was veryexciting because the robot was attempting to statically descend a significant distance in asingle step. Fortunately, testing in the simulation environment was sufficient for developmentof functional software. Running the code on the robot for the first time went without a singleaccident. Figure 4.15(c) shows the robot successfully transitioning down from an obstacleusing the θ = 60◦ transition.

Entering the Descending Transition went relatively smoothly compared to the ConstantAngular Velocity and Equivalent Spoke Length Transitions. The problems encountered inthe other transitions, i.e. hub gear train backlash and spoke compliance, had been magnifiedby large spoke lengths. When descending an obstacle, the back spoke is so short thatthese factors were less pronounced. However, it was found in initial experiments that thefront spoke did not extend all the way to the ground. The robot rotated to about 70◦

before touching the ground. This may have been due to some measurement error or somecompliance in the spokes. This problem was easily fixed by virtually increasing the height ofthe obstacle so that the robot would extend the front spoke further. For the 12 inch obstacleshown in Figure 4.15, the robot was actually told that the height of the obstacle was 14inches.


(a) Starting at top of obstacle (b) Hanging over obstacle

(c) Transition initiated θ = 60◦ (d) Transition complete

(e) Next transition

Figure 4.15: The transition case for θ = 60◦. The front spokes touch the ground beforeθ = 0◦ because of compliance in the spokes and backlash in the hub gear train.


An additional problem was encountered following the transition. Right after the DescendingTransition finished, the back spoke is almost at full length entering the next transition, shownby Figure 4.15(d). In the experiment, soon after the Descending Transition was complete thehub rotated through almost 20◦ freely- taking up the hub gear train backlash and bendingthe spoke.

If the CG had not been far enough back, the robot could have easily flipped over at thispoint. The effect of the placement of the CG on descending and ascending capabilities wasdiscussed in the Platform Development chapter, but did not include the dynamic aspectsfound in this experiment. The robot could be statically stable for the entirety of the step,but still fall forward from the momentum created during the free hub rotation. Even with arobot that will not fall because of a rear located CG, the peak torque required by the motorsto stop the robot’s momentum is particularly high. At this point in the experiment, thecurrent draw from the robot almost doubled.

Unfortunately, there are no easy software solutions to circumvent the free rotation describedin this experiment. One potential option is to try and reverse the motors when the backlashis being absorbed to catch the robot’s fall earlier. This would be very difficult to time, butcould potentially help the problem. The motor reversal was not investigated for this paper,but could be a good update in future software revisions.

C: Ascending Transition, Adjacent and Non-Adjacent Spokes, θ = 0◦

Ascending an obstacle has many geometric similarities to the descending case, but is funda-mentally different in that there is less focus on stability and more on torque. One constraintthat was imposed on the descending case will be removed for the ascending case. The ref-erenced constraint is that the spokes must contact the ground on the bottom of the foot asopposed to the side. Here we allow the robot to use the front spoke partially as a “hook”to pull itself onto the obstacle. Removing the normal contact constraint also opens up thepossibility of using non-adjacent spokes to climb. This gives two transitions, the AdjacentAscending Transition and the Non-Adjacent Ascending Transition. Adjacent spokes are de-fined as two spokes that are separated by angle β around the hub. By using non-adjacentspokes, we can greatly increases the size of obstacles that can be climbed. In order to ensurethat the foot would be able to stick, the edges were coated with a silicone based non-skidsubstance.

The first case that was investigated was transitioning over an obstacle with the bottom ofthe front spoke touching the ground. Here we can draw lessons from the descending case.The maximum ascending potential, AP, is obtained at θ = −60◦ with a vertical groundlink. This is 180◦ degrees from the location of the maximum DP. The equation for AP isessentially the negative of the DP, given by


AP (θ) = | ~rAz| − | ~rBz| = | ~rA| · Cos[θ]− | ~rB| · Cos[π

3− θ

](4.15)

The maximum AP configuration does not meet the criteria that the bottom of the feet musttouch the ground. Therefore, we are limited to θ > −30◦. As discussed in the DescendingTransition section, a vertical spoke provides greater stiffness and more reliable support. Wewill therefore consider θ = 0◦ for the Adjacent Spoke Ascending Transition. This switchingangle allows a maximum obstacle height of 19.75 inches, which is the same height that therobot can descend with the Descending Transition.

For obstacles over 19.75 inches, the robot must use the side of the front spoke as opposedto the bottom of the foot. The feet have been shaped to be able to hook terrain featuresand have been outfitted with a non-skid surface. However, this ascending case is likely toexperience some slip on the front foot contact point as the robot climbs.

Again here we find that the maximum climbing height is achieved with a vertical ground link,as was the case for the descending transitions. Using the Non-Adjacent Spoke Transition,the robot is able to theoretically climb 2

√3 times the nominal walking height. Here the

nominal walking height is defined as half the overall spoke length.

The theoretical height is not achievable for several reasons stemming from the implementa-tion of platform. For one, the full length of a spoke cannot be used on one side of the hub.The feet cannot travel into the hub. The full length of the spoke, L, is 25 inches, while themaximum spoke length,lmax is 21.5 inches. In addition to the spoke length constraint, wemust take into account the front spoke must be extended beyond the obstacle to hook theedge. The magnitude of the vector between the hub center and the contact point, ~rB, willend up being less than the spoke length needed to hook the obstacle. The distance thatthe front spoke must be extended past the contact point will be named lhook. The value oflhook must be determined conservatively since a slip while climbing could cause a great dealof impact to the robot. In the experimentation section, the value of lhook for the currentIMPASS prototype will be discussed.

Due to the physical constraints listed in the previous paragraph, the maximum climbingheight is reduced. The AP with θ = −30◦ is

AP = (lmax − lhook)2√

3 (4.16)

When lhook is taken into account, −30◦ is no longer the maximum possible obstacle height.This switching angle forces both | ~rA| and | ~rB| to be the same length, which is lmax − lhook.Really, only | ~rB| must be set to lmax − lhook. A larger obstacle height can be reached if | ~rA|is set to lmax. Using the law of cosines we can determine the maximum possible height

AP 2 = l2max + (lmax − lhook)2 − 2(lmax)(lmax − lhook)Sin(2β) (4.17)


Figure 4.16: This figure shows the Non-Adjacent Ascending Transition configuration chosenfor climbing large obstacles. θ is set equal to zero such that the back spoke is vertical,minimizing the effect of compliance in the spokes.

where 2β is the angle between the non-adjacent spokes. While this gives a maximum climbingheight for the ideal condition, we must consider that in reality the robot will not encounterideal obstacles and will have trouble with exact foot placement in the bottom corner of theobstacle. Additionally, one factor not mentioned above is the material properties of thespokes. Due to the compliance in the spokes, the rear spoke will become shorter under loadfurther reducing the maximum climbing height.

The amount of compliance in the rear spoke can be minimized while allowing for easy footplacement of the back spoke by using θ = 0◦. This climbing configuration positions IMPASSwith the back spoke vertically. This will minimize bending of the compliant material in therear spoke. Figure 4.16 shows the geometry for this configuration. The height that the robotcan climb, AP, using this transition is described by the equation

AP = | ~rA|+ Sin(30)(| ~rB + lhook) (4.18)

In this section we have discussed two configurations for the Ascending Transition. The firstis the Adjacent Ascending Transition, which ensures that all spokes will contact the groundwith the bottom of their feet. The Non-Adjacent Ascending Transition on the other handand uses the side of the front spoke to pull the robot onto the obstacle. The Non-AdjacentAscending Transition is capable of climbing obstacles much higher than the Adjacent As-cending Transition. Both Ascending Transition configurations use a switching angle of θ = 0because it minimizes the compliance in the back spoke and allows the robot to stand backfrom the obstacle.

Experiments with Ascending Transition As with the other transitions, the AscendingTransitions were run first in simulation. The results from these tests indicated that thealgorithms were ready for implementation on the robot. Running these algorithms on therobot was not only going to be a test for the software, but also for the hardware. The Non-


(a) Starting at θ = 0◦ (b) Raising the body (c) Front spokes initiate contactwith the obstacle

(d) Back feet leave the ground (e) IMPASS pulls itself onto theobstacle

(f) On top of the obstacle

Figure 4.17: IMPASS climbing a 12 inch obstacle with the Adjacent Ascending Transition,θ = 0◦. The front spokes touch the ground before θ = 0◦ because of compliance in the spokesand backlash in the hub gear train.

Adjacent Ascending Transition provides the ultimate test of the hub drivetrain’s availabletorque.

The first experiment was run on the Ascending Adjacent Spoke Transition, which is shown inFigure 4.17. The experiment was a success; IMPASS is capable of stepping onto an obstaclewith this transition. The height of the obstacle traversed was 12 inches- IMPASS’s nominalwalking height. One notable difference between simulation and physical testing was that theswitching angle was slightly less than 0◦ in the physical testing. This is most likely due toa hub positioning error in the motion planning software. As the body changes angle withrespect to the terrain, the Motion Planner must apply a correction to the hub position readfrom the encoder to achieve the absolute inclination of the body. The simulator’s model forthe body was not extremely accurate, which most likely caused a false correction value tobe sent during the climb. This correction error, in addition to hub gear train backlash andspoke compliance, is the reason that the robot transitioned a little early.

The Ascending Non-Adjacent Transition was tested on an obstacle 18 inches high. Therobot successfully climbed the obstacle. Perhaps the most impressive part of the climb iswhen the back spokes leave the ground an the robot’s weight is supported on the edges ofthe front spoke feet. The non-skid substance applied to the sides of the feet were able toprovide enough friction to hold most of the weight of a 15 pound robot. The robot couldthen proceed to pull itself onto the obstacle and continue walking.

One important finding from the experiments was the minimum distance that the spokeneeded to be extended beyond the lip of the obstacle. It was found that the robot’s frontspoke would always slide back to approximately 2 inches past the lip. It was at this position


(a) Starting at θ = 0◦ (b) Raising the body (c) Front feet initiate contact withthe ground

(d) Back feet leave the ground (e) IMPASS pulls itself onto theobstacle

(f) On top of the obstacle

Figure 4.18: IMPASS climbing a 18 inch obstacle with the Non-Adjacent Ascending Transi-tion, θ = 0◦. The compliance of the spokes is noticeable in Figure 4.18(d) [8].

the feet could get enough traction on the ground to support the bodies weight. From thiswe can conclude that the experimental minimum of lhook is 2 inches. It is more prudent toplace the spoke somewhat beyond the minimum lhook value and let the spoke slide down theobstacle edge into climbing position.

An observation that would be useful for reactive programming was noticed during theseexperiments. If the front spoke of the robot overshoots the obstacle, it will slide back untilit reaches a fairly consistent length of 2 to 3 inches past the lip of the obstacle. A climbingbehavior could be created that would fully extend the front spoke before coming down ontoa suspected obstacle. The robot would climb with consistent performance.


D: Default Transition, θ = 30◦

The previous two sections have discussed ascending and descending configurations. However,many obstacles that IMPASS encounters will be fairly small in size, not requiring any gaitadaptation. The gait that best balances moderate obstacles, both positive and negative, isa constant θ = 30◦. In this configuration, the slope of both front and back contact spokes isthe same. That means that the robot can achieve equal AP and DP during any step. Themaximum height change that can be traversed in the Default Transition, |AP | = |DP | is

|AP | = |DP | = lmaxSin(60◦)− lminSin(60◦) (4.19)

Having such flexible height change characteristics is very valuable. Sensor data can changeat a moments notice, springing a positive or negative obstacle into the robot’s path thatrequires changes to the current step. With the Default Transition, the robot is equally readyfor a height change in any direction.

This transition also has good stability characteristics. The hub can only extend past the backcontact point by distance lmax

2at full height. For the current robot, the Default Transition

is always stable. In general for a centrally located CG, this transition should be static formost, if not all, of its operating range.

In unstructured terrain, where future contact points are not guaranteed to be stable, it isvery beneficial to have a stable switching angle. Say the front foot stepped on a piece ofloose rubble, the back spoke would have to recover the robot’s stability. With a switchingangle greater than θ = 30◦ stability becomes a serious concern.

On the other hand, as θ is reduced from 30◦, the DP is decreased. Having a good DP valuefor the standard transition is critical, since it is more likely that a negative obstacle won’tbe detected than a positive one. An perception suite on a low lying robot will have troubledetecting negative obstacles from any significant distance.

The Default Transition provides a good balance of the characteristics that make up a goodstandard walking transition. In this configuration, the robot is equally able to ascend ordescend obstacles. This transition has good stability characteristics. On the current proto-type, the Default Transition is stable across the entire operating range. While maintainingthis stability, the robot is still able to descend fairly large obstacles that may appear latedue to perception difficulties. The Default Transition has the right properties to be usedthroughout much of IMPASS’s operation.

Experiments with the Default Transition The results from experimenting with thistransition showed the same results as the Constant Angular Velocity Transition and theEquivalent Spoke Length Transition. The robot in its physical form is not ideal like the sim-ulator. Examples of the robot running with θ = 30◦ in simulation are shown in Figure 4.19.


(a) Starting at θ = 30◦ (b) Approaching obstacle (c) Transition over obstacle

(d) Next transition

Figure 4.19: The robot traversing moderate obstacles in the simulation environment withthe Default Transition, θ = 30◦.

Physical problems including gear train backlash and spoke compliance led to issues with theaccuracy of the the hub rotation measurement. The robot tended to over-rotate at the endof a step causing early contact with the ground. This caused some problems with foot slip,forcing future contact points to be offset.

This transition worked well for climbing moderate obstacles, but would have trouble withlarge obstacles. For these we must look at other transitions.

4.2 Two-Point Contact Step Transitions

On terrain that is more unstable, the two-point contact gait is preferable. In this situationthe robot will have five points of contact (two feet from each hub and the tail) and alarger support polygon. The downside is that the kinematics of this gait constrained themotion to one DOF. Motion planning therefore become much less flexible. Planning aheadto accommodate future steps is of the utmost importance.

Figure 4.20 shows the solution space for a given step length. It is very possible for the robotto enter a step in which it cannot rotate continuously through the full range of motion. Forthe dimensions of the current IMPASS, the maximum step distance at which the robot canrotate continuously through its range of motion is 16.5 inches. Any step distance above thisvalue should be avoided unless it is only being used specifically for a short range of motion.One particular time that this could apply is in preparing for obstacle traversal.

Unlike with the one-point contact transitions, which can occur in a number of configurations,the two-point contact transitions are completely constrained by the contact points thathave been chosen. Figure 4.21 shows the possible trajectories of the hub center for threeconsecutive steps. Transitions between these steps must occur at the intersection of the arcsto obey the two-point contact kinematics.

Since the two-point contact transitions have no flexibility once contact points are chosen,


0 5 10 15 20 25

0

5

10

15

20

25

rA

r B

Figure 4.20: This graph plots the length of the back spoke against the length of the frontspoke during a two-point contact step. A dashed box has been included which shows themaximum and minimum spoke lengths. This particular step length of 17 inches enters andleaves the solution space six distinct times.

Figure 4.21: The arcs seen in this figure shows the possible position of the hub center forthree consecutive steps. The points at which the arcs intersect is where the transition mustoccur.


(a) Starting at θ = 60◦ (b) (c) Equivalent Spoke Lengths

(d) (e) (f) Switching at θ = 60◦

Figure 4.22: IMPASS walking in the two-point contact gait. The transitions are determinedby the contact points.

the transition geometry from one step to the next simply requires finding the intersection ofthe arcs and placing the next spoke down at that time.

4.2.1 Experimenting with Two-Point Contact Transitions

The two-point contact transitions were tested in the simulator. Frame captures from thesimulation are seen in in Figure 4.22. The motion commands for the two-point contact arethe same as the one-point contact, so the same robot driver could be used. IMPASS took itsfirst two-point contact steps, looking just as it had in simulation. The physical conditionsthat facilitated problems in the one-point contact gait did not effect the two-point contactgait. Because the robot could distribute its weight over more members, the spokes were lesscompliant and the hub gear train backlash was not freely released. The two-point contactcase closely mirrors the theoretical results.

Chapter 5

Motion Planning

Manual control of all the spokes on IMPASS during a maneuver would be a daunting taskfor any operator. For IMPASS to be practical for real world applications, automated motionplanning of the spokes is absolutely necessary. Instead of manual control, a user friendlysoftware interface was developed, shown in Figure 5.1. This software provides the user withindicators of the motions and intensions of the robot. Currently, the user only needs to inputa ”START” command and a ”Desired Forward Velocity”.

Due to the many degrees of freedom that IMPASS possesses, there are infinite solutionsto the motion planning problem. Research has uncovered many avenues that arrive at anacceptable solution, but there are a few methods that particularly stand out. This paperwill discuss the various motion planning methods that have been devised.

For the scope of this paper, motion planning will be defined as the manner in which the robottransitions from one stance to another stance, where the second stance can be, and often is,with different ground contact points. This motion planning component would answer to ahigher level supervisory program, such as a way-point follower for autonomous operation ordirectional control from a human operator.

(a) Supervisory Control (b) Motion Planner (c) Simulator Design Module

Figure 5.1: The software interfaces for controlling IMPASS.

53

Shawn C. Kimmel Chapter 5. Motion Planning 54

As discussed in the background chapter, there are a couple of ways to approach the AIproblem of autonomous navigation. Assuming sensor data is fairly accurate, the deliberativeapproach to path planning makes a lot of sense for IMPASS. This approach allows the robotto build a map of the terrain. With this information, IMPASS can plan an optimal pathbased on its hierarchy of needs. Again, this all hinges on a fairly reliable perception suite.

In reality, however, sensor data is never perfect. There is a significant amount of filteringand processing required to provide accurate data for a world map, and redundant sensors areoften used, further complicating the situation. To reduce the cost of extra sensors and theunreliability of perception data, the reactive approach to navigation can be implemented.This approach dictates a close coupling between sensor data and actions. The robot wouldbe equipped with simple sensors such as inclinometers, tactile sensors, and torque sensors.IMPASS’s actions would be dictated by a set of predefined behaviors selected based on thesituational data. This approach tends to suffer much less from sensor noise and unplannedencounters, but lacks the optimality provided by the deliberative approach. The deliberativeand reactive based methods of operating IMPASS will both be discussed in more depth inthis chapter.

5.1 Deliberative Motion Planning

In this section, we will delve into the aspects of motion planning from a deliberative stand-point. During development of the software, it was assumed that an accurate world map wouldbe provided by a perception component. However, this world map need not be accurate formore than a few steps, since the motion planning software will re-plan frequently.

The kinematic properties of IMPASS predicate an interdependence between the path of therobot body and the ground contact points. For the two-point contact gait, the robot bodypath and contact points are directly linked due to the one DOF mobility of the system.However for the one-point contact gait, one does not uniquely define the other. In thissection we will focus on the one-point contact gait because its flexility makes it an attractiveoption.

The problem of deliberative path planning can be approached from two angles. We canstart by either determining the body path first or the contact points first. Initial ContactPoint Selection (ICPS) focuses on choosing optimal foot contact points that ensure the bestchance of success. For example, IMPASS would avoid dangerous areas of the terrain andmaximize stability. On the other hand, Initial Body Path Selection (IBPS) allows the robotto determine an optimal body path catered towards particular goals, such as minimizingimpacts experienced by a payload.

While ICPS and IBPS are used to focus on particular motion planning goals, they are bothcapable of considering secondary goals. For example, with the ICPS method, once contactpoints are chosen there is still a great deal of freedom in choosing the body path of the robot


Figure 5.2: The architecture for the Initial Contact Point Selection (ICPS) implementationof the deliberative approach follows the sense-plan-act sequence. The ICPS method has threebasic steps within the contact point planner: Critical Contact Region (CCR) identificationand refinement, Critical Contact Point (CCP) determination, and Intermediary ContactPoint (ICP) determination.

to minimize payload accelerations. For the IBPS, once the body path is chosen, selection ofcontact points is still flexible. Selection of a motion planning approach for a given applicationwould be based on factors such as terrain type and the robot’s hierarchy of needs.

In this paper, we will focus on an implementation of the ICPS approach. The architecture isshown in Figure 5.2. The ICPS approach is broken down into two types of planning, contactpoint selection and step execution. Contact point selection sets up the transitions that therobot will need to take. Step execution determines how to get from transition to transition.Before the ICPS implementation can be discussed, we must first define the representation ofthe robot’s position and motion, and the terrain within the motion planning software.

5.1.1 Representation of the Environment and the Robot

Before the robot can make any decisions on what action to take, it must first acquire terrainand localization information. Both sets of information must be communicated in a modularmanner that can be useful to multiple decision making components. Here we discuss therepresentation of the terrain and the robot.

The terrain is represented as discrete segments of the surface in the two dimensional sagittalplane. Here we are assuming that the left and right hubs will encounter identical terrain.Estimating the terrain as discrete segments allows for easy separation of the terrain intotraversable and non-traversable sections. The resolution of the terrain segments can also


Table 5.1: Terrain Classifier Message.

Message Fields VariableContact Point (Inches) (xi, zi)Traversable? (Boolean) Travi

Non-Adjacent Spoke Required? (Boolean) Climbi

be tuned to accommodate different terrains. More irregular terrains would use a higherresolution. For the algorithms discussed in the deliberative portion of this paper, it has beenassumed that the length of the terrain segments is greater than the length of the foot contactarea. This allows the contact points to be mapped onto a single terrain segment. It was alsoassumed that the terrain could be expressed as a function of x, i.e. there are no overhangingterrain features. Even if overhanging terrain features existed, there is no apparent advantageto stepping under the overhang. Therefore, the overhang would be modeled as a verticalwall. Vertical walls are modeled as almost vertical, with a very slight x distance added tokeep the terrain a function of x. This section is not focused on the computation of the terrainprofile, but rather what form it is passed in, and how it is handled by the motion planningsoftware.

The IMPASS software architecture is set up such that a perception component compilessensor data and provides an array of terrain points. These points are in the form (xi, zi).The array of terrain segments is then fed into a platform specific software component thatprocesses the terrain information. This component is called the “Terrain Classifier”. Eachterrain segment is defined as traversable or non-traversable based on the slope of the segmentand a friction model of the robot’s feet. If a segment is non-traversable, it is also tagged witha rating that defines whether the robot can walk over it with adjacent spokes or if climbingis necessary with the Ascending Non-Adjacent Spoke Transition. The Terrain Classifier thenoutputs a message in the form described in Table 5.1. Here the “Traversable?” and “Non-Adjacent Spoke Required?” fields apply to the segment between terrain points (xi, zi) and(xi+1, zi+1). This output is much more useful to the Motion Planner than just the terrainarray.

Robot localization is based on the inertially fixed coordinate system. The pose of the robotis completely defined by the following variables α

� (xc, zc) = Hub center position with respect to the inertially fixed coordinate system

� α = Inclination of the body

� θ = Rotation of the hub (Absolute, with 0 being at the beginning of a mission).

� θ = Rotational velocity of the hub


Table 5.2: Experimental Motion Profile Message.

Message Fields VariableAbsolute Angular Position of Hub (Radians) θLength of Spoke 1 (Inches) l1Length of Spoke 2 (Inches) l2Length of Spoke 3 (Inches) l3

� R1, R2, R3 = Lengths of three adjacent spokes

� R1, R2, R3 = Linear velocities of three adjacent spokes

With this information, the robot can be positioned with respect to the terrain.

5.1.2 Defining Motion of the Robot

In addition to common protocol for the inputs to motion planning, we must have a conventionfor the outputs of motion planning. Motion planning must communicate its intended motionto the Robot Driver component. For the experiments referenced in this paper, the robot wascontrolled using position control. The motion planner outputs a four part message definedin Table 5.2. Only three spoke lengths are needed since each hub only has three motorscontrolling the spokes. The values l1, l2, and l3 are attached to specific spokes throughoutoperation. This means the motion planner must keep track of the orientation of the spokescontinuously.

It was found during experimentation that position control resulted in a somewhat jerkymotion. The communication rate with the motor controllers was not high enough to getcontinuous motion. Solving this problem requires a new motion profile message. There wereinsufficient resources available to re-design the Motion Planner around a new communicationprofile during the tenure of my research. However, an improved protocol is suggested forfuture research.

There are several descriptors that help to constrain IMPASS’s motion. Some of these thatare particularly useful include:

� Foot trajectories

� Spoke lengths and velocities

� Hub center trajectory and rotation


Table 5.3: Improved Motion Profile Message.

Message Fields VariableAbsolute Angular Position of Hub (Radians) θAngular Velocity of Hub (Radians/Sec) ωLength of Spoke 1 (Inches) l1Velocity of Spoke 1 (Inches/Sec) v1

Length of Spoke 2 (Inches) l2Velocity of Spoke 2 (Inches/Sec) v2

Length of Spoke 3 (Inches) l3Velocity of Spoke 3 (Inches/Sec) v3

� Center of gravity trajectory

The motion of IMPASS can be defined by many combinations of these descriptors. Dependingon the approach to the motion planning solution, certain combinations become more or lessconvenient. To keep things modular, a standard convention is chosen that is convenient forthe Robot Driver.

The Robot Driver is responsible for actuating four pairs of motors. One pair of motorscontrol the hub rotation for both hubs, and three pairs of motors control the spoke lengthsfor each hub. Therefore, a convenient representation of IMPASS’s pose would include motionprofiles for each of these motors. By adding a desired velocity to the previous motion profilecommand in Table 5.2, the motion profile paints a more complete picture of how the robotshould be moved. This “position at velocity” profile is shown in Table 5.3. Position atvelocity control should provide the Robot Driver with sufficient information to producesmooth motion.

5.1.3 Initial Contact Point Selection (ICPS)

Any robot traveling in unstructured terrain will encounter hazards. These hazards cancome in various shapes and sizes. Many mobile robots attempt to avoid terrain hazards bysteering around them. IMPASS has the advantage of being able to intelligently step overhazards by choosing suitable contact points. Examples of hazards IMPASS could avoid inthis manner are terrain instabilities, sharp points, and obstacle ledges. This section discussesthe approaches to motion planning from a contact point selection point of view.

There are three steps the motion planning software runs through to choose contact points:Critical Contact Region (CCR) identification and refinement, Critical Contact Point (CCP)determination, and Intermediary Contact Point (ICP) determination. The goals of this


software are first and foremost to avoid non-traversable terrain, secondly to undergo minimalvertical displacement, and thirdly to maintain a walking height as close to the nominalwalking height as possible to enable maximum adaptability to sudden changes in terrain.

Critical Contact Region (CCR) Identification and Refinement

The main focus of this research is the advanced mobility capabilities of the actuated spokewheel design. Therefore, the foremost concern with IMPASS is surmounting obstacles. Herewe define an obstacle based on the Terrain Classifier message: an obstacle is any set ofconsecutive non-traversable segments preceded and followed by a traversable segment. Apositive obstacle is one in which the terrain after the obstacle is more positive in the z axis,and a negative obstacle is one in which the terrain after the obstacle is more negative in thez axis.

In order for IMPASS to traverse an obstacle, there is a certain region preceding and followingthe obstacle in which the robot is required to step in order to climb the obstacle. Theseranges, hereto referred to as Critical Contact Regions (CCR), are determined by the physicaldimensions of the robot. Figure 5.3 shows the CCR for a given positive obstacle. Precedingthe obstacle is CCR1i, and following the obstacle is CCR2i, where i is the index of the firstnon-traversable segment in the obstacle. The points in the two-dimensional sagittal planefor a pair of CCR associated with an obstacle are as follows:

� (xCCRa, zCCRa) = furthest back point from which IMPASS can still pass the obstacle

� (xCCRb, zCCRb) = closest point preceding the obstacle from which IMPASS can make astep past the obstacle

� (xCCRc, zCCRc) = closest point on the far side of the obstacle to which IMPASS canstep from the front side of the obstacle

� (xCCRd, zCCRd) = furthest forward point to which IMPASS can step from a footholdpreceding the obstacle

The robot transitions which determine the CCR in Figure 5.3 are not the only configurationsthat solve for CCR of obstacles. Depending on the terrain before and after the obstacle, aswell as the dimensions of the obstacle, non-adjacent or adjacent spokes may reach the moreextreme point. For example, we need three configurations to determine the CCR for theobstacle in Figure 5.4. The height of the obstacle in Figure 5.3 prevents the non-adjacentspoke from reaching a far point on top of the obstacle because the hub and middle foot wouldcollide with the terrain.

There will be times when obstacles will be located in close proximity to each other. Justsuch a case is shown in Figure 5.5. Here the CCR will span other non-traversable segments,


Figure 5.3: When navigating non-traversable segments, IMPASS must step in certain crit-ical contact regions (CCR) before and after the obstacle. These regions are geometricallydetermined from the dimensions of the robot.

Figure 5.4: When navigating non-traversable segments, IMPASS must step in certain crit-ical contact regions (CCR) before and after the obstacle. These regions are geometricallydetermined from the dimensions of the robot.


as long as the step is still possible. However, the points that define the CCR are expanded todefine the boundaries of the other obstacles. All points in CCR1i previous to (xCCRb, zCCRb)are named (xCCRak

, zCCRak) where k = 1 is the boundary of CCR1i and k increases as the

points move towards the obstacle. For CCR2i, all the points after (xCCRc, zCCRc) are named(xCCRdk

, zCCRdk) where k = 1 is the boundary of CCR2i and k increases as the points move

towards the obstacle.

CCR are calculated very similarly for positive and negative obstacles. The method forcalculating CCR for a positive obstacle is as follows

1 Determine (xCCRc, zCCRc): The end point of the non-traversable segments (edge ofobstacle) is equal to (xCCRc, zCCRc).

2 Determine (xCCRa, zCCRa): Draw a circle of radius lmax√

3 (the maximum step distanceof a non-adjacent spoke step) around point (xCCRc, zCCRc) and find the furthest pointback on the terrain which it contacts. Beginning here, iteratively search forward in theterrain for the first valid IMPASS configuration that connect the current search pointto (xCCRc, zCCRc).

3 Determine (xCCRb, zCCRb): Because we have assumed that the terrain can be expressedas a function of x, we can conclude that the beginning point of the non-traversablesegment that start the obstacle is equivalent to (xCCRb, zCCRb).

4 Determine (xCCRd, zCCRd): We must investigate two possible points for (xCCRd, zCCRd),one for an adjacent spoke step and one for a non-adjacent spoke step. The adjacentspoke step point is found by drawing a circle of length lmax (maximum step distance ofan adjacent spoke step) around point (xCCRb, zCCRb); beginning at the most forwardpoint, we iteratively search backward in the terrain for the first valid IMPASS con-figuration that connects the current search point to (xCCRb, zCCRb). The non-adjacentspoke step point is then located by iteratively moving forward from the point just foundand checking for valid steps between (xCCRa, zCCRa) and (xCCRb, zCCRb) with the non-adjacent spoke configuration. If a distance of lmax

√3 from (xCCRb, zCCRb) is reached,

the search is stopped and the adjacent spoke step point is taken as (xCCRd, zCCRd).

The valid step check mentioned in the list was developed as a fairly simple lower levelprogram that checks through IMPASS’s known configurations for a geometrically possiblestep without terrain interference. Stability is absent from this analysis. This is to keep thesoftware more modular. With a moving CG being considered for future robots, stabilitycould require a more involved analysis. The stability will be taken into account later in thecontact point selection process. The method for determining the CCR for a negative obstacleis only slightly different from the positive obstacle determination. The steps are as follows

1 Determine (xCCRb, zCCRb): The begining point of the non-traversable segments (edgeof obstacle) is equal to (xCCRb, zCCRb).


(a)

(b)

Figure 5.5: The critical contact regions (CCR) for obstacle i can include other non-traversable segments.


2 Determine (xCCRd, zCCRd): Draw a circle of radius lmax

√3 (the maximum step distance

of a non-adjacent spoke step) around point (xCCRb, zCCRb) and find the furthest pointforward on the terrain which it contacts. Beginning here, iteratively search backwardsin the terrain for the first valid IMPASS configuration that connect the current searchpoint to (xCCRb, zCCRb).

3 Determine (xCCRc, zCCRc): Because we have assumed that the terrain can be expressedas a function of x, we can conclude that the end point of the last non-traversablesegment in the obstacle is equivalent to (xCCRc, zCCRc).

4 Determine (xCCRa, zCCRa): We must investigate two possible points for (xCCRa, zCCRa),one for an adjacent spoke step and one for a non-adjacent spoke step. The adjacentspoke step point is found by drawing a circle of length lmax (maximum step distance ofan adjacent spoke step) around point (xCCRc, zCCRc); beginning at the furthest backpoint, we iteratively search forward in the terrain for the first valid IMPASS config-uration that connects the current search point to (xCCRc, zCCRc). The non-adjacentspoke step point is then located by iteratively moving back from the point just foundand checking for valid steps between (xCCRc, zCCRc) and (xCCRd, zCCRd) with the non-adjacent spoke configuration. If a distance of lmax

√3 from (xCCRc, zCCRc) is reached,

the search is stopped and the adjacent spoke step point is taken as (xCCRd, zCCRd).

Essentially the difference between the positive and negative CCR is that they are mirrorimages if flipped over a vertical axis. This is reflected in the determination of the CCR.

Now that the full range of points which can be used to traverse an obstacle is known, wewill develop a Working Contact Range (WCR) which reduces the CCR to a range of pointsthat are practical for the robot. The WCR reduces the CCR in 2 ways. First, all unstabletransitions are thrown out. Second, the WCR eliminates contact points within a distance oflobsmin from the beginning or end of any non-traversable segment. Both of these constraintsrefine the potential contact regions to something the robot can actually be expected toexecute.

Figure 5.6 shows the terrain from Figure 5.5 reduced from CCR to WCR. The non-adjacentstep is unstable in most configurations for the current prototype. This significantly reducesthe WCR. A lobsmin of 2 inches is used, further reducing the WCR. We now have arrived ata set of practical contact range that account for the robot’s physical capabilities.

Critical Contact Point (CCP) Determination

Once the WCR have been determined, we can determine where best to put the contactpoints. Here is where IMPASS can capitalize on the deliberative approach. Figure 5.7 showsan obstacle which could be traversed two different ways, one of which is significantly more


(a)

(b)

Figure 5.6: The working contact regions (WCR) for obstacle i can include other non-traversable segments. The WCR may be significantly reduced from the critical contactregion (CCR).


Figure 5.7: When challenged by certain obstacles, IMPASS will have multiple configurationsavailable with which to traverse the obstacle. Some of these configurations will be moreintelligent than others.

efficient than the other. The algorithm used for choosing contact points will determine theapparent intelligence of the robot.

The first step in determining CCPs is to further refine the WCRs. Specifically, combiningand removing WCRs. For example, the dark IMPASS in Figure 5.7 has removed WCRs fromadjacent obstacles and the greyed IMPASS has combined WCRs. The purpose of combiningWCRs is to eliminate unnecessary steps. The combined segments can then be searched forpossible omissions.

First, the software combines all the overlaps. In this analysis, each traversable segment isevaluated independently of the WCR it is associated with. For example, (xCCRa1 , zCCRa1) to(xCCRa2 , zCCRa2) from Figure 5.6(b) is independent from (xCCRa3 , zCCRa3) to (xCCRb, zCCRb).All traversable segments are added to a new array of segments. Whenever two segmentshave an overlap, the common range to both is accepted as the new segment.

The combining of segments must be done from the beginning of the terrain to the back. Thereason being that as traversable ranges are combined and therefore shrunk, it will changethe WCR on the other side of the obstacle. This concept is shown in Figure 5.8. Thecombination occurs from the from the current position in the terrain array to the furthestaway because it is assumed that the quality of perception data will degrade with distance.This should minimize motion planning mistakes based on poor perception data. The newarray is called the combined working contact region (CWCR) array.

Once the WCRs have been combined, the motion planning can begin to search for contactregions that can be bypassed. Here, as with the combining of contact regions, we start


Figure 5.8: Working Contact Ranges (WCR) for multiple obstacles can be combined toreduce the number of steps required. This can reduce the WCR for future obstacles.

at the beginning of the terrain array where perception data will be more accurate. Anynon-adjacent range in the CWCR array that can be connected by a geometrically valid andstable step will result in the exclusion of the intermediary contact regions.

Using the methods of combining and bypassing contact point regions, we have filtered outWCRs that would have otherwise impeded efficient operation. This final step to reform thecontact regions gives us our finalized contact region (FCR) array.

The algorithm used for contact point selection is designed to maximize the success rate ofthe robot. Most importantly, this involves successful traversal of obstacles. The worst casescenario is slipping off the ledge of an obstacle. Therefore, contact points must be chosenthat step well beyond the top of an obstacle, for both positive and negative obstacles. Toaid in this, the contact points below obstacles can be chosen close to the obstacle wall.

These concepts are applied to a algorithm that builds contact points as it steps through theFCR array. The evaluations are as follows:

1 If the contact range is in a depression, meaning it has a negative obstacle behind anda positive obstacle in front, the contact point is chosen at the middle of the range.

2 Else if the contact range has a positive obstacle in front, the contact point is chosenat the front of the range

3 Else if the contact range has a negative obstacle behind, the contact point is chosen atthe rear of the range

4 Else choose the contact point in the middle of the range

The result of this algorithm is an array of contact points that provide a means of traversingthe obstacles. To complete the information for these contact points, each must have a


transition associated with it. Here, we consider the groundwork laid in the transitionschapter. There are four basic transitions we will use: Non-Adjacent Ascending Transition,Adjacent Ascending Transition, Descending Transition, and Default Transition. The Non-Adjacent Ascending Transition is used whenever the Adjacent Ascending Transition is notsufficient. The ascending and descending transitions are used for obstacles above a heighthobs1. To recap, the ascending and descending transitions use a switching angle of θ = 0◦ andθ = 60◦ respectively. The Default Transition uses a constant switching angle of θ = 30◦ forany obstacle of lesser height than hobs1. For the experiments performed on the DeliberativeMotion Planning, a value of 9 inches was used for hobs1.

Applying the transition determination criteria finalizes the critical contact point array. Thenext step is to determine the contact points in between non-adjacent critical contact points.

Intermediary Contact Point (ICP) Determination

The primary goal of contact point selection, which is to successfully traverse the obstacles inthe terrain, has been satisfied by the process presented in the CCP Determination section.With the selection of the ICPs, we can now factor in the secondary goal of minimizing heightchange and keeping the height relative to the terrain close to the nominal walking height.These two goals can compete, which the ICP determination method takes into account.

Each pair of CCPs that cannot be connected with a valid step must have at least onetransition in between. The first step in finding those intermediary contact points is todetermine the height of the robot’s hub center at the the bordering CCPs. The hub centerpoints are (x0,i, z0,i) and (x0,i+1, z0,i+1). Calculating a height requires a reference. It is notimportant what reference is used, just that it is consistently used to calculate all heights.Here we will use the z coordinate for the back contact point, CCPi. The heights of the backand front CCPs are given as hCCPi

and hCCPi+1respectively. These heights are averaged to

find hCCPiavg.

The ICP algorithm strives for linear interpolation of the hub center position. However, thisis a difficult proposition with uneven terrain. The ICP algorithm accounts for the contoursof the terrain by adapting the desired hub center position found using linear interpolation.To describe the features in the terrain, we will map a Lagrangian interpolation of the terrainonto the linear interpolation of the hub center path.

The degree of the Lagrangian polynomial is determined by the estimated number of stepsthat will be taken. The initial estimation for the number of steps is determined using anaverage step length taken at height hCCPiavg, defined as dgavg . These variables are shown inFigure 5.9. The distance between CCPi and CCPi+1 is divided by dgavg . This number isthen rounded according to hCCPiavg. If hCCPiavg is larger than the nominal walking heightthen we round up, otherwise it is rounded down. The resulting value n is the number ofsteps. The process of calculating n is summarized in the equation


Figure 5.9: The first step in the ICP selection algorithm is to estimate the number of stepsthat will need to be taken. This is done by calculating an average step distance dgavg basedon the average height of the robot between hCCPi

and hCCPi+1.

n =| ~CCPiCCPi+1|

dgavg

Rounded up if hCCPiavg > hnominal otherwise round down. (5.1)

The roots of the Lagrangian polynomial are determined by vertically projecting n−1 evenlyspaced points along the terrain between CCPi and CCPi+1. The distance that separatesthese contact points is d′gavg

. All together this gives n + 1 roots for the polynomial. Beforethe Lagrange polynomial is solved, the roots must be mapped onto the linear interpolationof the hub center path. This is done by applying a z axis correction factor, ∆z, accordingto the slope of two lines: the hub center linear interpolation and the contact point linearinterpolation. The lines are described by equations

OiOi+1 : z = m1x + z1 (5.2)

CPPiCCPi+1 : z = m2x + z2 (5.3)

For the purposes of this analysis, we translate the OiOi+1 line down to CCPi, making z1

and z2 equal. CCPi is set as the origin of a temporary coordinate system to make the zintercepts zero. These two lines are illustrated in Figure 5.10. ∆z is calculated by

∆z(xk) = (m1 −m2)xk (5.4)

Applying the correction gives a new set of coordinates, where x is unchanged and the new zcoordinate for root k, given as z′k, is calculated by the equation


Figure 5.10: To find the roots for the Lagrange polynomial, n − 1 evenly spaced contactpoints are placed between CCPi and CCPi+1. These must be vertically adjusted by ∆zk

based on the difference in slope between the hub center line (z = m1x + z1) and the contactpoint line (z = m2x + z2)

Figure 5.11: The vertically adjusted contact points (x′k, z′k) are used as the roots for a

Lagrange polynomial. The function is translated by hCCPiin the positive z direction to be

used as a hub center path.


z′k(xk) = ∆z(xk) + zk (5.5)

In determining the Lagrange polynomial, we must first solve for the Lagrange basis polyno-mials given by the equation

lk(x) =n∏

j=0,j 6=k

x− xj

xk − xj

(5.6)

These basis polynomials can then be combined to give the Lagrange polynomial using thefollowing equation

L(x) =n∑

k=0

zklk(x) (5.7)

we then translate the Lagrange polynomial to the hub centers, simplify to a n degree poly-nomial in the form

L(x) = anxn + an−1xn−1 + . . . + a0 + hCCPi

(5.8)

Figure 5.11 shows both the original and translated Lagrange polynomial function. TheLagrange equation is not necessarily an accurate representation of the terrain, it is insteada particular representation of the terrain that focuses on terrain features in the proximity ofthe ICPs. We don’t care so much about the terrain in between contact points since we havealready determined it is traversable, and therefore without drastic elevation changes. Thisis an advantage of the rimless spoke wheel design.

Now that we have a desired hub center path, the next step is to iteratively solve for thecontact points. The solver starts with the initial contact point, CCPi, and uses the geometryof the θ = 30◦ configuration to determine the set of contact points along the terrain thatcorrespond to the given L(x). Figure 5.12 demonstrates the set of contact points for a giventerrain and L(x). The contact points are named (xk, zk) where k is 0 for the first contactpoint and is incremented by one for each subsequent contact point. The contact points arefound until one is equivalent to CCPi+1 or goes beyond it. The end contact point will rarelyline up with CCPi+1 on the first run. To converge on a solution in which the final contactpoint aligns with CCPi+1, a vertical adjustment, δz, is applied to L(x). The algorithm thatis applied for determining δz after each run is

1 If any of the steps are out of range for the θ = 30◦ configuration, then a δz that bringsthe robot back into its physical range will be permanently applied. For example, if astep is too high, then δz will be negative for all subsequent iterations.


Figure 5.12: The first iteration of contact points for the Lagrange polynomial function, L(x),will most likely not converge on the end contact point, CCPi+1. Here the first iterationexceeds the physical limitations of the robot. According to algorithm 5.1.3, δz will benegative for every successive iteration until a solution is reached.

Figure 5.13: IMPASS is shown here reaching a solution for the ICP of a given terrain andbounding CCP.

2 If the conclusion of the iteration is reached and has not converged, and the robot hasnot broken a physical constraint at any time during any iteration, then we direct δzbased on the proximity of CCPi+1 to the last two contact points. If the last contactpoint is closer to CCPi+1 then a negative δz is applied. Otherwise a positive δz isapplied.

3 The solution is reached such that CCPi+1 aligns with the last contact point.

The solution for the terrain given in Figure 5.12 is given in Figure 5.13.

The set of contact points produced by this solver are the ICPs for a given section oftraversable terrain. Applying this solver to the rest of the non-adjacent CCP gaps will give acomplete contact point array in which each pair of contact points is adjacent. Additionally,each pair of adjacent contact points has been assigned one of four transition configurations:


high climb, ascending, descending, or moderate obstacle transition. This completes the con-tact point selection process. The robot has a separate motion planning component thathandles the execution of the steps. This will be discussed in the next section.

5.1.4 Step Execution

Once contact points have been found, the robot must determine how to get from point topoint. The step executer uses a path planning algorithm to accomplish this. Since the hubcenter position for each step has been determined by the contact point planner, the stepexecuter solves a hub center trajectory for the robot.

There are many mathematical approaches to solving a path for multiple points, so we mustfirst consider the factors that surround this problem. On a platform which is traversingunstructured terrain, look-ahead distance is a consideration. It is beneficial to plan ahead,but it is inevitable that the perception data will be updated on a regular basis, causing thecontact points and path of the robot to be replanned on a regular basis. Therefore, we donot need a large look-ahead distance.

A second consideration is physical constraints. Even though it is known that each transitionconfiguration is geometrically possible, it would be possible to plan a hub center path thatexceeds the geometric constraints of the robot. If a step execution algorithm allows forsolutions outside of the physical constraints of the robot, then clipping will occur. Thisconcept is presented in Figure 5.14.

In this section we discuss the initial path planner developed for IMPASS and outlines theimplementation of an improved path planner. The first path planner developed for IMPASS,which was implemented in robot experimentation, is linear interpolation between adjacenthub center points. This algorithm has a single point “look ahead” distance which would begood for short range sensors. It is possible for clipping to occur, so a check is preformed toensure that no motor commands are sent that cannot be realized. An example of the linearhub trajectory can be seen in Figure 4.4. A clear benefit of this method is that it is verycomputationally simple. The downside is that there is no smoothing between steps. A newpath planner is proposed to help with this smoothing.

A potential improvement to the path planner would use a Lagrange polynomial to connectthe hub center positions for multiple steps with a high order polynomial. This implemen-tation would be very similar to the contact point selection process described in the ICPDetermination section. The first root of the polynomial would be the previous hub centerposition. The robot would have a look-ahead distance of n steps, with the hub center fromeach step being a new root. Using Equations 5.6 and 5.7 we arrive at a polynomial whichdescribes a desired hub center path. This implementation has not been tested, but wouldalmost certainly provide a smoother hub center trajectory than the linear interpolation.


Figure 5.14: It is possible to choose a hub center trajectory that is physically impossible.This will cause clipping, shown by the hashed line.

Non-Contact Spokes

Up until this point, we have only discussed the motion planning of the contacting spokes.Unfortunately, we cannot forget about the non-contact spokes. The main consideration hereis terrain interference. An simple obstacle avoidance technique is described in this sectionwhich keeps the non-contact spokes from running into terrain features.

Behaviorally, the obstacle avoidance can be classified as a subsumption architecture. Thereare two behaviors, standard spoke motion and obstacle avoidance. If an “imminent collision”warning is triggered, then the obstacle avoidance behavior takes over.

The standard spoke motion behavior simply uses a constant velocity to go from one positionto another. Once the spoke leaves the ground, the length is referred to as lnc1, and theopposite side of the same spoke is L− lnc1, as shown in Figure 5.15. The name for the lengthof the given spoke changes to lnc2 as soon as the next spoke leaves the ground. Finally, whenthe opposite end of the given spoke touches the ground it is referred to as lnc3. The standardspoke motion behavior changes the position of the spoke from lnc1 to lnc3 with a constantvelocity.

The obstacle avoidance behavior is triggered by an “imminent collision” warning. This warn-ing is determined by a software component that calculates the foot paths of non-contactingspokes. The simulator can be used to display these foot paths, shown in Figure 5.16.

If a future collision is detected, the path of the spoke will be adapted to keep a certain


Figure 5.15: It is possible to choose a hub center trajectory that is physically impossible.This will cause clipping, shown by the hashed line.

(a) Foot paths for flat terrain.

(b) Foot paths for obstructed terrain: Without Collision Avoidance

(c) Foot paths for obstructed terrain: With Collision Avoidance

Figure 5.16: A program is used to calculate IMPASS’s footpaths. These paths are used todetect future collisions.


distance, lcolavoid, from the terrain. For the software tested in the simulator, shown in Fig-ure 5.16(c), a distance of 1 inch was used. This same distance was used in the experimentsdiscussed in the next chapter.

5.1.5 Experimenting with Deliberative ICPS Motion Planning

Deliberative motion planning software was tested in two phases. The first phase was insimulation. A custom built simulator was developed for analysis of IMPASS in the two-dimensional sagittal plane. The second phase of testing was conducted on the prototype.There were unforseen interactions with the hardware and the motion planning for the pro-totype. Finally, this section concludes with what adjustments could be made to the motionplanning algorithms.

Testing in simulation provided feedback on two critical aspects of motion planning. First, thesimulator showed the hub rotation and position of all the spokes. Simultaneously controllingmultiple spokes and keeping track of them throughout operation was an initial challengewhich the simulator helped with. Any glitches in position control could be detected withthe visual interface. Secondly, terrain interference could be checked for in the simulatorby putting in the test track geometry. The current obstacle avoidance software was tunedusing this method. See Figure 5.16 for a visual of the simulator’s assistance with obstacleavoidance.

The finalized deliberative software showed good results in simulation and also providedinsight into the limitations. A couple of examples of simulation tests can be found in theAppendix in Figures A.1 and A.2. It can be seen that the chosen contact points allow foreffective traversal of the terrain. Where CCR overlap, the software was able to combinethe regions thereby minimizing unnecessary steps. One major limitation of the software isthe foot trajectory planning. Since the obstacle avoidance of the feet is in the lower levelmotion planning, it is unable to plan ahead for obstacles. The result is that the collisionavoidance will sometimes require a very quick moment of the spoke that is outside of thephysical capabilities. One solution is to place a foot trajectory planner in the higher levelmotion planner. This will add computation time, but as long as this is not deemed to betoo slow the foot trajectory planner could add significant value to the robot.

After checking the software in simulation, it was implemented on the robot. Figure 5.17shows a test for the robot on uneven terrain. To truly test the motion planning software,perfect terrain information was read in. It would be hard to track errors in the perceptionsoftware, and be sure it was not motion planning’s error.

It was already known that gear train backlash in the hub drivetrain and spoke compliancecaused problems with the motion planning software- this was expected. The solutions tothis problem have been discussed in the experimental results for the Transitions chapter.One new problem did surface during this multiple step test. It was found that the robot’s


(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k) (l)

(m) (n) (o)

Figure 5.17: IMPASS traversing obstacles using the Deliberative ICPS Motion Planner. Aperfect terrain model was fed into the software to eliminate extraneous variables.


contact points could not be modeled as point contacts. The foot actually rolls over theterrain surface, gaining ground that is not accounted for in the motion planning software.This can be seen in Figure 5.17(j), where the robot’s spoke runs into the terrain feature.Here, the foot placement is supposed to be 3 inches in front of the obstacle, but instead endsup right against the wall. Figure 5.17(k) shows the foot in the corner preceding the obstacle.

Future software revisions of the motion planning component should include a model of thisrolling contact point. This mainly effects the contact point planner. In Figure 5.17 the robotwas about 3.5 inches past the expected position after 7 steps, giving us a forward creep ofabout half an inch per step.

5.2 Reactive Motion Planning

The downside to the deliberative motion planning approach is that it is dependent on reliableand complete sensor data. Should bad or incomplete sensor data be used to make a decision,the deliberative approach could fail. The reactive approach provides a more simple androbust method of motion planning. Here we discuss implementations that focus on a closelink between sensors and actions that produce failure resistent code.

For a robot with as many degrees of freedom as IMPASS, there are inevitably many ways toimplement the reactive paradigm. A reactive approach is closely tied to the sensors that arebeing used. Attractive sensors for using on a reactive IMPASS are an inclinometer, tactilesensors for the feet, and torque sensors. The inclinometer can be used to provide a generalidea of the slope of the terrain. The binary functionality of tactile sensors makes themattractive for just about any reactive robots. Torque sensors can be used to detect whetherthe robot has encountered an obstacle and is climbing.

Currently the robot is equipped with an inclinometer, but not tactile or torque sensors.Equipping the feet with tactile sensors is a challenge because of wiring. With the spokesmoving through the hub center, there is no conventional solution to communicating anelectrical or mechanical signal between the feet and the hub. Wireless technology may beable to provide a telemetry solution, but this has yet to be investigated. Torque sensorscould be added within the body. However, getting accurate indication of collisions fromthese devices may be difficult.

The first algorithm presented here is completely based on the inclinometer readings. It wasdeveloped by Eric Russell, a member of the IMPASS senior design team. The algorithm isbased around the Equivalent Spoke Length Transition. On flat ground this is described byθ = 30◦. As the terrain increases in pitch, θ is reduced by the terrain angle α. The same istrue for a descending terrain. This algorithm can be described by the equation:

θ = 30◦ − α (5.9)


This algorithm also uses a different non-contact spoke velocity curve. The spoke is extendedpast its desired length, L− lnc3, and then retracted to its desired length as the robot nearsthe contact point. This allows IMPASS to be stable when a descent is ahead, but cannotbe detected. Since there is no active terrain sensing, it does not matter if the feet slip.Figure 5.18 shows this algorithm implemented on the robot. In the pictures, IMPASS istraversing a bean bag. This dynamic terrain would be very difficult for the deliberativemotion planning, but poses no problem for the reactive motion planning.

Both deliberative and reactive approaches have their strong points. The deliberative isstronger is a more structured, static environment. However, for unstructured and dynamicenvironments, the reactive motion planning is clearly superior. There is still much moredevelopment that can go into developing algorithms for both approaches. This paper haspresented a couple of algorithms that are successful and have been implemented on theIMPASS prototype. As the IMPASS platform is further developed, these algorithms can beused in the testing of other hardware and software features.


(a) (b) (c)

(d) (e) (f)

(g) (h)

Figure 5.18: IMPASS traversing a dynamic terrain (a beanbag) using the reactive algorithmbased on inclinometer readings for the body position.

Chapter 6

Conclusion

In the field of mobile robotics, it has proven difficult to design a simple, yet highly mobilerobot. Wheeled robots have the advantage of being very simple and efficient. However theyhave trouble in unstructured terrains where the height of the ground can change suddenly.Leg-wheel robots are more mobile than wheeled robots because of increased DOF and discretecontact points. The downside to legged robots is that they require more complex mechanismsto realize their high mobility.

The advantages of wheels and legs can be combined into a single mechanism to create asimple and highly mobile platform. IMPASS is a leg-wheel hybrid robot with good mobilitycharacteristics and a simple actuation concept. The IMPASS mobility platform is a rimlesswheel with individually actuated spokes. Each spoke is capable of changing its length,allowing IMPASS to coordinate motion over complex terrains. The discrete contact pointsof the spokes allow IMPASS to “step” over obstacles significantly larger than the nominalwalking height.

In this paper, we discussed the hardware development that has resulted in the constructionof a functional IMPASS prototype. The prototype has two rimless wheels with six spokeseach supporting a body and tail. The spokes travel through the hub center effectively actingas two spokes separated by 180◦. This design only requires three spokes total for each wheel.The spokes are compliant, providing built-in shock absorbtion and allowing kinetic energyto be passed from step to step. A chain belt and sprocket mechanism is used to actuatethe spokes. Both wheels are driven by motors inside of the body. All of the actuatorsare controlled by an on-board PC-104 computer running Labview. The robot is capableof gracefully traversing obstacles over 1.5 times the nominal walking height and descendingobstacles greater than the nominal walking height. The prototype has proven an excellenttest bed for development of intelligence software.

IMPASS is not a practical platform without some form of automated motion planning.Controlling each independent degree of freedom of the system in coordinated motion is a

80

Shawn C. Kimmel Chapter 6. Conclusion 81

task most appropriate for a computer. Two gaits are possible with the IMPASS platform.Each is described by the number of feet in contact with the ground during a step. IMPASShas a one-point contact gait and two-point contact gait. Of these, the one-point contactis more attractive as a mobility platform because it has two DOF as opposed to the singleDOF possessed by the two-point contact. This paper has laid the ground work for intelligentmotion control, and discusses two very different philosophies in developing the motion controlsoftware.

Preceding any discussion of motion planning, we first seek understand the transitions thatthe robot undergoes from step to step. Transitions can be described by the angle that theback spoke makes with the z-axis, given as θ. There are infinite number of angles that therobot can switch at. Climbing and descending large obstacles is best done with angles of 0◦

and 60◦ respectively, because the extended spoke is oriented vertically. The compliance inthe spokes is least prevalent with vertically oriented spokes. There are two configurations inwhich IMPASS can climb, one using an adjacent spoke and the other with a non-adjacentspoke. In the non-adjacent spoke configuration, the forward spoke is used to actually pullIMPASS up onto the obstacle. This configuration can be used to climb much higher obstaclesthat the adjacent spoke climbing case. For normal walking, we have chosen a switching angleof 30◦ because it can achieve a future contact point that is either higher or lower with equalability. Using these transitions assists IMPASS in determining an intelligent method fortraversing terrain.

The two philosophies that can be used to approach the motion planning problem are thedeliberative approach and the reactive approach. Using the deliberative approach involvesthe construction of a world model. Decisions are made based on this world model and anya-priori knowledge, allowing a fairly comprehensive path planning solution. On the otherhand, a much simpler approach is reactive in nature- meaning that the actuator control isclosely linked to the sensor inputs.

There are two basic methods of deliberative motion planning for IMPASS, Initial ContactPoint Selection (ICPS) and Initial Body Path Selection (IBPS). This research focuses on theICPS method. Contact point selection can be broken down into three steps, Critical ContactRegion (CCR) identification and refinement, Critical Contact Point (CCP) determination,and Intermediary Contact Point (ICP) determination. The CCR step determines where therobot must step to traverse obstacles; the CCP step determines where in those ranges is mostideal for IMPASS; and finally the ICP step calculates the best way to get from one obstacleto the next. Each pair of adjacent contact points is assigned a transition configuration(i.e. ascending, descending, or default). Once contact points have been selected, a bodypath is chosen for the robot that moves it from one transition state to another. Motionplanning must also consider the the non-contact spokes in that they must avoid any terraininterference. To this end, collision avoidance software has been developed that plans thefoot trajectories. When a future collision is detected, the spoke path is altered to maintaina safe distance (one inch for the current prototype). The deliberative motion planning wastested with obstacles and ramps using ideal sensor data. The approach proved to be effective


in simulation but in reality was unable to account for the distance that the robot advanceseach step as the foot rolls over the ground. With minor modifications, this algorithm couldbe effective in a physical environment.

Deliberative planning is dependent on reliable and complete sensor data. While this as-sumption may be sufficient for the two-dimensional sagittal plane, it is a much more difficultproblem in unstructured three-dimensional environments. The reactive approach to motionplanning focuses on direct sensor data rather than a world map. For IMPASS, this caninclude, but is not limited to, inclination of the body, touch sensors on the feet, and torquesensors on the hubs. Information from these sensors can be directly tied to simple motionplanning behaviors. The algorithm discussed in this chapter uses inclinometer data to plansteps using the Equivalent Leg Length transition. This approach has proven effective overobstacles less than the nominal walking height of the robot and dynamic terrain. However,it is not yet capable of handling larger obstacles.

The considerations for motion planning in the two-dimensional sagittal plane have beendiscussed. Also, implementations of the deliberative and reactive approaches to motionplanning have been presented with their associated advantages and limitations. There isconsiderably more research that can be done with the IMPASS platform in light of thefindings from this thesis.

6.1 Future Work

As this paper has shown, IMPASS is an excellent mobility platform. There is still much workthat can be done to further understand the full capabilities of this robot. This paper hascovered the considerations for motion planning, but has only shown two implementations.For the deliberative approach, the ICPS method has been fairly well covered. An adjustmentneeds to be made that accounts for the forward creep of the feet during a step. Also a model ofthe spoke compliance and hub gear train backlash would provide for more accurate executionof the motion. The other deliberative approach, the IBPS method, has still yet to be delvedinto. This method could be very valuable for payload impact management.

There is still much work that can be done with reactive motion planning paradigms. Cur-rently the robot can climb moderate obstacles. Once the robot is outfitted with tactile andhub torque sensors, new algorithms can be tested that will advance the mobility of the robotwith reactive software.

All the experiments conducted for the deliberative software were done with ideal terraininformation. IMPASS is not yet capable of Simultaneous Localization and Mapping (SLAM).This is a difficult problem on any mobile robot in an unstructured terrain. Currently a LaserRange Finder (LFR) and cameras are outfitted on the robot. These sensors can be integratedinto a reliable perception suite using current SLAM techniques.


Lastly, there is a whole range of research topics for IMPASS in three dimensions. Themotion planning and perception problems in particular become far more complicated withthe addition of a third dimension. However, three-dimensional operation is where IMPASSmust succeed to become a practical mobility platform for real world applications. It remainsto be seen whether the deliberative or reactive approach will prevail for this robot, or whethera hybridization of the two will be the answer. With effective motion planning and perception,the applications for this platform are endless. The world would greatly benefit from suchwork.

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Appendix A

Deliberative Motion PlanningSimulations

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j) (k)

Figure A.1: IMPASS climbing a set of three six inch steps and down an eleven inch obstacle.The Default Transition is used throughout most of the simulation with the exception of thelarge negative obstacle. Here the Descending Transition is used.

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(a) (b) (c)

(d) (e) (f)

(g) (h) (i)

(j)

Figure A.2: IMPASS climbing a set of terrain features in simulation. CCR and CCP areshown as dots on the surface of the terrain.

considerations for and implementations of deliberative and

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