simplified motion control of a two axle compliant framed wheeled mobile...

TRO A05-34/A2003-291 Accepted by the IEEE Transactions on Robotics 1

Abstract—Kinematic models and motion control algorithms

for a two-axle compliant frame mobile robot are examined. General kinematics describing the compliantly coupled nonholonomic kinematics are derived using velocity constraints that minimize traction forces and consider foreshortening of the frame. Given the complexity of these equations, steering ratio, a, is defined to describe the relative heading angles of the front and rear axles. Simplified kinematic models are developed based upon a (Types I, II, and III) and the reference point used to guide the robot. Physical limitations and performance metrics (lateral mobility and maneuverability per unit of traction force) are derived to evaluate the models. Six groups of simulations and 24 experimental tests consisting of 120 trials evaluate the performance of the algorithms on carpet, sand, and sand with rocks. Results indicate that Type I (curvature based steering) provides superior maneuverability and regulation accuracy whereas Type II provides excellent lateral mobility at the cost of high traction forces, reduced accuracy, and potential singularities. Both models offer significant reductions in complexity for simplified control using standard curvature based unicycle control algorithms. These results support expectations derived from performance metrics and physical limitations. Experimental results also demonstrate efficacy of the robot to adapt to and maneuvering over extremely rugged rocky terrain.

Index Terms— Compliant, Kinematics, Motion Control, Mobile Robot.

I. INTRODUCTION inematic models and motion control algorithms for a two-axle Compliant Framed wheeled Modular Mobile

Robots (CFMMR), Fig. 1, are the subject of this research. The CFMMR concept is unique in two ways. First, it uses a novel yet simple structure to provide suspension and highly controllable steering capability without adding any additional hardware to the system. This is accomplished by using

Manuscript received September 2, 2003 and resubmitted January 24, 2005. Conditionally accepted August 7, 2005 and resubmitted October 10, 2005. Accepted November 24, 2005.

M. A. Minor is with the University of Utah, Mechanical Engineering, 50 S. Central Campus Dr. Room 2110, Salt Lake City, UT 84112 USA; phone: 801-587-7771; fax: 801-585-9826; email: [email protected].

B. A. Albiston was a graduate student at the University of Utah, Mechanical Engeineering. He is now with Orbital Sciences Corporation, Chandler, AZ 85248 USA (e-mail: [email protected]).

C. W. Schwensen was a graduate student at the University of Utah, Mechanical Engineering. He is now with Rosetta Inpharmatics, subsidiary of Merck & Co., Seattle, WA. (e-mail: [email protected]).

compliant frame elements to couple rigid differentially steered axles. In this study, a partially compliant frame provides roll and yaw Degrees of Freedom (DOF) between the axles. Relative roll provides suspension capability in order to accommodate uneven terrain, and yaw allows the axles to independently change heading for steering. Steering and maneuvering of the system are thus accomplished via coordinated control of the differentially steered axles. Since each axle can be steered independently, the system provides the capability to control the shape of its frame and thus enhance maneuverability in confined environments.

A second unique aspect of the CFMMR is its predisposition for modular mobile robotics. Reconfigurable modular robotic systems have been of keen interest to researchers during the last decade due to their improved ability to overcome obstacles and perform more tasks using a single hardware platform. Towards this goal, researchers have investigated homogenous robotic modules for reconfigurable manipulation [1, 2], mobility [3, 4], or combinations therein [5-7]. Homogeneity is argued to reduce maintenance, offer increased robustness through redundancy, provide compact and ordered storage, and increase adaptability [3, 8]. The CFMMR allows these concepts to be extended to wheeled mobile robots. Several modular configurations are shown in Fig. 2 which includes two-axle scouts, a four-axle train, and a four-axle moving platform. The scout is suited to reconnaissance and exploration, the train is tailored to transporting payloads extended distances, and the platform is adapted to moving large objects. The utility of the system is greatest for resource limited applications, such as space exploration or military operations. Potential civilian applications include farming, forestry, and mining. This study focuses in particular on the two-axle scout configuration.

A third unique aspect of the CFMMR is the simplicity of its mechanical design. At the most fundamental level, the axle

Simplified Motion Control of a Two Axle Compliant Framed Wheeled Mobile Robot

Mark A. Minor, Member, IEEE, Brian W. Albiston, and Corey L. Schwensen

K

Fig. 1. CFMMR two-axle scout experimental configuration.


modules are basic differentially steered mobile robots; they are rigid structures providing an interface for the frame that support two independently controlled wheels. Other than the wheel drive systems, there are no other moving parts in the axle modules. The compliant frame then provides flexible coupling between the axles to allow them to steer independently and conform to terrain variations. This alleviates the need for complicated linkages and associated hardware typical of steering and suspension systems. The compliant frame thus significantly reduces complexity and cost of the mechanical structure. Since the only moving parts of the CFMMR are the wheel drive systems, very few components are subject to wear. Those that do wear are similar throughout the entire system, which simplifies design-life and maintenance issues. Thus, the simplicity of the CFMMR design allows straightforward reduction of mechanical failure probability.

While the CFMMR provides new capabilities in steering, mobility, and reconfiguration, it also introduces new challenges that must be addressed in motion control [9, 10], dynamic control [11], sensor instrumentation and data fusion [12]. While each of these is important to the implementation of the system, this paper addresses issues related to controlling the motion of the robot. Frame compliance allows a wide range of steering algorithms to be applied, but it also complicates the kinematic model appreciably and provides significant challenges to motion control algorithms. To simplify the motion control task, steering constraints are established in this research by the ratio of the front and rear axle headings. These constraints can provide significant simplification of the kinematics and allow existing unicycle motion control algorithms to be applied. Hence, we first develop general kinematic models of the system and then illustrate the affect of these simplifications when considering performance metrics such as mobility, maneuverability, traction requirements, and control simplicity.

The structure of the paper follows. The CFMMR will first be compared to other similar robots in Sec. II. General kinematics are derived in Sec. III and three simplified models are proposed in Sec. IV. Performance metrics and limiting factors are discussed in Sec. V. Motion control algorithms are described in Sec. VI, which are applied in simulation and experiment to evaluate model performance in Sec. VII. Concluding remarks are provided in Sec. VIII.

II. BACKGROUND A limited number of compliant vehicles can be found in the

literature, and none possess a similar highly compliant frame whose deflection is controlled by coordinated actuation of the wheels. Earliest found reference is a system proposed for planetary exploration using compliant members to provide roll and pitch DOF for suspension of the axles [13]. This concept was later extended [14] to the frame of a vehicle composed of helical spring(s) with hydraulic cylinders used to control deflection. In each of these cases, compliance was introduced

for accommodating terrain. The CFMMR uses passive compliance in a similar spirit to provide independent suspension and advanced steering control between the axles without additional hardware or actuators. This vastly reduces the number of components required to construct a system, reduces probability of component failure, and allows aspects of modularity to be exploited.

More recent research has introduced compliance for accommodating measurement error and preventing wheel slip from occurring between independently controlled axle units on a service robot [15]. This robot is similar in spirit to the CFMMR in that it allows relative axle yaw, but this is provided by rotary joints connected to the ends of a frame with limited prismatic compliance. Other flexible robots use actuated articulated joints to provide relative motion between axles, as in the case of the Marsokhod rover [16] and other six wheeled research rovers with high relative DOF [17]. These actuated kinematic structures provide more direct control of their shape than the CFMMR, but it is accomplished at the expense of system complexity. The point is that the CFMMR provides similar capability to adapt to terrain, but does not require any additional hardware or mechanical systems.

Snake-like mobile robots and continuum type manipulators also bear resemblance to the CFMMR, although their applications are usually oriented towards search and rescue or material handling, respectively [18]. Each consists of a serial chain of modules. Trunk like continuum manipulators are similar to the CFMMR in that the interconnection is achieved by compliant beams or springs, but articulation is driven by tendons that exert bending moments on the modules [19, 20]. Hyper redundant manipulators replace the compliant member with rigid segments and active articulated joints [21]. This is similar to active-joint articulated snake-like mobile robots [3, 4] where the serial chain interacts with the ground to propel the robot. Many of these have wheels on the modules for reduced friction or to establish well defined kinematics. In some cases the articulated joints are active and the wheels are passive [22], and in other cases the wheels are active and articulation is either partially active [23, 24] or entirely passive [25]. Active wheels provide direct control over forward velocity and are better for traveling over terrain. Active joints allow direct control over robot shape, such as for climbing over very large obstacles, but they are usually slow

Fig. 2. Modular configurations.


due to high torque demands and limited space. Thus passive compliant joints have evolved for natural terrain adaptation, to reduce impact loads damaging to active joints, and to facilitate faster travel over rugged terrain [25]. Of the snake like robots, the CFMMR is similar to [25] in terms of rover type application and architecture, but complex and expensive mechanical joints with higher potential for failure are created for [25] to emulate the simple and cost effective compliance of the CFMMR. The CFMMR is also modular and facilitates numerous configurations and applications.

Control of mobile robots and nonholonomic systems has received a great deal of attention in recent years For a thorough survey of nonholonomic control techniques see the review in [26]. Many of these are well suited to unicycle type velocity constraints. While the kinematics of the compliant framed mobile robot are much more complex, we will show that they can be described in an equivalent coordinate frame that admits familiar unicycle motion control algorithms. In particular, we apply controllers discussed by Indiveri [27] and Tayebi [28] in conjunction with a dynamic extension to accommodate non-ideal initial conditions and provide drift free motion control. This is similar in spirit to the extension performed by Astolfi [29] in order to accommodate nonholonomic systems with drift.

III. GENERAL KINEMATIC MODEL Fundamentally, each axle module in the CFMMR is a

differentially steered unicycle type mobile robot, Fig. 3. Unlike unicycle robots that gain stability from additional castor wheels fore and aft of the axle, the CFMMR uses frame members to couple and stabilize multiple axles. Compliant coupling provides suspension to the axles and suits the system to uneven terrain since the axles can deflect to accommodate surface variations. Compliant axle coupling also implies that the kinematic model of the system is more complicated and that axle behavior must be coordinated.

Consider first the simplest component of the system: the single axle. Assuming no slip, the orientation angle φi and forward velocity, vi, are determined by the wheel velocities:

,1

,2

1 11 12

ii w

ii

v r

d d

ωωφ

⎡ ⎤ ⎡ ⎤⎡ ⎤ ⎢ ⎥= ⎢ ⎥⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

. (1)

where rw is the wheel radius, ,i jω is the angular velocity of the jth wheel, and i denotes the axle position in the system. The resulting nonhololomic Cartesian kinematic equations are thus,

cos( )sin( )

i i i

i i i

i i

x vy v

φφ

φ ω

=

=

=

(2)

where iω is the angular velocity of the axle about its center point, Ci. In the following developments, the control inputs to each axle will thus be vi and ωi, which can be used to calculate the individual wheel velocities in Eq. (1). Based upon these inputs and nonholonomic constraints, each axle imposes

displacement boundary conditions on the compliant frame member. These displacements may produce non-negligible reaction forces acting on the axles, which draw from available wheel traction. Hence, consideration of frame coupling in the motion control algorithms and kinematic models is critical. To consider this issue, we first define several steering strategies that simplify the kinematics of the system and then we will examine frame coupling in these situations.

A. Steering Configurations As Fig. 3 indicates, each axle can move in a forward

direction, with orientation determined by φi, and rotate instantaneously about its center point Ci. At any instant, these constraints are similar to the pin-slot type of boundary condition described in mechanics [30]. In order to consider axle spacing variations that result from changing axle headings, this analysis can be simplified without loss of generality if we consider one end of the robot to be pinned and the other end to be constrained by a pin-slot. This is due to the fact that a straight line can always be drawn between points C1 and C2, which can then be used to transform the orientation of the robot and the axle deflections relative to

1 2C C . Examples of the robot post transformation are shown in Fig.

4 with relative axle deflections, 1ψ and 2ψ , referenced to this horizontally oriented axis, x . These diagrams actually represent three steering configurations selected for their model reduction characteristics and similarity to existing steering systems. To simplify evaluation of these configurations, we introduce the steering ratio, a, expressed as the

Fig. 3. Single axle kinematic model.

Fig. 4. Boundary conditions and kinematic steering scenarios.


ratio of the relative axle headings, 2ψ and 1ψ , and angular rates, 2ψ and 1ψ , such that, 2 1aψ ψ= and 2 1aψ ψ= . (3) Type I Kinematics (a=-1) are defined such that each axle is steered in equal and opposite directions, Fig. 4(a). This configuration can most significantly reduce model complexity and traction forces, simplifies motion control, and provides good maneuverability. It can reduce the kinematics of the system to that of a simple unicycle defined by a curvature based path, and readily accepts such motion control algorithms for full control of robot posture. This steering arrangement is similar to that of an articulated vehicle, but no additional joints or actuators are required. Type II Kinematics (a=1) correspond to equal deflections of each axle such that the frame assumes a shape similar to that of a sinusoid, Fig. 4(b). This model provides simplified kinematics and excellent lateral mobility, but control of orientation is reduced and traction forces are higher. Thus, while this configuration is better for lateral mobility, it provides minimal simplification of motion control where orientation of the robot is considered and ultimately requires large traction forces. Type III Kinematics (a=0), Fig. 4(c), are similar to Ackerman (automobile) steering since the rear axle is always directed towards the center of the front axle and each wheel on the robot approximately travels about a common Instantaneous Center of Rotation (ICR) located along a line extended from rear axle axis. This varies from the strict Ackerman model, however, since the steering geometry of the CFMMR is produced by steering the entire front axle as opposed to using a complicated linkage to individually steer each front wheel. This model derives some kinematic simplification, but does not simplify motion control appreciably where strict control of robot posture must be considered. It requires higher traction forces, and provides a compromise of maneuverability and lateral mobility.

B. Frame Coupling To minimize compressive and tensile forces on the beam

along the x axis that would consume energy and available traction for steering, velocity constraints can be imposed on the axles such that the length of 1 2C C remains consistent with the frame length under pin-slot boundary conditions. This task is complicated by the fact that as the axles steer, the distance between points C1 and C2 must vary to accommodate the new shape of the frame. This shortening affect can be calculated by first considering the deflected shape of the frame. Modeling the frame as an Euler-Bernoulli beam in order to derive tractable velocity constraint expressions, we impose the pin and pin-slot boundary conditions to derive a third order polynomial describing the lateral frame deflection, u, as a function of the imposed angles 1ψ and 2ψ ,

3 21 2 1 222

2u x x x

LLψ ψ ψ ψ

ψ+ +

= − + (4)

where L is the undeflected length of frame and x is the position along the frame in the x direction. This is used to calculate the decrease in length of 1 2C C defined as foreshortening, Lδ [31] as,

( )2

2 22 1 2 10

1 2 22 30

L du LL dxdx

δ ψ ψ ψ ψ⎛ ⎞= = − +⎜ ⎟⎝ ⎠∫ (5)

The foreshortened length of 1 2C C is then denoted as Lf,

2 21 1 2 22 - 2

(1 )30fL L L L

ψ ψ ψ ψδ

+= − = − . (6)

The affect of foreshortening on the length of 1 2C C is illustrated in Fig. 5 for Type I, II, and III steering as functions of 1ψ . This figure compares foreshortening estimates provided by (5) with those determined experimentally for a flexible frame characterized by the parameters shown in Table 1. Experimental hardware consisted of fixtures maintaining

1ψ and 2ψ , where one fixture used a linear bearing to allow free axial deflection in order to measure foreshortening. Angular boundary conditions were measured by single turn potentiometers ( 2%± linearity) calibrated by precision fixtures ( 0.2± ° accuracy) across the range of motion. Axial displacements were measured by a Litton RVT K25-3 linear potentiometer with 4%± linearity.

As Fig. 5 indicates, for [ ]1 40 ,40ψ ∈ − ° ° , Eq. (5) predicts Lδ within 5.0, 1.7, and 0.2 mm for the Types I, II, and III

constraints, respectively. Even though larger errors are incurred with the Type I constraints, they are such that the frame is placed in slight compression, which requires significantly less force than when the frame is in tension. Errors incurred by Eq. (5) with the Type III constraints, in contrast, are such that the theoretical deflection is too small and the beam is in slight tension.

Axial forces produced by deviation from the actual foreshortened length of the frame are shown in Fig. 6. These

Fig. 5. Frame Foreshortening (L=0.350m).


have been experimentally determined using the fixture described above with axial tension and compression forces applied to the linear bearing by a Chattillon Model DPPH100 load cell. Regression lines closely matching experimental data are shown. As the data indicates, the stiffness of the frame is significantly higher when in tension (positive displacement) as compared to compression. The dead-band near zero displacement indicates a slight amount of stiction in the linear bearing. Thus, per Fig. 5, for a heading angle of

1 22.5ψ = ° , errors in the theoretical foreshortening

calculations are [ ]0.82 0.20 0.39− − mm, which per Fig. 6

are expected to produce axial forces of [ ]3.0 0.2 14.6− − N, respectively for the Type I, II, and III constraints. It is observed that a small amount of tension (positive error) produces significant forces, whereas a small amount of compression produces much smaller forces. This is attributed to the post-buckled configuration of the frame [11]. The figures further verify that Lδ is not negligible for 1 10ψ > ° , and hence velocity constraints maintaining axle spacing are critical to minimizing traction forces.

To assure that the axles maintain proper spacing, velocity constraints are established for the axles that satisfy (5). To determine the velocity constraint, the change in frame length

as a function of time is expressed as,

( ) ( )1 1 2 2 2 1

d4 4

dt 30 30fL L Lψ ψ ψ ψ ψ ψ= − − − − , (7)

which establishes the general velocity constraint,

1 1 2 2

dcos( ) - cos( )

dtfL

v vψ ψ = . (8)

Combining (7) and (8), the velocity constraint for Axle 2 is expressed in terms of v1 as,

( ) ( ) ( )

( )1 1 1 2 1 2 1 2

22

30 cos 4 430cos

v L Lv

ψ ψ ψ ψ ψ ψ ψψ

+ − + −= (9)

Next, this will be expressed entirely as a function of the velocity v1 and the kinematic state variables.

C. General Kinematics in Polar Coordinates Fig. 7 is a diagram of the robot in a general configuration

where 1ψ and 2ψ have not yet been specifically constrained or coupled. We strive to control the robot for purposes of posture regulation (position and orientation), path following, or general trajectory tracking by controlling the position of a point on the robot, such as C1 or O, and the angle, γ, of 1 2C C , which describes robot orientation.

Brockett’s Theorem [32] shows that a smooth time invariant control law cannot be used to provide globally asymptotic stability to continuous nonholonomic systems. This is easily circumvented, though, by introduction of a discontinuous polar coordinate description of the kinematics, which then admit a smooth time invariant control law [29]. In the polar description, Fig. 7, the location of a point along

1 2C C , chosen to be C1 for simplification, and the orientation of its velocity trajectory can be represented by the variables,

( )2 21 1 1 1

1 1 2 1 2

2 ,2( , )

e x y ATAN y xATAN y y x x

θα θ φ γ= + = − −

= − = − − (10)

Table 1. Prototype Parameters Parameter Value Units Description

rw 0.073 meters Wheel radius d 0.172 meters Axle width (half) L 0.350 meters Frame Length E 2.10x1011 Pa Frame Modulus of Elasticity I 5.47x10-13 m4 Frame second moment of

area about y axis

M 9.53 Kg Robot mass

Fig. 6. Axial Frame Forces when displacing from

foreshortened length.

Fig. 7. General steering kinematics. Frame omitted for clarity.


which result in the polar kinematic equations,

1

1 1

1

1 1 2 2

cossin

sin

sin sin

f

e v

ve

ve

v vL

ααα ω

αθ

ψ ψγ

= −

= − +

=

−=

(11)

that become discontinuous when e=0. Progressing towards a curvature based controller, consider that the angular velocity of Axle 1 can also be described as a function of path radius or curvature, r1 and 1κ respectively, and forward velocity v1:

11 1 1 1

1

vv

rω φ κ= = = . (12)

Imposing velocity constraint (9) in γ of (11) and using the steering ratio, a, to eliminate 2ψ , the rate of change of robot orientation is then,

( ) ( ) ( )( )

( )11 2

1 1 11

sin 1tan2 2

15 cosf f

aL aa a v

L L aψψ

γ ψ ψψ−

= − − + + (13)

where v1 and 1κ are now viewed as inputs to the system. 1ψ can then be expressed as a function of robot orientation, γ, and the polar coordinates such that, 1ψ θ α γ= − − (14) to provide, 1 1 1vψ κ γ= − (15)

given 1 1 1vφ θ α κ= − = . Substituting (15) into (13), solving for γ , and applying (14) such that the kinematic state equations are purely functions of the polar coordinates with velocity and path curvature inputs, v1 and 1κ , we have,

( )( )( )( )( )

1

1 1

1

11

cossin

sin

sin 11515 15 cosf

e v

ve

ve

abLv

L bL a

ααα κ

αθ

θ α γκγ

θ α γ

= −

⎛ ⎞= − −⎜ ⎟⎝ ⎠

=

⎛ ⎞⎛ ⎞ − − −= +⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟+ − −⎝ ⎠⎝ ⎠

(16)

where ( )( )21 1tan 2 2b a a aψ ψ= − − + and Lf is determined by

(6). The actual Axle 1 heading can then be evaluated by, 1 1φ θ α ψ γ= − = + , (17) which prescribes the angular rate to be, 1 1 1 1vφ ω θ α κ= = − = . (18) Given (3), we have the angle of Axle 2 described as, 2 2 1aφ ψ γ ψ γ= + = + , (19) where 1ψ can be eliminated by substitution of (14) such that only the polar coordinates and γ remain. The result is, ( ) ( )2 1a aφ θ α γ= − + − , (20) and the angular rate of Axle 2 is determined by differentiation to be a function of the kinematic variables,

( ) ( )2 1a aφ θ α γ= − + − , (21)

where (15) is imposed on (9) in conjunction with (3) such that the Axle 2 velocity constraint is,

( ) ( )21 1

2 1 11 1

cos 2 2cos 15

v Lv a a

a vψ γψ κ

ψ⎛ ⎞⎛ ⎞

= + − + −⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠⎝ ⎠. (22)

Hence, the forward and angular velocities of Axle 2 can be expressed purely in terms of the state variables by application of (14) and (16).

Hence, given a motion controller for (16) that prescribes v1 and 1κ as control inputs, the velocities for each axle can be determined. While (16) provides a generic description of the kinematics, the task of motion control is not simple, owing largely to the complexity of γ in (16).

IV. SIMPLIFIED KINEMATIC MODELS Simplifications based on selection of the steering ratio, a,

can significantly reduce the complexity of the kinematic model and facilitate motion control via standard algorithms.

A. Type I: Curvature Based Steering ( 1 2ψ ψ= − )

As indicated in Sect. III, Type I steering occurs if we constrain a=-1. This provides the most appealing description of the system since it simplifies the kinematics significantly, eliminating the need to consider γ as described in the next paragraph, and allows standard unicycle motion planners to provide full motion control. This is derived in part from the fact that 1 2ψ ψ= − imposes a constant moment across the frame, which ideally takes the shape of an arc segment. Geometric properties can then be used to easily describe the postures of the axles.

To facilitate simplification, Point O is defined at the midpoint of 1 2C C , Fig. 8. Point O is unique since its motion under Type I steering is determined to be that of a simple differentially steered axle and the orientation γ becomes equivalent to the velocity heading, Oφ , and ultimately

orientation α . Thus, the posture of 1 2C C at Point O ultimately possesses kinematics described by (2) where Ov and Oφ describe the velocity trajectory of the robot. This is apparent since the angle α describing the polar orientation of v0 also describes the robot orientation, γ, of the segment 1 2C C due to the symmetry of 1 2ψ ψ= − . This provides tremendous simplification of the motion control task since γ is eliminated and polar coordinates θ and α describe both the orientation of the robot and the velocity heading. The reduced kinematic model of the robot subject to Type I constraints is,

0

0 0

0

cossin

sin

e v

ve

ve

ααα κ

αθ

= −

⎛ ⎞= − −⎜ ⎟⎝ ⎠

=

(23)


where the velocity and path curvature of Point O, Ov and Oκ respectively, are the control inputs.

Towards implementation, note that at any instant, O is approximately traveling about an instantaneous center of rotation, indicated as Point I, with a path curvature 0κ and radius rO,

1 2cos2 cosO O

O

Lrr L

ψψ κψ ψ

= = = (24)

where ψ = ψ1 represents the relative angle of axles to the segment 1 2C C . While these equations must be solved numerically to determine ψ for a desired radius of curvature, note the simplicity of this task since length L is used rather than Lf because this equation is based on the actual arc length of the frame.

The control inputs, Ov and Oκ , and the polar coordinates can then be used to determine the desired trajectory of each axle. Based on Oκ , (24) determines the relative heading, ψ, to give the absolute axle headings, 1 Oφ φ ψ= + and 2 Oφ φ ψ= − (25) which can be represented in terms of the polar coordinates since 0φ θ α= − . Axle headings and rates are determined by, 1φ θ α ψ= − + and 2φ θ α ψ= − − (26)

1φ θ α ψ= − + and 2φ θ α ψ= − − (27) where the rate of relative heading change, ψ , is,

( )

2cos2 cos sin

L ψψ κψ ψ ψ

=+

(28)

as determined by differentiation of (24). The velocity magnitude, Ov , can then be related to that of the axles by,

d1

1 cos 2 dtfO Lv

vψ

= − and d1

2 cos 2 dtfO Lv

vψ

= + . (29)

where 1 2ψ ψ ψ= = − . Substituting (7), (3), and a=-1, the axle velocities are determined by,

11,2 cos 6

Ovv Lψψ

ψ= ∓ (30)

Hence, a reduced order kinematic model is obtained that fully describes the system posture and allows easy implementation of curvature based motion controllers. Even greater simplification can be achieved if velocity constraints are relaxed to neglect foreshortening for small steering angles, which provides,

1, 2 cosOv

vψ

= (31)

as was the case in earlier work [10].

B. Type II Kinematics: 1 2( )ψ ψ=

A different simplification of the kinematics can be obtained if 1a = such that 1 2ψ ψ= to produce the Type II kinematics, Fig. 9. The difference between the Type I and II models is readily apparent, however, after applying the constraint a=1 to (13) which results in,

1 1 11 tan( )5

γ ψ ψ ψ= − (32)

This non-zero component is due to the modified velocity that Axle 2 must assume relative to Axle 1 in order to compensate for foreshortening. If the velocity constraints are neglected, foreshortening is an issue and traction forces increase, but 0γ = . This nullifies the γ state variable and results in a reduced model similar to (23) but referenced to C1.

C. Type III Kinematics: 2( 0)ψ =

A third technique for simplifying the kinematic model can be achieved if we impose the constraint a=0. In this case,

2 0ψ = , and hence the rear axle is always directed towards C1. The kinematic model is simplified appreciably, but not entirely reduced since,

Fig. 8. Type I Steering Kinematics.

Fig. 9. Type II Steering Kinematics


11

sin

f

vL

ψγ = . (33)

Since the γ state variable is not entirely reduced as in the case of Type I steering, a simple unicycle type motion control algorithm will not drive γ to zero.

V. MOBILILTY AND MANEUVERABILITY In order to evaluate the performance of the steering modes,

we develop metrics based upon achievable mobility and maneuverability. Forces imposed by the frame play a critical role in evaluating these capabilities since they directly impact wheel traction forces and achievable steering angles. These capabilities are also affected by physical interference of components of the robot that may limit steering angles and potentially unstable configurations of the system. In the process of developing these factors and performance metrics, they are evaluated relative to the experimental platform characterized by the parameters shown in Table 1.

A. Limiting factors 1) Physical Interference: Depending on the proportions

of the robot, interference can occur in two different scenarios: the wheels on one side of the robot may touch, or one of the wheels may contact the frame. Wheel-wheel interference is apt to limit Type I steering, which provides the following equation based upon the geometry shown in Fig. 6,

2

tan2

wrL d

ψψ

ψ=

− (34)

which must be solved numerically for ψ. Based upon the robot parameters, Table 1, the limit is 36.5iψ ≤ ° .

Wheel-frame interference can occur in any of the steering modes if the frame is sufficiently long. Since the deflected shape of the frame and collision point is very difficult to evaluate analytically under these circumstances, this boundary is approximated by the angle of the axle when the leading or trailing edge of the wheel interferes with 1 2C C . This results in the following limit,

arctaniw

dr

ψ ≤ (35)

which corresponds to 67.0iψ ≤ ° . 2) Traction Forces: Quasi-static behavior is assumed

since dynamic interactions are not considered. Thus, traction forces required to impose boundary conditions on the frame are evaluated. Assuming foreshortening has been considered in the velocity constraints, boundary condition forces are described by lateral reactions, Ri, and moments, Mi, of the frame that result from axle orientations [35],

( ) ( )1 1 2 2 1 22 22 2

f f

EI EIM ML L

ψ ψ ψ ψ= + = − + (36)

1 21 2 1

f

M MR R R

L+

= − = − (37)

where E is the Young’s Modulus of Elasticity of the frame and I is its cross sectional moment of inertia of the frame

about the bending axis [30]. These forces and moments can then be used to calculate wheel traction forces,

( ), ,ˆ ˆ ˆcos sin2 2

i iM i i i i i R i

M RF x y F y

dψ ψ= + = , (38)

where ,M iF and ,R iF , respectively, represent the moment and reaction forces on a tire on the ith axle oriented with respect to the frame coordinates. The net traction force on a tire on the ith axle (- and + representing the left and right wheels respectively) is then the vector sum, ,T iF , , , ,T i M i R iF F F= ± + (39)

and the maximum traction forces, max,T iF , on the ith axle is,

max, , , 2

T i M i R iF F F= + (40)

The maximum of these axle forces then determines the maximum tire force on the robot: ( )max max

,max , 1..T T iF F i n= = . (41) Ultimately, the achievable traction force is limited by the

weight of the robot and the wheel slip characteristics [36]. Assuming that the robot has an evenly distributed mass m, a normal force mg/4 will be supported by each wheel. Thus, the maximum traction force can be approximated by,

max 4mgF µ

= (42)

given a coefficient of friction, µ, corresponding to the tire-surface interaction [33, 34]. Eqs. (40) and (42) can then be solved to determine the ideal maximum steering angle, ψ1, achievable for a steering ratio, a. Based upon the parameters in Table 1, Fig. 10 illustrates max

TF as a function of steering ratio for several relative axle headings. These results indicate that ideally Type I steering requires minimum traction force, followed by Type III and then Type II.

Based upon published tire-surface interaction, these results estimate that all steering modes should function effectively on

Fig. 10. Ideal required maximum wheel traction forces and available traction based on typical surfaces [33, 34].


sandy loam [34] and even snow covered ice [33]. An important point to remember, though, is that these traction force estimates do not include forces introduced by error in the foreshortening calculations, errLδ . If we include these forces, it is evident that the maximum traction forces including foreshortening error, err

TF , are larger and dependent upon foreshortening predictions, Table 2. In situations where

0errLδ > , the frame is in tension and traction forces may increase appreciably, as in the case of the Type III steering, and it is evident that the wheel slip will occur on slippery surfaces. In the case of the Type I and II steering, though, the traction force with foreshortening error is still sufficiently small that the robot should function well on slippery surfaces.

3) Configuration Instability: Instability could occur in the form of tip-over if the axles are co-linear. In the quasi-static case, this occurs when axle orientations satisfy,

1 2πψ = ± and 2 2

πψ = ±

As the physical interference limits indicate, tip-over instability is not a limiting factor in the case study.

B. Performance Criteria 1) Maneuverability: The capability to manipulate robot

orientation, and thus maneuver the robot around obstacles, can be evaluated by γ , which is generally described by (16). Note that this expression is a complex function of steering ratio, a, relative axle heading, ψ1, and path curvature κ. In general ψ1 is actually determined by the current configuration of the robot as determined by the evolution of the kinematics (16).

2) Lateral Mobility: In confined environments, the ability of the robot to move laterally without significant maneuvers is important. This mobility can be quantified by the average relative heading, avgψ , described by,

( )1avg 1 22ψ ψ ψ= + (43).

3) Scaled Performance Metrics (SPM): Since force is required to deflect the frame, wheel traction and energy are necessary. Objective evaluation of the performance criteria is accomplished by examining the magnitude of γ and avgψ per unit of maximum traction force required, represented respectively by Pγ and Pψ, as,

,maxT

PFγ

γ= and avg

,maxT

PFψ

ψ= . (44)

Evaluation of the SPM was performed based upon the parameters shown in Table 1. Larger performance factors are

generally desirable for maximizing the use of available wheel traction, but traction requirements, Fig. 10 and Table 2, and kinematic simplifications must be considered.

The SPM plot for maneuverability is shown in Fig. 11 for a relative steering angle 1 30ψ = ° and several path

curvatures, [ ]3,0,3κ = − , as a function of steering ratio, a. For

smaller ψ1, the variation between the curves becomes smaller and less illustrative of the SPM sensitivity to path curvature. Considering the three case studies, note that Pγ is largest for a=-1 (Type I) and smallest for a=0 (Type II). The actual maximum of the SPM is at about a = -2, but the traction forces are actually much larger and the kinematic simplifications are not significant.

In the case of 0κ > , note that Fig. 11 indicates 0Pγ < at a=1. This indicates potential destabilization since the robot orientation, γ, is decreasing while axle heading angles, φ1 and φ2, are increasing, Fig. 9. Thus, ψ1 increases and γ becomes even more negative. If 0κ > is maintained for sufficient time,

1 90ψ → ° and (22) becomes singular. Prior to this point, though, destabilization can be remedied if 0κ < is applied. Similar destabilizing and restabilizing scenarios occur when

0ψ < , and 0κ < or 0κ > , respectively. For robot configuration stability we thus desire ψi to remain

bounded and to converge to a steady state value proportional to the path curvature. Given 1 0ψ > this is achieved by 0γ > , and 0γ < for 1 0ψ < . Thus, we can guarantee configuration stability if, min maxa a a< < (45) where [ ]max 0.91,1.14a ∈ is derived numerically based upon

Fig. 11 for [ ]3,3κ ∈ − at 1 30ψ = ± ° . Note that max 1a → as

ψ1 approaches zero, but the aforementioned range of amax is given to indicate the nominal range of amax that is dependent upon κ. In contrast, we find that amin is a function of physical limitations (35) and is determined by,

Fig. 11.The Scaled Maneuverability Performance Metric indicates the ability to change robot orientation per unit of

traction force. 1 30ψ = ° is shown.

Table 2. Ideal maximum wheel traction force, maxTF , expected

foreshortening error, errLδ , and expected maximum wheel traction forces with foreshortening error, err

TF . 1 11ψ = ° 1 22.5ψ = °

Steering Mode

maxTF (N)

errLδ (mm)

errTF

(N)

maxTF (N)

errLδ (mm)

errTF

(N) I (a=-1) 0.56 -0.18 1.51 1.24 -0.82 2.42 II (a=1) 1.10 0.02 2.29 2.26 -0.02 2.35 III (a=0) 0.79 0.06 6.53 1.68 0.39 8.74


maxmin

1 1

67aψψ ψ

°≈ − < , (46)

based on (3) and (35), which results in min 2.23a ≈ − for

1 30ψ = ± ° . This lower limit actually becomes more negative for smaller ψ1, allowing a greater range of steering ratios, but this provides a conservative static boundary. Steering outside of the range (45) is certainly allowable for short periods, but observing (45) is generally desired. Steering near the boundaries of (45) is possible, as in Type II steering, but this may lead to stability issues in extreme maneuvers as we indicate in the evaluation section of the paper.

The SPM plot for lateral mobility is shown in Fig. 12 for 1 11 and 30ψ = ° ° as a function of steering ratio, a. Type II

steering (a = 1) provides the best lateral mobility, while Type I (a = -1) is the worst. Pψ actually is larger for a>1, but stability becomes an issue. When 1a < − , Pψ becomes negative, indicating behavior counter productive to lateral mobility, which results from large rear axle steering maneuvers that are actually better for maneuvering the robot as described above.

As the SPM indicate, the Type I steering (a=-1) is the best for maneuverability, while Type II (a=1) is the best for lateral mobility. Type III (a=0) provide a nominal mix of lateral mobility and maneuverability, but this kinematic model cannot be simplified to the same degree as the others and it introduces significant traction forces in lieu of foreshortening error. The Type I and II kinematics, though, each provide specific performance specialties with reduced kinematic models that vastly simplify motion control to provide control of full posture regulation with simple unicycle motion controllers.

VI. MOTION CONTROL ALGORITHM Utilizing the polar coordinate representation and the

simplified models, a pose regulation controller is developed based upon the Lyapunov function,

( )2 21 .2

V hα θ≡ + (47)

Indiveri [27] suggest the use of the smooth control law for the velocity v and the curvature κ ,

1

2 3sin sin

O

O

v k e

k ke e eα θ α ακ

α

=

= + +, (48)

where the parameters 1 2 3, , and k k k are constant gains. Inserting (48) and (23) into (47) the Lyapunov derivative becomes, 2

1 3 0V k k α= − ≤ , (49) where it may be shown that the origin is the solitary system equilibrium point; hence by LaSalle’s Theorem [37] the states ( , , )e θ α will be asymptotically driven to the origin [35].

A dynamic extension must be performed to implement the control law on real systems with drift and non-ideal initial conditions. The model meets the requirements for a cascade system as defined by Bacciotti’s Theorem 19.2 [38] and

therefore the smooth stabilizers,

( )( )

da v d a

a d a d

v k v v vkκκ κ κ κ

= − +

= − +, (50)

may be added with the overall system being smoothly stabilizable, where the subscript d represents the desired values from (48) and the subscript a represents the actual values. The state equations of the extended system with the control inputs (48) inserted now become,

2 3

1

sin sin

cossin

sin

( )

.

a

a v a d

a d

a

a a

a

v k e

k ke e e

e v

ve

ve

k v v

kκα θ α α κ

αα

αα κ

αθ

κ κ⎛ ⎞+ + −⎜ ⎟⎝ ⎠

= −

⎛ ⎞= − −⎜ ⎟⎝ ⎠

=

= − +

= +

(51)

This algorithm works well in simulation, but in experiments difficulties are encountered in a small neighborhood of the origin (~5-7cm) where drift causes (48) to command steering angles that the CFMMR cannot achieve. Thus, in a small neighborhood of the origin, (48) is modified to 1Ov k x= and

0Oκ = . As a result, final convergence of e, θ, and α are not perfect as indicated in the experimental results.

The dynamic extension also provides the system with the ability to track a desired trajectory by replacing the control inputs with any desired velocity and curvature trajectory generator. Further details of this and other similar proofs can be found in [10, 27, 29, 35, 39]. Application to the Type II and III kinematics with velocity constraints considered is easily achieved, but γ is not actively controlled. While the controller

A similar controller may be developed for the path following case where the path is expressed as a smooth directed function s as suggested by Tayebi [28]. Since path following is not used to evaluate performance of the robot here, the reader is referred to previous publications to further

Fig. 12. The Lateral Mobility Performance Metric indicates the ability to translate the robot laterally per unit of traction

force without complex maneuvers.


examine implementation of path following with this robot [11, 35].

VII. EVALUATION OF SIMPLIFIED MOTION CONTROL

A. Methods and Procedures The steering strategies discussed above were tested via

simulation and experiments on the CFMMR platform, Fig. 1, at the University of Utah. The system was controlled via tether by a dSpaceTM 1103 DSP from the MatlabTM Simulink environment via Real Time Workshop and Control Desk. Wheel odometry was used for feedback with simple servo-type PID wheel controllers. Manually obtained final posture measurements with an accuracy of +/- 0.1 cm are presented in Table 3 to compare experimental results. Video footage overlaid with trajectories according to odometry is used to indicate and document system performance, Fig. 16.

Simulation of the motion controllers applied to the CFMMR using the Type I, II, and III steering ratios were conducted to evaluate their efficacy while performing posture

regulation. Thorough experimental evaluation of posture regulation was conducted to evaluate the ability of each of the steering modes to accurately regulate the final posture of the robot given their inherent traction requirements and predispositions for controlling orientation, γ. In particular, Type I steering relative to the reference frame C1 versus O was considered. An Initial Condition (IC) (a) [x(0),y(0), φ(0)] =[-1.445 m, -1.221 m, 0°] was selected for these tests via simulation since it produced large steering angles that would be prone to traction loss in experiments. Since this IC actually caused the Type II steering to drive the robot to an unstable configuration in simulation, an IC(b) of [-1.445 m, -0.500 m, 0°] was also used to evaluate Type II steering under more nominal conditions. In all cases, the desired final posture was at the origin with the robot aligned with the x-axis., which corresponds to [e,θ,α,γ]=[0,0,0,0].

Carpet and sand surfaces were used to evaluate the affects of wheel slip on final posture error resulting from steering mode traction requirements. The carpet was a dense short nap that provided high traction. Sand tests were conducted with a

Table 3. Final posture error with IC (a) [x(0),y(0), φ(0)]=[-1.445 m, -1.221 m, 0°] and (b) [-1.445 m, -0.500 m, 0°]. Test No.

Steering Mode Ref. IC Surface

Beam vel. Constraint

e ±σe (cm)

θ ±σθ (deg)

α ±σα (deg)

γ ±σγ (deg)

Ref. Test

∆γ (deg)

%e incr.

1 O (a) Simulation ON 0.0 0.0 0.0 0.0 - - - 2 Type I O (a) Carpet ON 10.7 ±0.7 139.6 ±1.0 148.5 ±1.0 -7.7 ±1.6 1 -7.7 - 3 (a=-1) O (a) Sand ON 24.8 ±9.7 134.7 ±4.3 145.6 ±6.2 -1.5 ±11.7 2 6.2 132% 4 O (a) Sand OFF 31.1 ±1.8 134.4 ±2.9 146.4 ±2.8 -7.9 ±1.9 3 -6.4 25% 5 (a) Sand/Rock ON 66.2 ±4.8 116.8 ±1.1 129.6 ±9.1 -9.6 ±11.9 3 -8.1 167% 6 O' (a) Simulation ON 0.0 0.0 0.0 0.0 - - - 7 Type I O' (a) Carpet ON 2.8 ±0.4 -26.0 ±49.2 -21.1 ±49.4 -1.7 ±4.1 6 -1.7 - 8 (a=-1) O' (a) Sand ON 8.4 ±2.4 86 ±73.3 90.6 ±76.7 -1.8 ±6.9 7 -0.1 200% 9 O' (a) Carpet OFF 7.2 ±1.4 -56.7 ±2.4 37 ±126.6 -4.7 ±0.8 7 -3.0 157% 10 O' (a) Sand/Rock ON 66.1 ±7.4 118.8 ±3.2 129.8 ±8 -8.1 ±10.7 8 -6.3 687% 11 C1 (a) Simulation ON 0.0 0.0 0.0 16.4 - - - 12 C1 (a) Carpet ON 11.8 ±0.1 10.7 ±16.8 16.6 ±16.4 8.4 ±4.9 11 8.0 - 13 Type I C1 (a) Sand ON 13.0 ±3.8 68.8 ±32.2 72.1 ±37.6 5.1 ±5.1 12 -3.3 10% 14 (a=-1) C1 (a) Sand OFF 17.7 ±6.0 80.1 ±11.9 85.9 ±16.7 3 ±6.8 13 -2.1 36% 15 C1 (a) Sand/Rock ON 44.6 ±4.6 90.8 ±18 93.3 ±26.3 4.1 ±9.7 13 -1.0 243% 16 C1 (a) Sand/Rock OFF 51.5 ±3 102.1 ±2.6 113 ±12.2 -2.8 ±9.7 15 -6.9 15% 17 C1 (a) Simulation ON 0.0 0.0 0.0 18.0 - - - 18 Type II C1 (a) Carpet ON 41.2 ±4.5 115.4 ±1.2 142.1 ±1.2 3.0 ±3.9 17 -15.0 - 19 (a=1) C1 (a) Sand ON 77.6 ±21.4 106.1 ±30.7 137.4 ±28.7 -7.1 ±10.7 18 -10.1 88% 20 C1 (a) Sand OFF 99.8 ±8.6 106.1 ±4.9 137.6 ±10 -20.1 ±4.7 19 -13.0 29% 21 C1 (a) Sand/Rock ON 82.7 ±5.6 112.5 ±19.5 125.9 ±61.9 -8.8 ±12 19 -1.7 6.6% 22 C1 (a) Simulation ON 0.0 0.0 0.0 29.7 - - - 23 Type III C1 (a) Carpet ON 12.2 ±1.5 74.8 ±7.2 76.7 ±11.3 11.0 ±2.8 22 -18.7 - 24 (a=0) C1 (a) Sand ON 42.2 ±4.4 105.2 ±0.4 127.6 ±2.2 -13.3 ±6.8 23 -24.3 245% 25 C1 (a) Sand OFF 52.4 ±2.9 104.8 ±1.5 134 ±3.6 -16.4 ±2.5 24 -3.1 24% 26 C1 (a) Sand/Rock ON 87.5 ±3.5 100.3 ±1.1 129 ±10.1 -19.3 ±3.9 24 -6.0 107% 27 C1 (b) Simulation ON 0.0 0.0 0.0 0.0 - - - 28 Type II C1 (b) Carpet ON 19.1 ±1.6 67.6 ±2.7 76.4 ±3.6 -0.6 ±2.7 27 0.6 - 29 (a=1) C1 (b) Sand ON 40.4 ±2.4 75.1 ±0.5 85.2 ±2.2 -5.9 ±2.0 28 10.2 112% 30 C1 (b) Sand OFF 49.4 ±5.9 78.3 ±0.2 105.2 ±4.9 -15.6 ±4.5 29 -16.3 22%


1cm thick layer of sand spread evenly over a plastic sheet to simulate low traction. The sand was leveled between each trial and was sufficiently thick that the robot did not dig through unless significant obstacles were present. The ability of the frame foreshortening velocity constraints to reduce traction forces and wheel slip was evaluated on sand for each of the steering modes. The steering modes were also evaluated on sand with scattered rocks (1 - 2 cm thick flat hard sandstone nominally 4 cm x 4 cm spaced ~10cm) to evaluate the performance of the robot traversing obstacles in a low traction environment. Rocks were ~13%-27% of the wheel radius in order to provide non-trivial obstacles. During each trial, ~12 rocks were traversed by the tires and these were nominally distributed amongst the wheels and axles. For each test, five trials were performed to estimate repeatability. The robot was also evaluated on extremely rugged rocky terrain (undulating rock piles as large as the wheel radius) to evaluate its operability.

Typical challenges near the final posture caused by small path radius steering maneuvers were initially encountered while implementing the motion controllers experimentally [10]. This occurs with all steering modes and is caused by drift and phase lag resulting from the low-level wheel controllers that becomes problematic as the motion controller (48) attempts to compensate aggressively near the final posture. This is easily resolved by commanding the robot to follow a straight line as indicated in Section VI once it is within 7.5 cm of the origin until the x position error converges to zero. At the switching point, the configuration variables, α and θ, have converged nearly to zero and as the robot drives x asymptotically to zero, there is inevitably small error remaining in y. Thus, error in the configuration variables α and θ typically increases near the final posture.

In the particular case of Type I steering by Point O, two methods of eliminating this phenomenon near the origin were examined. The first was as described above where the posture and velocity of Point O were determined algebraically from the axle data. The second method forced a unicycle kinematic model (23) to describe the motion of Point O, where the axle velocities along and perpendicular to 1 2C C were used to determine the forward and angular velocity of the coordinate frame attached to O. This method of tracking Point O denotes the point as O' in the results.

B. Results and Discussion: Table 3 indicates the final posture error data and respective

standard deviations after pose regulation. Based on simulation results (Tests 1, 6, 11, 17, 22, and 27) note that only the Type I steering (relative to Point O or O', Tests 1 and 5) is capable of completely regulating the posture of the robot from the IC (a). Type II steering is capable of maintaining γ=0 if initial posture error is small (Test 27, IC (b)), but not in general given larger initial error (Test 17, IC (a)). Type III is unable to regulate γ (Test 22), which is also true of Type I steering via C1 (Test 11). In all simulations, though, position error of the reference point (O, O', or C1) characterized by e and θ converges completely to zero. These results are all predicted based upon previous analysis.

Experimental evaluation indicates nonzero final posture error in all tests, Table 3. In general, distance error, e, was smallest for Type I steering (Tests 2-4, 7-8, and 12-14), larger for Type III (Tests 23-25), and largest for Type II (Tests 18-20). Given that Type I traction forces were expected to be minimum, the first trend correlates well with predictions. At first glance, however, it is counterintuitive that the Type II steering produced the largest error since Type III was expected to induce the maximum traction forces for a given steering angle, ψ.

The source of the discrepancy is recognized upon examination of steering angles, Fig. 13, and traction forces, Fig. 14, derived from simulations. These show that the Type II steering encounters a discontinuity as ψ1 approaches 80°, and this produces estimated traction forces peaking at 9 N, which does not include foreshortening error that would be appreciable at these extreme deflections. These angles and forces are significantly larger (nearly three times) than those of the Type III steering, which has ψ1<30° and FT<2.5 N. The large steering angle results from potential Type II steering instability as described in Sect. V.B. In the case of IC(a) (Tests 18-20), 1 0κ > is large enough sufficiently long such that 1ψ increases until the singularity occurs. Path plots, Fig. 16 (d)-(f), indicate this discontinuity as a noticeable source of slip that causes a large net final error. Given the smaller IC (b) (Tests 27-30) requiring smaller path curvature, 1ψ only increases slightly and then decreases when 1 0κ < in the later part of the trajectory (Test 27). Thus, orientation γ converges

Fig. 13. Simulation of Axle 1 steering

angle per Tests 1, 11, 17, and 22.

Fig. 14. Simulation of maximum traction

force per Tests 1, 11, 17, and 22.

Fig. 15. Simulation of axle spacing per

Tests 1, 11, 17, and 22.


back to zero, which is also true of IC(b) after the singularity has occurred. Owing to the potential instability of Type II steering, it thus must be applied with caution. Such effects are not present in the other steering modes where the relationship between ψ, γ, and κ is stable.

Tests performed on Sand versus Carpet in general indicate an increase in e that is more characteristic of the expected traction forces. This is quantified by the percent increase in error, denoted as %e in Table 3, with respect to the appropriately indicated Reference Test (Ref. Test). As expected, Type I possesses the least percent increase (10% using C1 to guide the robot) whereas the percent increase for

Type II is larger (88%), and Type III is largest (243%). Note that the Type I steering guided by O or O' actually yields a much larger percent increase. This is attributed to the fact that the unicycle kinematics of the coordinate frame attached to O are entirely dependent on maintaining exact ratios of v1, v2, ψ1, and ψ2, which is difficult to achieve given the servo-type wheel controllers implemented here. Thus, while Type I steering via O or O' provides simplified kinematics by eliminating γ, the challenge is synchronously controlling v and ψ for each axle.

Tests performed on Sand/Rock indicate similarly expected

Fig. 16. Robot paths during posture regulation on carpet for Type I (a)-(c) (Test 2), Type II (d)-(f) (Test 18), and Type III (g)-(i)

(Test 23) steering kinematics, and on sand/rock for Type I (j)-(l) (Test 15). [x(0),y(0), φ(0)] = [-1.445 m, -1.221 m, 0°].


trends of increased error, e. Tests guided by C1 indicate best performance for Type I (Test 15) and worst for Type III (Test 26). Tests guided by Point O and O' (Tests 5 and 10, respectively) indicate nearly identical error that is much larger than when guided by C1 (Test 15). This again follows from difficulties controlling the exact ratios of v and ψ, which are exacerbated by the rocky terrain.

Deviations in final orientation, γ, from expected results, expressed as ∆γ in Table 3 with respect to the indicated Ref. Test, also agree well with performance predictions based upon traction forces. Namely, ∆γ was smallest for Type I steering and largest for Type III steering.

Also note that guiding the robot with the forced unicycle kinematics produced the smallest final error, e, on Carpet and Sand (Tests 6 and 7, respectively). While these results are outwardly very promising, the subtlety is that the posture of O estimated by the axle postures and posture estimated by the forced unicycle kinematics differed appreciably in their estimates. For example, the posture estimates attained from the axles were nominally 10.5 cm and 18.5 cm larger on Carpet (Test 7) and Sand (Test 8), respectively, than those derived from the forced unicycle model. As such, usage of the forced unicycle kinematic model has potential to vary appreciably from the posture of the robot. Modeling this slip and more accurately accounting for the kinematics of Point O is the subject of future work.

The effectiveness of the velocity constraints to account for frame foreshortening is amply demonstrated on the sand surface for each of the steering modes. Foreshortening on the order of 5-7 mm was typical, Fig. 15. In Tests 4, 14, 20, and 25, the beam foreshortening velocity constraints were disabled and the resulting error, e, was compared to the error observed during the initial Sand tests. As the results indicate, disabling beam foreshortening velocity constraints increased error between 24% and 36% for IC (a) on Sand. Tests on the Sand/Rock surface with Type I steering demonstrated that disabling the beam velocity constraints increased error by only 15%, which is attributed to the larger error already characteristic of the rock field. It can thus be concluded that the frame velocity constraints are quite effective for decreasing traction forces and wheel slip.

The robot path during posture regulation, Fig. 16, final

posture error, Table 3, and predicted traction forces, Table 2, indicate effectiveness of the Scaled Performance Metrics to estimate robot performance for different steering ratios, a. As Fig. 16 (a)-(c) indicate, Type I steering clearly requires the minimum space and is most effective for maneuvering the robot along a curvature based path with minimum final error. In contrast, the Type II kinematics require more space to maneuver, but they are best adapted to translating the robot laterally small distances (e.g. lateral maneuverability), although the motion controller (48) is not optimized for this. Type II traction forces are higher, though, and as a result final posture is more prone to error. Type III kinematics provide a compromise between Type I and II characteristics.

All things considered, it is concluded that the Type I steering guided by O or O' provide the most simplified control of robot posture, but that Type I kinematics guided via C1 provide the most consistent behavior given the odometry measurements and servo type wheel controllers implemented here. Overall, the SPM provide a good first estimate of these behaviors, but it is necessary to consider variations in traction forces resulting from foreshortening errors and the influence of larger relative steering angles, ψ, and particular steering ratios, a, that cause the robot to approach its limitations.

Evaluation of the Type I kinematics on extremely rocky terrain demonstrates the capability of the robot to conform to terrain variations, Fig. 17. In frame (b), opposite wheels can be observed (Right Front, RF, and Left Rear, LR) ascending obstacles simultaneously. In frame (c), LR continues its ascent and RF descends. In (d), the Left Front (LF) wheel ascends and LR descends. In (e), both LF and LR descend. Finally, RF ascends in (f) and descends in (g). As this sequence of images illustrates, the CFMMR is very adept to conforming to rugged terrain that would leave traditional rigid mobile robots teetering on two wheels. While this terrain is more severe than that used in the Sand/Rock tests (Tests 10, 15, 21, and 26), it does illustrate typical compliant frame undulations observed in those trials.

VIII. CONCLUSIONS General kinematics for a two axle compliant frame mobile

robot have been presented and characterized by steering ratio, a. Based upon three special cases of a, simplified kinematic

Fig. 17. Robot traversing extremely rugged undulating rocky terrain.


models have been derived and evaluated. Of these models, only Type I steering guided by Point O provides full control over posture regulation. Types I and III steering guided by C1 provide full position control with limited control over orientation, γ. Type II steering provides control over position, but orientation, γ, became unstable in simulation given initial conditions requiring aggressive maneuvers. Experimental evaluation corroborates that the Type I steering requires minimum traction, provides the most accurate posture regulation, and provides maximum maneuverability. The ability of the robot to adapt to rugged terrain and maneuver was confirmed by posture regulation on sand with scattered rocks and the ability to traverse extremely rugged rocky terrain was also demonstrated. Future work is on modular configurations, dynamic control laws, and improved sensor instrumentation.

ACKNOWLEDGEMENTS The authors gratefully acknowledge the financial support of

the National Science Foundation provided by grant number IIS-0308056 and the University of Utah.

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BIOGRAPHIES

Mark A. Minor (S’99-M’00) received the B.S. degree in Mechanical Engineering from the University of Michigan and the M.S. and Ph.D. degrees in Mechanical Engineering from Michigan State University.

He is currently an Assistant Professor at the University of Utah in the

Department of Mechanical Engineering where he has been a faculty member since 2000. Research interests include motion control of mobile robots, terrain adaptive mobile robot locomotion, sensing strategies and fusion structures, and nonlinear robust control of distributed parameter systems.

Brian W. Albiston received the B.S.

degree in Mechanical Engineering from Utah State University in 1999. He worked for the Williams Companies from 1999-2002 and then joined the University of Utah, Department of Mechanical Engineering, where he completed his M.S. degree in 2003. He joined Orbital

Sciences Corporation from 2003-2005 and is currently employed with Sagetech Corporation.

Corey L. Schwensen received the B.S.

degree in Mechanical Engineering from Rose-Hulman Institute of Technology in 1995. From 1999 to 2001 he attended the University of Utah and received the M.S. degree in Mechanical Engineering. He is currently a senior engineer at Rosetta Inpharmatics, a research division of

Merck & Co., Inc.

simplified motion control of a two axle compliant framed wheeled mobile...

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