design and optimization of an eight-bar legged walking … · design and optimization of an...

DESIGN AND OPTIMIZATION OF AN EIGHT-BAR LEGGED WALKING MECHANISMIMITATING A KINETIC SCULPTURE, “WIND BEAST”

Daniel Giesbrecht, Christine Q. Wu, Nariman SepehriDepartment of Mechanical Engineering, University of Manitoba, Winnipeg, Manitoba, Canada

E-mail: [email protected]

Received August 2011, Accepted November 2012No. 11-CSME-63, E.I.C. Accession 3303

ABSTRACTLegged off-road vehicles exhibit better mobility while moving on rough terrain. Development of legged

mechanisms represents a challenging problem and has attracted significant attention from both artists andengineers. In this paper, we present the design of a single-degree-of-freedom legged walking mechanismusing the mechanism design theory and optimization to imitate a well-known kinetic sculpture, a WindBeast. The optimization is set up to: i) minimize the energy input and ii) maximize the stride length. Theoptimization is based on the dynamic force analysis. A prototype of the optimized walking mechanism with6 legs was built to demonstrate its smooth motion. The success in designing a legged mechanism capable ofimitating the well-known kinetic sculpture using the engineering design theories is a small step bridging thegap between art and engineering.

Keywords: legged walking mechanism; mechanism design; type synthesis; dimension synthesis; optimiza-tion; kinematics and dynamics analysis.

CONCEPTION ET OPTIMISATION D’UN MÉCANISME MARCHEUR À HUIT BARRESIMITANT LA SCULPTURE CINÉMATIQUE "WIND BEAST"

RÉSUMÉLes véhicules tout-terrain à pattes sont généralement plus mobiles que leurs équivalents à roues en terrain

accidenté. Le développement de mécanisme à pattes représente un défi ambitieux, et a suscité de l’intérêtd’artistes et d’ingénieurs. Cet article présente la conception d’un mécanisme marcheur à un degré de libertéutilisant les théories des mécanismes et de l’optimisation pour imiter une célèbre sculpture cinématique,“ Wind Beast ”. L’optimisation vise à i) minimiser l’apport énergétique et ii) maximiser la longueur dupas. L’optimisation est basée sur l’analyse des forces dynamiques. Un prototype du mécanisme de dépla-cement optimisé à six pattes a été construit pour démontrer la souplesse du mouvement. Le succès de laconception d’un mécanisme à pattes capable d’imiter la célèbre sculpture cinématique utilisant les théoriesde conception technique est un petit pas de l’ingénierie vers l’art.

Mots-clés : mécanisme de mouvement à pattes ; conception mécanique ; synthèse de type ; synthèse dedimension ; optimisation ; analyse cinématique et dynamique.

Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 4, 2012 343

1. INTRODUCTION

It has been established that legged off-road vehicles exhibit better mobility, obtain higher energy efficiencyand provide more comfortable movement than those of conventional tracked or wheeled vehicles whilemoving on rough terrain [1]. In the last several decades, a wide variety of legged mechanisms have beenresearched for the applications of legged locomotion, such as planetary exploration, walking chairs for thedisabled and for military transport, and rescue in radioactive zones for nuclear industries or in other hostileenvironments.

From a design viewpoint, legged robots can be divided into two groups, single/low-degree-of-freedom(SDOF/LDOF) and multi-degree-of-freedom (MDOF) mechanisms [2]. The latter is the focus of the recentdevelopment of legged robots due to the advancement in control theories and actuation technologies. SuchMDOF robots have the advantages of being simple in the leg structure, but are demanding to the actuationand control. On the other hand, the SDOF/LDOF legged mechanisms have the advantages of requiringlittle control and simple actuation, yet can still create versatile walking motion with a low demand on theenergy input. It has been discussed that unlike a ground-based manipulator that can be operated with anoff-board power supply, a walking machine has to carry the entire power supply in addition to the externalpayload and the weight of the machine body. Thus it is desirable to use a small number of actuators toreduce the body weight and to simplify the motion coordination [3]. A number of six or seven-link SDOFleg mechanisms have been designed [3–7]. Rigorous research has been carried out on their mobility andenergy loss through kinematic and structural analysis. Two important findings have been documented: (1) acrank as an input link with continuous rotation motion should be used to achieve fast motion with minimumcontrol [3–5,8] and (2) an ovoid foot path is necessary to step over small obstacles without significantlyraising the body [3,4,8]. The challenge in developing a SDOF legged walking mechanism is the requirementof a large number of links required to provide high mobility. Thus, the type selection and the dimensionalsynthesis for such legged mechanisms are challenging.

Type synthesis has been the main focus for the early research on design of SDOF legged mechanisms,where slider-crank mechanisms [9] and multiple cam mechanisms [10] have been used. It was recommendedto use only revolute joints for legged walking machines due to the difficulties in lubrication and sealingof the sliding joints, which is essential for the machines to walk outdoors [9]. Many pin-joined leggedmechanisms have been designed, which are often compound mechanisms consisting of a four-bar linkageand a pantograph [3,5,6,9,11]. The potential advantages of such compound mechanisms are fast locomotion,minimal energy loss, simplicity in control design, and the slenderness of the leg [4].

Although legged walking mechanisms have a high potential in mobility and energy efficiency on roughterrain, they often involve a large number of geometrical dimensions, which makes it necessary to resortto optimization to achieve a high quality design. Reducing the energy loss has always been the interestin designing legged mechanisms [3,4,7]. In some research, springs were added to store the energy and toreduce the actuating torque [4]. In the process of energy optimization, the force analysis is needed. Due tothe complexity of the mechanisms, in previous research, the force analysis has been restricted to the staticanalysis [3,4,7]. It is known that the dynamic analysis of the mechanism has important impact on energyoptimization especially when fast locomotion is to be created.

While developing legged walking robots have been actively pursued by engineers, they have also attractedattentions from art fields. Mr. Theo Jansen, a Dutch kinetic sculptor, created a series of kinetic sculptures,“Wind Beasts”, shown in Fig. 1(a), which are multi-legged walking mechanisms powered by wind [6]. WindBeasts are able to walk gracefully (smooth motion with a long step length) on the beach of Netherlands. Theyare created by the pure artistic instinct, yet they are governed by basic mechanics. For examples, the legmechanism has many advantages from the design viewpoint, such as, SDOF, a crank as an input link andan ovoid foot path. On the other hand, Wind Beasts have certain unique artistic features, which represent

344 Transactions of the Canadian Society for Mechanical Engineering, Vol. 36, No. 4, 2012

significant design challenges. For examples, the two four-bar linkages with the common input link, A0ABB0and A0AEB0 shown in Fig. 1(b), are identical and the four-bar linkage, B0CDE also shown in Fig. 1(b), isa parallelogram mechanism. The most amazing artefact from an engineering point of view is the use of thewind power, which demands the leg mechanism be designed with low energy-input, yet still maintains along step length and smooth gait motion. Mr. Jansen’s Wind Beasts are a fusion of Art and Engineering,which inspires us to explore the feasibility of designing a Wind Beast using the mechanism design theoryand optimization.

In this work, we intend to demonstrate the feasibility of the classical engineering theories to create aleg mechanism similar to Wind Beasts [6], which requires a low energy input yet still exhibits smoothlocomotion motion with a long step length. The mechanism design theory and optimization will be usedin this work. Several challenges are encountered: (1) the special features included in the leg mechanism ofWind Beasts and the large number of the links make the solution process tedious, even infeasible, and (2)minimizing the energy-input requires dynamics analysis, which needs the information about the interactionsbetween the feet and the ground. Such information is not available. We will address the above challengesand present the design and the analysis of our legged mechanism imitating a Wind Beast. Finally, we builtour designed mechanism to demonstrate its smooth motion.

(a)

3""

mechanisms powered by wind [6]. Wind Beasts are able to walk gracefully (smooth motion with a long step length) on the beach of Netherlands. They are created by the pure artistic instinct, yet they are governed by basic mechanics. For examples, the leg mechanism has many advantages from the design viewpoint, such as, SDOF, a crank as an input link and an ovoid foot path. On the other hand, Wind Beasts have certain unique artistic features, which represent significant design challenges. For examples, the two four-bar linkages with the common input link, shown in red in Fig. 1(b), are identical and the 4-bar linkage, shown in blue in Fig. 1(b), is a parallelogram mechanism. The most amazing artefact from an engineering point of view is the use of the wind power, which demands the leg mechanism be designed with low energy-input, yet still maintains a long step length and smooth gait motion. Mr. Jansen’s Wind Beasts are a fusion of Art and Engineering, which inspires us to explore the feasibility of designing a Wind Beast using the mechanism design theory and optimization.

In this work, we intend to demonstrate the feasibility of the classical engineering theories to create a leg mechanism similar to Wind Beasts [6], which requires a low energy input yet still exhibits smooth locomotion motion with a long step length. The mechanism design theory and optimization will be used in this work. Several challenges are encountered: (1) the special features included in the leg mechanism of Wind Beasts and the large number of the links make the solution process tedious, even infeasible, and (2) minimizing the energy-input requires dynamics analysis, which needs the information about the interactions between the feet and the ground. Such information is not available. We will address the above challenges and present the design and the analysis of our legged mechanism imitating a Wind Beast. Finally, we built our designed mechanism to demonstrate its smooth motion. 2 DESIGN OF A LEGGED WALKING MECHANISM

2.1 Mechanism Description The planar SDOF mechanism inspired by Mr. Theo Jansen’s kinetic sculpture, Wind Beasts, is an eight-bar mechanism shown in Fig. 1(b), which consists of a pair of identical upper and lower four-bar mechanisms, A0ABB0 and A0AEB0, augmented with the parallelogram mechanism, B0CDEF where DEF forms one rigid foot-link. The eight-bar linkage is equivalent to a six-bar mechanism from a design viewpoint since the upper and lower four-bar linkages A0ABB0 and

"

Fig. 1. (a) A Wind Beast created by Mr. Theo Jansen, and (b) a schematic figure of the leg.

A0"B0"

C"

D"E"

F"

Coupler"Coupler"

"(a) (b)

A"

B"Upper"45bar"mechanism"

Lower"45bar"mechanism"

Foot5link"

Parallel""mechanism"

(b)

Fig. 1. (a) A Wind Beast created by Mr. Theo Jansen, (b) a schematic figure of the leg.

2. DESIGN OF A LEGGED WALKING MECHANISM

2.1. Mechanism DescriptionThe planar SDOF mechanism inspired by Mr. Theo Jansen’s kinetic sculpture, Wind Beasts, is an eight-

bar mechanism shown in Fig. 1(b), which consists of a pair of identical upper and lower four-bar mecha-nisms, A0ABB0 and A0AEB0, augmented with the parallelogram mechanism, B0CDEF where DEF formsone rigid foot-link. The eight-bar linkage is equivalent to a six-bar mechanism from a design viewpointsince the upper and lower four-bar linkages A0ABB0 and A0AEB0 are identical in dimensions as used in theWind Beast. Note that in the design procedure, the linkage, B0CDE, was not restricted as a parallelogrammechanism to impose fewer constraints to the solution procedure.

To design the legged walking mechanism, shown in Fig. 1(b), A0A serves as an input link and DEF servesas the foot-link with F as the tracer point. In our design, link A0B0 is fixed. The mechanism is designedsuch that the trajectory of the tracer point is an ovoid, as shown in Fig. 2(b), for two reasons: (1) the ovoid


path enables the walking mechanism to step over small obstacles without significantly raising its body orapplying an additional DOF motion and, (2) it can also minimize the slamming effect caused by the inertialforces during walking as discussed in [4,5]. The path of the tracer point is composed of two portions duringeach step. The first portion is the propelling portion, between F1 and F2 as shown in Fig. 2(b), where thetracer point F is in contact with the ground. The second portion is the returning portion, where the tracer-point, F, is not in contact with the ground. The distance between F1 and F2 is the stride length, which isproportional to the step length, and the height H (Fig. 2b) is the maximum height of an obstacle that thewalking machine can step over. Since the trajectory of the tracer point relative to the upper body (AA0)is a closed curve and A0A is located outside the curve, a crank-rocker mechanism must be designed asdiscussed in [9]. Note that the stride length (F1F2) is different from the step length in that during the design,the “hip” is fixed and the stride length is the propelling distance of the “hip” in the actual walking, whilethe step length is the distance between the two subsequent contact points of “foot” (the tracer points) andthe ground. However, the stride length and the step length are linear proportional, i.e., a longer stride lengthleads to a longer step length.

2.2. Mechanism Synthesis

4""

A0AEB0 are identical in dimensions as used in the Wind Beast. Note that in the design procedure, the linkage, B0CDE, was not restricted as a parallelogram mechanism to impose fewer constraints to the solution procedure.

To design the legged walking mechanism, shown in Fig. 1(b), A0A serves as an input link and DEF serves as the foot-link with F as the tracer point. In our design, link A0B0 is fixed. The mechanism is designed such that the trajectory of the tracer point is an ovoid, as shown in Fig. 2(b), for two reasons: (1) the ovoid path enables the walking mechanism to step over small obstacles without significantly raising its body or applying an additional DOF motion and, (2) it can also minimize the slamming effect caused by the inertial forces during walking as discussed in [4,5]. The path of the tracer point is composed of two portions during each step. The first portion is the propelling portion, between F1 and F2 as shown in Fig. 2(b), where the tracer point F is in contact with the ground. The second portion is the returning portion, where the tracer-point, F, is not in contact with the ground. The distance between F1 and F2 is the stride length, which is proportional to the step length, and the height H (Fig. 2b) is the maximum height of an obstacle that the walking machine can step over. Since the trajectory of the tracer point relative to the upper body (AA0) is a closed curve and A0A is located outside the curve, a crank-rocker mechanism must be designed as discussed in [9]. Note that the stride length (F1F2) is different from the step length in that during the design, the “hip” is fixed and the stride length is the propelling distance of the “hip” in the actual walking, while the step length is the distance between the two subsequent contact points of “foot” (the tracer points) and the ground. However, the stride length and the step length are linear proportional, i.e., a longer stride length leads to a longer step length. 2.2 Mechanism synthesis

Fig. 2. (a) notation of the mechanism, and (b) the trajectory of the tracer point. (a) (b) (a)

4""

A0AEB0 are identical in dimensions as used in the Wind Beast. Note that in the design procedure, the linkage, B0CDE, was not restricted as a parallelogram mechanism to impose fewer constraints to the solution procedure.

To design the legged walking mechanism, shown in Fig. 1(b), A0A serves as an input link and DEF serves as the foot-link with F as the tracer point. In our design, link A0B0 is fixed. The mechanism is designed such that the trajectory of the tracer point is an ovoid, as shown in Fig. 2(b), for two reasons: (1) the ovoid path enables the walking mechanism to step over small obstacles without significantly raising its body or applying an additional DOF motion and, (2) it can also minimize the slamming effect caused by the inertial forces during walking as discussed in [4,5]. The path of the tracer point is composed of two portions during each step. The first portion is the propelling portion, between F1 and F2 as shown in Fig. 2(b), where the tracer point F is in contact with the ground. The second portion is the returning portion, where the tracer-point, F, is not in contact with the ground. The distance between F1 and F2 is the stride length, which is proportional to the step length, and the height H (Fig. 2b) is the maximum height of an obstacle that the walking machine can step over. Since the trajectory of the tracer point relative to the upper body (AA0) is a closed curve and A0A is located outside the curve, a crank-rocker mechanism must be designed as discussed in [9]. Note that the stride length (F1F2) is different from the step length in that during the design, the “hip” is fixed and the stride length is the propelling distance of the “hip” in the actual walking, while the step length is the distance between the two subsequent contact points of “foot” (the tracer points) and the ground. However, the stride length and the step length are linear proportional, i.e., a longer stride length leads to a longer step length. 2.2 Mechanism synthesis

Fig. 2. (a) notation of the mechanism, and (b) the trajectory of the tracer point. (a) (b)

(b)

Fig. 2. (a) Notation of the mechanism, (b) the trajectory of the tracer point.

To simplify the notations of each link, a new convention of labelling is shown in Fig. 2(a), where Zi is avector representing each link. The mechanism, shown in Fig. 2(a), is synthesized in two steps. First is thesynthesis of the four-bar linkages Z1Z2Z3Z4 and Z1Z2Z5Z6. Since both linkages are identical, the synthesisof the linkage, Z1Z2Z3Z4, is presented. Such a linkage is treated as a function generator with Z2 as the inputlink (a crank) and Z4 as the output link (a rocker). The relationship between the input motion, θ2 and outputmotion, θ4, is described by a sinusoidal function, i.e.:

θ4 = Asin(θ2−B)+C , (1)

where ∆θ2= 360◦ and∆θ4= 2A. The selection of a sinusoidal function for the function generator is basedon the consideration that the human hip motion can be approximated as a sinusoidal function. To designa mechanism with low vertical movement and to a certain extent similar to the smooth human hip motion,


it was desirable to have the input and output motion to satisfy Eq. (1) as discussed in reference [9]. Threecoefficients A, B and C are used as free choices in the synthesis. Their selections will be discussed later.

In the synthesis of the function generator, the well-known Freudenstein’s method [12] is used with threeprecision points, determined by Chebyshev spacing [13]. With the free choice of the ground spacing Z1, thelengths of Z2, Z3 and Z4 are obtained.

The second step is the design of mechanism Z6Z8Z9Z11Z12 as a path generator using four precisionpoints, F1, F2, F3 and F4, where the dyad Z6Z12 and the triad Z8Z9Z11 are synthesized separately. Thesolution is obtained by solving the complex equations derived using the complex number method [14] whereone of the free choices for each dyad and triad is the rotation of the link Z6 or Z9 between the first andsecond precision points. Furthermore the rotation of the coupler (Z10Z11Z12) is not prescribed, thereforethe rotations of the foot-link, θ12, from the first to the remaining three precision positions are also the freechoices.

One challenge of the design is that the path generator must be compatible with the function generator inthat Z6 have already been determined during function generation. Therefore, the path generator synthesizedmust returning the same vector of Z6. Furthermore, to ensure that an acceptable result can be obtained, thefour precision points of the path generator are determined by assuming the crack angles and the length ofZ12 such that they form an ovoid path.

Together with the design of the function generator, there are a total of 16 free choices including the threeparameters, A, B, and C shown in Eq. (1), describing the function to be generated, the ground link Z1 of thefunction generator, the input link direction at the first precision point, θ2, the coupler angle θc, the lengthof Z8, the four angles defining the position of the foot-link, θ12, at each precision point, the length of Z12and the four crank angles corresponding to the four precision points for the path generator. Once these freechoices are selected, the dimensions of the legged mechanism, shown in Fig. 2(a), can be determined usingthe previously mentioned methods. In this work, an optimization scheme is used for the selection of the freechoices. A set of constraints must be satisfied in order to exhibit acceptable motion, which is discussed inthe following section.

2.3. ConstraintsA set of constraints imposed on the leg mechanism are presented below for determining acceptable mech-

anism.

2.3.1. Grashof CriteriaThe two four-bar function generators (Z1Z2Z3Z4 and Z1Z2Z5Z6) shown in Fig. 2(a) must be crank-

rocker mechanisms. This is guaranteed by (1) satisfying the Grashof criterion [15] where the sum of theshortest and longest link must be lower than the sum of the remaining two links and (2) ensuring that thecrank is the shortest link:

C1 = x1 + x2 < x3 + x4 , (2)

where x1 and x2 are the shortest and longest links, x3 and x4 are the remaining two links, and:

C2 = Z2 ≤ min{Z1,Z3,Z4} (3)

2.3.2. Stride LengthThe stride length is the distance between F1 and F2 as shown in Fig. 2(b). To produce a long step length,

the stride length must be above a specified value (HC1):

C3 =| F1−F2 |≥ HC1 . (4)


2.3.3. Parallelogram Mechanism AnglesAlthough in the design, Z6Z8Z9Z10 is not restricted to be an exact parallelogram mechanism, it is selected

to be close to it. It is noticed that certain parallelogram mechanisms (Z6Z8Z9Z10) exhibit jamming whenthe angles between the four links exceed certain ranges, HC2 and HC3. Therefore, the interior anglesof the parallelogram mechanism are examined throughout the gait cycle and if any angle goes beyond anacceptable range, the mechanism is rejected.

C4 = HC2≤ {θ6,8 ,θ8,9 ,θ9,10 ,θ6,10} ≤ HC3 , (5)

where θ6,8, θ8,9, θ9,10 and θ6,10 are the angles between links Z6 and Z8, Z8 and Z9, Z9 and Z10, and Z6 andZ10, as shown in Fig. 2(a).

2.3.4. Trajectory of the Tracer PointThe trajectory of the tracer point needs to be checked to ensure that two events occur: (1) the trajectory

along the ground, F1F2, is flat and (2) the tracer point does not come into contact with the ground duringits returning path. To ensure (1), the lowest point F3, shown in Fig. 2(b), is first identified, and the verticaldistance between F3 and an arbitrary point within F1 and F2 are determined, and if such distance is lowerthan a pre-determined value, HC4, the tracer point is considered along the ground, i.e.:

C5 =| YF −YF3 |≤ HC4 , (6)

where YF is the vertical coordinate of an arbitrary point between F1 and F2. For the returning path, if anypoints are in contact with the ground then the solution is rejected. Note that the coordinate system is attachedto the ground at A with vertical axis upward.

C6 = YF ≥ YF3 +HC4 . (7)

2.3.5. Parallelogram MechanismIn the Wind Beast, the linkage equivalent to Z6Z8Z9Z10 is a parallelogram mechanism. However, it is

not feasible to obtain such solutions due to the fact that the mechanism Z6Z8Z9Z10 must be compatiblewith the function generator Z1Z2Z5Z6 and it is dependent on the precision points of the foot path. Thefree choices of the rotation θ6 and θ8 between the first and second precision points used for the synthesisof the path generator, Z6Z8Z9Z11Z12, will directly affect the solution. The following constraints are setup. The satisfaction of such constraints will ensure the compatibility of the link Z6, determined from bothfunction generation, Z6 desired and path generation, Z6 actual and Z6Z8Z9Z10 to be close to a parallelogrammechanism:

C7 =| Z6 desired−Z6 actual |+ | θ6desired−θ6actual |≤ HC5 , (8)

C8 =| Z6 desired−Z9 actual |+ | Z8−Z10 |≤ HC6 . (9)

2.4. OptimizationDue to the large number of the links involved and the constraints imposed, it is extremely challenging

to obtain a set of acceptable solutions, much less about optimization. In this work, a constrained multi-objective optimization approach [16] is used to optimize the design.

The first objective of the optimization is to minimize the energy over the cycle, which is represented byintegrating the squared torque over one cycle (complete rotation of the crank), shown below:

O1 =∫

τ

0T 2dt , (10)


where τ is the amount of time for completing a full rotation of the crank and T is the torque applied to thecrank. Equation (10) is an integral of the control effort during each cycle. It has been discussed that reducingthe control effort indicates the low energy consumption and has often been used in the energy optimizationin bipedal walking robots [17].

To calculate the torque throughout the cycle, the complete kinematics need to be determined for each linkbased on the position, velocity and acceleration of the crank. Furthermore, the torque applied to the crankand the forces applied at the joints can be determined using inverse dynamics [18,19]. In this work, thefollowing assumptions are used for the dynamic analysis: (1) the links are simplified as rigid bodies withuniform distributed mass; (2) the friction at the joints is neglected, but the friction between the tracer pointand the ground is sufficiently high so that no slipping occurs during the contact phase between the tracerpoint and the ground, and (3) when the leg comes into contact with the ground, it is assumed to have a zeroimpact. The ground reaction forces are critical for dynamic analysis. Since the force sensors for measuringground reactions are not available, the following approximations are made. Considering the Wind Beastwalking on a relatively flat sand beach, there are three forces applied on each leg, the gravity, wind forceand ground reactions. Since the wind force is relatively low, it was neglected in the dynamics analysis.We estimated the ground reactions by using the accelerations of the mass center of the entire leg, since theground reactions are the only forces in addition to the gravity applied on the leg. The accelerations of theoverall mass center can be estimated using the kinematics of each link [20].

In legged locomotion, the step length, which is proportional to the stride length is as important as theenergy consumption. However a small increase in the step length can cause higher energy consumption. Tofind the stride length, F1F2, shown in Fig. 2(b), the following equations are used:

Y _POS = Z6 sinθ6 +Z12 sin(θ12) , (11)

X_POS = Z6 cosθ6 +Z12 cos(θ12) (12)

and the stride length is approximated as:

O2 =| F1−F2 | . (13)

Two objectives are combined to form a minimization problem. This was done by dividing objective one byobjective two seen in the following:

O3 =

∫τ

0 T 2dt| F1−F2 |

. (14)

Note that the combination of the two objectives shown above is rather arbitrary and the multi-objectiveused in this work is one of the many ways for optimization. Our multi-objective functional is intuitive inthat the decrease in the control effort and the increase in the step length are simultaneously achieved by min-imizing the proposed multi-objective functional. Other multi-objective functional, such as the summationof the individual objective functions with assigned weights, can also be used for optimization.

Due to the large number of the free choices and constraints, it is extremely challenging to obtain an accept-able legged mechanism. We first selected the free choices by trail-and-error with the goal to achieve a longstep length while keeping the 4-bar linkage Z6Z8Z9Z10, shown in Fig. 2(a), as a parallelogram mechanismand maintaining smooth motion. The process of trial-and-error enables us to gain insights into the effectsof the free choices on the motion of the mechanisms. We then applied for an optimization process usingthe free choices from the trial and error as the initial parameters for the optimization process. In addition,certain free choices from the trial-and-error results were set as constants to reduce the computational time.Furthermore, for the four-bar function generator, originally there are four free choices including the functionto be generated. Instead we decided to have three free choices holding the ground spacing (Z1) at a constant


throughout the optimization to exclude the effects of scaling. The trial-and-error results demonstrated thatnumerous solutions can be found. The same crank angles and the angle of the foot-link corresponding toeach precision point of the path generator from the trial-and-error search were used in optimization. Withsetting these variables to constant values, the optimization problem is greatly reduced in complexity leavinga more realistic optimization problem that uses 6 variables (Z2,Z3,Z4,Z8,Z12,θc).

We found that there exist many local minima during optimization. We performed a coarse exhaustivesearch over defined regions of the variables; with the best configurations found, based on the objective func-tion Eq. (14), we performed a local optimization at those configurations. The function fmincon (constrainednon-linear minimization) in the optimization toolbox in Matlab was used to perform the local optimizationat the best configurations found in the coarse exhaustive search. The function fmincon can accept a largeamount of variables with the ability to define certain constraints and regions of interest for each variable.

2.5. Physical PrototypingTo demonstrate that the legged mechanism imitating a well-known kinetic sculpture, but designed based

on engineering design theories, can produce smooth walking motion with a long step length, a low-costprototype was built based on the dimensions determined from the optimal design. The mechanism consistsof 3 pairs of legs with the overall dimensions of 149.9 cm long, by 66.0 cm wide and 64.8 cm high. A framewith wheels is used to support the motors. This is only necessary due to the limited available power fromthe existing motors.

The mechanism is made of 1.91 cm electrical metal conduit metal tubing. All links are connected withpin joints by inserting 0.48 cm metal rods into the pre-drilled holes at the ends of the tubes except for thelinks attached to the frame, which have a connector glued on the end which slides over the tubing. To spacethe leg pairings evenly apart, the crank of each pair is offset by 120◦ to have smooth motion when beingpropelled forward. To increase the friction at the feet, rubber pads were added to the foot. This also addssome cushioning to reduce impact as the foot contacts the ground. Furthermore, the lengths of the legs werechosen to be scaled 1.4 times as the optimized dimensions. A DC motor with a gear reduction and a variablespeed control is attached at each end of the crank. The motors are reversible which allows the mechanismto walk in forward and backward directions.

3. RESULTS AND DISCUSSION

A number of challenges are encountered to design a legged mechanism imitating the Wind Beast. Someof the challenges are stemmed from artistic effects and others are due to design limitations. Such challengesmake it challenging to obtain a reasonable solution. The goal is to demonstrate that a legged walkingmechanism imitating Wind Beast can be designed based on engineering theories. Such a mechanism shouldexhibit graceful walking motion, i.e., smooth motion with a long step length, while requires low energyinput.

A coarse exhaustive search was first performed. The trial-and-error portion was performed by selecting16 free choices as discussed in Subsection 2.2. This procedure provides significant insights into the effectsof the changes in the free choices on feasibility and the quality of the legged mechanism. The followingvalues were chosen: the minimum stride length (HC1) of 9 cm, the range of the angles between links ofthe parallelogram mechanism of 5◦ (HC2) and 175◦ (HC3), the vertical offset used to defined the stridelength (HC4) of 0.002 m. To avoid the scaling effect, Z1 was chosen as 15 cm. Through the search, thebest obtained function is θ4 = 0.524sin(θ2−1.134)+1.833(rad). Next, the coupler angles, θC and Z8 werechosen as 60◦ and 10 cm. For the design of the path generator, a 19 cm was chosen for Z12 and the crankangles of−98◦,−102◦,−74◦ and−46◦ were selected corresponding to the four precision points for the pathgenerator, After a substantial amount of time searching, an acceptable mechanism with the desired tracer


point trajectory was found. Figure 3(a) shows the designed legged mechanism in dashed lines and Table 1lists the lengths of the links of the trial-and-error mechanism. As a result the stride length, F1F2, is 11.68 cmand the highest obstacle that can be clear, H, is 6.5 cm as shown in Fig. 3(b).

The Matlab optimization software was then used on the best found configurations from the trial and errorto find the local minimum at each configuration. Each solution used the same parameters discussed aboveas the initial parameters with the additions of the followings: (i) the links are uniform with a density of0.5 kg/m; (ii) a friction coefficient of 0.5 is experienced at the foot and (iii) a constant crank velocity of180°/sec (π rad/s). Using these parameters the objective function, shown in Eq. (14), was calculated foreach solution. The main dimensions are shown in Table 1 and the mechanism is shown in Fig. 3(a) insolid lines. It was found that it had a stride length of 10.51 cm and the highest obstacle that can be clearedis 2.11 cm (Fig. 3b). Note that it also noticed during the procedure seeking for solutions, the motion ofBB0 and EB0, shown in Fig. 1(b), has significant effects on the linkage, B0CDE, being a parallelogram andhaving the same motion of BB0 and EB0 is crucial for obtaining a parallelogram.

It can be seen from Table 1 that some links have minor changes in lengths while others are quite noticeable.Figure 3(b) shows the foot trajectories of both designs, which have similar flat profiles along the ground,however the major difference between the two trajectories is the return path. The return path of the optimizeddesigned mechanism has the lower height and the stride length. Thus less energy is used to overcome gravity.

Link Lengths (cm) number Trial-and-error Optimized

Z1 15 15 Z2 4.17 2.78 Z3 20.33 20.02 Z4 12 12.05 Z5 20.33 20.02 Z6 12 12.07 Z7 11.14 12.30 Z8 10 7.01 Z9 11.89 12.08 Z10 10 6.84 Z11 26.11 22.31 Z12 19 19.54

Table 1 Table 1. Trial-and-error and optimization results.

Detailed kinematic and dynamic analyses were carried out on both mechanisms. Figure 4 shows the inputtorque for each mechanism, respectively. It can be seen that the torque from the optimized mechanism islower in magnitude throughout the entire gait cycle, and thus has a lower energy input. Figure 4 also showsthat the largest magnitudes of torque are found during the return phase of the foot path. This is where thegreatest improvements made through optimization in the demand of the low input torque are found.

When comparing the dynamics of both mechanisms, the peak torque of the optimized mechanism de-creases by 55.4 % and its stride length decreases by 10 %. Examining the calculated objective function,shown in Eq. (14), it was found that the value of the objective function, O3, from the optimized mechanismwas 84 % less than that of the trial-and-error. Furthermore, the energy over the cycle, shown in Eq. (12), wasfound to be 85.6 % lower. An undesirable component of the results is that the stride length of the optimized


leg decreased. This occurs because the mechanism determined by trial-and-error aims at having a longerstride length without any consideration of energy consumption, while for the optimized mechanism, energyminimization is a dominant factor in the optimization used here.

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Detailed kinematic and dynamic analyses were carried out on both mechanisms. Figure 4 shows the input torque for each mechanism, respectively. It can be seen that the torque from the optimized mechanism is lower in magnitude throughout the entire gait cycle, and thus has a lower energy input. Figure 4 also shows that the largest magnitudes of torque are found during the return phase of the foot path. This is where the greatest improvements made through optimization in the demand of the low input torque are found.

When comparing the dynamics of both mechanisms, the peak torque of the optimized mechanism decreases by 55.4% and its stride length decreases by 10%. Examining the calculated objective function, shown in equation (14), it was found that the value of the objective function, O3, from the optimized mechanism was 84% less than that of the trial-and-error. Furthermore, the energy over the cycle, shown in equation (12), was found to be 85.6% lower. An undesirable component of the results is that the stride length of the optimized leg decreased. This occurs because the mechanism determined by trial-and-error aims at having a longer stride length without any consideration of energy consumption, while for the optimized mechanism, energy minimization is a dominant factor in the optimization used here.

Fig. 3: (a) Comparison of Leg, (b) Foot trajectories (cm).

""Trial5and5error""""""""""""""""""optimized" """""a)

"""""b)

(a) (b)

Fig. 3. (a) Comparison of leg, (b) foot trajectories (cm).

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To demonstrate that the designed legged mechanism imitating a Wind Beast using engineering design theories, can produce smooth walking motion with a long step length, a low-cost prototype, shown in Fig. 5, was built. The motion is video recorded and the trajectory of the tracer point is closely observed. Figure 6 shows progressive shots of a single leg motion. It can be seen that the trajectory of the tracer point is similar to the desired one with a flat portion of the tracer point trajectory. This is important for smooth and close to horizontal motion of the “hip”. It was also observed that no-slipping occurred at any feet, the motion of all links and the whole mechanism is smooth with the step length close to the desired one. Figure 7 shows a progressive still shot of the entire mechanism motion. Overall, the above results demonstrate that the legged mechanism imitating a Wind Beast and designed using engineering design theories can produce smooth walking motion showing the success in the designed legged mechanism.

Fig. 4: Torque curves Fig. 4. Torque curves.

To demonstrate that the designed legged mechanism imitating a Wind Beast using engineering designtheories, can produce smooth walking motion with a long step length, a low-cost prototype, shown in Fig. 5,was built. The motion is video recorded and the trajectory of the tracer point is closely observed. Figure 6shows progressive shots of a single leg motion. It can be seen that the trajectory of the tracer point is similarto the desired one with a flat portion of the tracer point trajectory. This is important for smooth and closeto horizontal motion of the “hip”. It was also observed that no-slipping occurred at any feet, the motion ofall links and the whole mechanism is smooth with the step length close to the desired one. Figure 7 shows aprogressive still shot of the entire mechanism motion. Overall, the above results demonstrate that the leggedmechanism imitating a Wind Beast and designed using engineering design theories can produce smoothwalking motion showing the success in the designed legged mechanism.


Fig. 5. Physical prototype of the leg mechanism.

Fig. 6. Progressive still shot of a single leg motion.

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4 CONCLUSIONS Legged off-road vehicles have better mobility, higher energy efficiency and are easier to

control as compared with those of conventional tracked or wheeled vehicles while moving on rough terrains. Development of such vehicles represents a challenging problem, and has attracted attentions from both engineering and art fields. Mr. Theo Jansen, created amazing kinetic sculptures, Wind Beasts, which are actuated by wind power, walking gracefully on the beaches of Netherland. Wind Beasts are the fusion of art and engineering.

We presented the design of a SDOF eight-bar legged walking mechanism imitating the Wind Beasts using the mechanism design theory. Our mechanism was designed to produce smooth walking motion with a large step length and to require a low energy input. The mechanism consists of a function generator and a path generator, and the equations for both generators were derived and 16 free choices were identified. The mechanism is further optimized to reduce the input energy while keeping a large stride length. Two acceptable mechanisms were successfully designed. One was through trial-and-error to achieve a satisfactory stride length, and the second one was further optimized to reduce the energy input and to maximize the stride length. The kinematics and dynamics simulations showed that both mechanisms exhibit desired ovoid trajectories of the tracer point with satisfactory stride lengths. The input torque of the optimized leg mechanism is consistently lower than the one from the mechanism via trial-and-error. However, kinematics analysis shows that the stride length of the optimized mechanism is 10% lower than the stride length of the original mechanism. This is because for the original mechanism the energy input was not concerned. To demonstrate the proposed design is feasible, a prototype of the leg mechanism based on the optimal design was built to demonstrate that the desired motion was achieved.

Fig."7:"Progressive"still"shot"of"the"leg’s"motion"Fig. 7. Progressive still shot of the leg’s motion.


4. CONCLUSIONS

Legged off-road vehicles have better mobility, higher energy efficiency and are easier to control as com-pared with those of conventional tracked or wheeled vehicles while moving on rough terrains. Developmentof such vehicles represents a challenging problem, and has attracted attentions from both engineering andart fields. Mr. Theo Jansen, created amazing kinetic sculptures, Wind Beasts, which are actuated by windpower, walking gracefully on the beaches of Netherland. Wind Beasts are the fusion of art and engineering.

We presented the design of a SDOF eight-bar legged walking mechanism imitating the Wind Beasts us-ing the mechanism design theory. Our mechanism was designed to produce smooth walking motion witha large step length and to require a low energy input. The mechanism consists of a function generator anda path generator, and the equations for both generators were derived and 16 free choices were identified.The mechanism is further optimized to reduce the input energy while keeping a large stride length. Twoacceptable mechanisms were successfully designed. One was through trial-and-error to achieve a satisfac-tory stride length, and the second one was further optimized to reduce the energy input and to maximizethe stride length. The kinematics and dynamics simulations showed that both mechanisms exhibit desiredovoid trajectories of the tracer point with satisfactory stride lengths. The input torque of the optimized legmechanism is consistently lower than the one from the mechanism via trial-and-error. However, kinematicsanalysis shows that the stride length of the optimized mechanism is 10 % lower than the stride length of theoriginal mechanism. This is because for the original mechanism the energy input was not concerned. Todemonstrate the proposed design is feasible, a prototype of the leg mechanism based on the optimal designwas built to demonstrate that the desired motion was achieved.

Art and engineering have often been considered to be separate fields and the resistance to merge comesfrom both sides. This work is an initial attempt to imitate an art work using the engineering design theoriesand to cross the border between art and engineering.

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