DESIGN AND OPTIMIZATION OF AN EIGHT-BAR LEGGED WALKINGMECHANISM

IMITATINGA KINETIC SCULPTURE,“WIND BEAST”

Daniel Giesbrecht,Christine Q.Wu,Nariman Sepehri

Department of Mechanical Engineering,University of Manitoba,Winnipeg,Manitoba,Canada

E-mail:cwu@cc.umanitoba.ca

Received August 2011,Accepted November 2012

No.11-CSME-63,E.I.C.Accession 3303

ABSTRACT

Legged 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 and

engineers.In this paper,we present the design of a single-degree-of-freedom legged walking mechanism

using the mechanism design theory and optimization to imitate a well-known kinetic sculpture,a Wind

Beast.The optimization is set up to:i) minimize the energy input and ii) maximize the stride length.The

optimization is based on the dynamic force analysis.Aprototype of the optimized walking mechanismwith

6 legs was built to demonstrate its smooth motion.The success in designing a legged mechanismcapable of

imitating the well-known kinetic sculpture using the engineering design theories is a small step bridging the

gap between art and engineering.

Keywords:legged walking mechanism;mechanismdesign;type synthesis;dimension synthesis;optimiza-

tion;kinematics and dynamics analysis.

CONCEPTION ET OPTIMISATION D’UN MÉCANISME MARCHEUR À HUIT BARRES

IMITANT LA SCULPTURE CINÉMATIQUE"WIND BEAST"

RÉSUMÉ

Les 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’intérêt

d’artistes et d’ingénieurs.Cet article présente la conception d’un mécanisme marcheur à un degré de liberté

utilisant 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 du

pas.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 la

conception d’un mécanisme à pattes capable d’imiter la célèbre sculpture cinématique utilisant les théories

de 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 de

dimension;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 efﬁciency

and provide more comfortable movement than those of conventional tracked or wheeled vehicles while

moving on rough terrain [1].In the last several decades,a wide variety of legged mechanisms have been

researched for the applications of legged locomotion,such as planetary exploration,walking chairs for the

disabled and for military transport,and rescue in radioactive zones for nuclear industries or in other hostile

environments.

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 recent

development of legged robots due to the advancement in control theories and actuation technologies.Such

MDOF robots have the advantages of being simple in the leg structure,but are demanding to the actuation

and control.On the other hand,the SDOF/LDOF legged mechanisms have the advantages of requiring

little control and simple actuation,yet can still create versatile walking motion with a low demand on the

energy input.It has been discussed that unlike a ground-based manipulator that can be operated with an

off-board power supply,a walking machine has to carry the entire power supply in addition to the external

payload and the weight of the machine body.Thus it is desirable to use a small number of actuators to

reduce the body weight and to simplify the motion coordination [3].A number of six or seven-link SDOF

leg mechanisms have been designed [3–7].Rigorous research has been carried out on their mobility and

energy loss through kinematic and structural analysis.Two important ﬁndings have been documented:(1) a

crank as an input link with continuous rotation motion should be used to achieve fast motion with minimum

control [3–5,8] and (2) an ovoid foot path is necessary to step over small obstacles without signiﬁcantly

raising the body [3,4,8].The challenge in developing a SDOF legged walking mechanismis the requirement

of a large number of links required to provide high mobility.Thus,the type selection and the dimensional

synthesis 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 cammechanisms [10] have been used.It was recommended

to use only revolute joints for legged walking machines due to the difﬁculties in lubrication and sealing

of the sliding joints,which is essential for the machines to walk outdoors [9].Many pin-joined legged

mechanisms have been designed,which are often compound mechanisms consisting of a four-bar linkage

and 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 rough

terrain,they often involve a large number of geometrical dimensions,which makes it necessary to resort

to optimization to achieve a high quality design.Reducing the energy loss has always been the interest

in designing legged mechanisms [3,4,7].In some research,springs were added to store the energy and to

reduce the actuating torque [4].In the process of energy optimization,the force analysis is needed.Due to

the complexity of the mechanisms,in previous research,the force analysis has been restricted to the static

analysis [3,4,7].It is known that the dynamic analysis of the mechanism has important impact on energy

optimization especially when fast locomotion is to be created.

While developing legged walking robots have been actively pursued by engineers,they have also attracted

attentions 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].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

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,A

0

ABB

0

and A

0

AEB

0

shown in Fig.1(b),are identical and the four-bar linkage,B

0

CDE also shown 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 mechanismto demonstrate its smooth motion.

(a)

(b)

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

2.DESIGN OF A LEGGED WALKINGMECHANISM

2.1.MechanismDescription

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 mecha-

nisms,A

0

ABB

0

and A

0

AEB

0;

augmented with the parallelogram mechanism,B

0

CDEF 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 A

0

ABB

0

and A

0

AEB

0

are identical in dimensions as used in the

Wind Beast.Note that in the design procedure,the linkage,B

0

CDE,was not restricted as a parallelogram

mechanismto impose fewer constraints to the solution procedure.

To design the legged walking mechanism,shown in Fig.1(b),A

0

Aserves as an input link and DEFserves

as the foot-link with F as the tracer point.In our design,link A

0

B

0

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

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 345

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 F

1

and F

2

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 F

1

and F

2

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 (AA

0

)

is a closed curve and A

0

A is located outside the curve,a crank-rocker mechanism must be designed as

discussed in [9].Note that the stride length (F

1

F

2

) is different fromthe 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.MechanismSynthesis

(a)

(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 Z

i

is a

vector representing each link.The mechanism,shown in Fig.2(a),is synthesized in two steps.First is the

synthesis of the four-bar linkages Z

1

Z

2

Z

3

Z

4

and Z

1

Z

2

Z

5

Z

6

.Since both linkages are identical,the synthesis

of the linkage,Z

1

Z

2

Z

3

Z

4

,is presented.Such a linkage is treated as a function generator with Z

2

as the input

link (a crank) and Z

4

as the output link (a rocker).The relationship between the input motion,q

2

and output

motion,q

4

,is described by a sinusoidal function,i.e.:

q

4

=Asin(q

2

B) +C;(1)

where q

2

=360

andq

4

=2A:The selection of a sinusoidal function for the function generator is based

on the consideration that the human hip motion can be approximated as a sinusoidal function.To design

a mechanism with low vertical movement and to a certain extent similar to the smooth human hip motion,

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

it was desirable to have the input and output motion to satisfy Eq.(1) as discussed in reference [9].Three

coefﬁcients 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 three

precision points,determined by Chebyshev spacing [13].With the free choice of the ground spacing Z

1

,the

lengths of Z

2

,Z

3

and Z

4

are obtained.

The second step is the design of mechanism Z

6

Z

8

Z

9

Z

11

Z

12

as a path generator using four precision

points,F

1

,F

2

,F

3

and F

4

,where the dyad Z

6

Z

12

and the triad Z

8

Z

9

Z

11

are synthesized separately.The

solution is obtained by solving the complex equations derived using the complex number method [14] where

one of the free choices for each dyad and triad is the rotation of the link Z

6

or Z

9

between the ﬁrst and

second precision points.Furthermore the rotation of the coupler (Z

10

Z

11

Z

12

) is not prescribed,therefore

the rotations of the foot-link,q

12

,from the ﬁrst to the remaining three precision positions are also the free

choices.

One challenge of the design is that the path generator must be compatible with the function generator in

that Z

6

have already been determined during function generation.Therefore,the path generator synthesized

must returning the same vector of Z

6

.Furthermore,to ensure that an acceptable result can be obtained,the

four precision points of the path generator are determined by assuming the crack angles and the length of

Z

12

such that they forman ovoid path.

Together with the design of the function generator,there are a total of 16 free choices including the three

parameters,A,B,and C shown in Eq.(1),describing the function to be generated,the ground link Z

1

of the

function generator,the input link direction at the ﬁrst precision point,q

2

;the coupler angle q

c

,the length

of Z

8

,the four angles deﬁning the position of the foot-link,q

12

,at each precision point,the length of Z

12

and the four crank angles corresponding to the four precision points for the path generator.Once these free

choices are selected,the dimensions of the legged mechanism,shown in Fig.2(a),can be determined using

the previously mentioned methods.In this work,an optimization scheme is used for the selection of the free

choices.A set of constraints must be satisﬁed in order to exhibit acceptable motion,which is discussed in

the following section.

2.3.Constraints

Aset of constraints imposed on the leg mechanismare presented belowfor determining acceptable mech-

anism.

2.3.1.Grashof Criteria

The two four-bar function generators (Z

1

Z

2

Z

3

Z

4

and Z

1

Z

2

Z

5

Z

6

) shown in Fig.2(a) must be crank-

rocker mechanisms.This is guaranteed by (1) satisfying the Grashof criterion [15] where the sum of the

shortest and longest link must be lower than the sum of the remaining two links and (2) ensuring that the

crank is the shortest link:

C1 =x

1

+x

2

<x

3

+x

4

;(2)

where x

1

and x

2

are the shortest and longest links,x

3

and x

4

are the remaining two links,and:

C2 =Z

2

minfZ

1

;Z

3

;Z

4

g (3)

2.3.2.Stride Length

The stride length is the distance between F

1

and F

2

as shown in Fig.2(b).To produce a long step length,

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

C3 =j F

1

F

2

jHC1:(4)

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 347

2.3.3.ParallelogramMechanismAngles

Although in the design,Z

6

Z

8

Z

9

Z

10

is not restricted to be an exact parallelogrammechanism,it is selected

to be close to it.It is noticed that certain parallelogram mechanisms (Z

6

Z

8

Z

9

Z

10

) exhibit jamming when

the angles between the four links exceed certain ranges,HC2 and HC3.Therefore,the interior angles

of the parallelogram mechanism are examined throughout the gait cycle and if any angle goes beyond an

acceptable range,the mechanismis rejected.

C4 =HC2 fq

6;8

;q

8;9

;q

9;10

;q

6;10

g HC3;(5)

where q

6;8

,q

8;9

,q

9;10

and q

6;10

are the angles between links Z

6

and Z

8

,Z

8

and Z

9

,Z

9

and Z

10

,and Z

6

and

Z

10

,as shown in Fig.2(a).

2.3.4.Trajectory of the Tracer Point

The trajectory of the tracer point needs to be checked to ensure that two events occur:(1) the trajectory

along the ground,F

1

F

2

,is ﬂat and (2) the tracer point does not come into contact with the ground during

its returning path.To ensure (1),the lowest point F

3

,shown in Fig.2(b),is ﬁrst identiﬁed,and the vertical

distance between F

3

and an arbitrary point within F

1

and F

2

are determined,and if such distance is lower

than a pre-determined value,HC4,the tracer point is considered along the ground,i.e.:

C5 =j Y

F

Y

F3

jHC4;(6)

where Y

F

is the vertical coordinate of an arbitrary point between F

1

and F

2

.For the returning path,if any

points are in contact with the ground then the solution is rejected.Note that the coordinate systemis attached

to the ground at A with vertical axis upward.

C6 =Y

F

Y

F3

+HC4:(7)

2.3.5.ParallelogramMechanism

In the Wind Beast,the linkage equivalent to Z

6

Z

8

Z

9

Z

10

is a parallelogram mechanism.However,it is

not feasible to obtain such solutions due to the fact that the mechanism Z

6

Z

8

Z

9

Z

10

must be compatible

with the function generator Z

1

Z

2

Z

5

Z

6

and it is dependent on the precision points of the foot path.The

free choices of the rotation q

6

and q

8

between the ﬁrst and second precision points used for the synthesis

of the path generator,Z

6

Z

8

Z

9

Z

11

Z

12

,will directly affect the solution.The following constraints are set

up.The satisfaction of such constraints will ensure the compatibility of the link Z

6;

determined from both

function generation,Z

6 desired

and path generation,Z

6 actual

and Z

6

Z

8

Z

9

Z

10

to be close to a parallelogram

mechanism:

C7 =j Z

6 desired

Z

6 actual

j +j q

6desired

q

6actual

jHC5;(8)

C8 =j Z

6 desired

Z

9 actual

j +j Z

8

Z

10

jHC6:(9)

2.4.Optimization

Due 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 by

integrating the squared torque over one cycle (complete rotation of the crank),shown below:

O1 =

Z

t

0

T

2

dt;(10)

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

where t is the amount of time for completing a full rotation of the crank and T is the torque applied to the

crank.Equation (10) is an integral of the control effort during each cycle.It has been discussed that reducing

the control effort indicates the low energy consumption and has often been used in the energy optimization

in bipedal walking robots [17].

To calculate the torque throughout the cycle,the complete kinematics need to be determined for each link

based on the position,velocity and acceleration of the crank.Furthermore,the torque applied to the crank

and the forces applied at the joints can be determined using inverse dynamics [18,19].In this work,the

following assumptions are used for the dynamic analysis:(1) the links are simpliﬁed as rigid bodies with

uniform distributed mass;(2) the friction at the joints is neglected,but the friction between the tracer point

and the ground is sufﬁciently high so that no slipping occurs during the contact phase between the tracer

point and the ground,and (3) when the leg comes into contact with the ground,it is assumed to have a zero

impact.The ground reaction forces are critical for dynamic analysis.Since the force sensors for measuring

ground reactions are not available,the following approximations are made.Considering the Wind Beast

walking on a relatively ﬂat sand beach,there are three forces applied on each leg,the gravity,wind force

and 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 the

ground reactions are the only forces in addition to the gravity applied on the leg.The accelerations of the

overall 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 the

energy consumption.However a small increase in the step length can cause higher energy consumption.To

ﬁnd the stride length,F

1

F

2

,shown in Fig.2(b),the following equations are used:

Y_POS =Z

6

sinq

6

+Z

12

sin(q

12

);(11)

X_POS =Z

6

cosq

6

+Z

12

cos(q

12

) (12)

and the stride length is approximated as:

O2 =j F

1

F

2

j:(13)

Two objectives are combined to form a minimization problem.This was done by dividing objective one by

objective two seen in the following:

O3 =

R

t

0

T

2

dt

j F

1

F

2

j

:(14)

Note that the combination of the two objectives shown above is rather arbitrary and the multi-objective

used in this work is one of the many ways for optimization.Our multi-objective functional is intuitive in

that 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 summation

of 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 long

step length while keeping the 4-bar linkage Z

6

Z

8

Z

9

Z

10

,shown in Fig.2(a),as a parallelogram mechanism

and maintaining smooth motion.The process of trial-and-error enables us to gain insights into the effects

of the free choices on the motion of the mechanisms.We then applied for an optimization process using

the 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 function

to be generated.Instead we decided to have three free choices holding the ground spacing (Z

1

) at a constant

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 349

throughout the optimization to exclude the effects of scaling.The trial-and-error results demonstrated that

numerous solutions can be found.The same crank angles and the angle of the foot-link corresponding to

each precision point of the path generator from the trial-and-error search were used in optimization.With

setting these variables to constant values,the optimization problemis greatly reduced in complexity leaving

a more realistic optimization problemthat uses 6 variables (Z

2

;Z

3

;Z

4

;Z

8

;Z

12

;q

c

).

We found that there exist many local minima during optimization.We performed a coarse exhaustive

search 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 (constrained

non-linear minimization) in the optimization toolbox in Matlab was used to perform the local optimization

at the best conﬁgurations found in the coarse exhaustive search.The function fmincon can accept a large

amount of variables with the ability to deﬁne certain constraints and regions of interest for each variable.

2.5.Physical Prototyping

To 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-cost

prototype was built based on the dimensions determined from the optimal design.The mechanism consists

of 3 pairs of legs with the overall dimensions of 149.9 cmlong,by 66.0 cmwide and 64.8 cmhigh.Aframe

with wheels is used to support the motors.This is only necessary due to the limited available power from

the existing motors.

The mechanism is made of 1.91 cm electrical metal conduit metal tubing.All links are connected with

pin joints by inserting 0.48 cm metal rods into the pre-drilled holes at the ends of the tubes except for the

links attached to the frame,which have a connector glued on the end which slides over the tubing.To space

the leg pairings evenly apart,the crank of each pair is offset by 120

to have smooth motion when being

propelled forward.To increase the friction at the feet,rubber pads were added to the foot.This also adds

some cushioning to reduce impact as the foot contacts the ground.Furthermore,the lengths of the legs were

chosen to be scaled 1.4 times as the optimized dimensions.ADC motor with a gear reduction and a variable

speed control is attached at each end of the crank.The motors are reversible which allows the mechanism

to 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.Some

of the challenges are stemmed fromartistic effects and others are due to design limitations.Such challenges

make it challenging to obtain a reasonable solution.The goal is to demonstrate that a legged walking

mechanismimitating Wind Beast can be designed based on engineering theories.Such a mechanismshould

exhibit graceful walking motion,i.e.,smooth motion with a long step length,while requires low energy

input.

A coarse exhaustive search was ﬁrst performed.The trial-and-error portion was performed by selecting

16 free choices as discussed in Subsection 2.2.This procedure provides signiﬁcant insights into the effects

of the changes in the free choices on feasibility and the quality of the legged mechanism.The following

values were chosen:the minimum stride length (HC1) of 9 cm,the range of the angles between links of

the parallelogram mechanism of 5

(HC2) and 175

(HC3),the vertical offset used to deﬁned the stride

length (HC4) of 0.002 m.To avoid the scaling effect,Z

1

was chosen as 15 cm.Through the search,the

best obtained function is q

4

=0:524sin(q

2

1:134)+1:833(rad).Next,the coupler angles,q

C

and Z

8

were

chosen as 60

and 10 cm.For the design of the path generator,a 19 cm was chosen for Z

12

and the crank

angles of 98

,102

,74

and 46

were selected corresponding to the four precision points for the path

generator,After a substantial amount of time searching,an acceptable mechanism with the desired tracer

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

point trajectory was found.Figure 3(a) shows the designed legged mechanism in dashed lines and Table 1

lists the lengths of the links of the trial-and-error mechanism.As a result the stride length,F

1

F

2

,is 11.68 cm

and the highest obstacle that can be clear,H,is 6.5 cmas shown in Fig.3(b).

The Matlab optimization software was then used on the best found conﬁgurations fromthe trial and error

to ﬁnd the local minimum at each conﬁguration.Each solution used the same parameters discussed above

as the initial parameters with the additions of the followings:(i) the links are uniform with a density of

0.5 kg/m;(ii) a friction coefﬁcient of 0.5 is experienced at the foot and (iii) a constant crank velocity of

180°/sec (p rad/s).Using these parameters the objective function,shown in Eq.(14),was calculated for

each solution.The main dimensions are shown in Table 1 and the mechanism is shown in Fig.3(a) in

solid lines.It was found that it had a stride length of 10.51 cm and the highest obstacle that can be cleared

is 2.11 cm (Fig.3b).Note that it also noticed during the procedure seeking for solutions,the motion of

BB

0

and EB

0

,shown in Fig.1(b),has signiﬁcant effects on the linkage,B

0

CDE,being a parallelogramand

having the same motion of BB

0

and EB

0

is crucial for obtaining a parallelogram.

It can be seen fromTable 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 optimized

designed mechanismhas the lower height and the stride length.Thus less energy is used to overcome gravity.

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

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 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,O

3

,fromthe optimized mechanism

was 84 %less than that of the trial-and-error.Furthermore,the energy over the cycle,shown in Eq.(12),was

found to be 85.6 %lower.An undesirable component of the results is that the stride length of the optimized

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 351

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.

(a)

(b)

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

Fig.4.Torque curves.

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 mechanismis smooth with the step length close to the desired one.Figure 7 shows a

progressive still shot of the entire mechanismmotion.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.

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

Fig.5.Physical prototype of the leg mechanism.

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

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

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 353

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.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 us-

ing 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 satisfac-

tory 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

mechanismis consistently lower than the one fromthe mechanismvia trial-and-error.However,kinematics

analysis shows that the stride length of the optimized mechanismis 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.

Art and engineering have often been considered to be separate ﬁelds and the resistance to merge comes

fromboth sides.This work is an initial attempt to imitate an art work using the engineering design theories

and to cross the border between art and engineering.

REFERENCES

1.Bekker,M.G.,Off-the-Road Locomotion,University of Michigan Press,Ann Arbor,MI,USA,1960.

2.Moreckie,A.A.,Bianchi,G.and Kedzior,K.,“Theory and practice of robots and manipulators”,Proceedings

of RoManSy 1984:The ﬁfth International Centre for Mechanical Sciences-International Federation for the

Promotion of Mechanism and Machine Science Symposium (Artiﬁcial Intelligence),Hermes Publishing,Kogan

Page,London,1985.

3.Shieh,W.B.,Tsai,L.W.and Azarm,S.,“Design and optimization of a one-degree-of-freedomsix-bar leg mech-

anismfor a walking machine”,Journal of Robotic Systems,Vol.14,No.12,pp.871–880,1997.

4.Shieh,W.B.,Tsai,L.W.,Azarm,S.and Tits,A.L.,“Optimization-based design of a leg mechanismvia combined

kinematic and structural analysis”,American Society of Mechanical Engineers,Design Engineering Division

(Publication),DE,Vol.69,No.1,pp.199–209,1994.

5.Williams,R.P.,Tsai,L.W.and Azarm,S.,“Design of a crank-and-rocker driven pantograph:A leg mechanism

for the University of Maryland’s 1991 Walking Robot”,Proceedings of 2

nd

National Conference on Applied

Mechanisms and Robotics,Paper No.VIB.2,Cincinnati,OH,USA,1991.

6.Jansen,T.,The Great Pretender,010 Publishers,Rotterdam,2007.

7.Shieh,W.B.,Tsai,L.W.,Azarm,S.and Tits,A.L.,“Multiobjective optimization of a leg mechanismwith various

spring conﬁgurations for force reduction”,Journal of Mechanical Design,Transactions of the ASME,Vol.118,

No.2,pp.179–185,1996.

8.Song,S.M.,Waldron,K.J.and Kinzel,G.L.,“Computer-aided geometric design of legs for a walking vehicle”,

Mechanism and Machine Theory,Vol.20,No.6,pp.587–596,1985.

9.Funabashi,H.,Ogawa,K.,Gotoh,Y.and Kojima,F.,“Synthesis of leg-mechanisms of biped walking machines

(Part I,Synthesis of ankle-path-generator)”,Bulletin of the Japan Society of Mechanical Engineers,Vol.28,

No.237,pp.537–543,1985.

10.Todd,D.J.,“Evaluation of mechanically co-ordinated legged locomotion (the Iron Mule Train revisited)”,Robot-

ica,Vol.9,No.4,pp.417–420,1991.

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

11.Funabashi,H.,Ogawa,K.,Honda,I.and Iwatsuki,N.,“Synthesis of leg-mechanisms of biped walking machines

(Part II,Synthesis of foot-driving mechanism)”,Bulletin of the Japan Society of Mechanical Engineers,Vol.28,

No.237,pp.544–549,1985.

12.Waldron,K.J.and Kinzel,G.L.,Kinematics,Dynamics,and Design of Machinery,John Wiley and Sons Inc.,

New York,NY,USA,1999 (pp.260–262).

13.Uicker Jr,J.J.,Pennock,G.R.and Shigley,J.E.,Theory of Machines and Mechanisms,Third ed.,Oxford

University Press Inc.,New York,NY,USA,2003 (pp.341–343).

14.Erdman,A.G.and Sandor,G.N.,Advanced Mechanism Design:Analysis and Synthesis Vol.II,Prentice-Hall

Inc.,Englewood Cliffs,NJ,USA,1984 (pp.180–183).

15.Myszka,D.H.,Machines & Mechanisms:Applied Kinematic Analysis,Third ed.,Prentice-Hall Inc.,Saddle

River,NJ,USA,2005 (pp.27–28).

16.Andersson,J.,“A survey of multiobjective optimization in engineering design”,Department of Mechanical

Engineering,Linköping University,Technical Report LiTH-IKP-R-1097,Linköping,Sweden,1999.

17.Chevallereau,C.and Aoustin,Y.,“Optimal reference trajectories for walking and running of a biped robot”,

Robotica Vol.19,No.5,pp.557–569,2001.

18.Cleghorn,W.L.,Mechanics of Machines,Oxford University Press Inc.,New York,NY,USA,2005.

19.Erdman,A.G.and Sandor,G.N.,Mechanism Design:Analysis and Synthesis Vol.1,Third ed.,Prentice-Hall

Inc.,Upper Saddle River,NJ,USA,1997.

20.Giesbrecht,D.,Design and Optimization of a One-Degree-of-Freedom Eight-Bar Leg Mechanism for a walking

Machine,M.Sc.Thesis,Department of Mechanical and Manufacturing Engineering,University of Manitoba,

Canada,2010.

Transactions of the Canadian Society for Mechanical Engineering,Vol.36,No.4,2012 355

## Comments 0

Log in to post a comment