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REFERENCES

H. G. Barrow, J. M. Tenenbaum, R. C. Bolles, and H . C. Wolf,

Parametric correspondence and chamfer matching: Two new tech-

niques for image matching, In Proc. Fifth Int. Joint Conf. Artifi-

cial Intell.

(Cambridge, MA), 1977, pp. 659-663.

G. Borgefors, Distance transforms in arbitrary dimensions, Com-

p u t .

Vision Graphics Image Processing, vol. 27, pp. 321-345,

1984.

G. Borgefors, Distance transforms in digital images, Comput .

Vision Graphics Image Processing, vol. 34 , pp. 344-371, 1984;

FOA Rep. C

30401-EI.

Nat. Def. Res. Ins. Linkoping, Sweden.

T . E. Boult. On the complexity of constrained distance transforms

and digital distance map updates in two dimensions, Tech. Rep.

Dept. Comput. Sci., Columbia Univ., June 1989.

P. E . Danielsson. Euclidean d istance mapping,

Comput. Vision

Graphics Image Processing, vol. 14, pp. 227-248, 1980.

L.

Dorst and P.

W .

Verbeek, The constrained distance transforma-

tion: A pseudo-Euclidean, recursive implementation of the lee-al-

gorithm, in I. T . Young et al. (eds.), Signal Processing

III:

Theories and Applications.

S Kambhampati and L.

S .

Davis, M ulti-resolution path planning for

mobile robots, Proc. DARPA Image Understanding Workshop

(DARPA), Dec. 1985, pp. 421-432.

C . Y. ee, An algorithm for path connections and its applications,

IRE Trans. Electron. Comput. , pp. 346-365, Sept. 1961.

U. Montanari, A method for obtaining skeletons using a quasi-

Euclidean distance,

J .

A C M , vol. 15, pp. 600-624, 1968.

A. Rosenfeld and J. Pfaltz, Sequential operations in digital picture

processing,

J .

A C M , vol. 13, pp. 471-494, 1966.

A. Rosenfeld and J. Pfaltz, Distance functions on digital pictures,

P a t . R e c o g . ,

vol.

1,

no.

1.

pp. 33-61, 1968.

T. Soetadji, Cube based representation of free space for the naviga-

tion of an autonomous mobile robot, in L.O. Hertzberger and

F.C.A. Groen (Eds.), Proc. Int.

Intell.

Autonomous Syst. Conf.

(Amsterdam, Netherlands), Dec. 1986, pp. 546-561.

P. W. Verbeek, L. Dorst, B. J . H . Verwer, and

F.

C. A. Groen,

Collision avoidance and path finding through constrained distance

transformation in robot state space, in

L . 0

Hertzberger and F. C.

A. Groen (Eds.), Proc. Int. Intell. Autonomous Syst. Conf.

(Amsterdam, Netherlands), Dec. 1986, pp. 627-634.

New York: Elsevier, 1986.

A Free Gait

for

Generalized Motion

P.

K .

PAL A N D K. JAYARAJAN

Abstract-A

method has been presented for the generation of a

locally optimal free gait for 2-D generalized motion of a quadruped

walking machine. It employs a heuristic graph search procedure based

on the A * algorithm. The method essentially looks into the conse-

quences o f a move to a certain depth before actually committing to it.

Deadlocks an d inefficiencies are thus sensed well in advance and avoid ed.

I. INTRODUCTION

The control of a walking machine requires making decisions as to

which leg to lift and when, and also when to place a leg and where,

so

that the machine may continue to move stably and efficiently

along its prescribed route. The periodic solutions to the problem

[1 ]- [3 ] for a straightline motion of the m achine prove ineffective for

Manuscript received July 27, 1988 ; revised May 30, 1990 .

P. K. Pal is with the Computer Division, Bhabha Atomic Research

K. Jayarajan is with the Division of Remote Handling and Robotics,

IEEE Log Number 9038855.

Centre, Bombay, India.

Bhabha Atomic Research Cent re, Bombay, India.

YO

Y

U

xo

Fig. 1.

Machine coordinate system X,,, ) t x ,

y )

and at an orientation

0

with respect to the ground coordinate system A,, o ) . F is the foot

position.

a generalized motion on a terrain with regions that are not stepwor-

thy. The problem of generation of a terrain-adaptive aperiodic gait,

which is commonly referred to as a free gait

[ 5 ] ,

was first formal-

ized and solved by Kugushev and Jaroshevskij [4] and subsequently

improved upon by M cGhee and Iswandhi [5 ] . Their algorithm tries

to maintain stability and maximizes the minimum kinematic margin

over all legs through the lifting and placing of legs. Though

brilliantly conceived, the absence of any look-ahead feature in the

method may result in inefficiency of the generated gait, or worse, in

deadlocks. Hirose applied a similar method for the generation of a

free gait for a quadruped machine, adding certain guiding rules for

the selection of a foothold [6] that tends to generate efficient gait

while avoiding deadlock. These guiding rules, which are specific to

the quadruped machine, may not be easily extended to the case of

other legged machines nor do they assure optimality of the gener-

ated gait in all circumstances. In addition, Hiroses machine, while

following the prescribed route, maintains a constant orientation and

resorts to crab walking.

Thus, the existing methods for the generation of free gait lack in

two respects-they have

not

been applied to the case of a general-

ized motion, and they do not employ a general set of principles or

techniques for the generation of an optimal free gait. Even for a

regular terrain, even though there has been an attempt to determine

the foot motions required for a generalized motion through the

introduction of a crab angle [7], the underlying gait is assumed to be

one optimized for straight-line motion.

In this paper, we present a method for the generation of a locally

optimal free gait for a

2-D

generalized motion of a quadruped

walking machine. It employs a heuristic graph search procedure

based on the

A*

algorithm

[8 ] .

The method essentially

look5

into

the consequences of a move to a certain depth before actually

committing to it. Deadlocks and inefficiencies are thus sensed well

in advance and avoided.

11.

FOOT

MOTION

OR

A GENERALIZEDOTION F THE

WALK ING ACHJNE

Let X o ,Y o ) represent the ground coordinate system with its

origin at 0 (Fig. l), and let ( X , , Y , ) represent the machine

coordinate system with its origin at C, hich is the center of gravity

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( C G ) of the machine. F indicates a foot position. Then, at any

instant, if the machine is at

x ,

y at an orientation 8 with respect

to the ground coordinate system, the coordinates of the foot F in the

two coordinate systems will be related as

[ ;I [ sein 0 - s i n eos

0

y

1[: l ]

YI

0 1

which may be written as

where , A 1 is the matrix that transforms position from the machine

to the ground coordinate system, F, is the column matrix represent-

ing the foot position in the ground coordinate system, and F , is that

in the machine coordinate system.

F, = , A * F , 1)

Equation (1) may be rewritten as

F , ,A * F,

(2)

where A o ( , A ) - .

At a later instant, suppose the machine moves to a position

x 6 x ,

y 6y and that its orientation changes to

0 +

6 0 . Then,

the foot coordinates in the displaced machine coordinate system may

be written, following 2), as

F2

= 2 A o * F o

3 )

where 2Ao s obtained from

, Ao

by substituting 0

+

68 for 0 ,

x 6 x

for x , and y 6y for

y

Using

l ) ,

3)

may be written as

F2

= 2 A o * , A * F , .

The foot motion required for a differential motion of the machine

is then

6 F ,

F2 F ,

[ 2 A o * , A Z ] * F , .

(4)

Equation (4) gives the foot motion required in the machine

coordinate system to achieve a generalized motion of the machine

from x , y , 0) to x 6 x ,

y

+ 6y ,0 +

60)

in the ground coordi-

nate system. The equation holds irrespective of the number of legs

or their arrangement in the machine.

111. MODEL F THE MACHINE

The walking machine has been modeled as shown in Fig. 2. The

legs are numbered clockwise starting from the first quadrant of the

machine coordinate system

( X I , , ) .

C is the CG of the machine

and the origin of the machine coordinate system. The small square

around C is representative of the body of the machine. The reach-

able areas of each of the legs are shown as dashed squares. All

distances are measured in the un it of a side of these squares.

IV. A MEASUREF

GENERALIZED

OTION

We need to have a measure of motion that is uniformly sensitive

to both translation and rotation of the machine. The sum of foot

travels of all the legs is one such measure, but it depends on the

positions of the feet. Thus,

it

would be appropriate to measure a

motion in terms of the magnitude of the required foot motion (see

4)) averaged over the entire reachable area of the machine. How-

ever, to reduce computation, we determine the average of the

magnitudes of foot motions that would be necessary had the foot

been at the center of each of the reachable areas. We will refer to

motion measured in this way as the travel of the machine. The

generalized motion of the machine may now be seen as a linear

motion in the travel space.

V . STATESND

TRANSITIONS

The input to our gait generator program will be the desired

positions and orientations of the machine at closely spaced time

intervals. We will refer to the motions between successive positions

Fig. 2.

the

and orientations of the machine as motion segm ents. Our treatment

of the problem will be discrete in the sense that decisions regarding

lifting or placing of legs will be taken only after completion of a

motion segment and not during such a motion. Hence, we will be

concerned about the possible states of the machine only at these

points of motion.

A leg state primarily describes whether the foot

is

on the ground

or in the air. If it is on the ground, the description also includes the

coordinates of the foot in its reachable area.

A machine state is described in terms of its leg states, whereas a

walking state includes a description of the machine state as well as

the position and orientation of the machine in the ground coordinate

system.

For the machine at a certain walking state, let us consider the

possible motions of a given leg. If it is touching the ground, then it

must satisfy the foot motion implied in (4). It may as well be lifted,

provided that does not make the machine unstable. On the other

hand, if it is in the air, then it may continue as such or be placed on

the ground at one

of

several points within its reachable area. These

choices determine the leg states, to which, transition from the

current leg state may be made. To limit the number of possible

transitions (in order to reduce the number

of

nodes in the search

tree), we introduce the following two rules governing the lifting and

placing of legs:

1) A leg in stance is not considered for lifting until the machine

completes a prespecified length

of

travel during the stance. During

this period, the foot has no choice other than to follow the motion

prescribed by (4). After it becomes eligible for lifting, it has the

choice of continuing to remain on the ground or to be lifted for a

transfer. At this stage, if the leg is still on ground and moves out of

its reachable area, then, of course, it has to be lifted.

2) A leg in the air must continue as such until the machine

completes a prespecified length of travel since the leg was lifted

off

the ground. After this, the leg has the choice of continuing in the air

or making contact with the ground at one of the points decided by a

foothold selection algorithm.

A machine state will in general have a number of connecting

states such that a transition to any of these states is stable and effects

the next motion segment. A machine state transition is brought

about by a compatible set of leg state transitions. Compatibility of

the foot motions is ensured by the use of 4). Hence, the leg state

transitions must be such that the machine remains stable during the

transition.

The support pattern for a machine state is the convex hull of the

point set in a horizontal plane, which contains the vertical projec-

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2

.,s 4

Fig. 3 .

S, . S, arc the candidate footholds when the instantaneous

direction of foot motion is SIP.

tions of the feet

of

all supporting legs [ 5 ] . If the support pattern

includes the vertical projection of the CG of the machine, then its

absolute stability margin is the minimum distance between the

vertical projection of the CG and the boundary of the support

pattern. A machine state is statically stable only if the absolute

stability margin of its support pattern exceeds a certain prespecified

value.

A transition from machine state

n ,

to

n 2

will be stable if

n I

and

n 2

are separately stable. Here the support pattern of n , is formed

by excluding any leg that is in the air in

n ,

and the support pattern

of n 2 is formed by excluding any leg that is in the air in n I .

VI. SELECTIONF

CANDIDATE

OOTHOLDS

The placement of a foot determines its kinematic margin, i.e. , the

travel the machine can have along its prescribed route before the

foot moves out of its reachable area. It also affects the stability of

the machine in a way that is difficult to determine in adv ance.

Hence, we allow a foot to be placed at one of four preselected points

within the reachable area. These points are

so

chosen that among

themselves, they try to satisfy varying stability requirements while

assuring a reasonably high kinematic margin.

Fig.

3

shows the candidate footholds. S , is the center of the

reachable area and is one of the candidate footholds. The line

segment S I P represents the direction of foot motion at the time of

contact had the foot been placed at

SI .

S I P

s extended backwards

to meet the boundary of the reachable area at S , , which is another

candidate foothold. The other two footholds are S , and S , , and they

lie on the perpendicular bisector of the line segment S , S , at a

distance

of

0.35 on either side. T he distance

0.35

(i.e.,

/4)

has

been selected

so

that these points lie within the reachable area for

any direction of foot motion. If the instantaneous center of rotation

of the machine is close to the center of the reachable area, the

footholds are selected on a circle around the center.

If the terrain has regions that are not stepworthy, then the

candidate footholds (SI

.

. .

S,)

have to be checked for suitability.

If any of them is found to be unsuitable, a predefined array of points

around it are checked for suitability, and the nearest suitable point,

if any, is passed as the candidate foothold. If no substitute is found,

that particular foothold is dropped from the set of candidate

footholds.

VII . THENOTIONOF PROGRESS

In a graph search, it is necessary to measure the progress made

by the machine towards its goal at any point in the course of its

motion. The measure should be sensitive to both the travel and the

state of the machine, thereby accounting for the motion that

is

realized as well as the potential for ocward motion.

Walking involves two distinct processes-the support phase, dur-

ing which a leg pushes the body forward, and the transfer phase,

through which a leg restores its kinematic margin. These two

processes must go hand in hand for the machine to advance. H ence,

for a straightline motion of the machine, we define its progress as

the mean of the distance covered by its CG and the average distance

covered by its legs from the beginning of motion or any reference

walking state. Following this definition, the progress from a starting

state

s

to the current state

n

may be shown' to be c + K ,

K , ) / 2 ,

where is the distance covere d by the

CG

of the machine, and K ,

and K , are the average kinematic margins of the legs in states n

and s, espectively.

In the case of a generalized motion, we carry forward the same

expression for progress with a slight reinterpretation of the first

term. c is now the travel of the machine, where K , and K , are

now the actual kinematic margins.

VIII . THEGRAPHSEARCH

TRATEGY

We have broken up the entire motion of the walking machine into

small motion segments. After completion of every motion segm ent,

the machine is given a choice of lifting or placing som e of its legs in

the form

of

connecting states. However, there is

no

judicious way to

choose from these states. What is required is a control structure that

will allow

us

to probe deeply into the future before deciding

on

which of the connecting states should actually be taken up for a

transition. A graph search

[8]

provides a method for doing this.

A graph consists of a set of nodes connected to each other

through a set of arcs. The nodes in this case will correspond to the

walking states of the machine, and the arcs will represent transitions

between such states. The g raph search procedure, which is based on

the A* algorithm

[8]

for generating an optimal route to the goal,

starts with the initial walking state as the sole node of the graph.

This node is expanded to form its offspring nodes,

which are

essentially its connecting states. These are included in a list of open

(i.e., unexplored) nodes. The open nodes are evaluated for their

promise or worthiness to be included in the solution, and the most

promising open node is removed from the open list and expanded.

This process continues, and the graph grows until the goal node is

reached.

The evaluation of the promise of a node proceeds on a heuristic

line. For example, in the gait generation problem, we try to

minimize the number of transfers of the legs to reach the goal.

Hence, a node

looks

most promising if the path passing through that

node seems to require the least number of transfers. The estimated

cost of reaching the goal through the node n may be given by

f n )

g n)

h ( n ) , where g n ) is the number of transfers from

the start node

s

to the node n and is directly available from the

generated graph, whereas

h n )

is a heuristic estimate of the number

of transfers necessary to reach the goal from node n. It is reason-

able to assume that h n ) should be approximately proportional to

the progress to be made to reach the goal from node n. The

proportionality constant is chosen to correspond to the quadruped

wave gait with zero stability margin. When h n ) is thus underesti-

mated, optimality of the solution is assured

[8].

To reduce the search effort. we limit the search tree to a certain

If

k,,

and

k,,

are the kinematic margins

of

the ith leg at the initial

walking state

s

and current state

n ,

respectively, the distance covered by the

this leg from states

s

to

n

will

be

c k n i

k,, ,

where c is the distance

covered by the CG of the machine during this period. The average distance

covered by a leg is then c ( K , K , ) , where K , I , k , , ) / 4 is the

average kinem atic margin of the machine at state

n

and likewise for K , . The

associated progress

of

the machine then evaluates

to

[ c

+ (c + K ,

K , ) ] / 2 ,

. e . ,

c + ( K , K , ) / 2 .

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I

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~

Fig.

4.

Successive frames of the machine as it moves straight, takes a

sharp turn to its left, and then proceeds straight again. Only the states in

which a leg has either been lifted or placed have been shown. A leg,

which is not shown, is in the air.

depth (in terms of progress) with respect to the root. W henever the

graph tends to grow beyond this limit, the machine makes a

transition from the root towards the deepest open node in the graph,

its newly acquired state is redefined as the root, and the tree is

accordingly pruned. Because the search is confined to a limited

depth, the generated gait is only locally optimal.

IX. THEGAIT

GENERATIONLGORITHM

The algorithm is summarized below.

1

A

search graph consisting solely of the start node s (corre-

sponding to the initial walking state) is created, and s s put on a list

called

OPEN.

2)

If OPEN is empty, the program exits with the message cannot

advance. Otherwise, the first node from

OPEN

is removed. Let it be

n .

3) Expand n, generating offspring nodes corresponding to its

connecting states (see Section V) allowed by the terrain and include

them in OPEN. If no such state is found, loop back to step

2).

4 The progress to each of the nodes, which are generated in step

3),

are evaluated relative to the root. The node with the maximum

progress ( m u x p r o g ) is labelled as m ux n. If m ux p ro g exceeds a

prespecified value, or if the total number of nodes in the search tree

exceeds a certain number, the machine makes a transition from the

root towards m u x n . The node now occupied by the machine is

redefined as the root. The branches of the previous root that were

left out are removed from the tree, and their open nodes are

removed from

OPEN.

5)

Loop back to step

2).

The output of the program will be the sequence of walking states

through which the machine actually makes transitions.

The nodes in the OPEN list are maintained in the ascending order

of the cost evaluation function f n ) , nd a node is removed for

expansion only from the top. A new node is added at the appropriate

place in the list to maintain order.

X . COMPUTER

I M ULATI ON

The gait generation algorithm for generalized motion has been

implemented on a Norsk Data Computer ND-570 in Pascal. In the

test runs, the search tree was limited to a depth of unit progress and

to a size of 2000 nodes. The commanded motion segments corre-

sponded to a travel of about

0.05,

and the absolute stability margin

demanded was 0.025, where all these were measured in the unit of a

side of the reachable area of a leg. A leg was not allowed to be

lifted until the machine completed a travel

of

0.5.

Likewise, a leg in

transfer was not allowed to be grounded until the machine made a

travel of 0.1.

Fig. 4 hows the walking machine as it moves straight, takes a

sharp

turn

to its left around its

CG,

and proceeds straight again.

Only the states in which the leg has just been lifted or placed have

Fig.

5.

Successive frames of the walking machine as

it

takes a slow turn

to

its right along a circular arc on a terrain in which the marked regions are

not stepworthy. Only the states in which a leg has just been placed have

been shown.

been shown in the figure. The element of anticipation, which com es

through the graph search, is made obvious at frame 5 when leg is

placed in preparation of the coming rotation, or say at frame 11,

where, aga in, leg 1 has been placed keeping in mind the coming

translation.

Fig. 5shows the machine as it takes a slow turn to its right along

a circular arc on a terrain in which the marked areas are not

stepworthy. He re, only the states in which a leg has just been placed

have been shown.

Fig. 5 also shows how the machine selects

footholds from the available regions of the terrain to execute the

commanded motion with as few transfers as possible. The CPU time

used to compute the states in the figure was slightly more than a

minute.

XI.

CONCLUSI ON

We have developed the basic framework for the generation of

free gait through a graph search. Graph search, as a technique, is

fairly old and well known, and its relevance to gait generation is

obvious. Howev er, it remained unexplored, possibly because of the

associated computational complexity. In the present work, we have

demonstrated the feasibility and usefulness of this approach.

Our

treatment of generalized motion is simple and general. The

notion of progress should also prove useful in this area of research.

Improvem ents may be sought mainly in two areas-the foothold

selection algorithm, through a better understanding of the role of

stability and kinematic margin in such matters, and the leg state

transition rules, where stronger heuristics may reduce the search

time. The latter will prove

to

be crucial if this technique has to be

applied in the case of a hexapod machine.

REFERENCES

R . B. McGhee and

S . S

Sun, On the problem of selecting a gait for

a legged vehicle, in Proc. VI IFAC Symp. Automat ,

Contr.

Space (Armenian S . S . R . , USSR), Aug. 1974.

A. P. Bessnov and N. V. Umnov, The analysis of gaits in six legged

vehicles according to their static stability, in Proc. Symp. Theory

Practice Robots Manipulators (Udine, Italy), Sept. 1973.

R . B . McGhee and A. A. Frank, On the stability properties of

quadruped creeping gaits, Math. Biosci. , vol. 3 , no. -3 /4 , pp.

331-351, Oct. 1968.

E . I. Kugushev and V . S Jaroshevskij, Problems of selecting a gait

for an integrated locomotion robot, in Proc. Fourth I n t . Conf.

Artificial Intell. (Tbilisi, Georgian SSR, USSR), Sept. 1975, pp.

R. B. McGhee and G . I . Iswandhi, Adaptive locomotion of a

multilegged robot over rough terrain, IEEE Trans. Sysf . Man

Cybern, vol. SMC-9, no.

4

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176-182,

Apr.

1979.

S Hirose, A study of design and control of a quadruped walking

vehicle, Int. J . Robotics Res., vol. 3 , no. 2, pp. 113-133,

Summer 1984.

D

E. Orin, Supervisory control of a multilegged robot,

I n f . J .

Robotics Res., vol. 1, no. I pp. 79-91, Spring 1982.

N. J . Nilsson, Principles

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