Dynamic Control of a Quadruped Standing Jump

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DYNAMIC CONTROL OF A QUAD RUPED STANDING J U M P

WO

Cheung (Eric) Wong

FANUC Robotics North America, Inc.

2000 South Adams Road

Auburn Hills, Michigan 48326

Abstract

In this paper control strategies are developed for a

quadruped standing jump ouer irregular terrain ob-

stacles. Sim ple open-loop leg forces are planned to

remoue the large linear and angular momentum of

the body during landing. Super real-time simulati on

which involves predicting landing conditions based on

simulat ion using

a

simplified model is used to select

leg touchdown angles. Using the principle of sym-

metry the

l e g

forces during take08

are

derived from

those predicted

f o r

landing.

Using these strategies the

quadruped is able to clear vari ety of obstacles includ-

ing isolated walls terra in steps and ditches.

1 Introduction

During the past several decades, researchers have built

numerous legged machines to study the principles

of

legged locomotion [ l ,

2,

31.

It is of particular in-

terest to demonstrate their superiority over wheeled

or tracked vehicles during locomotion over obstacles

which are comparable in size to their dimensions.

Some examples are: ditches, isolated walls, and verti-

cal steps [4].

In statically stable machines, the general approach

h a s

been designing gaits

for

crossing these terrain con-

ditions [4]. These gaits typically involve complex ma-

neuvers of the body and legs. T hus , the mot ion often

tends to be rather slow. Further, they may not be ap-

plicable for certain obstacles such as wide ditches, thin

isolated walls, or large vertical steps. The limitation is

mainly due to the restrictions imposed upon the body

and leg motions by the condition of stat ic stability.

On th e other hand, it was observed tha t animals

usually jump over obstacles [ 5

61.

This suggests that

an alternative and potentially better strategy t o nego-

tiat e obstacles is to jum p over them , thus ignoring the

local effects of terrain irregularities between successive

footholds.

Over the last decade, significant progress has been

made in controlling dynamically stable legged ma-

chines to hop, run, and perform somersaults [3,

7,

81.

Studies of jumping have mainly been focused on

David E. Orin

Department of Electrical Engineering

The Ohio State University

Columbus, Ohio 43210

monopods and bipeds where the leg actuators are ar-

ranged such that the linear and angular motions of the

body are mostly decoupled, thus simplifying the con-

trol. On the other han d, the legs of a quadruped are

located far from the center

of

mass of the body, causing

coupled angular and linear body motions, complicat-

ing the control of jumping. To more

fully

realize the

potential of a quadruped in negotiating obstacles, the

control

of

one type of quadruped jump, namely the

standing jump

as

illustrated in Fig.

1,

will be investi-

gated in this paper.

2 Simulation Model

Raibert et al. pioneered the development of a number

of legged machines which balance actively [3,9, 10, 111.

Telescoping and springy legs are used to conserve en-

ergy and simplify control. Similar leg mechanisms as

in [I21 are used in this study. Since the components

of force and motion of an animal perpendicular to the

plane of motion during jumping are usually relatively

small [13], a planar model as shown in Fig.

2

is chosen

to study jumping only in the sagittal plane. The pa-

rameters of the model are chosen to simulate a large

dog and they are given in [14].

The Decoupled Tree-Structure ( DTS ) Approach [15]

is used to derive the dynamic equations of motion for

the planar quadruped and they are given

in

[14]. Full

inertial effects of each leg as well

as

the compliant

effects

of

the terrain-foot interaction are simulated

3 Phases

of

Standing Jum p

A computer graphic simulation

of

a quadruped stand-

ing jump over an isolated wall of 0.9 m is shown in

Fig.

1.

The simulation is run off-line on

a

SPARCsta-

tion and then displayed on a

G E

Graphicon worksta-

tion. Motion traces of the body center of mass and the

feet are also shown. The terrain is divided into 0.5m

by 0.5m grids.

The basic motion sequence of a standing jump may

be divided into three phases:

takeoff flight

and

land-

ing. Both the takeoff and the landing phases may

fur-

ther be divided into a single contact stage (frame b

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0

1993 EEE

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(a) Takeoff Front Leg Thr us t

b)

Takeoff Back Leg Thr us t

m

(c)

Flight

d) Flight

f ) Landing

Figure

1:

Standing

Jump Over

an Isolated Wall of 0.9m.

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t

= l

Figure 2: Planar Quadruped Model

during takeoff and frame e during landing) and a dou-

ble contact stage (frame a during takeoff and frame f

during landing) depending on the number of legs on

the ground.

During takeoff, it is impor tan t to thrust the

quadruped into the air with initial flight conditions

such that it not only clears the obstacle but the body

does not under or over-rotate during flight. These con-

ditions are the

body takeoff angle body angular take-

off veloci ty body vertica l takeofl velocit y

and the

body

horizontal takeoff veloci ty .

They will determine the

body angle during landing. If the body is pitched too

much, then control of landing may be difficult and loss

of balance will result . Although the body angular ve-

locity during flight may be adjusted by swinging the

legs or varying their lengths, such motion may be lim-

ited because the legs have other funct ions to perform

such as obstacle clearance and preparation for landing.

Two important parameters th at characterize th e

flight phase a re th e jumping height and the f l ight span.

Th e jum ping height is the vertical distance of the body

c.0.m. at the top of flight from the takeoff level, while

the flight span is the horizontal distance the body

c.0.m. travels from the time the back leg lifts off during

takeoff to the time the front leg touches down during

landing . Leg motions are planned t o maximize clear-

ance during flight and stability during landing. The

detailed planning of leg motions during flight is dis-

cussed in [14].

During landing, the large amounts of linear and an-

gular momentum of the body have to be removed as

it moves to a stable standing posture on the same

footholds on which it lands. Further, t he landing con-

trol strategy must be robust enough to accommodate

a moderate variation in the landing level which affects

the actual body landing angle.

4

Landing

Control Strategy

In this paper, the landing control strategy involves

planning open-loop and simple nominal leg forces and

leg touchdown angles to remove the large linear and

angular momentum

of

the body. During the

s ingle

contact s tage

the front leg force removes the linear

momentum of the body along the front leg axis, re-

gardless of its effects on th e body angular momen tum.

Then, during the

double contact s tage

the back leg

force removes the body angular momentum while the

front leg force maintains a constant front leg length.

To simplify force planning, no hip torque is used when

a leg is on the ground. In fact, it was observed in

goat locomotion studies that the knee plays a more

significant role in jumping than the hip

[13].

In any

event, hip torque may be used in future approaches

and should make the control even more robust.

The natural dynamics of the system are considered

when planning the leg forces. However, the equations

of motion of the plana r quadruped are still rather com-

plex for control development.

As

the Nubian goat lo-

comotion studies indicate that the inertial effects of

the torso dominate those of the lower extremities [13],

a simplified model shown in Fig. 3 is used for control

synthesis. It consists of a rigid body with two point

masses at the hips, thus ignoring the inertial effects of

the legs.

4

Figure

3:

Simplified Planar Quadruped Model.

4 1 Single Contact Stage

If the body linear momentum along the front leg axis

is known, the front leg force may then b e planned to

remove this linear momentum over a defined period of

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time. That is,

/ l dt =

A(mw)

where

ront leg actuator force, and

ehange in body linear momentum

along the front leg axis.

A mi\

An outline of the force planning is given below, with

the force profile and its parameters illustrated in

Fig. 4 .

1. Upon touchdown, the front leg force is allowed to

build up to a threshold value

f l t h

as the body

compresses the leg spring/damper.

2 . The leg force is then maintained a t t he threshold

value for a period of time A t 2 = 2 l to remove

a major portion of the linear momentum.

3 . The remaining momentum will be removed as the

actuator force is ramped down to a value

f c l

over

a period of time

A t 3 = t 3 2 .

4 . The leg force is then held at the value fft to main-

tain a relatively constant leg length until the back

leg touches down.

DoubleContact

I

I

I

Single

Contact

f ,

syinghnasdhperdynmi~~

I

c

f

f , i i

I------+

;pf h;.

t

Figure 4: Leg Actuator Force Profiles for the Two

Contact Stages of Landing.

During th is stage, the front leg force also creates a

moment on the body, increasing the angular momen-

tu m of the body. This momentum will be removed in

the double contact stage.

4.2 Double Contact Stage

In th is stage, the front leg force maintains a relatively

2 . Both the front and the back leg forces are held a t

their constant values over a period of time, A t 6 =

t 6 5 , to remove a major portion of the body

angular momentum while maintaining

a

relatively

constant front leg length.

constant leg length, thus pivoting the body about the

front hip. The back leg force then removes the angular

momentum over a defined time. That is,

/ M dt

= A(I )

(2)

where

M = moment of the back leg actuator

A(I ) = change in body angular momentum

force about the front hip, and

about the front hip.

The actuato r forces of both legs are planned as fol-

lows and illustrated in Fig. 4 :

1. The back leg actuator force is allowed to build up

to

a

threshold value

f 2 1 h

as

the body compresses

on the leg spring/damper. Meanwhile, the front

leg actuator force is ramped to

a

value,

fez,

to

maintain a relatively constant front leg length.

3 .

The remaining portion of the angular momen-

tum

is

removed as both the front and the back

leg forces are ramped down to a nominal fi-

nal force value of 0.5m

g

over a period of time

A t 7 = 7 - t 6 . Each leg then supports half the

quadruped s weight.

The force levels and time intervals of the force pro-

files in Fig. 4 are derived in [14].

4.3 Landing Preparation Stage

During flight, the legs are servoed to the appropri-

ate touchdown angles in preparation for landing. T his

stage is called the landing preparataon stage. The leg

touchdown angles are chosen to remove the forward

momentum of the body. However, an analytical solu-

tion is not readily available. The concept of super real-

time simulation is used. In par ticular, simulations of

landing, using the simplified model with different leg

touchdown angles, are performetl in super real-time

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during flight, and their predicted effects are evaluated

before landing.

Perhaps the best time to perform super real-time

simulation to select leg touchdown angles is when the

body is at the top of flight. While at the to p, the body

is clear of obstacles and the latest information

on

the

landing terrain level may be taken into consideration,

thus allowing the quadruped to adapt to variation in

the terrain level.

5

Takeoff Strategy

The control of takeoff is complicated by the fact tha t it

takes place over two stages. During the double conlacl

stage of takeoff, the front leg exerts thrust to throw the

body upward until the leg lifts off the ground. How-

ever, the desired conditions of the body at that in-

stant are unknown. They depend on the desired take-

off conditions of the body when the back leg lifts off

the ground and the amount of back leg thrust during

the single contact s tage of takeoff, both of which are

unknown during the double contact stage of takeoff.

One solution is to use the principle of symmetry [16].

In particular, it may be noted that the leg actuator

forces during takeoff and landing are symmetrical if

the takeoff and landing conditions are symmetrical.

The principle of symmetry

[16]

is used to simplify

the planning of leg forces during takeoff of a stand-

ing jump.

Consider the top of Fig. 5.

A

,quadruped

lands with an initial angular velocity of Bi and lin-

ear velocities of x and with the motion proceeding

from left to right. Th e front and the back leg actuator

forces, fi and

f 2

are planned according to the landing

strategy developed in Section

4

to reduce the veloci-

ties to zero. Consider the bot tom figure of Fig. 5. The

body initially has conditions equal to the final condi-

tions of landing in the figure above. If the same force

profiles are applied, reversed in time, to the body over

the same period of time , the body motion will proceed

from right to left with the final conditions symmetrical

to the initial conditions of landing. Thus, if the forces

applied to the quadruped during landing are known,

the same forces, reversed in time, may be used for

takeoff.

However, since landing occurs after takeoff, the

actuator inputs at landing are unknown when the

quadruped takes off. An approximate solution is used

here. Th e landing forces are predicted based on the

simplified model. Given the desired takeoff conditions

of the planar quadruped, a set of symmetrical con-

ditions are used as the initial landing conditions for

the simplified model. Before the actual takeoff, super

real-time simulation of landing is performed to deter-

mine the force profile parameters and touchdown an-

f2 ) -

fl

Y = O

1

t

fl

Figure 5: Takeoff Strategy Using the Principle of Sym-

metry.

gles. These values are then applied, reversed in time,

for takeoff.

6 Simulation Results and Discussion

The results of the actual and planned leg forces are

shown in Fig. 6. Note that any negative values of

the leg actuator forces result from-the lower limb seg-

ments hitting their hard end stops [14]. The gen-

eral force profiles as discussed in Section 4 are used.

It may be noted that the takeoff and landing forces

are relatively symmetric. The slight deviations of

the desired and actual actuator forces are due to

the errors in open-loop modulation of the leg spring

compression/decompression and the approximations

in planning the force profiles. As a result of this,

the quadruped tends to over-rotate when the planned

forces are used completely open loop. Thus, a simple

prediction scheme is used. The body stat es are con-

stantly monitored during the single contact stage of

takeoff. When the projected body angle at the to p of

flight is about zero, the back leg thrust is termi nated.

7

Summary and Conclusions

In this paper, a control strategy for a quadruped

standing jum p has been presented. The linear and an-

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2000

,

I I

Dcsind

FrontLcg

Force

Actual

Front

Lcg

h r c c

. . .

~ ~

. ..

Desired

BackLcg Force

.~~~~~.....c b l d B a c k L c g F m

r

... . . ... .... . .. . .

I

0

03

0 4 0 6

0.8

1 3

1.4

1 6

T im sec)

Mol

Figure 6: Leg Actuator Forces.

gular momentum of the body are removed during dif-

ferent stages of landing. Super real-time simulation is

used to select appropriate leg touchdown angles. With

the speed of development of todays computer technol-

ogy, the effective realization of super real-time simu-

lation is well within range. Leg force profiles during

takeoff are predicted using the principle of symmetry.

These strategies allow a quadruped to jump over

obstacles with substantial height. Further, the strate-

gies are versatile so tha t they allow the quadruped to

negotiate a wide variety of terra in obstacles such as

isolated walls with different takeoff and landing levels,

terrain s teps, and ditches. A number of cases have

been tested in

[14]

with good results and data for one

of these cases

h a s

been presented here. Hopefully, the

approach to control

as

proposed here will provide the

foundation for jumping machines that are developed

in the future.

8

A c k n o w l e d g m e n t s

This research was supported in part by the Defense

Advance Research Projects Agency under Contract

No. MDA972-88-K-0003 and The Ohio State Uni-

versity. The permission of Prof. Vijay Kumar and

Mr. John Bradley of the University of Pennsylvania to

use their facilities in preparing the initial manuscript

of this paper is gratefully acknowledged.

9

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Dynamic Control of a Quadruped Standing Jump

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