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7/25/2019 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
1050-4729/93 3.00
0
1993 EEE
346
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7/25/2019 Dynamic Control of a Quadruped Standing Jump
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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.
347
7/25/2019 Dynamic Control of a Quadruped Standing Jump
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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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7/25/2019 Dynamic Control of a Quadruped Standing Jump
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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
349
7/25/2019 Dynamic Control of a Quadruped Standing Jump
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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-
350
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7/25/2019 Dynamic Control of a Quadruped Standing Jump
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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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I