Proceedings of the 2002 IEEE
International Conference on Robotics & Automation
Washington, DC • May 2002

Development of a Biologically Inspired
Hopping t obot "Kenken"
S.H.Hyon,

T.Mita[2]. They also succeeded in acrobatic motions [3][4].
Buehler and his co-workers have developed an electrically driven one-legged robot, which has spring at the
hip joint [5], and realized an energy-efficient running
using modified Raibert's controller [6].
Since Raibert's controller is based on the uncontrolled dynamics of the springy leg, it does not require much control effort and yields quite natural running gait. To embed the uncontrolled dynamics into
the control algorithm, Raibert made an assumptionof decoupling of the rotary motion of the body fromthe telescopic motion of the leg. Therefore, for his
controller applicable to one-legged hopping, the mechanical model itself should be carefully designed tosatisfy the assumption: the leg should be sufficiently
light compared to the body and the C.M. (center ofmass) of the body should be positioned closely to the
hip joint. For example, a novel design can be found in
Zeglin's "Bow Leg Hopper" [8], where the leg dynamics
is "mechanically" decoupled from the body dynamics
during stance and the body attitude is passively stabilized because the C.M. of the body is below the hip
joint.
Animals leg in the nature, however, is anarticulated-type and the C.M. of the body is set off
the hip joint. As their dynamics have a strong nonlinear coupling, analysis of uncontrolled dynamics isvery difficult and there are few theoretical studies [7].
Moreover, to the best of our knowledge, there havebeen no successful experiments in running of this typeof robots, except for two examples: "Monopod" [9] and"Uniroo" [10], studied by Raibert and his co-workers in
90's. In order to realize robotic running as animals do,it goes without saying that we should establish generalcontrol theory for running. Nevertheless, we think it isalso important to invent several new mechanical m o d els, inspired by biological or biomechanical studies, as
the examples above, and to demonstrate successful experimental results.
In this paper, as our first step toward the goal of dynamic quadruped running, we propose a new simple
mechanical model of hindlimb and the hardware design of a one-legged hopping robot named "Kenken".
This robot h a s an articulated leg composed of threelinks and uses two hydraulic actuators as muscles andlinear springs as a tendon. Using an empirical con-

Abstract m
As many biomechanists indicate~ tendon
plays important role for running or jumping motion. It
stores the kinetic energy as a potential energy during
stance and also absorbs the impulse at touch down. Inspired by such biomechanical studies~ we propose a new
simple mechanical model for hindlimb of a dog to realizea robot that imitates dog running~ and the hardware design of one-legged running robot, "Kenken'. The robot
has an articulated leg and uses two hydraulic actuators as muscles and a tensile spring as a tendon. The
spring being attached like gastrocnemius or plantaris
enables the robot to produce sufficient propulsion force
by virtue of the "energy transfer" from the knee, even
if there is no actuator at the ankle joint. Using an empirical controller based on the characteristic dynamics
of the model~ the robot has succeeded in planar onelegged hopping. Although the problem related to thestability at the higer speed remains~ the experimental
results demonstrate that the proposed hindlimb mechanism is effective for legged running.
Keywords--Hopping
robot~ Running robot~ Legged
locomotion, Biomechanics, Tendon.

I. INTRODUCTION

Matsuoka may be the first researcher in running
robot. He analyzed a linearlized two-link model with
mass-less leg, and proved the foot placement(and its
velocity) is essential for the gait stability, and realized
stable hopping motion in an inclined gravity field [1].
In a gravity field, animals must bear about morethan two-times larger ground force than their weight.
Moreover, they suffer from large impulse at touch
down. As many biomechanists indicate, tendon plays
an important role for running or jumping motion. It
stores the kinetic energy as a potential energy during
stance and also absorbs the impulse at touch down.
To realize a running robot, introducing such springy
characteristics into the leg design is quite natural. The
pioneering work was done by Raibert and co-workers.
They developed one-legged, biped, and quadruped
running robots, all of of them have springy legs oftelescopic type, and realized various running motions
S.H.Hyon is with the Department of Mechatronics and Precision Engineering, Tohoku University, Sendai 980-8579, Japan.
E-maih sangho@ieee.org
T.Mita is with the Department of Mechanical and Control
Engineering, Tokyo Institute of Technology, Tokyo 152-8552,
Japan. E-mail: mita@ctrl.titech.ac.jp
This work is a part of the Grant-in-Aid for COE Research
Project supported by the Ministry of Education, Science, Sports
and Culture, JAPAN

0-7803-7272-7/02/$17.00 © 2002 IEEE

3984

Thigh (Femur)

troller based on characteristic dynamics of the model,
the robot has succeeded in hopping in a plane.

Gastt
Plan

This paper is organized as follows. A new mechanical model of hindlimb is proposed in Section II, design and hardware overview of the hopping robot"Kenken" is described in Section III. Details of thecontroller are shown in Section IV. Experimental results and discussion are given in Section V.

Sole~

Shank (Tibia)

Foot

Heel lFig. 1. Muscle group of ankle joint

II. P R O P O S A L OF A NEW MECHANICAL MODEL OFHINDLIMB

A. Role of tendon, articulated leg
Alexander modeled a muscle and tendon system ofleg as a serial connection of inelastic actuator and aspring, and studied the role of tendon during running.
He described in [11], there exists unique running speed
where the length of muscle is maintained constant, andthe mechanical energy is preserved. In this case, themechanical work for running is not done by the muscle,
but by the elastic tendon [12]. Achilles tendon, inparticular, is known to have large elasticity and ability
to store up to 35% of mechanical energy in human or
kangaroo running [13].

HI~

Knee-actuator~k'~"~---

H~p-actuator
Thigh

~

KNEE

Leg-spring
Heel

ANKLE(passive)

TOE

Fig. 2. One-legged robot with n e w hindlimb mechanism

In design of a robotcapable of running which needs
more energy than walking, it is not a clever idea toassign the mechanical work to the actuators only. Instead, it is quite effective to use energy storing mechanism like spring. If we intend to do so, the mostdirect and simple implementation will be the linear
mass-spring model adopted in the Raibert's robots[2]. For studying the mechanism of animal running,
this model is very tractable, and many biomechanics
researcher dealt with this model. For example, McMahon uses this model to discuss relationship between
running speed and the spring constant in real animal

B. New simple hindlimb mechanism
There are several design solutions to make an articulated leg springy. Arranging the mechanism of serial
connection of inelastic actuator and a spring to eachjoint is one possibility. By using this design, P r a t t realized natural walking of a biped robot [17]. However,
how to link "multiple" passive joints with meaningful
periodic running motion is so difficult problem that
there is no successful examples so far.
Instead of considering multiple passive joints, here
we forcus on only one tendon; the ankle tendon. Fig.1
shows extensors of an ankle joint. We aimed at thearrangement of the muscle groups, named "gastrocnemius" and "plantaris", and proposed new leg model
shown in Fig.2, in which the leg spring is attached tothe same position as those muscle groups.
It is a planar one-legged robot that has articulated
leg composed of three links. It has two active joints,
hip and knee, and one passive joint, ankle, and there
is no actuator at foot (this means the toe can rotate
on the ground during stance phase, acting like a freepivot).
The most distinctive feature of this model is thearrangement of the leg spring. The leg spring is attached between thigh and heel parallel to the shank.
The arrangement of the leg spring in this way yields
the following two important effects during hopping:• During stance, holding the knee joint enables the

However, there certainly are some reasons that realanimals adopt not such a telescopic leg, but articulated
one and we summarized the practical advantages ofarticulated-type leg below.

(1) Large reachablity range ~ more clearance between foot and ground.
(2) Passive leg retraction during flight (provided if
the link parameters are appropriately chosen).
(3) Simple structure, because it connects two endsof links with rotary joint --~ easy to build.
In particular, (2) is helpful for energy-efficient runningand has already seen in passive walking robot [16].
Therefore, we are interested in an articulated leg forrunning robot, not only from its reality, but also fromits utility.

3985

oxton,io'("ddaN--

Spring extension ~

I ,-,

I

8q
Touch-down

BottomL~ft-off

Stance phase

Apex

Touch-down

Flight phaseFig. 3. Active energy pumping mechanism
Fig. 5. Leg operation during one stride of hopping
Swin

Fig. 4. Passive leg retraction

leg spring to absorb large impulse at touch-down
and to transfer its kinetic energy to potential energy for the next stride. Extending the knee yields
an extra displacement of the spring, and hence it
adds potential energy to the spring (Fig.3).
• During flight, the spring constitutes a member of"parallel four-bar linkage", because compressive
force to the spring is not so large at that duration. Consequently it enables passive retractionand extension of the leg, provided if the inertia ofthe links are chosen appropriately (Fig.4).
Prilutsky and Herzog show by experiments with catthat there is a transfer of energy between the ankle andknee joints via gastrocnemius and plantaris, and conclude this serves for energy-efficient running motion[18][19]. Using this model, we can see how the ankle
tendon operates on the run, in combination with theproximal joint motions. On the other hand, the springarrangement in "Uniroo" [i0] is the same as "soleus"in Fig.l.
Fig.5 illustrates one stride of hopping motion, together with each instant of phase transition (touchdown, bottom, lift-off, apex).

Fig. 6. One-legged hopping robot -"Kenken"

0.43[kg], respectively. The C.M. offset from the hip
joint is about 0.10[m]. For the leg spring, two tensile
coil springs are installed in parallel between thigh andheel. Initial value of the spring constant was determined by a simple energy analysis.
Since running requires relatively high energy and always accompanies the shock against the ground, thejoint actuation is the most critical problem in hardware realization. We introduced a powerful hydraulics
and developed a small and lightweight servo actuator,
directly mounted with an industry servo-valve. In thisTABLE I
MAIN SPECIFICATIONS OF "KENKEN"

III. ONE-LEGGED HOPPING R O B O T - - " K E N K E N "Parameter
total mass
body mass (incl. boom)
leg mass
body length
thigh, shank, foot length
toe length
leg length (max.)leg length (min.)maximum stride
body inertia around hipleg inertia around hip (max.)
leg inertia around hip (min.)leg spring coefficient (each)length of moment arm
rated actuator force @14MPa
rated actuator speed @I4MPa

Fig.6 shows "Kenken", the one-legged hopping
robot which was designed and built based on the leg
model proposed in the previous section. Main specifications are given in Table I.
We relied on some articles related to biomechanics
to choose the link parameters. A medium-sized dog
about 0.5[m] in hip height was selected as a target
model. Length of all links (including foot) was chosen
to be the same 0.18[m], for tractability of kinematics.
This corresponds to the range of the leg length from0.31[m] to 0.54[m] for ideal parallel four bar linkage.
The mass of the leg is relatively large, 3.6[kg]. Individual mass of thigh, shank, and foot is 2.42[kg], 0.75[kg],

3986

Unit
kg
kg
kg
m
m
m
m
m
m
kgm 2
kgm 2
kgm 2
N/m
m
N
m/s

Value
13.26
9.66 (0.5)
3.60
0.85
0.18
0.05
0.52
0.31
0.52
0.46
0.13
0.07
10000
0.06
2200
2.21

This is why we control each variables "intermittently"
using FSM.
In this section, after introducing characteristic dynamics of our model in Section IV-A, we propose thecontrol law for the stance phase in Section IV-B, andfor the flight phase in Section IV-C. They are combined to the FSM in Section IV-D.

sence, we gave priority to the stiffness over the autonomy. For the passive leg retraction during flightmentioned in Section II-B, the actuator force must be
zero. But this is difficult for a commercial flow-control
servo-valve and we have abandoned this at the prototype stage.
The experimental setup is like a Raibert's one. A
tether boom constrains the robot to sagital plane andmeasures the horizontal position, vertical position, andpitch angle of the body via three optical encoders. It
also carries hydraulic hose, signal line, and DC line.
Aluminum box attached to the rear includes interface
circuit, servo amplifier, and signal conditioning circuit,which were hand-made with taking the impulses andvibrations into account. The control program is written in MATLAB/SIMULINK code and runs in a single
timer task with sampling period 0.5[ms].
IV.

A. Characteristic dynamics of the model
Raibert derived the controller based on the uncontrolled dynamics of simple telescopic type leg [2].
For the mechanical system in general, "uncontrolled"means zero applied force or torque. This is a problem for our robot. A flow-controlled hydraulic servo
actuator has less force controllability (we should consider the active joints as very stiff) 4. That is why wedefined the control input as the input currents to theservo amplifier, instead of the actuator force. In thiscase, zero input implies holding joints with actuators(since hydraulic servo valve has a very high pressure
gain, it is easy to hold joints even under a relatively
heavy load). Note that holding joints during hoppingis allowed because the leg spring absorbs large amountof touch-down impulse.
Hence, we derive the controller based on "characteristic" dynamics of our model, which means, in ourcase, the dynamics which appears according to the input currents to the actuators. From simulation, we
obtained the following basic characteristics:(a) Dropping with both control inputs zero (il --- 0,
i2 - 0), negative pitching(q~ < 0) occurs at touchdown to the bottom, and after that, it turns to a
positive one (q5 > 0). Then, the robot eventuallylift off with positive angular momontum (Fig.8).
(b) Control inputs that extends either hip joint orknee joint (il ~ 0 or i2 > 0), produces a spring
extension, as well as a positive or negative body
pitching (Fig.9).
These characteristics come from the C.M. offset fromthe hip joint and the articulated leg design.

AN EMPIRICAL CONTROLLER FOR STABLE
HOPPING

In order to see the effectiveness of our leg mechanism, we tried to achieve a one-legged hopping of therobot, although it is a challenging task to stabilize it.
For this, we designed the controller in the form of "finite state machine (FSM)". T h a t is, we divide one
stride of hopping into several discrete states, then describe the transition (switching) law and design thecontrol law for each state. Then, the continuous statesof the system transit according to this FSM and we obtain periodic hopping gait. Each of the control laws
are derived based on the characteristic dynamics ofour model. The values of the control parameters are
determined empirically.
The coordinates of the robot are defined in Fig.7.
The controlled variables are (2, ~, ¢): the forward seed,vertical speed, and attitude of the body in the sagital
plane. The "virtual leg" length r and angle 0, which
are used in the control at flight, are also defined in
the figure. Our control inputs are il and i2, the input
currents to the hip actuator, and the knee actuator respectively (for reason as described below). Clearly thismodel is under-actuated, as there are only two inputs,
while the number of the controlled variables is three.

B. Control during stance
Since our robot has only two inputs, two variables
can be controlled independently. Based on observationof the characteristic dynamics at stance phase above,
we control the attitude and the vertical speed of the
body as follows.

q1~2-

q3~~vJ virtual?" leg{Cd),0,

if ¢ _< Cd
else

(1)

4We refer the readers to [20] for the force control using flowcontrol hydraulic servo valve. We shall also refer to [21], whereinteresting force controllable actuator, composed of spring andhydraulic actuator in series, is given.

Fig. 7. Coordinates ( ~ denotes actuated joints, O denotes
passive joint)

3987

Negative reaction moment

the gait stability of the robots that cannot change leg
length arbitrarily and have no actuator at foot [22] [23].
Although our robot has no actuator at foot, it hasa knee actuator that can be used to control the leg
length. But we have already used the knee actuator
for the vertical speed control. Hence, we use the touchdown leg angle for the forward speed control, following
Raibert's algorithm [2].
That is, we choose the desired touch down angle 01
of virtual leg as:

Touch-down

Free fall ~oO pL°Csiked
j°ii~

~o:om

[Ground'force~L / f l _ oreactionf f P ° smoment
itive

tension0 s - 0* + Ks(~c - kd) + Oo

(3)

2Ts 1
7-

(4)

where

Fig. 8. Free motion
Hip extension

0*

Positive .reaction moment• Activejoit
0 Lockedjoit
0 Passive joitAround force

nsion

il

/ Groundforce
Fig. 9. Actuated motionselse
if q'3 _~ 0, k _~ kd, r < rmax

arcsin(1Here, £d is a desired forward speed, and Ts is the
stance time which decreases as the forward speed increases [15]. The constant r0 is the nominal lengthof the virtual leg, 00 is introduced empirically to reduce the coupling effects from the offset of the C.M..
Control parameters are a feedback gain Kf and 00.
Having determined the touch-down angle Of, the hip
actuator swings the leg so that it tracks a smooth reference trajectory O(t) which reaches Of. Also, for leg
retraction, we give a smooth reference trajectory r(t)which reaches the nominal length r0. Then, the simple
local feedback law can be applied.

Negative reaction momenti2 -- { iC,o,-:-Kl(ql - q-i-)(5)i2 = -K2(q2 - -~)(6)Here, ql -- ql(r(t), O(t)), q-2 -- q2(r(t), O(t)) are thedesired joint trajectories calculated via inverse kinematics of the parallel four-bar linkage mechanism and
K1, K2 are the position gains.

(2)

D. Implementation of the controller

Where, Kp is a position gain, Cd is a desired attitude,
Zd is a desired vertical speed, ic is a constant current,
and rmax is the maximal virtual leg length.
Eq.(1) and (2) mean that the hip actuator controls
body pitch by a feedback law which is executed "onlywhen" the pitch angle is lower than specified value,
while the knee actuator controls vertical speed andalso suppresses the positive body pitch by giving a
constant input, which is exerted "only when" the bottom occurs.

F i g . 1 0 shows the FSM which combines the controllers described in Section IV-B and Section IV-C.
Each discrete state and the corresponding control law,as well as the events and the corresponding switching conditions, are summarized in the table, where sw
represents O N / O F F state of the foot switch. The situation that does not obey these transition rules implies
falling down. We used SIMULINK/Stateflow for programming the FSM.

V. EXPERIMENTAL RESULTS AND DISCUSSION

C. Control during flight

Using the controller described in Section IV, theexperiments for one:legged hopping were carried out.
T a b l e I I gives a list of initial conditions and control
parameters. Since we have not yet applied optimization or learning, the control parameters as well as theDuring flight, the robot swings the leg to prepare forthe next touch down. It also retracts and extends theleg to reduce the inertia for fast leg swinging and not
to stub against the ground. We note that the touchdown angle (foot placement) of the leg is critical to3988

TABLE II
PARAMETERS IN EXPERIMENTS

I

Flight phaseI start
,

//

/

'

Stance phase

I

I

•

;

Value
Kp
-10
ic
10
K/
0.47
0o
0.55
K1
Cd
-5
/(2
#d
2
Fig.10
Fig.11
Xd ]Unit
0.5
m/s I 0.0
¢o
0o
ro
zo

\

phineg\,//<:) Ibottom(SWING)ilift off ~

'. (

State
HOLD
PITCH
EXTEND
POSE
SWING
Event
td_postd_negphi_negbottomext_max
lift offbosom k

)

/

¢d),

i2 = oCd), or 0, i2 = ic
il -- O, i2 = 0
il = - K l ( q l)-i-), i2 = K2(q2Condition when event occurs
il = - K p ( ¢

¢

Value
7
0.5
0.1
10
100
100
Fig.13
1.5

and obtained the following observations. Stability andtracking performance is better at lower speed (below
l[m/s]) than higher speed. Actually, at faster speed
than 1.5[m/s], the speed does not regulate to desired
one, as seen from Fig.14. This instability at higher
speed seems mainly by the deviation of pitch angle attouch-down.
We can see from the plots, that the faster desired
speed, the larger leg angle at touch-down because ofEq.(3), which results in more nose-down body pitch
angle at touch-down. Because of the control law forthe body pitch (Eq. (1)), different body pitch angle attouch-down results in different motion during stancephase, even if the leg angle at touch-down is identical.
There are two ways to avoid such a deviation of the
body pitch angle at touch-down. The simplest way will
be replacing the leg by more lightweight one. It justreduces the amplitude of the body pitching, relatively
to the leg swinging during flight phase. The other way
is to actively control the body pitching, by installing an
tail mechanism into the body as "Uniroo" [10] does, orintroducing some nonholonomic attitude controls [24].

i2 = 0il = - / ( ; ( ¢s w = l,s w = l,Unit
mA/radmAs/mdeg
mA/radmA/radFig.121.0Jl:,OS3 t_max

Control actioni l = 0,

Unit
deg
deg
m
m
deg
m/s

¢
¢

Cd > O
Cd < OCd<043>0r = rmax, Or ~

Zd>0

8W=0

Fig. 10. Controller implemented as FSM

leg spring stiffness are tuned empirically. The robot is
controlled to 2[m/s] vertical s p e e d , - 5 [ d e g ] pitch angle, and 0 ~ 1.5[m/s] forward speed at lift off. F i g . l l~ F i g . 1 4 depict the corresponding time evolutions.
Fig.15 is the snapshots of a hopping motion. We feel
it looks very similar to the motion of dog hindlimb.
In all experiments, the robot is initially set to be
0.1[m] away from the ground, and then dropped. Notethat ~ (denoted by x D in the figures) represents thevelocity of the hip, not of the C.M.. The dashed linein the graph of r and 0, represents a "dummy" trajectory, i.e. a situation where the angles of ankle andknee are the same (q3 = q2). The pressure supply
for the actuator was set to 7[MPa], half of the ratedpressure, because we have a plan to extend this onelegged robot to the biped and quadruped robot(weight
becomes larger) in the near future. The experimental
results shows that our hindlimb mechanism can produce enough propulsion force for hopping at various
speed.
We also tried parameters different from Table II,

VI. CONCLUSION
Inspired by biomechanical studies on animal running, we proposed a new simple mechanical model forhindlimb of a dog and developed the one-legged hopping robot, "Kenken". The leg spring being attachedlike gastrocnemius or plantaris, enables the robot to
produce sufficient propulsion force, by virtue of an energy transfer from the knee, even if there is no actuatorat the ankle joint.
Using an empirical controller based on characteristic
dynamics of the model, the robot has succeeded inplanar hopping, although the problem related to thestability at the higher speed remains.
Therefore we conclude that our proposed mechanism
is actually suited for the hindlimb of the dog-like robot
which we plan to develop. Additionally, we feel it is thefirst time that such a realistic gaits as shown in Sec-

3989

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.0.,

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