Proceedings of 2004 IEEE/RSJ International Conference onIntelligent Robots and Systems
September 28 - October 2, 2004, Sendai, Japan

Passive Running of Planar 1/2/4-Legged Robots
Sang-Ho Hyon

Xin Jiang

Takashi Emura

Tetsushi Ueta

Department of Bioengineering and Robotics
Tohoku University
Sendai 980-8579, Japan
Email: sangho@ieee.org

Abstract— In this paper, we report on passive runningof planar one-legged, biped, and quadruped robots. The
topic includes the analysis of passive running gaits and theirorbital stabilization. For one-legged robot, two stabilizing
controllers that asymptotically stabilize periodic passive gaits
are derived. In particular, the second controller is based on“energy-preserving principle” and its original form generates
interesting quasi-periodic running gaits, which can be seenin Hamiltonian system. The controller is extend to a planar
biped robot with torso, which does not have any passive
running gaits. Combining simple attitude controller at stancephase generates stable periodic running gaits. For a planar
quadruped robot, with an advanced gait searching algorithm,
some fundamental proterties of the passive running gaits, suchas stability or symmetry, are observed.

I. I NTRODUCTION
After the Raibert’s excellent works [1], running robots
have been widely studied both experimentally [2][3][4][5]and theoretically [6][7][8]. In the fast running control,
energy-efficiency is especially crucial for autonomous
robots (including biped humanoid robots or quadruped
robots) because it directly extends their operation time.
In this connection, there are some remarkable researches
on passive running, where the passive running means“unforced” periodic running. Tompson and Raibert showedthat spring-driven one-legged hopping robot can hop without any inputs, provided if the initial conditions wereappropriately chosen [9]. Ahmadi and Buehler applied
Raibert’s algorithm to this robot and realized energyefficient hopping in simulation and experiment [10][11].
François and Samson derived a rather systematic controller
based on linearization of the periodic orbit [12].
In this paper, we report on our research motivated fromthe above works: passive running of planar one-legged,
biped, and quadruped robots. The robots are installed withlinear springs at the telescopic knee joints and torsional
springs at the hip joints. These springs assign strong passive dynamics to the robots. Doing so, we expect the robot
exhibits passive periodic running gaits for appropriately
chosen parameters and initial conditions.
However, it is almost impossible to obtain analyticalsolution even for simple 2 DOF case [7]. Threfore thesolution can be found numerically. If we can “luckily”
find the passive gaits, the next stage is their stability
analysis. Even if they are unstable, when the system hassome controllability, we can design the controllers, which
asymptotically stabilize the gaits. This is a stabilization

0-7803-8463-6/04/$20.00 ©2004 IEEE

Center for Advanced Information Technology
Tokushima University
Tokushima 770-8506, Japan
Email: tetsushi@is.tokushima-u.ac.jp

problem of nonlinear hybrid oscillator, which has notbeen well discussed. Thus obtained limit cycle is utmost
important for energy-efficient locomotion because it does
not require any control input when the solution lies on thelimit cycle. Moreover, if the controller extends the regionof attraction, it directly links to the application to more
realistic robot models. If we cannot find any passive gaits
unfortunately, we can find some controllers that generate
stable limit cycle, in some cases.
Our approach to the analysis and control of passive
running robots is different from previous studies in thefollowing sense:
1) We do not rely on some approximations or linearized
equations2) We do not assume the leg has neglgible mass, andconsider the impact between the leg and the ground3) We design feedback controllers based on “energypreserving principle”
The first point is important for stability analysis and forgeneration of control inputs that converge to zero whenstabilization is achieved. Otherwise, we will have steadystate error. The second point is to reveal the true behavior,
which the real robot exhibits. In a simple model, many
literature neglect the mass of the leg, and hence, the impact
between the foot and the ground. Mass-less leg hides
the behavior of the counter-oscillation of the body, andexclusion of impact phenomenon leads to incorrect stability
analysis and the controller design. The third point is toderive energy-efficient controllers: energy-preservation isthe essential necessary condition of stable passive running.
We will show the above approach by three specific examples; planar one-legged, biped, and quadruped running
robots. In Section II, the analysis and control of a passive
one-legged running robot is provided. With new results
on stability analysis, a local feedback controller is derived
and our previously derived stabilization controller, which
asymptotically stabilize unknown (multi) periodic passive
gaits, is discussed.
Section III extends the controller of one-legged model
to a planar biped model having a torso. This biped modelis the example that any passive gaits are not found. Combining a simple attitude controller at stance phase, we try
to stabilize unknown periodic biped running gaits.
Section IV shows numerical search on passive runninggaits of planar quadruped robot. Planar quadruped runningis composed of several phases and this complicates the3532

gait analysis. To this end, numerical algorithm in onelegged model is modified. We show two interesting passive
running gaits and their properties.

hip spring Kh
BODYτ

A. ModelWe consider one-legged passive running robot shown in
Fig. 1. The robot is attached with not only a leg spring but
also a hip spring. The generalized coordinates are definedas the position of center of gravity (CoG) (x, z), the leg
length r, and the attitude of the body and the leg (φ, θ).
The control inputs are the leg force f and hip torque τ .
These are applied parallel to the springs. Table I shows
the physical parameters, together with the values used fornumerical analysis.
The following assumptions are imposed on the model:

LEG Jurθ
leg spring
Kl

Z

f
FOOT
(no-mass)

X

Fig. 1.

Passive one-legged hopper

TABLE IP HYSICAL PARAMETERS OF ONE - LEGGED MODEL(A) Mass of the foot (unsprung mass) is negligible(B) The foot does not bounce back, nor slip the ground(inelastic impulsive impact)(C) The springs are mass-less and non-dissipating
Assumption (A) means the foot of the robot is mass-less,and hence, there is no inertial force due to the foot mass.
Of course, this assumption cannot be satisfied exactly inreal machine, but it is rather easy to make foot lightweightbecause the foot is a simple rod. Thanks to this assumption,
there is no impulse along the leg axis.
On the other hand, when the robot receives the impactof perpendicular direction to the longitudinal axis of theleg, energy less occurs due to the inertia of “upper part”
of the leg. If the foot is too repulsive, it may bounce backdue to this impulse and the robot cannot touchdown the
ground appropriately. However, for our robot, it is easy
to make the foot lightweight, its restitution low. More
importantly, our controller minimizes touchdown impulse
normal to the longitudinal axis of the leg, as shown later.
Therefore, assumption (B) on inelastic impulse model
represents touchdown dynamics of our one-legged robot
most effectively. Assumption (C) is not restrictive one
because it is easy to compensate energy loss by applying
leg force f .
The equations of motion are presented in [13]. A runningmotion is composed of successive phase transitions; Stance→ Flight → Stance → · · ·. Between Stance and Flight,
thre are discrete events; Touchdown and Lift-off. Table IIdefines the phase-indicating subscripts for variables. For
example, ẋlo is the forward velocity of C.M. at Liftoff, θ̇td+ is the angular velocity of the leg just before
Touchdown, θ td is the leg angle of just before, or, justafter Touchdown, and so on.

φJb (x,z)M

II. O NE - LEGGED RUNNING ROBOT

g
M
r0
Jb
Jl
Kl
Kh

Meaning
gravity acceleration
total massnatural leg length
body inertia
equivalent leg inertia
leg spring stiffness
hip spring stiffness

Unit
m/s2
kg
m
kgm2
kgm2
N/m
Nm/rad

Value
9.8
12
0.5
0.5
0.11
3000
10

With the state variable x = [x, θ, φ, ẋ, ż, θ̇, φ̇]Tand the cross section h(x) := z − r 0 sin θ = 0 (just after
Touchdown), the Poincaré map can be represented as:
xn+1 = P (xn ),(1)

where n is the number of strides. We want to find a solution
x of (1) that maps onto itself, i.e. a solution satisfies theequation:G(x) = x − P (x) = 0(2)

The search space is 7-dimensional. We use NewtonRaphson method for root seeking, where an initial guess
for the fixed point is given and then updated based on thefollowing scheme.
For small changes in the state variables, the change in
P is approximated by its Taylor series,
P (x + ∆x) = P (x) + DP (x)∆x + O(x),where DP (x) =term.

∂P (x)∂xand O(x) represents higher order

TABLE IIP HASE - INDICATING SUBSCRIPTS

B. Passive running gaits
To search the passive running gait, Poincaré map isconstructed because the existence of the fixed point ofPoincaré map means that of passive running gait. To findthe fixed points, the Newton-Raphson method is employed
as follows.

Subscript
tdtd+
td
lo

3533

(3)

Meaning
just before Touchdown
just after Touchdown
just before, or after Touchdown
Lift-offVertical height [m]

START
k=k+1

k=1

x = P(x )
k

k

n+1

n

Update the Guess
k

k

n

n

substitution

k

n

n

x

6

7

8
Horizontal distance [m]

9

10

11

k+1
n

, where τ1 and τ2 are constant values, t is the time after
the lift-off, and T v := 2ẋlo /g means “expected” flight time.
Then, we obtain the closed loop system that is stabilizable:


τ1
,
(7)
xn+1 = DPx xn + DPu
τ2

k

n

END

Fig. 2.

0

Fig. 3. Subsequent two steps of 1-periodic passive running gait. Runningspeed is 5 m/s. The robot moves from left to the right.

-1

x + (I-DP(x ) ) [P(x ) - x k ]x - x k <10-12
n+1

0.5

5

Initial Estimate

Evaluation of
the Return Map

1

Gait searching algorithm

Therefore, we haveDG(x)∆x = −G(x)⇒ ∆x = (I − DP (x))−1 (P (x) − x),(4)where x is the value of the states of the return map calculated at the nth stride. Therefore, we have the following
update scheme with a given initial guess x0 ,xk+1 = xk + (I − DP (xk ))−1 (P (xn ) − xn ),(5)where the index k corresponds to the number of iterations.
The algorithm is summarized in Fig. 2. Note that inaddition to the solution x(t), DP should also be calculated
numerically. Also, if there are jump in the solution x due
to the impact, DP also should be processed accordingly.
See [14] for the details.
For the one-legged model, this algorithm converges
rapidly, and we can obtain multiple passive running gaits.
The solutions exist for every admissible initial condition
and they are actually loss-less, i.e. no energy is dissipated.
For example, a fixed point x = [−0.3129, 0.6763,−0.1198, 5.0000, −2.0760, −5.2004, 1.2818] corresponding to high speed passive running as shown in Fig. 3.
Its stability can be found from the eigenvalue of DP . In
this case eig(DP ) = [−5.6574, 2.0839, −0.1800, 0.0070,
0.0000, 0.0000, 0.0000]. Since the first two elements lie
outside the unit circle on the complex plane, we canconclude the passive gait is orbitally unstable. All the othersolutions are found to be unstable, except for trivial vertical
hopping.
C. Local stabilization by linear feedback
Having obtained passive gaits and checked their stability,
we should find appropriate stabilizing controller. Here weshow two kinds of controllers in the sequel.
The first one is local stabilization by linear feedback
controller. In the above example, we have two unstablemanifolds regarding above two unstable eigenvalues. For
example, we can apply control input τ at the flight phase:
if 0 ≤ t < Tv /2
τ1 ,
(6)
τ =
if Tv /2 ≤ t < Tv
τ2 ,where DPx is the same as DP in the previous section and∂P ∂P
DPu = ∂P∂u = ( ∂τ1 , ∂τ2 ) is newly appeared derivative byintroducing control input (6).
If we place the two unstable pole to zero, we canstabilize all of the unstable passive gaits obtained in theprevious section. However, the region of attraction is foundto be quite narrow because the controller is local. Actually,
it is found by simulation that the local controller does notallow even 0.05 m/s error in initial velocity ẋ 0 .
D. Stabilization based on energy-preserving principle
We show an alternative controller based on its energy
analysis, firstly proposed in [15]. The underlining principle
is energy-preservation. This means the controller preserves
system energy as much as possible. The most importantreason why we use this principle is; if the system energy
is preserved, it is expected that the system autonomously
generate natural periodic gaits, just as some class ofHamiltonian system exhibit natural periodic orbit. Insteadof depending on some pre-calculated periodic solutions, ortarget (desired) dynamics, analysis on energy change ofthe original nonlinear hybrid system are utilized as shown
below.
First, we choose desired touchdown angle θ d and angularvelocity θ̇d at the moment of lift-off to meet the following
energy non-dissipation condition:µtd− := ẋtd− cos θtd + żtd− sin θtd + r0 = θ̇td− = 0 (8)Since the energy change between just before and after
touchdown is calculated to be:
M Jl
µ2 ,
Etd+ − Etd− = −(9)2(Jl + M r02 ) td−
the condition (8) means there is no energy exchange
between the robot and the ground, provided no control
input applied during stance phase. Having determined θ dand θ̇d , finally we can apply the same controller (6), which,in this time, becomes linear dead-beat controller.
As a result, interesting quasi-periodic orbits [16], which
can be seen in some Hamiltonian system, are found [15].
Fig. 4 depicts such a simulation result. Herein, the freeparameters of initial values are chosen to be θ 0 = 0.25
[rad] and ẋ0 = 2 [m/s] (off the fixed point). The region

3534

5

106
104

0

E [J]

τ [Nm]

E [J]

108
τ1
τ
2

102
100
0

−5
0

20
40
time [s]

50
steps

100

108

4

107

2
τ [Nm]

110

106
105

−2

104

−4

103
0

−6
0

20
40
time [s]

2.6

2

50
steps

100

0.47

0.475
z(k)

0.48

−0.2

0
0.2
φ(k)

0.4

−1.7
−1.8

−0.2
−0.1
x(k)

0.44

0.46
z(k)

1.9
−0.18

0.48

0
−2.95
−3

−0.5

−3.1

−1.95
−2
−2.05

−0.16
x(k)

−0.14

−2.4

0.5

−2.5

0

θD(k)

−2.6

φD(k)

−3.05

2.1
2

−1.9
2
−0.3

zD(k)

−1.9

−1.6

φD(k)

2.2

2.2
xD(k)

2.4
zD(k)

xD(k)

τ1
τ

−1.85
−1.5

θD(k)

0

−0.5

−2.7
−2.8

−1

−3.15

−1

−2.9
0.25 0.3 0.35 0.4 0.45
θ(k)

−0.1

0

0.1 0.2 0.3
φ(k)0.25 0.3 0.35 0.4
θ(k)Fig. 4. Simulation result of Energy-Preserving Touchdown Control: The
top two graph shows time evolutions of energy level (E) and control
inputs (τ1 and τ2 ). The lower four graphs represent the selected imagesof the Poincaré map, where notation “D” represents the time derivativesof preceding variables. Please note that the position x is manually resetat each iterative crossing of the section, for visibility. Although all the
points are bounded in some regions, there are no fixed points appeared.
From this, we can see the gait obtained is quasi-periodic one.

of attraction is very large if we do not specify forward
velocity. It is quite different point from the above local state
feedback controller. Since the gaits obtained are nonlinear
flows on an invariant manifoldĖsys := Ė − τ ψ̇ = 0,(10)

if there is a sudden change of energy level, the gait jumps to
new quasi-periodic orbits, hence we can say the controller
has some robustness against disturbance.
Thanks to the similar recurrent property between quasiperiodic orbits and chaotic orbits, we can apply a delayed feedback-like controllers for chaotic system [17], toasymptotically stabilize quasi-periodic gaits to (unknown)
periodic ones “having desired period”. Specifically, θ d canbe chosen as follows.
 1− 2 (θlo (k) + θlo (k − p)),
if k > p(11)
θd (k) =else,−θlo (k),where k > 1 is the iteration step and p > 1 is a
desired period. A general form of delayed feedback and its
limitation can be found in e.g. [18]. Note that the desired
angular velocity θ̇d (k) is automatically determined by (8)accordingly. Especially for 1-periodic gait (p = 1), with an−1.5

Fig. 5.
Simulation results of the orbital stabilization controller andadaptive enegy controller. The top two graph shows time evolutions ofenergy level (E) and control inputs (τ1 and τ2 . The lower four graphs
represent the selected images of the Poincaré map, where notation “D”
represents the time derivatives of preceding variables. Please note thatthe position x is manually reset at each iterative crossing of the section,
for visibility. The input minimization via adaptive energy controller
is activated from the 60th step. You can see the images of Poincaré
map asymptotically converge to one fixed point and the control inputs
eventually converge to zero!

additional adaptive energy controller, the robot eventuallyhops without any control inputs, that is, complete passive
running is obtained. Fig. 5 is an example of simulation
results, which shows the complete passive running. The
detailed description and results can be found in the literature [13].
III. BIPED RUNNING ROBOTIn this section, a planar biped running is presented. The
biped model is the example that we cannot find passive
solution. As described in Section I, even we cannot findany passive gaits, we may find some controllers, which
generate stable limit cycle, where the control inputs canbe made small enough although it cannot be zero.
A. ModelFigure 6 shows a planar biped robot considered here. The
robot has two springy telescopic legs swinging around hip
joints. Leg actuators are mounted parallel to the leg spring.
The generalized coordinates are defined as the positionof CoG, x = (xg , zg )T ∈ R2 , the attitude of the torso, φ ∈

3535

φ

Lb

C. Stabilizing controller
In this section, we derive a controller similar to onelegged model to find stable “unknown” running gaits. First,
we should overcome the instability caused by the torso
at the stance phase. Here we remove hip springs andtemporarily introduce simple pitch control:

Torso
Mb, I

τ1 = −K1p φ − K1d φ̇,(xg,zg)
Leg2

m, J

ψ2 −ψ
1
θ2

Z

−θ1

where K1p ≥ 0 and K 1d ≥ 0. The supporting leg isassumed to be Leg1 indicated in Fig. 6.
For swinging leg (Leg2), we are also temporarily controlling it by

L

Leg1

m, J
r0

τ2 = −K2p (ψ2 + ψ1 ) − K2d (ψ̇2 + ψ̇1 ).

Lateral view
Biped running robot

R1 , and joint angles, ψ = (ψ 1 , ψ2 )T ∈ R2 . Table III shows
the physical parameters, together with the values used inlater simulations. This model is highly nonlinear because it
has massive legs and torso, whose CoG are located away
the hip joint. We suppose the same assumption as onelegged case. The equations of motion are shown in [19].
B. Passive running gait searching
We have applied almost the same numerical algorithm(Fig. 2) to find passive biped running gait. Only the numberof the phase is increased because the robot has two legs.
Wherein, the hip springs are installed as in the case of onelegged model to make the leg swung passively. However,no passive running gaits, except for trivial solution (vertical
hopping), were found. The reason seems to lie on the torso
located above the hip joint. It is known that a torso mounted
below the hip joint has passive stability [2]. Therefore, incontrast to one-legged robot, in which no control inputs
are applied, the biped robot cannot hold its torso upright
posture without pitch control, because of reaction forcesfrom hip joints. Our analysis implies that just installing
hip spring between the torso and the leg is not enough tokeep the torso upright.

This equation means we can control ψ̈ arbitrarily by newcontrol inputs u 1 and u2 .
Next, we define a target dynamics. Conservation law of
the angular momentum around CoG can be expressed as
J0 φ̇ + J1 ψ̇1 + J2 ψ̇2 = P0 ,(φ̇0 is the initial value of φ̇ at flight phase), we get thefollowing target dynamics.
φ̇ = φ̇0

Unit
m
m
m
kg
kg
kgm2
kgm2
N/m

Value
0.08
0.16
0.2
4.11
2.13
0.05
0.015
3000

(17)

This means “another first integral of motion is created by a
feedback control”. The control input u 2 can be calculated
by combining (16) and the original equations of motion.
The final task is to determine the control input u 1 . The
control objective is to dead-beat the absolute angle of theswing leg θ1 and its velocity θ˙1 to some desired values
at given fixed time T v (flight time). As we explained in
Section II, desired values are chosen to preserve energy at
touchdown. For the biped robot case, condition of energy
preservation is given by:

TABLE III
Meaning
natural leg length
position of the leg CoG
position of the torso CoG
mass of torso
mass of leg
inertia of torso
inertia of leg
leg spring stiffness(15)

where Ji (i = 0, 1, 2) is nonlinear inertia terms and P 0means initial angular momentum of flight phase. If we
chose
1
ψ̇2 =
(P0 − J0 φ̇0 − J1 ψ̇1 )(16)
J2

λ̂p = 0,P HYSICAL PARAMETERS OF BIPED MODEL
r0
L
Lb
Mb
m
I
J
Kl

(13)

Under this controller, counter oscillation of each leg is
expected during running.
Next, we apply the controller of one-legged model tothe biped at the flight phase. To do so, decoupling control
and target dynamics are introduced. By new control inputs
u1 and u2 , a part of the equations of motion is decoupled:
 


ψ¨1
u1
=
(14)
u2
ψ¨2

X

Fig. 6.

(12)

(18)

where λ̂p is the constraint impulse imposed at touchdown,
which can be calculated explicitly.
For a given flight time T v , however, there are many pairsof θd and θ̇d , as in the case of one-legged model. To find“unknown” periodic running gait, we set θ d according to(11). Since (14) is a trivial second order linear ODE, we
can easily dead-beat ψ 1 and ψ˙1 , by applying the same
controller as (6) to u 1 .

3536

Vertical height [m]

1
0.8

Z

0.6

L

0.4

L

3.5

4

4.5
5
5.5
Horizontal distance [m]

6

6.5

M, Jb
Kh

Fig. 7. Stick animation of subsequent three steps of 1-periodic bipedrunning, corresponding to Fig. 8. The robot moves from left to the right.
2

0.1

1

0.05
φD [rad/s]

zgD [m/s]

φ

z

0

0
−1

m, Jl

θ1

m, Jl

Kh

θ2
r2

FORE LEG
Kl

r1

HIND LEG
Kl

X
x

O

0

−0.05

Fig. 9.
−2
0.4

0.5
zg [m]

0.6

−0.1
−2

10

10

5

5

ψ2D [rad/s]

ψ1D [rad/s]

BODY

0.2

0
−5

−5
−10
−1

0
ψ1 [rad]0.5

1

2
φ [rad]4

6

Planar quadruped running robot
TABLE IV

−3x 10

P HYSICAL PARAMETERS OF QUADRUPED MODEL

0

−10
−1

−0.5

0

−0.5

0
ψ2 [rad]0.5

1Fig. 8. Phase portraits of stable 1-periodic running gait of 2 m/s inhorizontal speed, where the control parameters are Kp1 = 200, Kd1 =50, Kp2 = 100, Kd2 = 10. Notation “D” represents the time derivativesof preceding variables. Note that the scale of the abscissa of right top is10−3 and small horizontal curve running form φ = 0 [rad] to 2 × 10−3[rad] does not mean an instantaneous jump, but indicates the result ofhigh-gain PD-feedback about body pitch.

D. Simulation
Using the above controller, stable one-periodic runninggaits are investigated. Fig. 7 shows the stick animation of
2 m/s running and Fig. 8 depicts the corresponding phase
portraits. Stable running in different horizontal speed isalso obtained. Interestingly, dynamics about Leg2 (zerodynamics of decoupling controller) is found to be stable. Actually, the motion of Leg2 indicates the counter
oscillation to Leg1. Although the running gait seems tohave symmetry, the motion of torso is slightly asymmetry,
as recognized from the right top graph of Fig. 8. Thisasymmetry become more significant if the feedback gainsof (12) become smaller. Without attitude control, the robot
falls down after a few steps. Stabilization to (unknown)
period 2 gaits (p = 2) is also succeeded [19].
IV. Q UADRUPED RUNNING ROBOTIn this section passive running of a planar quadruped
robot is presented. Gait searching becomes more complicated than one-legged case and only the outline is shown.
A. ModelThe model of a planar quadruped robot is shown in
Fig. 9. This is the extended model of Fig. 1. That is, it

r0
L
Mb
m
Jb
Jl
Kl
Kh

Meaning
natural leg length
distance between the legs
mass of torso
mass of leg
inertia of torso
inertia of leg
leg spring stiffness
hip spring stiffness

Unit
m
m
kg
kg
kgm2
kgm2
N/m
Nm/radValue
0.5
0.4 or 0.25 ∗20
1 or 3 ∗1.67
0.017 or 0.051 ∗8000
25* L, m and Jl differ for each Profile A or B (Fig. 14).

has springy telescopic legs rotating around hip joint withhip springs. All of the assumptions are the same as thosein the one-legged or the biped. The physical parameters
are summarized in Table Table IV.
The model can be completely described by 7 coordinates, we define a generalized coordinate q =[θ1 θ2 φ x y r1 r2 ]T and the state vector x̄ = [q q̇] T , where(θ1 , θ2 ) are the leg angles, φ is the pitch angle, (x, y) are
the Cartesian coordinates of the body’s CoG, and (r 1 , r2 )are the length of the fore and hind leg respectively.
Note that in planar quadruped model, each pair of fore
legs and hind legs moves together, and hence, only thebounding gaits can be generated (gallop does not appear).
A complete bounding cycle is showed in Fig. 10. We cansee from the figure that each of these events initiates thecorresponding phase.
Related to the quadruped passive running, there is apreceding study [21]. The main differences of our modelto their model are; consideration of mass of legs and hip
springs. As described in Section I, the important physicalphenomena that flowed from this consideration are: passive
swinging of the leg, the counter-oscillatory motion of thebody, and energy-loss due to the impact between the footand the ground.
The equation of motion is represented by the following
set of differential equations and algebraic equations:• Continuous dynamics:x̄˙ = fi (x̄)

3537

(i = 1, · · · , 4)(19)

Touchdown

fft-o

Hind Leg Stance

Double Support

f

Li

n
ow

d

ch

u
To

Lift-off
Touchdown

Lift-off
Touchdown

Lift-off

1

1

0.8

0.8

0.6

0.6

0.4

0.4

0.2

0.2

0

0

−0.2
−0.5

FlightLift-off

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(a) Gait A

Fore Leg StanceToudown

Fig. 10.

0

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(b) Gait B

Fig. 12. Two different passive running gaits: the hind leg touches theground in Gait A, while the fore leg touchdowns first in Gait B.

Phase transition of planar quadruped running gait

0.15

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Double Support

FlightUpdate guess ofreference point using
Newton-Raphson
method

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Is the reference point
reached ?

Fore Leg Stanceθ2

Hind Leg Stance

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(a) Gait A

END

Fig. 11.

•(b) Gait B

Fig. 13. Time trajectories of one stride of running: The graphs imply
the symmetry; φ(t1 ) = −φ(t2 ) = φ(t3 ), θ1 (t1 ) = −θ1 (t2 ) = θ1 (t3 )and θ2 (t1 ) = −θ2 (t2 ) = θ2 (t3 )Gait searching algorithm for quadruped model

Discrete dynamics:x̄+ = Hj (x̄− ) (j = 1, · · · , 4)(20)Here, f1,2,3,4 the vector fields corresponds to eachphase Hind Leg Stance, Double Support, Fore Leg Stance,
Flight respectively. H 1,2,3,4 corresponds to events Hind Leg
Touchdown, Fore Leg Touchdown, Hind Leg Lift-off, Fore
Leg Lift-off, respectively. The details of equations are given
in [20] and will be published elsewhere.
In continuous phase, the equation of motion can be represented by four different dynamical system, which takes
the form of autonomous system because we have no inputs.
Note that the DOF of each phase is different and they are all
bellow 7. Therefore, some of the variables in the vector x̄,do not participate in the dynamic equations in some phases.
In this case, they become the dependent variables, and the
update of these variables should be calculated based on theconstraints subjects to the corresponding phases.
On the other hand, the equation of motion at transition
phase can be represented by four different algebraic equations. The transition models mainly contribute to describe
the behavior of the impact phenomena, which occurs when
the leg strikes the ground (the other is to describe “reset”
of the position at the end of one stride). This consideration complicates the analysis greatly. However, they are
expected to be a more realistic system description.

B. Passive running gaits
Since the quadruped running is composed of multiple
phases, an advanced algorithm is implemented as shownin Fig. 11, where no order of phase transition is presumed.
Depending on which event occurs, the integrator selects
the appropriate equations of motion, which correspond to
the phase triggered by that event according to Fig. 10. The
fixed point to be found is x∗ = [θ1∗ , θ2∗ , φ∗ , θ̇1∗ , θ̇2∗ , φ̇∗ , ṙ1∗ ] at
the cross section of poincaré map selected as the event ofHind Leg Touchdown, or x∗ = [θ1∗ , θ2∗ , φ∗ , θ̇1∗ , θ̇2∗ , φ̇∗ , ṙ2∗ ]at Fore Leg Touchdown. Although we do not assume any
specific order of phase transition, only the gaits which tend
to converge will exhibit a repetition of ordered phases.
Thanks to this algorithm, two typical gaits, Gait A and
Gait B, shown in Fig. 12(a) and Fig. 12(b) respectively, areobtained. In Gait A, the Hind Leg Touchdown event occurs
first after Flight, while in Gait B, the Fore Leg Touchdown
event occurs first. From their analysis, the followings areconcluded.
First, different from other related studies, all the passive
running gaits we found are proved to be unstable. The
investigation of characteristic multipliers corresponding
to the periodic gaits revealed that their magnitude areoutside the unit circle. For example, Gait A is the periodic orbit with the fixed point x∗ = [0.1396, 0.1438,

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Profile A

Fig. 14.

Profile B

Two different profiles

0.0084, −1.7619, −1.5909, 0.2482, −1.4232], whosecharacteristic multiplier eig(DP )| x∗ is [7.1516, −0.7776,−0.1961+0.3112i, −0.1961 − 0.3112i, 0.2130, 0.0567,
0.0018] and Gait B is the periodic orbit with the fixed
point x∗ = [0.2800, 0.2778, −0.1350, −0.9629, −2.5999,−0.3145, −1.8308], where eig(DP )| x∗ is [−7.3902,
1.5585, −0.8970, −0.7272, −0.0019, −0.2030 + 0.0901i,−0.2030−0.0901i]. Although the motion starting from thefixed points can continue for some steps (8 and 25 steps as
maximum for Gait A and B respectively), it falls finally.
Secondly, our simulations reveal that both Gait A and
Gait B exhibit symmetric behavior. The time evolutionof angles shown in Fig. 13(a) or Fig. 13(b) implies such
symmetric properties: φ(t 1 ) = −φ(t2 ) = φ(t3 ), θ1 (t1 ) =−θ1 (t2 ) = θ1 (t3 ) and θ2 (t1 ) = −θ2 (t2 ) = θ2 (t3 ). This
property agrees with the result of [21].
Thirdly, thre is a strong profile dependency of passive
running gaits. Actually, Gait A and Gait B are generated
by two different profile (Table 9, Fig. 14). In our numericalanalysis, Profile A always converges to Gait A, and Profile
B always converges to Gait B.
V. C ONCLUSION
We reported on passive running of planar one-legged,
biped, and quadruped robot. The robots are installed withlinear springs at the telescopic knee joints and torsional
springs at the hip joints. In this paper we addressed theanalysis of passive running gaits and their orbital stabilization.
For a planar one-legged robot, passive running gaits were
found numerically. To this end, numerical search algorithm
was introduced. The analysis showed that the all passive
running gaits were unstable except for a trivial gait. Next,
orbital stabilizing controller was derived, based on energypreserving principle. Different from a local feedback controller derived from the linearized Poincaré map, the newcontroller demonstrated the best harmony with the passive
gaits. With this controller, interesting quasi-periodic orbits,
which can be seen in Hamiltonian system, were observed.
Moreover, with an additional adaptive scheme similar todelayed feedback in chaotic system, orbitaly stabilization
to (unknown) passive running gaits were achieved and thecontrol inputs converged to zero.
For a planar biped robot attached with torso above hip
joint, we could not find any periodic passive solution.
Nevertheless, with a simple attitude controller at stancephase and delayed feedback-like controller in flight phase,“almost symmetric” stable 1-periodic running gaits weregenerated.
For a planar quadruped robot, more advanced numericalalgorithm was developed. With this algorithm, two typicaldifferent passive running gaits were found. Although theresults are premature, we found the following propertiesof the passive running gaits: Firstly they were all unstable
except for trivial case, secondly they had symmetry, thirdly
they had a strong profile dependency.
Our ongoing task includes continual searching and analysis of passive running gaits for both of the biped andthe quadruped. The experiments with biped robot are also
performed in parallel.
ACKNOWLEDGEMENT
This research was partially supported by the Ministry of
Education, Science, Sports and Culture, Grant-in-Aid forYoung Scientists, 15760144, 2003.
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