Detection and Stabilization of Hybrid Periodic
Orbits of Passive Running Robots
Sang-Ho Hyon

Takashi Emura

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

Abstract— In this paper a new algorithm to detect hybrid
periodic orbits of autonomous hybrid dynamical system isdeveloped. Conventional Newton algorithm is modified so thatit suits to the analysis of Poincaré return map of hybrid
dynamical systems that include multiple phases (modes) anddiscrete jumps. Then, the algorithm is applied to a specificexample; planar one-legged robot model having a springy
leg and a compliant hip joint. With the algorithm, passive
running gaits of the one-legged robot are automaticallydetected for various parameter sets and initial conditions.
The analysis of the characteristic multiplier of the returnmap revealed the stability and the bifurcation of the passive
running gaits. Two kinds of controllers that achieve orbital
stabilization are presented. A similarity is found between
the detection algorithm and the stabilizing controller. The
algorithm can be applied to any kinds of the robots (e.g.
walking robot).

I. I NTRODUCTION
After the Raibert’s excellent works [1], theoretical aspects of basic one-legged running model has been deeply
studied by Koditschek and his coworkers [2] [3] [4]. In thefast 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 runningmeans “unforced” periodic running. Tompson and Raibert
showed that spring-driven one-legged hopping robot can
hop without any inputs, provided the initial conditions wereappropriately chosen [5]. Ahmadi and Buehler applied
Raibert’s algorithm to this robot and realized energyefficient hopping [6]. François and Samson derived a rather
systematic controller based on linearization of the periodic
orbit [7].
Our goal is to study the mechanism of energy-efficient
legged locomotion and to realize it in legged robots. To
this end, a general approach is established:
1) Explore the “passive dynamics” of the model (study
the periodic orbits if they exit),
2) Design a “main controller” that makes the systemtrajectory to be reccurent (prevent the robots fromfalling),3) Design “sub controllers” that adaptively stabilize therecurrent trajectories to optimal periodic orbits (passive orbits if they exist),

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

where “optimal” periodic orbits means passive (zero input)
periodic orbits for passively driven legged robots.
However, it is very difficult to obtain analytical solution
even for simple 2 degrees of freedom (DOF) case because
the system is not integrable [3]. Therefore, numericalmethod must be employed. If we can “luckily” find passive(unforced) periodic gaits, the next stage is their stability
analysis. When the system has some controllability, we
can design controllers, which asymptotically stabilize thegaits, even if passive gaits are unstable. This is a stabilization problem of nonlinear hybrid oscillator, which
has not been well discussed. Thus obtained limit cycle isutmost important for energy-efficient locomotion because
the control input becomes zero when the solution lies on
the limit cycle. Moreover, if the controller has large regionof attraction, it will be a good template controller for more
complex robot models.
In this paper, we develop a new simple algorithm todetect hybrid periodic orbits of autonomous hybrid dynamical system and apply it to one-legged robot [8]. In
Section II, the new detection algorithm is derived according
to the previous work [9]. Conventional Newton algorithm
for continuous dynamical system (e.g. [10]) is modified sothat it suits to the analysis of the Poincar é return map(simply called “return map”) of the hybrid system that
includes multiple phases (modes). Using this algorithm,
search space is reduced from 2n to 2n − 1 or less. This
leads to a great help for the analysis of the full-dimensional
hybrid system behavior. Also, a modification is made sothat the algorithm can deal with discrete jumps, which
naturally participate in the dynamics of legged robots. With
this algorithm, the stability of hybrid periodic orbits also
can be exactly identified (we mean “exactly” in numericalsense).
The algorithm derived in Section II is applied to a planar
one-legged robot model in Section III. We will show thepassive running gaits and their stability in terms of returnmap.
Based on the stability analysis, Section IV shows controllers that achieve orbital stability. First, a local linear
feedback controller is derived. Then, we will introduce
our previously derived Delayed-Feedback Controller, combined with Energy-Preserving Controller [12]. The controllers are derived along with our general approach 2)

and 3) listed above. They actually asymptotically stabilize
unknown (multi) periodic gaits, and additional adaptiveenergy controllers achieves stable passive running. Then, aclose relationship between the detection algorithm and the
Delayed-Feedback Controller is discussed.
II. D ETECTION OF HYBRID PERIODIC ORBITS
To search passive running gaits, we construct return map
with the control input being zero, because the fixed pointsof the map mean the existence of passive running gaits.
Newton-Raphson method is widely used for finding thesolution of algebraic equation. However, running robot isa quite complex hybrid system composed of stance phase,flight phase and other discrete events. Therefore, NewtonRaphson method should be modified appropriately.
A. Autonomous hybrid dynamical system
The locomotion dynamics of n-DOF legged model canbe represented by an autonomous (s-modal) hybrid dynamical system composed of differential equations andalgebraic equations:• Continuous transition:ẋi = fi (xi ) (i = 1, 2, · · · , s)
•

•

(1)

Discrete jump:
xi+ = hi (xi− ) (i = 1, 2, · · · , s)(2)

Πi := {xi |Si (xi ) = 0} (i = 1, 2, · · · , s)(3)

Events:Here, xi ∈ R2n is the state vector, i and s are theindex and the number of the phases (modes) respectively.
fi ∈ R2n is the continuous vector fields, and h i ∈ R2n is
the jump equation, by which the states reset to the initial
position of each phase. The variables xi+ and xi− mean
the states just after and before events respectively.
The equation of motion of continuous transition canbe represented by autonomous dynamical systems because
there are no control inputs, nor specific “clock time”. Notethat DOF of each phase could be different; it could be n or
less. Therefore, some of the elements in the state vector xi
do not participate in the dynamics in some phases. In thiscase, they become dependent variables, and the update ofthese variables should be calculated by kinematic constraint
equations subject to the corresponding phases.
On the other hand, the equations of motion at discrete
events can be represented by algebraic equations. They
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 endof one stride). This consideration complicates the analysis,
but they make the model more realistic.
The solution of this hybrid dynamical system evolves
as the following manner (see Fig. 1) : x i (t) starts fromthe initial condition xi (0) = xi+ and evolves along to
Eq. (1). When x i (t) reaches the hyper-surface Π (i+1) , itjumps according to Eq. (2), then it evolves in the next
phase.
B. The algorithm
The purpose of this section is to provide the searchingalgorithm of the hybrid periodic orbit, which is indicatedby γ in Fig. 1. Let’s begin with the single modal system,that is, s = 1 in Eq. (1) – (3) (We omit the mode-indicating
subscript 1 for simplicity).
By choosing Π as the local cross sections 1 , the return
map can be represented by:x(k + 1) = P (x(k)),

where k means the step of running. With this setup, what
we have to do is; finding the solutions of:G(x) := x − P (x) = 0.

P (x + ∆x) = P (x) + DP (x)∆x + O(x),where DP (x) = ∂xUsing (5), we have:

Πs

x−s-

(5)

To this end, we apply the conventional Newton-Raphson
method. The variation of P corresponding to a bit changeof x can be obtained from Taylor expansion:∂P (x)

x−3(4)

and O(x) is the higher-order term.

DG(x)∆x = −G(x)⇒ ∆x = (I − DP (x))−1 (P (x) − x),

x−s+

(6)

(7)

0

Π3

x−3-

x3+
−

x−2

γ

x−1-

Hybridperiodic orbit

Π2

x2+
−

Therefore, for a given initial guess x , the searchingalgorithm can be written as:

x−s

x−2-

Π1

xj+1 = xj + (I − DP (xj ))−1 (P (xj ) − xj ),x−1+where the superscript j is the iteration number of the
Newton algorithm.
In this algorithm, the update of the solution should beexecuted on Π, therefore DP must be calculated by:


1
∂S
f (x)
(9)
ΦT (x),DP (x) = I − ∂S∂x∂x f (x)x−1Fig. 1. Autonomous hybrid system and its hybrid periodic solution γ. xi
are the states of each phase (modes), and Πi are Poincaré’ cross sections.

(8)

1 The cross section Π must be chosen to meet the transversality
condition: DS
f (x) = 0. See [13].
Dxwhere ΦT (x) is the principal matrix solution [14], the solution X(t) of the following variational equation evaluatedat time t = T (T is the period; the solution starts from thesection at time 0 and returns after time T ):dX
dt

= Df (x)Xwith X(0) =(10)∂φt (x0 )
= I,
∂x0

(11)

where φt is the flow of the vector field f (x). Note that inautonomous hybrid system the period T depends on initialconditions and automatically updated as the algorithm
proceeds. Usually, the principal matrix solution Φ t (x) are
obtained numerically in parallel with x(t). For example,
constructing “extended state vector” composed of originalstates and the elements of identity matrix and evolve it atonce in the ODE solver.
In the case of continuous system, the above equations
are sufficient to find periodic orbits. If there is a jump in thesolution x, however, DP should be processed appropriately
as follows. Suppose the discrete jump is represented by analgebraic equation:x+ = h(x− ).

(12)Then, from the chain rule of differentiation, Φ changes
according to:
Φ+ =

∂h
Φ− ,
∂x−

(13)

where Φ− and Φ+ represents the value of Φ evaluated justbefore and after jump respectively.
Moreover, since we have defied the return map on thelocal cross section in full-dimensional space, thus obtained
principal matrix solution always includes characteristic
multiplier (eigen value of DP ) 1, whose associated eigen
vector coincide with the vector field f [14]. Since this“trivial” multiplier lead to a difficulty in further bifurcation
analysis, Φ must be reduced by:Φred =

∂η
∂η −1
·Φ·
,
∂x
∂y

(14)

where y is the local coordinates, the explicit expressionsof S(x) = 0, and η : x → y is the projection of 2ndimensional manifold to (2n − 1)-dimensional manifold.
Its inverse η −1 is embedding to 2n-dimensional manifold[15]. The examples of local coordinates are given in thenext section.
Finally, combining (9), (13) and (14) results in:


∂η ∂h
1
∂S
DP (x) =f (x)·· I − ∂S∂x ∂x−∂x∂x f (x)·ΦT (x) ·

∂η −1
,
∂y

(15)

which is used in the update law (8). This formula is much
simpler than Hisken’s one (compare Eq. (15) with Eq.
(57)–(60) in [11]), which focuses on the general representation of hybrid dynamical system. We need only thealgorithm that detects the periodic orbits of “autonomous”
hybrid dynamical system, where the jump is always autonomous. For the details of (9) and (14), see [9].
Now, it is very easy to consider the multiple modal case,that is, when s ≥ 2 in Eq. (1) – (3). In this case, we only
have to take the following procedure.
(1) Take the jump solution xi+ as the initial condition of
the current phase xi (0).
(2) Restore ΦiTi into the memory and take identity matrix
I as the initial conditions for the variational equationof each phase.
(3) At the Poincaré section Π 1 , combine Φi s by:
ΦT (x) = Φ1 T1 (x1 ) · Φ2 T2 (x2 ) · · · Φs Ts (xs ),(16)(4) Apply Eq. (15) to obtain DP (x).
Note that Eq. (16) comes from the fact:
P = P1 ◦ P2 ◦ · · · ◦ Ps ,(17)

where Pi is the return maps that brings x i (0) to x(i+1)− ,and from the chain rule. Note that the period of the orbit
is T = T1 + T2 + · · · + Ts .
C. Stability analysis
Once periodic solutions are obtained using the algorithm,
we can easily check their stability by the following theorem(see e.g. [14]).
Theorem 2.1: If all the eigenvalues of DP (x) havemodulus less than 1, then the closed orbit γ is asymptotically orbitally stable. If one of the eigenvalues of DP (x)have modulus greater than 1, then the closed orbit γ isasymptotically orbitally unstable.
III. A PPLICATION TO A ONE - LEGGED ROBOT
A. Model description
Figure 2 shows the model of the planar one-legged
running robot considered here. The robot is attached not
only with a leg spring but also a hip spring.
We impose the following assumptions on this model.
(A) The center of mass (C.M.) of the body leis on thelongitudinal axis of the leg (e.g. on hip joint in Fig. 2)(B) Mass of the foot (unsprung mass) is negligible(C) The foot does not bounce back, nor slip the ground(inelastic impulse assumption)(D) The springs are loss-less
The equations of motion are composed of two phases;
stance and flight phase, triggered by two discrete events;
lift-off and touchdown. We use event-indicating subscripts
for variables. For example, ẋ lo is the forward velocity of
C.M. at lift-off, θ̇td+ is the angular velocity of the leg
just before touchdown, θ td is the leg angle of just before,or, just after touchdown, and so forth. Table 1 shows thephysical parameters, together with the values used in later
simulations.
At stance phase, the dynamics is described as: M r̈ + Kl (r − r0 ) − M rθ̇2 = M g(1 − cos θ) + f ld(18)
J θ̈ + Jb φ̈ + dt
(M r2 θ̇) = rM g sin θ lJb φ̈ + Kh (θ − φ) = τ,•Hip spring Kh
BODYτ

xi+ = hi (xi− ) (i = 1, 2)

φ

(x,z)
Jb M

•

LEG

Ju

θ

r

Z

fl

Cross sections:x|S1 (x) = z − r0 sin θ = 0
(25)
Π1 :=


2
2
Π2 :=
x|S2 (x) = x + z − r0 = 0 (26)y = η(x) = [ xFOOT(no-mass)

X

Passive one-legged hopper

P HYSICAL PARAMETERS OF ONE - LEGGED MODEL
Unit
m/s2
kg
m
kgm2
kgm2
N/m
Nm/rad

Value
9.8
12
0.5
0.5
0.11
3000
10

where fl is the control force of the leg, and τ is the control
torque of the hip joint.
At flight phase, the dynamics is given by:ẍ = 0 z̈ = −g(19)
Jl θ̈ + Jb φ̈ = 0
Jb φ̈ + Kh (θ − φ) = τ,At touchdown, the velocities of the generalized coordinates change instantaneously at touchdown phase by
Assumption (C):cos θtdẋtd+ = ẋtd− − JJll +Mµ
r02 td−


J
sin
θ

l
td

 żtd+ = żtd− − Jl +M r02 µtd−
r0
(20)
µ
θ̇td+ = θ̇td− − JlM+M r02 td−





 φ̇td+ = φ̇td−
ṙtd+ = żtd+ cos θtd − ẋtd+ sin θtd ,whereµtd− := ẋtd− cos θtd + żtd− sin θtd + r0 θ̇td− .

ż

φ̇ ]T

θ̇

(27)

r0 cos θ

θ

φ

ẋ

θ̇

ż

T
φ̇ ] . (28)Using the above formulation, the algorithm can bedirectly applied to the passive running model. The algorithm converges rapidly (approximately 10 iterations are
sufficient until the error < 10 −8 ). The passive running gaits
are found for every admissible initial guess and they are
actually loss-less, i.e. no energy is dissipated.
Figure 3 shows the characteristic multipliers of passive
running gaits with its forward speed varied from 0 to
3 m/s (Two trivial unity multipliers are not shown). For
example, a fixed point y ∗ = [−0.1589, 0.3234, −0.0723,
2.0000, −2.0490, −2.4904, 0.5428] corresponds to a midspeed passive running (2 m/s), whose stick animation isdepicted in Fig. 4. The characteristic multiplier is calculated to eig(DP ) = [2.0098, −0.7682, −0.1230 + 0.1795i,−0.1230 − 0.1795i, 0.7382, 0.0000]. On the other hand,

2
1.5
1
0.5

3

Unit circle

5
1

2

0

−0.5

(21)

At lift-off, there are no discontinuous changes of states
except for:ṙlo = 0.

ẋ

C. Passive running gaits

Im [Eig (DP)]Meaning
gravity acceleration
total massnatural leg length
body inertia
equivalent leg inertia
leg spring stiffness
hip spring stiffnessφWith this transformation, the dimension of Φ red is (7 ×7) and search space becomes to 7. Furthermore, since wewant to parameterize passive running gaits according to
its forward speed ẋ, we can fix this parameter during theiteration of the Newton method. Then, the dimension of
the search space finally reduces to 6.

TABLE I

g
M
r0
Jb
Jl
Kl
Kh

θ

and its inverse map η −1 can be written as:
x = η −1 (y) = [ xFig. 2.

(24)

Note that Eq. (18) can be expressed by (23) using therelationship x = r sin θ and z = r cos θ. Also, the local
coordinate y in Eq. (14) is defined as:

Leg spring

Kl

Discrete jumps (Eq. (20) and (22)):

(22)

−1

6

−1.5

4

−2
−1

0

1

2

3

4

Re [Eig (DP)]B. Hybrid model representation
With the state variable x = [x, z, θ, φ, ẋ, ż, θ̇, φ̇]T ∈R , the above equations are summarized as:• Continuous transitions (Eq. (18) and (19)):8ẋi = fi (x) (i = 1, 2)(23)

Fig. 3. Locus of the characteristic multipliers of passive running gaits,
where the forward speed is varied from 0 to 3 m/s. The index numbers
indicate the elements of the multipliers in order. Only the multipliers ofvertical hopping (0 m/s) lie within the unit circle. Note that the second
multiplier cross the unit circle, which indicates the bifurcation of periodic
orbit.

Vertical height [m]

1
0.5
0
7

8

9
10
Horizontal distance [m]1112Vertical height [m]Fig. 4. Subsequent four steps of mid-speed passive running gait, where
the running speed is 2 m/s
1
0.5
0
5

6

7

8
Horizontal distance [m]

9

10

11

Fig. 5. Subsequent two steps of high-speed passive running gait, where
the running speed is 5 m/s

a fixed point y ∗ = [−0.3129, 0.6763, −0.1198, 5.0000,−2.0760, −5.2004, 1.2818] corresponds to a high-speed
passive running (5 m/s) (Fig. 5). The characteristic multiplier is calculated to eig(DP ) = [ 2.0755, −5.7708,−0.1933, 0.0220, 0.5940, 0.0000].
The stability of the gaits can be examined by Theorem
2.1. From Fig. 3, we can see bifurcations occur twice. At
the speed 0 m/s, all the multipliers lie within and on the unit
circle and the periodic orbit is neutrally stable. If the speed
exceeds 0 m/s, a relatively large unstable multiplier appears
in the right half plane. Then, the left multiplier crosses -1
when the speed exceeds 2.34 m/s. However, no remarkabledifferences between the gaits are observed (e.g. the period
does not change). In summary, we can conclude the passive
gaits are orbitally unstable (roughly speaking, although therobot starting from the fixed points can continue for several
steps, it falls finally), except for trivial vertical hopping.

where ξ := x − x∗ is the error from the fixed points andDPξ is the same as the DPx in the previous section. And∂P ∂P
DPu = ∂P∂u = ( ∂τ1 , ∂τ2 ) is the newly appeared derivative
due to the control inputs (29).
If at most two unstable multipliers can be made to bezero, all of the passive running gaits can be stabilized. Thisis actually possible because the closed-loop system is foundto be locally stabilizable with the control input (29).
We do not show the simulation results here, because
although the controller stabilizes all of the passive gaits, theregion of attraction is found to be quite narrow. Actually,
it is found by simulation that the local controller does notallow even 0.05 m/s error in initial velocity ẋ 0 . If we use
fl as the inputs, the region of attraction could be enlarged[7], but the controller is local and strongly depends on x ∗ .
B. Energy-Preserving Controller and Delayed-Feedback
Controller
An alternative controller does not require priori knowledge about passive running gaits [12]. The central control
strategy is energy-preservation, which aims to preserve thesystem energy as much as possible. We refer the reader to[16] for the details and give a brief outline here.
The essence of the controller is to design the touchdown
angle θtd and its velocity θ̇td− so as to meet the nondissipative condition:µtd− := ẋtd− cos θtd + żtd− sin θtd + r0 θ̇td− = 0.

This is a necessary condition for passive running because
no energy is lost if this condition holds. For given translational lift-off velocities (ẋ lo , żlo ), there is a region of thepair (θtd , θ̇td ) that meets the condition. Delayed-Feedback
Controller (DFC), widely used in chaos control [17], cangive a good selection. If we consider the touchdown angle
as a control inputs to the discrete dynamical system (returnmap), then we can use the adaptation law:

IV. S TABILIZATION
Having obtained passive running orbits and found they
are unstable, we must derive appropriate stabilizing controller. This is the most valuable stage of our approach to
the energy-efficient legged locomotion (see Section I).
As the control inputs, suppose using the hip torqe τ at
the flight phase only (for the reason see [16] ). We consider
the following piecewise constant input:
τ

=

τ1 ,
τ2 ,

if 0 ≤ t < Tv /2,
if Tv /2 ≤ t < Tv

(29)

where τ1 and τ2 are the constant, and t indicates the timeafter the lift-off, and T v := 2ẋlo /g represents “expected”
flight time.
A. Local stabilization by linear feedback controllerξ(k + 1) = DPξ ξ(k) + DPu

τ1
τ2

,

θ(k) =− 21 {θlo (k) + θlo (k − p)} ,−θlo (k),if k > p
else,

(30)

(32)

where θ(k) is the desired touchdown angle, and p > 1 isa desired period. That is, we try to control the period of
the gaits by adapting touchdown angle. General form of
discrete version of DFC and some discussion including its
limitation, can be found in [18]. Having determined θ andthe associated desired velocity θ̇, it is trivial to dead-beatsecond order linear system (19) by the control inputs (29).
Using this controller, “unknown” periodic gaits with “desired” period are orbitally stabilized. However, the control
inputs does not disappear, hence the orbits do not converge
to passive orbits. To solve this problem, in addition to the
DFC, the following adaptive energy controller is applied:fl (k) = γf · sign [∆τ (k − 1)] · ∆τ (k − 1),

The first controller is a conventional local feedback controller. Linearizing the system around the unforced periodic
orbits, we obtain the following closed-loop system:

(31)

(33)

where fl (k) is the stepwise leg force, ∆τ (k) := τ2 (k) −τ1 (k) means a “magnitude” of the control torque, and
γf > 0 is an adaptation gain. Figure 6 shows one of
the simulation results. From the figure, we can see thecontrol inputs eventually converge to zero; perfect passive

115

4
2

E [J]

τ [Nm]

110

0
τ1
τ

−2

105

2

−4
100
0

−6
0

20
40
time [s]

100

−1.8
zD(k)

2.2
xD(k)

50
steps

2.1

−1.9
−2
−2.1

2

−2.2
1.9
−0.18

−0.16
x(k)

−0.14

0.47

0.475
z(k)

0.48

−2.5
φD(k)

θD(k)

0

−2.6
−2.7
−2.8
−2.9

−0.5

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[19] S. Hyon, X. Jiang, T. Emura and T. Ueta, ”Passive running ofplanar 1/2/4-legged robots”, Proc. IEEE/RSJ Intelligent Robotics andSystems, 2004 (to appear).
[20] S. Hyon and T. Emura, ”Running control of a planar biped robot
based on energy-preserving strategy”, Proc. of IEEE Int. Conf. on
Robotics and Automation, pp.3791–3796, 2004.

−1.5
0.3

0.35
θ(k)

0.4

ACKNOWLEDGEMENT
This research was partially supported by the Ministry of
Education, Science, Sports and Culture, Grant-in-Aid forYoung Scientists, 15760144, 2003.

0.5

−2.4

Our ongoing task includes exploring passive runninggaits and stabilization of a biped and quadruped model[19], where two kinds of passive running gaits of planar
quadruped were found. For biped robot with a torso, we
have not found the passive running gaits so far. But,
using the property of DFC as a “real-time periodic orbits
detector”, we have successfully obtained “unknown” stable
periodic biped running gaits (not passive one) [20]. Since
the controller is written in a state feedback form, if we
express the closed-loop system as the autonomous hybrid
dynamical system, then the algorithm is applicable andwe will obtain “constrained periodic orbits”, which will
be presented near the future.

−0.2

0
0.2
φ(k)

0.4

Fig. 6. Simulation results of the orbital stabilization to (unknown) 1perioric gait and energy adaptation by leg force control: γf is set to 40 andthe adaptation is activated from 60th step. E represents the total energyof the robot. Notation “D” represents the time derivatives of preceding
variables.

running is achieved! From the adaptive update mechanismof Eq. (32), DFC can be considered as a kind of “real-time
periodic orbits detector”.
V. C ONCLUSION
In this paper a new algorithm to detect hybrid periodic
orbits of autonomous hybrid dynamical system was developed. The algorithm is very powerful tool and can beapplied to any kinds of the robots (e.g. walking robot).
Conventional Newton algorithm was modified so that it
can be applied to the analysis of return map of hybrid
dynamical systems that include multiple phases (modes)
and discrete jumps. With the algorithm, passive runninggaits of one-legged robot were automatically detected forvarious parameter sets and initial conditions. In particular,
the search space was reduced from 2n to 2n−1 or less. This
led to a great help for the analysis of the full-dimensional
hybrid system behavior. The analysis of the characteristic
multiplier of the return map revealed the stability and thebifurcation of the passive running gaits: all passive runninggaits of one-legged robot were orbitally unstable except fora trivial vertical hopping gait. Two kinds of controllers that
achieve orbital stabilization were presented; one was a local
feedback controller based on the linearized return map,and the other was DFC-like like controller. Since the latter
controller brought the system trajectories to the complete
passive orbits, a similarity was found between the detection
algorithm and the controller.