Invariant Manifold of Symmetric Orbits for
Globally Optimal Biped Locomotion
Sang-Ho Hyon ∗,∗∗ Kenji Fujimoto ∗∗∗∗JST, ICORP, Computational Brain Project, Kyoto, Japan
ATR, Computational Neuroscience Laboratories, Kyoto, Japan(e-mail: sangho@atr.jp)∗∗∗
Graduate School of Engineering, Nagoya University, Nagoya, Japan(e-mail: fujimoto@nagoya-u.jp)∗∗

Abstract: This paper proposes novel walking gait generation for a double-pendulum-like bipedwalking model based on the family of the symmetric orbits. By introducing an involution R
associated with the leg switching map, the ﬂow of the saddle-center dynamics of the uncontrolled
pendulum possesses time-reversal symmetry with respect to R, and the conjunction of the ﬂow
and R forms a family of symmetric orbits parameterized by the orbital energy and the stride. The
invariant manifold of the family of symmetric orbits is obtained numerically or approximately
using a perturbation method. The proposed methods are evaluated on numerical simulations.
By constraining the solution onto the invariant manifold using the control inputs, stable walking
gaits are generated in a semi-global manner in simulations. Based on the passivity of the closedloop system, a robust speed-controlled walking is achieved in a very simple way.
Keywords: Biped locomotion, Hamiltonian systems, Time-reversal symmetry, Symmetricorbits, Invariant manifold, Passivity, Numerical simulation
1. INTRODUCTION
The goal of this paper is to generate robust and energyeﬃcient locomotion gaits that achieve simple targets suchas a speciﬁed walking speed or desired locations. For
that purpose, we propose a novel motion control strategy
based on the useful properties of Hamiltonian systems.
As outlined in Hyon (2005), the strategy comprises threesteps:

Speciﬁcally, we utilize the symmetric orbit inherent in theuncontrolled system. The symmetric orbits (see Sevryuk(1986) or Lamb (1998)) are special orbits in (time-) reversal systems such as Hamiltonian systems. They exist as afamily parameterized by the initial conditions such as theEnergy level (imagine ﬂows around the center of a simple
pendulum dynamics), and form an invariant manifold in
the phase space.

S1. First disregard dissipativity of the system to extract
purely Hamiltonian (conservative) dynamics. Then,
apply an invariance controller that renders the closedloop system lossless and restricts the hybrid solutions
into invariant sets.
S2. Retrieve (and inject as necessary) dissipativity andapply tracking controllers to stabilize asymptoticallytarget states or periodic orbits passing through them.
S3. Apply adaptive or learning schemes to minimize control eﬀorts.

Symmetric orbits are not necessarily periodic. However, if
we allow switching between phase variables, they, together
with the switching map, can form some meaningful hybrid
periodic orbits on an invariant manifold. Consequently, byconstraining the solution curves to the invariant manifold
using control inputs, (semi-) global orbit generation would
be possible within the admissible region. In this case,external forces just change the internal energy of thesystem, which results in robust gait generation. This is
the motivating idea of this study.

Roughly speaking, S1 is introduced to make biped robot
behave as a ball rolling on the ground, by way of feedback
control. The resultant controller has a hierarchical structure in which the existence, stability, and boundednessof the solution are ensured by the lowest layer with ahigh priority; more advanced control goals are achieved
at higher layers.

The idea of constraining the solution to some parameterized family of orbits is not itself new in locomotion literatures. For example, Wieber (2006) proposed to use wholewalking trajectories computed oﬀ-line, and to select oneof them online. Although not mentioned about symmetricorbits, Morita (2004) proposed a ballistic walking based on
the symmetry. Hyon (2005) also studied symmetric orbits
parameterized by the step length and angular momentum,
by using numerical method based on Poincaré map. In
Hyon (2005), however, the authors had no idea how to represent and store the obtained inﬁnitely many symmetricorbits. Therefore, the authors picked up a simplest target
manifold (q1 + q2 = 0) and constrained the solution to
the manifold by control inputs. Although the controller

In this paper, we treat a double-pendulum biped modeland propose the semi-global 1 locomotion generation.
1 The “semi-global” means we consider stability at once. Since
configuration space is limited in biped locomotion (foot cannot
penetrate the ground), the stability is naturally limited to semiglobal.

was almost same as the past-proposed one Grizzle (2001),
we believe Hyon (2005) is the ﬁrst paper that addressed
its (semi-) global stability of the symmetric orbits on the
manifold, from a global dynamics point of view. The idea
was also validated on a real biped walking robot. Onceobtained the global stability, it was easy to obtain global
asymptotically stable periodic gaits, by using a very simple
feedback controller 2 .
This paper extends our previous work toward globally
optimal walking gait generation by the following two
novel approaches. First, we compute approximate, but
analytical invariant manifold in which symmetric orbits
are ﬁlled in. We do this because, once the invariant
manifold has been obtained analytically, it is easy toapply model-based controllers to constrain the solution
onto the manifold. Actually, in this paper, we employ
two kinds of continuous feedback controllers. One is a
simple trajectory tracking controller, and the other isa velocity-ﬁeld tracking controller in Duindam (2004),
resulting in the second contribution of this paper. The
eﬀectiveness of the proposed methods are evaluated on
numerical simulations.
This paper is organized as follows. In Section 2 we introduce the reversal symmetry and symmetric orbits ofHamiltonian systems and explain the close link to the periodic orbits which we are interested in. In Section 3, we clarify the relationship between biped walking and symmetricorbits, by introducing a double-pendulum (compass-like)biped model. We obtain the symmetric orbits numerically,and explain their properties. In Section 4 we approximately compute the invariant manifold of the symmetricorbits. In Section 5, we propose a sub-optimal gait generation based on the symmetric orbits and its invariant
manifold. In particular, we show the application of themethod by Duindam (2004), which maintains the energy
constant, while tracking to some given tangent vector
ﬁeld. In Section 6, we examine how the proposed method
is applicable to realistic walking model with dissipation
through simulations, then demonstrate that a very simple
passivity-based speed controller can make the robot walkstably at arbitrary speed. Section 7 summarizes the paper.
2. REVERSAL SYMMETRY AND SYMMETRICORBITS IN HAMILTONIAN DYNAMICAL SYSTEMS
Hamiltonian systems have a special symmetry, called(time-) reversal symmetry Devaney (1976)Lamb (1998).
Let F : M → M be a diﬀeomorphism on a manifold M,
and a map (called involution) R : M → M such thatR ◦ R =id (identity). If
R ◦ F = F −1 ◦ R(1)is satisﬁed, we say F has reversal symmetry with respect
to R Lamb (1998) (or simply say F is R-symmetric).
Let us consider a Hamiltonian system with the Hamiltonian

Its ﬂow is represented by φt : (q(t0 ), p(t0 )) → (q(t0 + t), p(t0 + t)).
Then, φt is R-symmetric since H0 is invariant with respect
to a map R : (q, p) → (q, −p). That is, we haveR ◦ φt = φ−t ◦ R.

(3)

The ﬁxed point sets, Fix(R) and Fix(R ◦φt ), are especially
important because a family of orbits, called symmetricorbits, pass through these sets and are mapped onto
themselves by R with their direction of ﬂow reversed.
Therefore, if we have a R-symmetric ﬂow φt , and if its
ﬁxed point sets Fix(R) and Fix(R ◦ φt ) are identiﬁed, we
obtain the associated symmetric orbits. The symmetricorbits exist as a family, which is parameterized by initialcondition such as initial energy.
The symmetric orbits are not necessarily periodic orbits.
However, if they are closed orbits, they pass Fix(R) exactly twice Devaney (1976). Such symmetric orbit we callsymmetric periodic orbit. The Reversible Lyapnov Center
Theorem is related to the existence of the symmetric periodic orbits. See Devaney (1976) or Sevryuk (1986).
However, note that a compass-like walking system shownin Section 3.1 has no double center equilibrium because thecenter equilibrium is out of the admissible conﬁguration
space. Instead, it has a saddle-center (a pair of saddle
and center) as the meaningful equilibrium. Therefore, adouble-pendulum has no meaningful symmetric periodic
orbits in nature. However, if we allow us to have a hybrid
dynamical system framework, where the continuous statesand discontinuous states coexist, hybrid symmetric orbits
can be newly deﬁned, and they play a central role in global
generation of periodic orbits. As we will see in Section 3.2,
this scenario is actually possible if R represents a leginterchanging map.
3. SYMMETRIC ORBITS OF BIPED WALKING
MODEL
3.1 Model description
Let us consider a walking model shown in Fig. 1. This isoften called “compass model”, and is a double pendulum
as long as its foot sticks to the ground all the way,
which we assume in this paper. We take the angles ofthe swinging leg and supporting leg as the generalized
coordinates, q = (q1 , q2 ),on the admissible conﬁguration
space N := {q ∈ S 2  |q1 | ≤ π/2}. The conjugate
momentum is p = M (q)q̇ ∈ M = T ∗ N , where
M (q) =



(m1 + 2m2 )L2 + m2 (L − b)2 −m2 bL cos(q1 − q2 )−m2 bL cos(q1 − q2 )
m2 b2



is the inertia matrix. The potential energy is given by
U0 (q) = (m1 + 2m2 )gL cos q1 − m2 gb(cos q1 + cos q2 )with g being the gravity acceleration. Then, the dynamics
of the double pendulum is 2-DOF Hamiltonian systemT1
H0 (q, p) = pT M (q)−1 p + U0 (q).
22Experiments and simulation
http://www.cns.atr.jp/ sangho/

moviesareavailable

(2)
from

∂H0 (q, p)
∂p


T
T
∂H0 (q, p)∂H0 (q, p)ṗ = −+B u−D
∂q
∂p

q̇ =

(4)

interchanging map 3 , (q1 , q2 , p1 , p2 ) → (q2 , q1 , p1 , p2 ), in
which the impulsive eﬀect is completely ignored. Suchorbits are R-symmetric orbits which pass the origin of the
conﬁguration space (included in the Fix(R)). Therefore,
it is easy to see we can obtain (ideal) symmetric walking
gaits by conjunction of R and R-symmetric ﬂow φt . For
this construction making sense, we need to consider ﬂows

m1
L

b

u2
m2

q2

m2
−q1

u1


F := φt t=T ,Fig. 1. A simple biped robot with the mass-less foot: q1
and q2 are the absolute angle of the supporting leg
and swinging leg respectively.

F = φT

Map:

State:

Initialq1(0) = − qr
q2(0) = qr

1 stepR

InitialR
F q (T) = q1
rq2(T) = − qr

not arbitrary ones. Wherein, T > 0 is the time which one
cycle of walking ﬁnishes (at the end q1 + q2 = 0 is met).
Therefore, the problem of ﬁnding symmetric walking gaits
is to ﬁnd the ﬁxed point set Fix(R ◦ F ).
The top graph of Fig. 3(a) shows the numerically foundsymmetric orbits (Hyon (2004)) associated with the ﬁxed
points of Poincaré maps (Guckenheimer (1983)) with discrete jump (Kousaka (1999)). The robot parameters are{m1 = 2, m2 = 1.2, b = 0.2, L = 0.4}. The stride is
ﬁxed to qr = 0.5 here, but similar results can be obtainedfor other values. Fig. 3(b) shows the bifurcation diagramof the eigenvalues of the linearized Poincaré map DP at
Fix(R◦F ), where the bifurcation parameter is the momentum p1 . As intuitively expected, the (hybrid) symmetricperiodic orbits for uncontrolled system without dissipation
are found to be neutrally stable or unstable (see Hyon(2005)). 4

−q1

q2

(6)

q1(0) = − qr
q2(0) = qr

Some notes are in order:
Fig. 2. Gait transition of symmetric walking. The map F isdeﬁned in (6). A family of steady symmetric walking
gaits is the ﬁxed point set of R ◦ F .
with the Hamiltonian (2), where D = is a friction coeﬃcient matrix (diagonal), u = [u1 , u2 ]T ∈ R2 is the control
input vector, and B = 10 −1
is a driving matrix which
1converts u to the generalized input associated with q. The
equilibrium (origin) is a saddle-center, and the eigenvalues
are {λ, −λ, iω, −iω} with real numbers λ, ω.
3.2 Numerical search of symmetric orbits
The model in the previous section represents the dynamics
of the single support phase. When the swinging leg landson the ground, the energy is lost and sudden change of thevelocity occurs. Energy will be lost due to the joint friction
too. Along with the strategy (S1) in Section 1, however,
we ignore the dissipation tentatively (we get back to thispoint in Section 6), and try to derive a gait generating
controller by focusing on the pure Hamiltonian systems.
For this purpose, let us consider a R-symmetric ﬂow φt of
the free system; (4) with D = 0, u = 0. Herein, we areinterested in the mirror map about p-axis:
R : (q1 , q2 , p1 , p2 ) → (−q1 , −q2 , p1 , p2 ),• The orbits approach a straight line q1 + q2 = 0 as theenergy increases and q̇2 becomes negatively large. In
this case, the walking speed is high, but, in reality,
is limited by unilateral constraint and friction cone,
which we ignore in this paper for simplicity.
• If the energy is high and q̇2 is positively large, theorbit is curved to the q2 -direction. A particular orbit
passes the singular point q1 = 0, q2 = π. Interestingly,
this “giant swing” orbit is free from the impulse at
touchdown (lies in the null space of the impulse map).
Of course, this is not a walking gait.
• In regions |q1 | ≥ π/2 and |q2 | < |q1 |, walking is
indeﬁnable.
Although the region is limited, the symmetric walking
gaits ﬁll a certain area of the conﬁguration space. In
the 4-dimensional phase space, they form a 3-dimensional
invariant manifold.
The global gait generation developed in this paper isachived by constraining all the solutions to this invariant
manifold using control inputs. Once the solutions are constrained on the manifold, the control inputs also converge
to zero (in ideal lossless system).
4. APPROXIMATION OF INVARIANT MANIFOLDOF SYMMETRIC ORBITS(5)

rather than the conventional involution R : (q, p) →(q, −p). As can be seen in Fig. 2, for some symmetricwalking gaits which start q1 (0) = −q2 (0) = −qr at
t = 0 (qr is some positive number representing thestride) and terminate at q1 (T ) = −q2 (T ) = qr at time
t = T , the involution (5) corresponds to the ideal leg-

Let us approximate the invariant manifold. We assume thefamily of symmetric orbits can be approximated by
3 We need this map when we define the coordinate as the angles forsupporting leg and swinging leg, not those for right leg and left leg
4 Asymptotical stability is impossible in Hamiltonian system innature. See Guckenheimer (1983).


1

q2

0

−0.5

0
q1

0.5

0.6

η2

0.2
0
−0.2
−0.4
−1

0
η1

1


=

γ −1
1 1




q1
.
q2

(8)

Now, we compute the velocity ﬁeld v(η) = [v1 (η1 ), v2 (η1 )]Ttangent to the invariant manifold. From reversal symmetry
this meets v(−η1 ) = v(η1 ). We take a similar procedure
for computing center manifold (Guckenheimer (1983))
up to terms of necessary order. The invariant manifold
is represented by a power series parameterized by thevelocity v10 = v1 |η1 =0 (alternatively, the energy anddirection) at which the curves pass q1 = −q2 = q0 and
q1 = −q2 = qr .

0.4

−2



By setting a limit A > 0, the symmetric orbits meet |q2 | <|q1 | in region |q1 | < qr . The orbits Fig. 3(a) is transformed
to Fig. 3(b) in the new coordinate (η1 , η2 ). It should be
noted that the controlled symmetric orbit in Hyon (2005)
is a single orbit η2 = q1 + q2 = 0, which corresponds to“scuﬃng gait”. Contrary, (7) meets η2 = q1 + q2 = 0
only at the origin or the leg interchanging conﬁguration
q1 = −q2 = ±qr , as long as A = 0. The uncontrolled
system does not have any solutions with A = 0 (i.e.
η2 = q1 +q2 = 0 is met uniformly). The amplitude A codes
the energy level, and is uniquely determined if q̇1 (0) orq̇2 (0) are given together with the stride and initial energy.

0.5

−1
−0.5

η1
η2

2

We ﬁrst expand (7) to yield(a) A family of symmetric orbits in the original conﬁguration space (top) and the transformed conﬁguration space(bottom)

Im [eig(DP)]2(9)

Then, its tangent vector is represented by
v1 = c0 + c2 η12 + c4 η14 + . . .

(10)v2 = d0 + d2 η12 + d4 η14 + . . .

(11)where c0 = v10 . By diﬀerentiation, we have1

Invariant poles
3

0
2

1
Unit circle

−2
−2

−1

0
1
Re [eig(DP)]2v̇1 = (2c2 η1 + 4c4 eta31 + . . . )(c0 + c2 η12 + . . . )(12)v̇2 = (2d2 η1 + 4d4 eta31 + . . . )(d0 + d2 η12 + . . . )(13)Then, these equations are substituted into the dynamics, then solved for the coeﬃcients {bi , ci , di } using theterminal conditions η2 (ηr ) = η2 (−ηr ) = 0. From thisprocedure we approximately obtain the vector ﬁeld v(η) =[v1 (η1 ), v2 (η1 )]T . We used Maple for the symbolic computation.

−1−3

η2 = b1 η1 + b3 η13 + . . .

3(b) Bifurcation of characteristic multipliers of Fix(R ◦ F )for (5) and (6)5. CONSTRAINT CONTROL TO THE INVARIANTMANIFOLD

Fig. 3. Numerical computation of the family of symmetricorbits for the simple biped model with the parameters:{m1 = 2, m2 = 1.2, b = 0.2, L = 0.4} and { q1∗ = −0.5,−10 ≤ p∗1 ≤ 0}

We generate semi-globally stable walking gaits by constraining the integral curves to the approximated invariant
manifold (4). If the approximation is precise enough, it isexpected that the resultant closed-loop solutions maintainthe energy level, and that the control inputs become close
to zero. The resultant walking gaits would be optimal withrespect to the criteria

η2 = A sin(ωη1 )(7)

via a transformation Y : (q1 , q2 ) → (η1 , η2 ), where ω = ηπrand ηr = (1 + γ)qr . This shows curves passing three points(η1 , η2 ) = (0, 0), (0, ±ηr ). This assumption is obtained
from the numerical results in the previous section. We
apply perturbation method up to terms of necessary order.
As the transformation Y we use

1 T
u 2 dt(14)
2 0
for given stride. However, the gaits would not be optimal if
we consider the energy dissipation due to the touchdown.
J=

Here we will show simulation results by using two kindsof controllers. The one is a simple trajectory tracking

0.5graph shows the result with q̇2 (0) = 5 and A computed
from

0.5
Time = 0.58

Time = 0.94

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.2

−0.1

0

0.1

0.2

0.3

−0.2

1

0
−0.5

0.2

0.3

ActualDesired
q1+q2=0

5.2 Full-actuated maneuver on tangent vector field ofinvariant manifold

−0.5

−0.1

0
q1

0.1

−1
−0.2

0.2

−0.1

0
q1

0.1

0.2

2
u1
u2

0.2
0
−0.2

0.1

0.2

0.3
0.4
Time [s]0.5

0.6

Joint torques [Nm]Joint torques [Nm]

0.1

0

0.4

−0.4
0

0

0.5
q2

q2

0.5

−1
−0.2

−0.1

1
ActualDesired
q1+q2=0

1 v10 + v20.
(17)ω γv10 − v20
In both case, the orbits are R-symmetric, and have symmetry about the walking direction. They always pass theorigin and do not fall. But only the former one lies on
the approximated invariant manifold, not the latter one(it is not uncontrolled symmetric orbit). The energetic
performance is clearly diﬀerent. We can see, however, thata bit control eﬀort is still required even for the former case,due to the approximation error.
A=−

u1
u2

1
0
−1
−2
0

0.2

0.4
0.6
Time [s]0.8

1Fig. 4. Controlled optimal and non-optimal symmetricwalking gaits. (Left): Starting from a numericallyobtained ﬁxed point. (Right): With the same initialconditions except for q̇2 (0), which is manually perturbed. Note that the control input in the left ﬁgure
is small, but not zero due to the approximation errorof the invariant manifold.
controller in the under-actuated case (Section 5.1), where
we don’t use the tangent vectors but use only the trajectories in conﬁguration space. The velocity is uniquelydetermined from the available energy of the system. The
other controller utilizes the tangent velocity ﬁeld, insteadof using the trajectory itself (Section 5.2). If the initialconditions are the same, and the precision is good, theboth results will be identical.
5.1 Trajectory tracking for under-actuated caseLet suppose the ankle torque is unavailable, u1 = 0, but
only the hip torque u2 is available. The controller is simple
as follows. First, we select a symmetric orbit representedby (7). Then, we compute the tangent vector by

If the robot is not under-actuated, we have additional
DOF for any other control purpose than the trajectory
tracking. Here we adopt a velocity ﬁeld tracking control
proposed by Duindam (2004). By using this method, we
can maintain the kinetic energy constant while making
the solution follow the desired velocity ﬁeld. We use thisbecause it satisﬁes our purpose of utilizing an important
property of Hamiltonian systems – energy preservation.
If we give the well-approximated tangent vector of theinvariant manifold as the target velocity ﬁeld, the closedloop solutions of this controller maintain the energy level,
and the control inputs converge to zero. In other words,
the solution evolves along the geodesic flow (see Marsden (1999)) of the original uncontrolled system, without
falling.
Let v(q) ∈ T N be the target velocity ﬁeld. The controller
comprises the following two steps. The ﬁrst controller isgiven byun = M (q)q̇, v 2q̇, v ∇v v, q̇
∇v v − M (q)∇v v. (18)v, v 2v, v 2

Wherein, v, w = v T M (q)w is the inner product on
Riemannian manifold. ∇v w is the covariant derivative(see Marsden (1999)) of w along with v, which can be
represented in local coordinate(∇v w)i = v j∂wi+ Γijk v j wk
∂q j

(19)

with the connection coeﬃcient Γijk . Note that Γijk =
1 il
2m

∂mkl
∂mlj
∂mjk
∂q j + ∂q k − ∂q lij

is the Christoﬀel symbol ofv2 (η1 ) = −Aω cos(ωη1 )η̇1 .
(15)Then, we transform (7) and (15) to the target trajectory
qd , q̇d by the inverse of (8). Finally, we apply a feedback
controller

the second kind (m is the element of the inverse of theinertia matrix). The controller (18) ensure q̇(t) = v(q(t))
as long as the initial velocity q̇0 is in parallel to v(q0 ), andthe cross product of the input and the velocity becomes tozero: un |q̇ = 0, that is, energy is preserved.

u2 = −K1 (q2 − q2d ) − K2 (q̇2 − q̇2d ),(16)
where K1 , K2 are tracking gains. In this setting, theapproximation error will appear in u2 .

A second controller is given by

Fig. 4 is the simulation result with qr = 0.2, where two
diﬀerent gaits are shown. The left graph shows the resultwith the initial conditions q̇1 (0) = 1.7086, q̇2 (0) = 3.4944,
which are picked up from the numerical solution. The right| q̇, v |ν, ν sign q̇, vv − βM (q)ν (20)|v||v|
with appropriate tracking gain β. Wherein, ν is a vector
such that 5
ua = βM (q)5q̇,v

we used α = v,v .

1

1ActualDesired
q1+q2=0

0
−0.5

−0.1

0
q1

0.1

0

−1
−0.2

0.2

0.3
−0.1

0
q1

0.1

0.2

0.2

0.1

2
u1
u2

0.5

0

0.1

0.2

0.3 0.4
Time [s]0.5

0.6

Joint torques [Nm]Joint torques [Nm]1

Rocking motion

Forward walking

0.4

−0.5

−1
−0.2

−0.5
0

0.5

ActualDesired
q1+q2=0

0.5
q2

q2

0.5

u1
u2

1

0

0

−0.2

0

0.2

0.4

0.6

0.8

1

−1
−2
0

0.05

0.1
0.15
Time [s]

0.2

0.25

Fig. 6. Stick animation of an under-actuated symmetricwalking on a level ground. The corresponding date isshown in Fig. 7.

Fig. 5. Two simulation results of the velocity ﬁeld tracking.
The initial conditions are the same as Fig. 4, left.
(Left) Tracking to the velocity ﬁeld of the invariant
manifold. (Right) Tracking to the “test” velocity ﬁeld.
q2

q̇ = αv + ν

(21)

0
−0.2
−0.2

The ﬁnal control input is given by2

By this controller, ν → 0 (t → ∞) is met for any initialstate (q0 , q̇0 ), which means that the velocity of the solution
asymptotically along with the desired tangent vector of the
manifold. (18) and (20) correspond to feedforward andfeedback term, respectively. The former suﬀers from themodeling error, which results in poor performance of theenergy preservation, while the latter does not because of
the nature of feedback. See Duindam (2004) for the details.
Note that, unlike in Duindam (2004), we do not needto compensate the potential term because the symmetricorbits are the solutions of the uncontrolled Hamiltoniansystem.
The left ﬁgures in Fig. 5 show the simulation results of thiscontroller with the approximated tangent velocity ﬁeld.
The right ﬁgures show the results with the target velocity
v = [1, −1]T , which corresponds to the conservative symmetric orbits that meet q1 + q2 = 0. The control inputs
for the latter case are larger than the former. However, we
can see the residual errors due to the approximation. Notethat the ankle torque is used, and the energy is preserved(not shown, but see Fig. 8). If we choose a straight gaitrather than the curved gait, the approximation becomes
better. We expect that the error will become smaller if weincrease the order of the expansion.
We refer the reader to Yamakita (2001), where a similar
walking gait generation is presented, but using anothervelocity ﬁeld control proposed by Li (1999). Diﬀerent fromthe authors method, they attempted to follow the velocity
ﬁeld of the “asymptotically stable” passive walking gait
(McGeer (1990)), which were approximated by an RBF
network.

−0.15

−0.1

−0.05

0
q1

0.05

0.1

0.15

0.2
u1
u2

1
0
−1
−2
0

1

2

3
Time [s]

4

5

6

1

2

3
Time [s]

4

5

6

14
Energy [J](22)

Joint torques [Nm]and v(q), ν = 0.

u = un + u a .

ActualDesired
q1+q2=0

0.2

13.5
13
12.5
12
0

Fig. 7. Simulation result of an under-actuated symmetricwalking on a level ground. The stride parameter is
ﬁxed to ±0.2 rad. The robot looses the kinetic energy
at every touchdown event and exhibits a few steps ofrocking motions before it ﬁnally stops. The walking
gaits are automatically chosen by the energy level,
which are diﬀerent by every step. See Fig. 6 for the
animation.
6. ASYMPTOTICALLY STABLE WALKING BYPASSIVITY-BASED CONTROL
So far, we considered ﬁctious lossless model to derive
the invariance controller in S1 in our control strategy(Section 1). However, (5) does not match to the actual
leg-interchanging map because there is a state transition
according to the impulse equation.
Nevertheless, the invariant manifold is still powerful because we can select the trajectories of a preferred stride

ActualDesired
q1+q2=0

0
−0.2
−0.15

−0.1

−0.05

0
q1

0.05

0.1

0.15

0.2

u1
u2

1
0
−1
1

2

3
Time [s]

4

5

−0.2

13.5

1

2

3
Time [s]45

6

Fig. 8. Simulation result of a full-actuated symmetricwalking on a level ground, in which the velocity is constrained to the tangential velocity ﬁeld of the invariant
manifold while the total energy is kept constant. Dueto the approximation error of the invariant manifold,the control performance deteriorates at the low-speed
walking.
and walking speed (energy). We can make one trajectorytransit to another simply by pumping energy. The stability
is ensured as long as we constrain the solutions onto the
manifold. To illustrete this point, we conducted a simulation where the symmetric orbit is automatically selectedaccording to the available internal energy, which decreases
whenever the touchdown event occurs. Fig. 6 shows theresult where the simple controller in Section 5.1 is used.
The simulation data is shown in Fig. 7. The robot looses
its energy when touchdown, then enters into the rocking
motion, and ﬁnally stops at double support contact withzero kinetic energy, which is similar to our previous result
by Hyon (2005). In this simulation, the norm of the input
at 6 sec, was 43.844.
On the other hand, Fig. 8 shows the simulation results ofthe controller described in Section 5.2. Since the approximation is not good, the robot fell within a few steps. We
didn’t detailed analysis to see how the performance could
be improved as the approximation order increases.
We give some comments on the rocking motion. The
symmetric orbit in this paper is the orbit that passes originof the conﬁguration space. Therefore, there is a minimum
energy such that the robot can pass the potential barrier tocontinue walk. Otherwise, we have to consider two-periodic
solution. From (3) (R ◦ φt ) ◦ (R ◦ φt ) = id. We ﬁnd a ﬂow
Fr such that (R ◦ Fr ) = −id , and the symmetric orbits

0
q1

0.05

0.1

0.15

0.2

u1
u2

−1

13.5

12
0

−0.05

0

14

12.5

−0.1

1

14

13

−0.15

2

−2
0

6

Energy [J]Energy [J]

−0.2

2

−2
0

0

Joint torques [Nm]Joint torques [Nm]−0.2

Actualq1+q2=0

0.2
q2

q2

0.2

1

2

3
Time [s]

4

5

6

1

2

3
Time [s]

4

5

6

13
12.5
12
0

Fig. 9. Simulation result of a simple under-actuated symmetric walking on a level ground, in which the velocity
is constrained to q̇1 + q̇2 = 0. The walking gaits are
not changing, but the speed is changing.
passing the ﬁxed point set Fix(R ◦ Fr ◦ R ◦ Fr ) . Then
we constrain the solution to these orbits. The hybrid limitset (Grossman (1993)) where the kinetic energy is zero, is{q1 = qr = −q2 } ∩ {q1 = −qr = −q2 } ∩ {p1 = p2 = 0}
Hyon (2005).
For comparison, we also did a simulation where the solution is forced to follow a single straight symmetric orbit.
The result is shown in Fig. 9 (same as in Hyon (2005)).
Compared to the previous results, the qualitative behaviors are not so diﬀerent, but the amount of the control
torque is larger. The norm of the input at 6 sec. was
147.71. Actually, the time when rocking motion occurs is
1 sec. earlier due toe the rapid decrease in the energy. This
result makes sense because the uncontrolled system does
not posses such a straight symmetric orbit.
Now we are in position to derive a passivity-based controller to asymptotically stabilize the periodic orbits withdesired walking speed. Similar to simple pendulum, we canstabilize periodic orbits by adding or deleting the energy.
This corresponds to our motion control strategy (S2) in
Section 1. As an example, let us consider stabilization by
the ankle torque. As shown in Section 5.1, the stability isensured even if the ankle torque is zero, we only have topump the energy into the system by applying the ankle
torque. As a simple controller, we can use a proportional
feedback law
u1 = −KE q̇1 (E − E)

(23)

ACKNOWLEDGEMENTSq2

The authors gratefully acknowledge the contribution ofProf. Kazuyuki Yagasaki (Gifu University, Japan) in computation of the invariant manifold.

ActualDesired
q1+q2=00.2
0REFERENCES−0.2

Joint torques [Nm]

−0.2

−0.15

−0.1

−0.05

0
q1

0.05

0.1

0.15

0.2

2
u1
u2

1
0
−1
−2
0

1

2

3
Time [s]

4

5

6

1

2

3
Time [s]456Energy [J]

14
13.5
13
12.5
12
0

Fig. 10. Simulation result of a stable walking. A simple
ankle actuation is used for controlling the energy.
for some target energy E with KE > 0. Fig. 10 shows thesimulation result. It can be seen that an asymptoticallystable walking has been achieved.

7. SUMMARY
This paper proposed novel walking gait generation for a
double-pendulum-like biped walking model based on thefamily of the symmetric orbits. The eﬀectiveness of the
proposed method was conﬁrmed through the numericalsimulations.
By introducing an involution R associated with the leg
switching map, we allowed the ﬂow of the saddle-center dynamics of the uncontrolled pendulum to possess the timereversal symmetry with respect to R, and the conjunctionof the ﬂow and R to form a family of symmetric orbits
parameterized by the orbital energy and the stride. The
invariant manifold of the family of symmetric orbits was
obtained approximately using a perturbation method. By
constraining the solution onto the invariant manifold using
the control inputs, stable walking gaits were generated
in a semi-global manner. Based on the passivity of theclosed-loop system, a robust speed-controlled walking wasachieved in a very simple way.
Our current work includes the detailed analysis on theerror propagation, learning optimal walkin control byusing iterative learning Satoh (2008), as well as the scaling
up the current method to realistic biped model shown inHyon (2005).

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