Proceedings of the 2005 IEEE
International Conference on Robotics and Automation
Barcelona, Spain, April 2005

Symmetric Walking Control: Invariance and Global Stability ∗Sang-Ho Hyon and Takashi Emura
Department of Bioengineering and Robotics
Tohoku University
Aramaki-Aza-Aoba 6-6-01, Sendai 980-8579, Japansangho@ieee.org

Abstract— This paper first presents a novel control strategy
for periodic motion control based on a Hamiltonian system. According to the strategy, hybrid symmetric orbits (ideal walking
gaits) are explored using reversal symmetry of the Hamiltoniansystem. Then, an invariance controller, a Symmetric Walking
Controller, is derived systematically to distribute the symmetricorbits densely throughout the entire phase space. Finally, a
new robust walking speed controller is formulated based on
the passivity of the controlled system. Consequently, solutions
starting from any point globally converge to a stable limit cycle
having a desired energy level. The controller has strong passivity
and robustness, thereby rendering it capable of using external
disturbances as energy for walking propulsion. It requires nomodel parameters and can be implemented in a very small
program size. Furthermore, it is applicable to any biped robot
without major modification. In this report, the effectiveness of
this controller is proved mathematically, validated numerically,
and confirmed experimentally.
Fig. 1.

Index Terms— Dynamic locomotion, Hybrid periodic orbit,
Invariance, Passivity, Controlled Hamiltonian, Symmetric orbit.

I. I NTRODUCTION
A. Background
The mathematical substance of walking is nothing but anonlinear periodic orbit. Nevertheless, this is not the periodic
orbit of a single continuous dynamical system that has beenstudied for long time in celestial mechanics. Rather, it is the
hybrid periodic orbit of a hybrid dynamical system comprising continuous dynamics and discrete dynamics, which has
not been discussed well in dynamical system theory [1].
Because walking robots must always actively be prevented
from falling, a theoretical guarantee of stability is of utmost
importance for controller designers. Regarding this point, theZMP criterion has played a major role in controller synthesis.
ZMP can be considered as a special constraint that ensures
the tracking of some pre-planned reference walking pathin real-world coordinates (X-Y-Z). Therefore, in a typical
ZMP-based walking controller, some walking patterns are
partially designed in real-world coordinates; subsequently, theremaining patterns are calculated off-line or in real-time sothat the ZMP criterion is met [2][3][4].
Because the location of ZMP restricts the walking gait,
design of ZMP as well as other pre-specified walking pat∗ This study is partially supported by the Ministry of Education, Science,Sports and Culture, Grant-in-Aid for Young Scientists (B), 15760144. The
authors thank M. Suzuki for machining the mechanical parts of the robots.

0-7803-8914-X/05/$20.00 ©2005 IEEE.

Experimental and simulation models of biped walking robots

terns have remained as salient obstacles. On the other hand,
momentum-based controllers that consider momenta insteadof ZMP have been proposed recently [5]. Although theyguarantee the regulation or tracking of a reduced inverted
pendulum model, their overall gait stability depends on thewalking pattern synthesis. An appropriate method to design a
globally stable and natural pattern remains unresolved, as in
the case of ZMP-based controller.
In that regard, passive bipedal walking robots have inspired
optimal and energy-efficient walking controllers [6]. Using
control inputs as the gravity effect, passive-like walking was
realized on level ground [7]. However, because the controlledwalking gait has the same property of the original uncontrolled passive walker on a slope, the region of attraction is
not large: improved stability remains elusive.
B. A novel control strategy based on Hamiltonian dynamics
We propose a controller that generates stable and robust
locomotion gaits autonomously without any pre-planned reference trajectories. To do so, we first propose a novel andgeneral control strategy for controlling oscillatory movement
based on Hamiltonian dynamics. The strategy comprises threesteps:
1. First disregard dissipativity of the system to extract
Hamiltonian (conservative) dynamics. Then, design aninvariance controller that restricts the hybrid solution
into invariant sets.
1455

2. Account for dissipativity and design some period and/or
energy controller to obtain asymptotic stable periodic
orbits (limit cycle) that pass through some desired target
states.
3. Design adaptive mechanisms to minimize control efforts.
The resultant controller has a hierarchical structure in which
the existence, stability, and boundedness of the solution are
ensured by the lowest layer with a high priority; moreadvanced control goals are achieved at the higher layer.
Because of the lack of space in this paper, we cannot describe details of this general control strategy. Instead we willshow one specialized application of this strategy for walking
control, along with an outline of the paper organization.
In Step 1, dissipativity is disregarded; thereby, the system
becomes purely Hamiltonian. Symmetry of the Hamiltoniansystem is utilized to find a symmetric (hybrid) periodic orbits(ideal walking gaits). Section II reviews reversal symmetryof Hamiltonian dynamical systems. This property reveals
the ideal walking gaits of the biped robot, as explained in
Section III. Moreover, the invariance controller is introduced
to enlarge the existence region of the periodic orbits. The
controller renders the system passive and the hybrid solution
invariant. Thus constructively derived controller based onsymmetry is called symmetric walking control (SWC) [8].
In Step 2, energy control is applied to the closed-loopsystem obtained in Step 1 to achieve asymptotically stable periodic orbits. Specifically, Section IV-B proposes apassivity-based robust speed-controller with an additional
control input. In particular, it is proved theoretically that,in principle, under SWC, the robot does not fall. Section V
evaluates the controller by simulation or experiments using
the dynamic biped robot Skipper II and Skipper III (Fig. 1).
The controller is simple enough that it can be implemented
on some inexpensive microcontrollers.
In Step 3, we apply an adaptive controller to minimize
control inputs. It includes the problem of re-design of theactuation method, in which some passive elements such as
springs may be introduced to the model. However, this is ourworking problem and is not applied in this paper. Section
VI summarizes the effectiveness of SWC and discusses someextensions including walk-to-run control.
Note that the above general control strategy was partiallyapplied by the authors to a 4-DOF monopedal and a 5DOF bipedal running robot in [9][10][11], in which anenergy-preserving controller actively removed the dissipation
at touchdown and revealed quasi-periodic orbits on torus
in Step 1, a delayed-feedback controller generated orbitallystable running gaits in Step 2, and a spring stiffness adaptation
minimized control inputs to obtain completely passive running(for the monoped case) in Step 3.
II. R EVERSAL SYMMETRY IN H AMILTONIAN SYSTEM
As explained in Step 1 of the previous section, our strategy
is to extract and investigate Hamiltonian dynamics. Hamiltonian mechanical systems possess a special symmetry called

time-reversal symmetry [12]. Consider a diffeomorphism F
on manifold M
F : M → M.

(1)

An involution R : M → M (R ◦ R = id) is called (involutive) reversing symmetry of (1) if the following relationshipholds.
R ◦ F = F −1 ◦ R(2)Let φt : (q(t0 ), p(t0 )) → (q(t0 + t), p(t0 + t)) be an flow
map of Hamiltonian “mechanical" system with Hamiltonian1 T
p M (q)−1 p + U (q).
(3)
2“Any" Hamiltonian mechanical system is reversible w.r.t.
involution R : (q, p) → (q, −p); that is,
H0 (q, p) =R ◦ φt = φ−t ◦ R.

(4)

The intersection of the fixed point sets F ix(R) and F ix(R◦φt ) is of special interest because an orbit, called a (R) symmetric orbit, passes through the intersection and theorbit is closely related to our “hybrid" periodic motion. Insummary, if there is a pair of the involution R and the Rreversible flow φt , and if we can find the intersection of
F ix(R) and F ix(R ◦ φt ), then we obtain a symmetric orbit.
This is an important part of our control strategy.
Time-reversal symmetry has been studied mainly in celestial mechanics [12]. The idea was firstly implicitly applied toa monoped hopping robot in [13], but recently, more concrete
treatment was done in [14]. In particular, the theory can proveexistence of the “controlled" quasi-periodic orbits of a 4-DOF
one-legged hopper discovered in [9][14]. But the purpose
here is different from these works; we derive an invariance
controller and a passivity-based controller in the following
two sections.
III. S YMMETRIC WALKING CONTROLThis section develops an invariance controller for bipedrobots according to Step 1 of the strategy in Section I-B.
A. Symmetric (hybrid) periodic orbit of biped robot
Consider a compass-like biped model, as shown in Fig. 2.
Let q = (q1 , q2 ) be generalized
coordinates on configuration

space N := {q ∈ S 2  |q1 | ≤ π/2 and |q1 | < |q2 |}, where
the solution can be defined. Let z = (q, p) be the canonicalcoordinate on phase space M = T ∗ N . The conjugate
momentum p is associated with q by p = M (q)q̇, where
M (q) =

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

Define the control input as u1 = τ ∈ R1 . Then the dynamics
of the biped model at the single support phase can be
represented by the 4-DOF Hamiltonian control system [15]ż µ = {z µ , H0 } − {z µ , H1 }u11456(µ = 1, 2, 3, 4),

(5)


.

m1

1
0.5
q

0.5

−1
−0.5

m2
−q1

0.3

−2
0.2

0
0

Fig. 2. A compass-like biped robot comprising a point mass: q1 is theabsolute angle of the supporting leg, whereas q2 is that of the swinging leg.
τ is the hip torque.

where

H1 (q) =1 T
p M (q)−1 p + U (q),2
q1 − q2

(6)

−4

(8)

Now, consider R-reversible “uncontrolled" flow φt of (5).
Instead of conventional involution (R := (q, p) → (q, −p)),here we are interested in the mirror image about q-axis, thatis, the involution
R : (q1 , q2 , p1 , p2 ) → (−q1 , −q2 , p1 , p2 ).

1

p0
2

4

p1

−1
6

8

10

−6
−8

2

10

(a) Partial plots of F ix(F ◦ R), where q2 isomitted because q2∗ = −q1∗ .

20

30

40

p

−2

1

(b) Symmetric orbit for
q1∗ = 0.5

Fig. 3. Numerical results of symmetric orbits (ideal walking gait) of acompass biped with the parameters: m1 = 2, m2 = 1.2, b = 0.2, L = 0.4.
The gaps result from roughness of the initial guess; they do not indicate
emptiness.

(7)

are the internal Hamiltonian and interaction Hamiltonian,
respectively, and {·} is the Poisson bracket [16]. The potentialfunction is defined as
U (q) = (m1 + 2m2 )gl cos q1 − m2 gb(cos q1 + cos q2 ).

0.5

1

0

0.1

H0 (q, p) =

0
q

0.4

p2

m2

L

0

−0.5

1

q2

0.6

q

τ

2

0.7

b

(9)

Because U (q) is symmetric (invariant) to R, R is reversing
symmetry of φt .
For hybrid dynamical systems such as legged robots, Rreversibility of the flow is especially important when the
involution R itself can be considered as the discrete jumpof the hybrid system. In this case, the R-symmetric orbit corresponds to the hybrid periodic orbit of the system(walking gait). The mirror image (9) is a natural involution
for the compass-like biped robot because it meets the ideal
touchdown condition (q1 + q2 = 0). Therefore, the connectionof the involution R and the R-reversible flow φt constitutes
an “ideal" symmetric walking gait.
To render this connection make sence, one should consider
the restriction of flow φt
F := φt t=T ,(10)where T is defined as the time when the touchdown condition
q1 +q2 = 0 holds. With this background and the assumption of“physical" realization of R, the problem is reduced to finding
the fixed point set F ix(F ◦ R). Define z ∗ = (q ∗ , p∗ ) ∈F ix(F ◦ R). The system is non-integrable; therefore, weshould introduce some numerical algorithm.
Figure 3(a), though coarse, depicts F ix(F ◦ R), which is
obtained using the algorithm developed in [17]. The result
indicates F ix(F ◦ R) forms a 2-dimensional surface. The

trivial one is the origin, which corresponds to the equilibrium
point. Figure 3(b) shows the phase portrait of symmetric orbits
starting from a fixed point in Fig. 3(a). The symmetric orbit ismapped onto itself by R with its direction reversed. A similar
orbits in the top of Fig. 3(b) appear in the “ballistic walking
gait" in [18]. Our examination newly obtains the following
results.
(1) Symmetric orbits are at best neutrally stable (the characteristic exponents [19] lie on S 1 ).
(2) Symmetric orbits have at most three-times bifurcations.
The details will be presented elsewhere. Just note that thesesymmetric orbits cannot be realized without control because
the actual impulse equation always perturbs the initial condition away from z ∗ .
B. Symmetric walking controller
We have seen that uncontrolled symmetric orbits are atbest neutrally stable. However, using invariance control, we
can actively construct another controlled symmetric orbit
reversible to R. Specifically, we can obtain controlled symmetric orbits by setting the output function,y = h(x) = q1 + q2 ,to zero and using a leg exchange scheme, S(x) := x  |q1 | − q = 0,

(11)

(12)

where q ≥ 0 is the switching angle (scalar value), which islater used for tuning convergence performance. We call thecontroller that makes (11) zero combined with the switching
logic (12) symmetric walking control (SWC).
The idea of output zeroing of the function (11) is notnew. Intuitively, in a steady walk, a human swings her leg
opposite to the supporting leg. Some walking toys employ
a similar concept, where only the hip “joint angles" can beconstrained. The idea was also introduced in [20][21], where
stability analysis via Poincaré map with an impulsive effectis performed.
1457

2

However, its systematic derivation from the viewpoint of
global stability has not been done before. These are developedin the next section. Moreover, the obtained walking controller
differs markedly from the past proposed ones in the following
points. 1) We ensure global stability, not local stability. 2)
We do not require Poincaré map nor its attendant stability
analysis. 3) An invariance controller keeps the robot fromfalling 4) A passivity-based controller enables control energy
and walking speed robustly, as explained in Section IV

Symmetric orbit
1.5

1

p

I
0.5

II
0

IV

C. SWC Properties−0.5This section formulates SWC properties.
Proposition 3.1: Under (11) the controlled system (5) is reduced to the new 2-DOF Hamiltonian system with controlled
Hamiltonian
1 2
Hc (q, p) =p + (m1 + 2m2 )gL cos q,(13)
2Mc
where Mc = (m1 + m2 )L2 − 2m2 bL.
Lemma 3.1: Let Fc be the Hamiltonian flow associatedwith (13). R is a reversing symmetry of Fc under (11).
Moreover, because of (12), the hybrid solution
 of the system(5) is bounded to the set Mi = {x ∈ T ∗ M  |q| ≤ q}.
This lemma leads to one main result of this paper:
Theorem 3.1: Under (11) and (12) the following holds.
F ix (Fc ◦ R) = Mi1 ,F ix (Fc ◦ R ◦ Fc ◦ R) = Mi2 ,

(14)
(15)

Therein, Mi1 and Mi2 are the semi-half  subspaces
 |p| >of
 Mi defined as: Mi1 = Mi ∩ {x2M
(m1 + 2m2 )gL(1 + cos q)} and Mi2 = Mi ∩ {x 
|p| < 2M (m1 + 2m2 )gL(1 + cos q)}.
Theorem 3.1 shows the global invariance of the solution inMi1 ∪Mi2 . That is, any solution starting from Mi1 ∪Mi2 lies
on a symmetric orbit passing through the initial point! Thisinvariance is the theoretical basis of passivity-based orbital
stabilization developed in Section IV.
Therefore, the ideal walking gaits are generated. However,
one important assumption is the realizability of R. Consider
the dissipativity by the impulse equation at the leg exchange.
The equation can be represented simply by a map ([6])
R1 : (q1 , q2 , p1 , p2 ) → (−q1 , −q2 , η(q)p),(16)

in which det(η(q)) ≤ 1 (equality hold only for q = 0).
Hence, R1 is not an involutive, but rather a contracting map.
Therefore, the controlled Hamiltonian Hc in (13) is found tobe piece-wise constant and meets the passivity relationship.
dHc
≤ 0.
(17)
dt
Thereby, we have obtained the second main result of thepaper:
Theorem 3.2: If there is an energy-dissipation at leg exchange, the invariant set of SWC is hybrid ω-limit set L+ =(q, −q, 0, 0) ∪ (−q, q, 0, 0), which is globally asymptoticallystable on Mi .

−1
−0.4

III

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

q

Fig. 4. Phase portrait of the reduced 2-DOF system under SWC with q =0.3. The solution starting from (q(0), p(0)) = (−0.3, 1.5) asymptoticallyconverges to the limit set (solid curve). If there is no dissipation at theleg exchange, it becomes a symmetric orbit (bold-dashed curve). The dashdouble-dot line indicates the separatrix.

Proof. From Lemma 3.1 the solution of the closed loop
system is bounded. Take Hc (13) as a positive semi-definite
function. Then, Lemma 3.1 and LaSalle’s invariance theorem[22] concludes the statement.
This is theoretical evidence of the statement in Section I-B;
under SWC, the robot does not fall in principle. However, if
we consider the slippage in contact between the foot and theground, Mi should be narrowed appropriately, otherwise, onemust consider the impulse reduction.
D. Numerical illustration
The phase portrait Fig. 4 illustrates Theorem 3.1 and Theorem 3.2. This figure shows the reduced (controlled) 2-DOF
Hamiltonian flow φt of (13) with and without considering
the impulsive effect. Note that the origin is an unstableequilibrium point (saddle). The separatrix and the switchingborder |q| = q divides the phase space M into Mi1 (region
I ∪ III), Mi2 (region II ∪ IV).
In region I, q is increasing monotonically (normal forward
walking). If there is no impulse effect, the solution returns
to the initial condition when it reaches the border. The
orbit is the ideal walking gait corresponding to R-symmetricorbits. Note that the symmetric orbit is not isolated, but isinstead surrounded by other symmetric orbits corresponding
to different initial conditions (Recall the flow around a center
equilibrium of a single autonomous system). This is a directconsequence of Theorem 3.1.
On the other hand, if the impulsive effect (16) at touchdown
is considered, the momentum is decreased stepwise. The
solution starting from region I eventually enters region II; thedirection of motion is changed. Once this happens, the flow
φt is “trapped" in region II ∪ IV. Then it finally converges to aω-limit set: L+ = (q, 0)∪(−q, 0). As the states become close
to L+ , infinitely many switchings occur. This phenomenon iscalled Zeno [1]. Physically, the robot loses its kinetic energy
1458

0

Angles [rad]Angles [rad]

q
1
q2

0.2

−0.2

0

m1
q5

−0.2

−0.4
0

1

2

3

22
H
Hc

20
18
16
0

1

2

c

−0.4
0
Hamiltonian [J]Hamiltonian [J]

q
1
q2

0.2

1

20

ρ2
1

2
Time [s]

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0

0

−0.1
0

−0.1
0

0.4

0.5

3

q2

m2

τ1
−q1
m2

ρ1

q4

4

b
q3

(b) Dissipative

0.5

0.3

τ2

18

(a) Conservative

0.2

4
H
Hc

Time [s]

0.1

3

22

16
0

3

2

0.6

(c) Conservative

0.7

0.8

0.9

Fig. 6. Model of a five-link biped robot with torso: (ρ1 , ρ2 ) are the leg
forces and (τ1 , τ2 ) are the hip torques.

0.1

0.2

0.3

0.4

0.5

0.6

(d) Dissipative

Fig. 5.
Simulation results of a compass-like biped walking with: theparameters M = 5, m = 1.2, L = 0.4, b = 0.2; the initial conditions
q1 (0) = 0.2 = −q2 (0), p1 (0) = 1.2, p2 (0) = −0.2; and switching angle
q = 0.2. (a),(c): symmetric walking gait. (b),(d): the robot finally stops at
L+ = (0.2, −0.2, 0, 0) ∪ (−0.2, 0.2, 0, 0). In both cases, the controlled
Hamiltonian Hc is piece-wise constant during one walking cycle.

and stops with its legs contacting the ground (double support)
even though the switching “logic" may occur infinitely manytimes. In this case, L+ is globally asymptotically stablebecause it does not depend on the initial condition.
Figure 5 shows simulation results. Figure 5(a) and 5(c)depict an ideal conservative case, whereas Fig. 5(b) and
Fig. 5(d) depict the dissipative case, where the impulsive
effect at touchdown is considered. The bottom two graphs
depict the internal Hamiltonian H0 in (6) and the controlled
Hamiltonian Hc in (13). In both cases, Hc is step-wise
constant. Note that the scuffing of the swinging leg is disregarded in this simulation. Because no type of compass-likebiped model can walk on a level surface without scuffing the
ground, the experimental model must have some leg retraction
mechanism.
IV. O RBITAL S TABILIZATION USING PASSIVITY
We obtained global stability and passivity under SWC in
the previous section. Consequently, we are ready to develop
several passivity-based orbital stabilizations according to Step
2 in Section I-B. For the SWC-controlled system, orbital
stabilization is equivalent to controlling the orbital energy or
the walking speed. Additional DOF is required to put energy

into the system.
The additional DOF is installed as in Fig. 6, which shows
the model of our planar biped walking/running robot Skipper
II [11]. The new coordinates from the compass-model is the
supporting leg length (q3 , q4 ) and the attitude of the torso q5 .
Let u = (ρ1 , ρ2 , τ1 , τ2 ) be the control input.
In “walking mode", the leg length is kept constant during
the supporting phase. Also, the torso inclination is kept
constant. That is, the following output function is constrained
to zero.
y = h(x) = (q1 + q2 , q3 − r0 , q4 − rd , q5 − q5d )

T

(18)

Where r0 is the natural length of the leg spring attached below
the knee joint, which is further utilized for dynamic runningmotion [11]. The desired swinging leg length rd is designed
simply so that the leg retracts or extends in proportion to q2 .
Any retraction scheme that meets rd = r0 when q2 = q is
possible for a robot that has light-weight feet, as Skipper II
does.
A. Speed control by leg extension
First consider the simple impulsive control that compensates energy dissipation at touchdown. For simplicity, supposethat c (torso length) = 0. Then, one can realize the inverse
map of R1
R2 : (q1 , q2 , p1 , p2 ) → (q1 , q2 , η −1 (q)p),(19)

by exact (model-based) impulsive controller û, which meets
R = R2 ◦ R1 .
Another option is a simple and robust control using the passivity of SWC. Thanks to the densely distributed symmetricorbit on Mi , one can use the impulsive controllerû = k (Γ(ẋ − ẋd ))if ẋ < ẋd ,(20)

where k(·) is an input transformation, Γ > 0 is the gain, ẋ
represents the walking speed, and ẋd is its commanded value.
This controller does not require exact information on η(q).
1459

0.6

0.6

0.5

0.5

0.4

0.4

0.3

0.3

0.2

0.2

0.1

0.1

0
−0.1

restoration. This affects the settling time of the walking speed.
The longer the torso CoG is, the more energy can be restored
during one walking cycle. It is also the same for larger q,
but large q leads to a high energy loss at touchdown, thereby
increasing the steady state error of the walking speed.

0

0

0.2

0.4

0.6

0.8

1

1.2

1.4

−0.1
−0.6

(a) Constant-speed regulation

−0.4

−0.2

0

0.2

(b) Variable-speed tracking

Fig. 7. Animations of two kinds of speed-controlled walking. Only theright leg motion is depicted for visibility. (a): Constant speed walking withẋd = 1.0 m/s. (b): Variable speed tracking with ẋd = 0.5 → −0.5 → 0.5
m/s. the robot first moves forward, then turns backward and turns forward
again. Thin solid lines indicate the virtual pendulum between ZMP and CoG(center of gravity). ZMP is shifted from the supporting point.
0.2

Attitude [rad]Speed [m/s]

2

0
−0.2
−0.4
0

2

4

0.5

−0.5
−1
0

6

30

0.2

29.5

0

28.5

−0.4
0

28
0

6

Time [s]Fig. 8.

xgD
xD
d

2

4

6

29

−0.2
4

Before applying the controller to an actual robotic system,a precise 3D simulator was constructed on LMS DADS R
connected with SIMULINK R and various simulations werecarried out. To make the controller robust, the simulation andlater experiment use a simple PD-feedback.
u = −Kp y − Kd ẏ

B. Simulation 2: Symmetric 3D walking

2

4

6

Time [s]Time evolution of the state variables corresponding to Fig. 7(b)Also, steady state error can be removed by adding integral
action.
B. Speed control by torso inclination
Non-zero inclination of the torso can be utilized for controlling the walking speed. Specifically, this paper proposes acontroller of the proportional feedback form
q5d = −Γ1 (ẋ − ẋd ),(22)

Therein, Kp > 0 and Kd > 0 are relatively high gains.
As one example, Fig. 9 and Fig. 10 show a robustness test.
In this simulation, the robot impacts with the wall during
steady walking of 0.5 m/s. Despite this disturbance, the robot
never falls. Instead, it merely stops. If the wall is removed,
then it starts to walk again. Robustness comes from theinvariance and the global stability of SWC (Theorem 3.1 andTheorem 3.2).

0

0.4

2

A. Simulation 1: Robustness against disturbance

1

θ1
θ

Energy [J]Leg angle [rad]0.4

V. A PPLICATION TO 2D AND 3D BIPED PROTOTYPES

0.4

(21)

where Γ1 > 0 is the gain. With this controller, the input
energy balances “automatically" with dissipated energy, whichleads to a limit cycle. The steady state error can be removed
by adding integral action.
The speed command ẋd can be set arbitrarily. For example, Fig. 7 and Fig. 8 show simulation results of speedregulating/tracking walking. The robot starts from a standstill;then the torso pitches forward and the speed increases. As
the speed approaches the desired value, the torso attitude
converges to a constant value. Thereby, asymptotically stable
walking gaits (limit cycles) with a specified walking speed
are achieved.
There is a limitation in acceleration because of the boundedness of the torso angle. The effective maximum torso
inclination is ±π/2, which results in the limitation of energy

The SWC can be applied to a 3D walking model without
any major modification. The only modification is the extension of the output function.

q1 + q2
 q3 + q4 
 q5 − r0 (23)
y = h(x) =  q6 − rd 
 q7 − q7d 
q8 − q8d
Therein (q1 , q2 , q3 , q4 ) are the pitch and roll attitudes ofthe legs, (q5 , q5 ) are the leg lengths, and (q7 , q8 ) are thepitch and roll attitudes of the torso. The desired swinging
leg length rd should be designed according to the attitudecos−1 (cos q2 cos q4 ). Figure 11 and 12 show one simulation
result. Except for the orientation of the walking, which isdifficult to maintain for the robot having point-contact foot,the stability and the behavior is the same as in the 2D walking
model. However, because of the collision avoidance between
the legs, some gain tuning is required.
C. Experiment: Speed-controlled walking
Because Skipper III is under preparation, only the experiments with Skipper II are shown here.
Figure 13 shows a typical gait of the symmetric walking.
The robot stands on a tread mill. It is constrained to thesagittal plane by two transparent acrylic plates. It carries anattitude sensor comprising a gyro and an inclinometer at thetorso; simple On/Off sensors at the feet detect the groundcontact. In the current experimental setup, the walking speed
is estimated from joint angles because position sensors are
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0.4

0.1

0.2

0.08

Attitude [rad]Leg angle [rad]Fig. 9. A robustness test: during steady walking of 0.5 m/s, the robot
impacts with the wall. However, the robot does not fall. It merely decreases
the energy and stops. If the wall is removed, then it begins to walk again.

0
−0.2

0.06
0.04

−0.4
−0.6

0.02
8

10

12

0

14

8

10

12

14

Impact!
1

0

−0.5
−1

0.8
0.6

Fig. 13. Snap-shot of the walking experiment: The robot initially stands
still. When the speed command is set to some positive value, the torso pitches
forward and then starts to walk.

0.4
0.2
0

8

10

12

14

8

10

Time [s]1214

Leg Angle [deg]0.5Foot switch

Speed [m/s]1Time [s]−20020

Speed [m/s]Sagittal motion
1
ActualReference

0.4

0

10

20

−1
0

30

0.2

10

20

30

0.2
ActualReference

ActualReference

0.1
q [rad]

0.15

0

8

0.1

7

q [rad]

25

5

10

15

20

25

0.5
ActualReference
5

10

15

20

25

−0.5

0.2

0.05
0
0

20
ActualReference

Fig. 14. Time evolution of some state variables during a long-distance andhigh-speed walking experiment on tread mill. The commanded speed is aramp signal of 0–0.75 m/s. Asymmetry in the top graph comes from theslippage of the legs and the dely of the PD-feedback.

ActualReference

0.5

y dot [m/s]x dot [m/s]

0.6

0
0

15

Lateral motion

1
0.8

10

0

0
0

3D animation of symmetric walking.

5

10

−10
0
1

Fig. 11.

q1
q2

0

Attitude [deg]Fig. 10. Time evolutions corresponding to Fig. 9: After impact at 9.23 s,
the walking speed decreases and finally converges to zero around 12.5 s (leftbottom). Because the commanded speed is fixed to 0.5 m/s in this example,
the torso attitude still increases (right top).

20

−0.1
10

20
Time [s]

30

−0.2
0

10

20

30

Time [s]Fig. 12. Time evolution of the state variables during 3D symmetric walking.
The commanded walking speed is 0.5 m/s to the forward direction.

not installed. Figure 14 shows the time evolution of the mainstate variables during long-distance and high-speed walking
experiments. The bottom graph indicates that the error isvery small. One can start a walking experiment from any
initial conditions. In this experiment, a variable feedback gain
Γ1 = |ẋ − ẋd | and variable step length q = Γ2 |ẋ − ẋd |are introduced to improve tracking performance. Because thebehavior is almost identical to that in simulations, we do
not repeat their explanations. But note that the information
regarding physical parameters such as mass distribution or
length is not required to obtain a stable limit cycle because
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SWC is not a conventional model-based controller.
VI. D ISCUSSION
A. Advantages of SWC
The SWC combined with passivity-based controller allows
restoration of the total mechanical energy and control of
the walking speed. Solutions starting from any point in thedomain Mi converge globally to a stable limit cycle or stablehybrid ω-limit sets. Global stability was proved theoreticallyusing the invariance principle. The controller is written in a
program of very small size: it can be easily implemented on an
inexpensive micro-controller. Moreover, the control parameteris actually the switching angle q only. There is no room for“trial-and-error". The SWC has strong passivity and robustness against disturbance. That passivity is especially useful forhuman-safety functions, which conventional trajectory-based
controllers do not support. For example, when the desired
speed is set to zero, the robot can stop even if an unknown
disturbance is applied. The settling time depends on themagnitude of the disturbance. Alternatively, the disturbancecan be transferred to the propulsion energy of walking: a sortof an externally-driven walking just like pushing a tire on
the floor. The SWC can be utilized as an “auto balancer"of bipedal humanoid robots, which may be activated at thelowest layer of the whole control system.
Its strong stability and robustness will encourage application of SWC to many kinds of bipedal robot without major
modifications. We have succeeded in simulating the stable 3D
walking of Skipper III. We have also succeeded in simulating
stable walking of a five-linked planar anthropomorphic bipedrobot with a “passive knee joint" using an inexpensive fourline controller in which the mechanical energy is compensated
not only by the torso inclination, but also by the ankle torqueat the double support phase [8].
B. Road to running: Impulse reduction, input optimization
Energy loss attributable to the impact is unavoidable instiff walking. This is a direct consequence of the impact
equation (16). However, the energy loss can be reduced if theleg is compliant. Moreover, kinetic energy can be restored
into the spring mechanism. Compliant leg actuation witha spring mechanism leads to energy preserving symmetricrunning controller implemented in [11]. We have obtained
speed controlled walking by torso inclination. For that reason,
we expect that speed-controlled running can be achieved in
the same context.
In this regard, the transition from walking to running will
be an interesting problem. Our precise simulations showed
that, for walking faster than 1.33 m/s, which corresponds to
Froude number = ẋ2G /(gLG) = 0.45, the robot slipped on the
ground; thereby, a very short period of flight occurred. This isalmost the same conclusion of a preferable walk-to-run transient speed, similar to that discussed in biomechanics studies(e.g. [23]). Faster locomotion without slipping is possible
only if a sufficient ground reaction force is provided by an“extending leg". In this case, the mechanical energy is restored

not only by leg extension, but also by torso inclination. Even
when some ankle torque is available, because of the ZMP
limitation, we ultimately rely on leg extension to increase theenergy. This phenomenon leads to hip-torque reduction. We
expect optimal walking and running gait to be realized using
a “single controller". All these expectations will be tested
during experiments using Skipper prototypes.
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