Proceedings of the 2002 IEEE/RSJ
Intl. Conference on Intelligent Robots and Systems
EPFL, Lausanne, Switzerland • October 2002

Quasi-Periodic Gaits of Passive One-Legged Hopper
12Sang-Ho Hyon , Takashi Emura

1 Tohoku University, Sendai 980-8579, Japan, sangho@ieee.org
2 Tohoku University, Sendai 980-8579, Japan, emura@emura.mech.tohoku.ac.jp

Abstract
In this paper, we present a novel controller for a passive one-legged hopping robot. First, based on the dynamics of this nonlinear hybrid system, we derive a
simple control law to ensure the total energy preservation and continuation. Simulation results showthat the robot can hop from the wide set of initialconditions. The generated hopping gaits are found to
be quasi-periodic orbits, which can be seen in some
Hamiltonian systems. Next, we propose a simple parameter adaptation law to asymptotically stabilize thequasi-periodic gaits to the periodic gaits of arbitrary
period, and spring sti ness adaptation law to minimize control inputs. Simulation results show that the
robot eventually hops without any control inputs, especially for 1-periodic gait. We believe that the controllers we invented have much potential for energyeÆcient control of legged running robot.

1 Introduction

After the Raibert's excellent works [1], one-legged
hopping robots attached with only the leg spring,
have been widely studied both experimentally [2, 3,
4, 5] and theoretically [6, 7, 8].
In addition to the leg spring, hip spring also plays
important role for animal running [9]. It enables theleg to be swung passively. Tompson and Raibert
showed that spring-driven one-legged hopping robot
shown in Figure 1, can hop without any inputs, provided if the initial conditions are appropriately cho-

Hip spring

Leg spring

Figure 1: Passive one-legged hopper

0-7803-7398-7/02/$17.00 ©2002 IEEEsen [10]. Therefore, this model is a good templatemodel for the purpose of studying energy-eÆcientrunning. Since this model is shown to be marginallystable and eventually falls without controls, somesuitable controller should be applied to ensure thestability.
Ahmadi and Buehler applied Raibert's celebrated
Foot Placement Algorithm [1] to this passive hopping
robot, in which the Neutral Point should be preapproximated. They realized energy-eÆcient hopping in simulation and experiment [11, 12]. On theother hand, Francois and Samson derived new controller di erent from Raibert's [13]. They applied
general control method used in nonlinear oscillatory
system. That is, rst constructing Poincare map(discrete system), then linearizing it around xed
points, and nally applying some linear feedback to
get asymptotically stable periodic orbit. However,since the model cannot be integrable, both examples
above need approximated models, or approximatedperiodic solutions, to derive the controllers. Therefore, there remains static error comes from the modeling error.
In this paper, inspired from Francois and Samson's
work [13], we present a novel controller to realize
energy-eÆcient one-legged hopping. Instead of depending on some pre-planned periodic solutions, ortarget dynamics, here we utilize the intrinsic dynamics of the original nonlinear hybrid system to makethe robot generate natural hopping gaits. A model
description is the same as those in [13] and reviewed
in Section 2. Since our goal is energy-eÆcient hopping, we explore the condition of energy-preservationand continuation in Section 3. Then, we derive acontrol law to ensure those conditions and show thesimulation resuts in Section 4. In Section 5, twoparameter adaptation laws, one for asymptotic stabilization to desired periodic gait, and the other for
input minimization, are given.
Simulation results shows that the controllers we invented have much potential for energy-eÆcient control of legged running robot.

2625

Table 1: Variables

hip spring Kh

φ

Body

z

x
z
r


f


E
Ts
Tv

τ
leg spring
Kl

θ

r

f
FOOT

MeaningUnithorizontal position of hip mvertical position of hipmleg lengthmbody angle
radleg angle
radapplied force of leg
Napplied torque of hip
Nenergy dissipation
total mechanical energyJstance timesight times
Table 2: Phase-indicating suÆxSuÆx
Meaningtdjust before touch-downtd+
just after touch-downlolift-ox
Figure 2: Model de nition

Table 3: Physical parameters

2 Model description of passive hoppingrobot

We consider exactly the same model of passive hopping robot as [13] in this paper. In this section, themodel description is reviewed.
2.1 Model de nition and notation

We consider the planar one-legged hopping robot
shown in Figure 2. The robot is attached with not
only the leg spring but also hip spring.
We impose the following assumptions on the model
as seen in many related literatures.
(A) The center of mass (COM) of the body is juston the hip joint and COM of the leg lies on theleg(B) Mass of the foot (unsprung mass) is negligible(C) The springs are mass-less and non-dissipating(D) The foot does not bounce back, nor slip theground (inelastic impulsive impact)
In addition to these assumptions, we further suppose,just for the simplicity, that the COM of the robot ison the hip joint without loss of generality. Table 1
summarizes the variables appear in this paper. The
equations of motion are composed of four phases;
stance phase, lift-o phase, ight phase, and touchdown phase. Table 2 de nes the phase-indicatingsuÆx of the variables. For example, x_ lo represents
the forward velocity of COM at the lift o . Table

g
M
r0
Jb
Jl
Kl
Kh

Meaning
Unit
Valuegravity acceleration
m=s2
9.8
total masskg
12
natural leg lengthm
0.5
body inertia about hip kgm2
0.5
leg inertia about hip
kgm2
0.11
leg spring sti ness
N/m
3000
hip spring sti ness
Nm/rad10

3 shows the physical parameters, together with thevalues used in later simulations.
2.2 Equation of motion

At the stance phase, the leg compresses and extends,
and the angular momentum of the robot around thecontact point evolutes under the gravity eld.
8
< M r + Kl (r r0 ) M r_2 = M g(1 cos ) + f d r2 _) = rM g sin (1): JJl ++JKb (+ dt (M) = sbhHere, f is the control force to the leg, while s isthe control torque to the hip joint, which is applied
during stance. The spring is initially loaded with thesame value of gravity force (M g).
At the ight phase, COM of the robot moves along
the ballistic ight path and the angular momentum2626

of the robot around the COM is preserved.
8x = 0>>< z = g>>: JJl ++JKb (= 0 ) =  :
b

h

(2)

v

Here, v represents the control torque to the hip
joint, which is applied during ight.
By the assumption (D) in Section 2.1, the velocities of the generalized coordinates change instantaneously at the touch-down phase, according to thefollowing equations.
8 x_ = x_
Jl cos td
td+
td>
Jl +Mr02 td
>
>
J
>
>
z
_
=
z
_
< td+ td Jll +sinMrtd02 td
Mr0(3)
_td+ = _td
Jl +Mr02 td
>
>
>
_
_
>
>
: td+ = td
r_td+ = z_td+ cos td x_ td+ sin td :Here,
td := x_ td cos td + z_td sin td + r0 _td :(4)
At the lift-o phase, there is no discontinuouschanges except for r_lo = 0.
3 Analysis on the energy preservationand continuation

For a passive running robot, the analysis on the energy is important because, during complete passive
running, the total mechanical energy is conserved.
According to (3), the energy change between justbefore touch-down and just after touch-down is calculated to be
Etd+

Etd


td+
= 21 M (x_ 2 + z_ 2 ) + 12 Jb _ 2 + 21 Jl _2
td
M Jl
2
= 2(J + M r2 ) td :
(5)
l

0

We call td \Energy Dissipation CoeÆcient" because, if the conditiontd = 0(6)
holds at touch-down and no input is applied to therobot, then total mechanical energy of the system
is preserved during hopping. Of course if we apply
some control inputs, then the internal energy is notconserved. But we can say that (6) is necessary condition for complete passive running.
Next, we consider the conditions for the robot tosustain hopping without falling to the ground. We
can see the condition is found to ber_td+ < 0:(7)This is the condition for the axial velocity of theleg just after touch-down to be negative. This iswell understood if we recognize from Section 2.2 thatthe necessary condition for the robot to be in the
stance phase is r < r0 . Unless this condition holds,
the hybrid system cannot switch to the stance phasefrom the ight phase, and hence, it cannot continue
time evolution. Note that (7) is not always satis ed.
4 \Non-Dissipative Touch-Down Control" and quasi-periodic hopping gait

4.1 Control goal

Due to the compactness of the phase space, if thecontinuous time evolution is ensured and the totalenergy is preserved, the solution of this hybrid nonlinear system lies on periodic orbits on the energyinvariant manifold. This invariant hybrid ow is thegait we are searching for. Therefore, we derive controller to ensure the both condition (6) and (7), andcall this new controller \Non-dissipative Touch-downControl ".
We can do this by applying control inputs only at theight phase, and allowing the robot freely to move atstance phase with zero inputs. The reason why weare doing so is that if no energy dissipation occurs at
touch-down (td = 0), there is no interaction between the robot and environment, even we use apply
control inputs at ight phase. Then, the total mechanical energy of the robot \including power sourceof the actuators", is exactly preserved. Energy preserving gait in this sense, is what we want.
Since we are using the control torque only at theight phase, we de ne the control input  as follows.
f = s = 0;
v =: :

(8)
(9)

Our control problem is to nd the control input that makes the robot land at the time Tv , with
(td ; _td ) satisfying (6) and (7), for any given lift-ostates (x_ lo ; lo ; _lo ; lo ; _ lo ). Though this is the deadbeat control to bring Tv , td, and _td to the desired
values, we have only to choose some Tv and td , because _td is automatically calculated by (6). There
are, however, a large number of such pairs (Tv ; td )satisfying (7). Here, we choose the simplest values
as follows.
First, we determine Tv so that the vertical velocity
at touch-down and lift o is the same in magnitude
and opposite in direction,
Tv
2627= 2zg_lo :(10)

Next, we choose desired touch-down angle  to besymmetric to the lift-o angle about vertical axis,
 =
lo :
(11)
Then, from (6) we obtain desired touch-down angularvelocity _ as,_= 1 (x_ cos  + z_ sin  )
td

=

r0

td

r0

lo

td

td

td

1 (x_ cos  + z_ sin  )lolo= _lo =: :_

lo

(12)

Additionally, consideringz_td+ = z_td = z_lo gTv(13)
x_ td+ = x_ td ;(14)
we obtain
r_td+ = (z_lo gTv ) cos td x_ lo sin td= z_lo cos lo + x_ lo sin lo= r_lo :(15)
Therefore, the condition (7) is ensured if and only ifr_lo > 0.
Again, note that (10) and (11) is no more than theone of the options.
4.2 Derivation of the controller

Using new variables:=  ;(16)= Jb  + Jl ;(17)
the dynamics of ight phase can be rewritten as,
8 x = 0>>< z = g(18)
 + 2h = K2h >>h:  = 0:

Since these are the independent second order linear
ODEs, we can easily dead-beat and _ , by onceswitching of the constant inputs.
Discretizing (18) using the piecewise constant inputs,1 ;if 0  t < Tv =2
 =
(19)
2 ;
if Tv =2  t < Tv ;where t is the time after the lift-o , and integrating(18), the control inputs can be calculated as follows.
   1lo1= B (Tv )A(Tv ) _; (20)
_2lowhere,

 


 
1 (J + J )  + C (T ) lo ; (21)
=
b
l
v
_
_
_ lo
Jb
and,

1 sin( h Tv ) cos(
h Tv )hA(Tv ) =cos( h Tv )h sin( h Tv )


h Tv
1
cos(
cos(
)
h Tv )
2
B (Tv ) =h Tvsin( h Tv )K
h sin( 2 )hh
1 Tv :C (Tv ) =0 1
Since det(B (Tv )) = Khh2 sin( h2Tv ) holds, control inputs (20) always exist, unless Tv = 0, or Tv = 2h .
4.3 Simulation results and discussion

We have simlulated the action of the Non-Dissipative
Touch-Down Control (19), (20) and (21), together
with (10)-(12), for wide range of initial conditions.
Figure 3 depicts the time evolutions of state variables (x;_ ; ; z ) and the energy level E , and the stepwise inputs (1 ; 2 ). The lower three small graphs
represent the selected images of the Poincare map.
Here, the free parameters of initial values are chosen
to be 0 = 0:25[rad] and x_ 0 = 2[m/s]. This corresponds to middle speed of hopping for the robot ofTable 3.
We can see from the images of the Ponincare mapthat the obtained gaits are quasi-periodic ones thatcan be seen in some Hamiltonian systems [14]. Theirperiods depend on the initial conditions.
5 Stabilization of periodic orbits
5.1 Stabilization to a desired periodic orbit
by adaptation of touch-down angle

As mentioned in the previous section, if the solution eventually converges to the 1-periocic orbits,
then the inputs also converge to the constant values. What should we do to change the period ? Remember Section 4.1, where we said the choice of thedesired touch-down angle is not unique.
We invented the following stepwise adaptation law:
 1
if k > p (22)2 (lo (k ) + lo (k p));(k ) =
lo (k );else:
Here, k  1 is the iteration step, and p  1 is thedesired period.
We also note that this method is similar to Pyragas's
method [15] to stabilize some chaotic orbits to thedesired periodic orbit, if we consider the touch-down
angle as a control inputs to the discrete dynamicalsystem (Poincare map of the system).
2628

5.2 Input minimization by adaptation ofspring sti ness
xD(t)

3
2

θ(t)

1
0
1

φ(t)

30

40

50

10

20

30

40

50

10

20

30

40

50

0

−0.5
0
1
z(t)

20

0

−1
0
0.5

0.5
10

20

30

40

50

10

20

30

40

50

105

τ1(k),τ2(k)

100
0
5
0

−5
0

10

20

30

40

50

−3

−0.5

60

70

80

φD(k)

0

θD(k)

−2.9

−3.1

−1

−3.2

0.3

0.4
θ(k)

0.5

−0.1

0

0.1 0.2 0.3
φ(k)

−1.4
zD(k)

where,> 0 is the adaptation gain.

5.3 Simulation results and discussion

0
0
110
E(t)

10

During walking or running, animal may adapt theirjoint sti ness by varying the tension of the antagonistic muscles to minimize energy consumption. In
order to simulate such a adaptation mechanism, we
examine an adaptation law for the sti ness of the hipspring to minimize the control inputs.
Since we are using piecewise constant inputs, thesimplest adaptation law will be given as:h (k + 1)(
1 (k )g ; if k > 2h (k ) + f2 (k )= qK ( 1 + 1 );if k = 1(23)
;
h Jb
Jl

Because of the rack of the space, only the result of thestabilization to 1-periodic gait is shown. Figure 4
shows the time evolution, where the touch-down angle adaptation is activated from the beginning, while
the spring sti ness adaptation is activated from the60th step for visibility. The initial conditions are thesame as those of Figure 3. We have drawn the line
between the points (images) of the Poincare map, forvisibility. We can see from the gure that the adaptation laws works well and control inputs eventuallyconverges to zero ! Thus, we have succeeded in complete passive running. It is not surprising that phasevolume is contracting because the system is not actually Hamiltonian.
We have also simulated stabilization to an arbitrary
periodic gaits and con rmed that the periods of solutions converge to the desired one. But the complete passive running has not yet been succeeded for
multi-periodic gait. We also observed there are signi cant di erence in convergence speed, dependingon the initial conditions.
6 Conclusion

−1.6
−1.8
−2
0.44

0.46
z(k)

0.48

Figure 3: Simulation result of \Non-Dissipative
Touch-Down Control". The lower four small graphs
represent the selected images of the Poincare map.
From this, you can see the gait obtained is quasiperiodic one.

We presented a novel controller for stabilization ofa passive one-legged hopping robot. First, based on
the dynamics of this nonlinear hybrid system, we derived a simple control law, Non-Dissipative TouchDown Control, to ensure the energy preservationand continuation. The generated hopping gaits were
found to be quasi-periodic orbits, which can be seen
in some Hamiltonian systems. Next, we proposeda simple parameter adaptation law to asymptotically stabilize the quasi-periodic gaits to the periodic
gaits of arbitrary period, and spring sti ness adaptation law to minimize control inputs. Simulation results demonstrated that the robot can eventually hop
2629

without any control inputs, especially for 1-periodic
gait. Our current study includes robustness analysis,
derivation of tracking controllers, and the application
to the one-legged robot \Kenken" [5], or to the bipedand quadruped robot under construction.

xD(t)

2.5

Acknowledgments

2

θ(t)

1.5
0
1

10

φ(t)

40

50

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

10

20

30

40

50

0

0.5

0
0
110
105

τ1(k),τ2(k)

100
0
5
0

−5
0

10

20

30

40

50

60

70

80

0.5

−2.5
φD(k)

θD(k)

−2.6

0

−0.5

−2.7

−1

−2.8

−1.5
0.3

0.35
θ(k)

0.4

−0.2

0
0.2
φ(k)

0.4

zD(k)

−1.8
−1.9
−2
−2.1
0.465

The rst author gives thanks to Claude Samson for
discussion about his paper.
References

−0.5
0
1
z(t)

30

0

−1
0
0.5

E(t)

20

0.47
0.475
z(k)

Figure 4: Simulation results of the stabilization to
1-periodic gait and the adaptation of spring sti ness.
Control inputs eventually converge to zero.

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