Advanced Robotics, Vol. 18, No. 4, pp. 357 – 381 (2004)
 VSP and Robotics Society of Japan 2004.

Also available online - www.vsppub.com

Full paper
Energy-preserving control of a passive one-legged running
robot
SANG-HO HYON ∗ and TAKASHI EMURA
Department of Bioengineering and Robotics, Graduate School of Engineering, Tohoku University,
Aramaki-Aza-Aoba 01, Sendai 980-8579, Japan
Received 24 June 2003; revised 3 October 2003; accepted 2 December 2003
Abstract—Fast and energy-efficient control is an increasingly important and attractive area of
research in legged locomotion. In this paper, we present a new simple controller for a planar onelegged passive running robot having a springy leg and a compliant hip joint. The most distinctive
advantage of the controller over previously proposed ones is it does not require any pre-planned
trajectories nor target dynamics. Instead, it utilizes exact non-linear dynamics. Our results are
summarized as follows. First, we propose an energy-preserving control strategy for energy-efficientand autonomous gait generation. This strategy is successfully implemented as a new touchdown
controller at the flight phase. Simulation results show that the robot can hop from a wide set of initialconditions. Moreover, the running gaits generated are found to be quasi-periodic orbits, which canbe seen in Hamiltonian systems. Since the controlled running gaits exist for every admissible energy
level, they have some robustness against disturbances. Next, it is shown that an adaptive control of
the touchdown angle, which is similar to a delayed feedback controller for a chaotic system, canasymptotically stabilize these quasi-periodic gaits to the periodic ones of the desired period, withsome limitations. In particular, for one-periodic gait, by using some additional adaptive controllers,
the robot eventually hops without any control inputs. Since our energy-preserving strategy is clear
and implementation of the controller is straightforward, we believe it can be easily applied to a wideclass of legged mechanisms.
Keywords: Legged locomotion; passive running; orbital stability; energy-preserving control; quasiperiodic orbit.

1. INTRODUCTION

After Matsuoka and Raibert’s works [1, 2], one-legged running robots filled witha leg spring have been widely studied both experimentally [3– 5] and theoretically [6– 8]. These studies contributed to biomechanics in investigating mechanisms of animal locomotion. On the other hand, it was found that in addition to∗ To whom correspondence should be addressed. E-mail: sangho@ieee.org

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S.-H. Hyon and T. Emurathe leg spring, a hip spring also plays an important role in animal running because
it enables the leg to be swung passively [9]. Tompson and Raibert showed thatthe spring-driven one-legged running robot depicted in Fig. 1 can hop without any
inputs, provided the initial conditions are appropriately chosen [10]. Therefore,
this is a good template model for the purpose of studying energy-efficient running.
However, a numerically determined solution was shown to be marginally stable andeventually fails without controls. Therefore, some suitable controller should be applied to ensure orbital stability.
The first successfull controller was found by Ahmadi and Buehler. They
applied Raibert’s celebrated Foot Placement Algorithm [2] to this passive running
robot model. They realized energy-efficient running in simulation [11] and evenperformed experiments [12]. However, this controller needs a neutral point, whichshould be approximated or pre-planned just as described in [2]. On the other
hand, François and Samson derived a new controller different from Raibert’s [13].
They applied a general control method used in a non-linear oscillatory system,
i.e. first constructing a Poincaré map (discrete system), then linearizing it aroundfixed points and finally applying a linear feedback controller to get asymptoticallystable periodic orbits. While François and Samson used approximated linear flow,
McGeer used numerically determined passive biped running gaits and extensively
studied their stability [14]. However, since the model cannot be integrable, theabove examples need approximated models, or approximated periodic solutions, toderive the controllers. Therefore, there remain various output errors arising frommodeling errors, and, hence, complete passive running gaits have not been obtained
through these controllers.
Motivated by these studies and to overcome the above problems, we present a
new simple controller to realize energy-efficient one-legged running [15]. Instead of
depending on some pre-calculated periodic solutions, or target (desired) dynamics,
here we utilize the dynamics of the original non-linear hybrid system to generate(unknown) natural running gaits.
For this purpose, we propose an energy-preserving control strategy. This means
the controller preserves the system energy as much as possible. The most importantreason why we use this strategy is that if the system energy is preserved, it isexpected that the system autonomously generates natural periodic gaits, just as some

Figure 1. Passive one-legged hopper.

Energy-preserving control of a running robot359

class of Hamiltonian system exhibit natural periodic orbit, even though our system
is a complex hybrid dynamical system.
In developing controllers, we chose a simple, but realistic model of a one-legged
hopper, the same as that of Ref. [13], which is reviewed in Section 2. In Section 3,
we explore the conditions of energy dissipation and normal phase switching for thismodel, along with our energy preserving control strategy. Then, a new control law
to ensure the above conditions, named the non-dissipative touchdown controller, isderived in Section 4. Simulation results and a discussion are also given. In addition,a new controller for asymptotic orbital stabilization to a desired period is proposed
and two kinds of adaptation laws for input minimization are presented in Section 5.
Section 6 gives a conclusion of the paper.

2. MATHEMATICAL MODEL OF A PASSIVE RUNNING ROBOT

We consider exactly the same model of passive running robot as in Ref. [13] in thispaper. In this section, the mathematical model is reviewed.
2.1. Model definition and notation
Figure 2 shows the model of a planar one-legged running robot considered here.
The robot is fitted not only with a leg spring, but also a hip spring.
We impose the following assumptions on this model as seen in much of the related
literature.

Figure 2. Model definition.

360

S.-H. Hyon and T. Emura(i) The center of mass (CM) of the body is just on the hip joint and the CM of theleg lies on the leg.
(ii) The mass of the foot (unsprung mass) is negligible.
(iii) The foot does not bounce back nor slip on the ground (inelastic impulsive
impact).
(iv) The springs are mass-less and non-dissipating.
We will explain assumptions (ii)–(iv).
Assumption (ii) means the foot of the robot is mass-less and, hence, there is noinertial force due to the foot mass. Of course, this assumption cannot be satisfied
exactly in a real machine, but it is rather easy to make the foot lightweight because
the the foot is a simple rod.
Thanks to this assumption, there is no impulse along the leg axis. Suppose thatthe robot performs vertical hopping and the foot touchdowns the ground along thedirection of the leg axis. Since the foot is mass-less, it sticks to the ground at the
moment of touchdown. In this case, there is no impulse nor kinetic energy loss.
Moreover, the robot has a leg spring, which produces a repulsive force once theleg is compressed. Therefore, even when the robot has a small mass on the foot,the spring pushes the foot to the ground. In this case, there may be some small
oscillation known as ‘chattering’. However, since the chattering can be rapidly
recovered by using low restitution material, we can assume the foot sticks to theground at the moment of touchdown. This is in accordance with observations of ourpreliminary experiment with our prototype robot, whose foot was made of thin tube
fitted with hyper-low restitution rubber.
On the other hand, when the robot receives impact in the perpendicular direction
to the longitudinal axis of the leg, energy loss occurs due to the inertia of the ‘upper
part’ of the leg. If the foot is too repulsive, it may bounce back due to this impulse
and the robot cannot touchdown the ground appropriately. However, for our robot,
it is easy to make the foot lightweight, its restitution low. More importantly, ourcontroller minimizes the touchdown impulse normal to the longitudinal axis of theleg, as shown later. Therefore, assumption (iii) on the inelastic impulse model
represents the touchdown dynamics of our one-legged robot most effectively.
Concerning slipping, we assumed large enough friction to simplify the derivationof the controller. It depends on the ground reaction force whether the foot slips
the ground or not. If the vector of the reaction force lies within the friction cone,
the foot does not slip. However, it is very difficult to predict the reaction force ingeneral, because the dynamics is not integrable. Nevertheless, the developmentof the controller to avoid slipping under a low-friction environment is not only
important, but also challenging work for legged robots and we will explore thisproblem in the future. Assumption (iv) is not a restrictive one because it is easy tocompensate for energy loss by applying leg force f .
Table 1 summarizes the variables that appear in this paper. The equations of
motion are composed of four phases: stance phase, lift-off phase, flight phase andEnergy-preserving control of a running robot

361

Table 1.
Variables

x
z
r
φ
θ
f
τ
µ
E
Ts
Tv

MeaningUnithorizontal position of hipvertical position of hip
leg length
body angle
leg angle
applied force of leg
applied torque of hipenergy dissipation coefficient
total mechanical energy
stance time
flight time

m
m
m
rad
rad
N
N
m/s
J
s
s

Table 2.
Subscripts indicating phases
Subscript

Meaning

td−
td+
td
lo

just before touchdown
just after touchdown
just before, or after touchdown
lift-off

Table 3.
Physical parameters

g
M
r0
Jb
Jl
Kl
Kh

Meaning

Unit

Value

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

m/s2
kg
m
kgm2
kgm2
N/m
N m/rad

9.8
12
0.5
0.5
0.11
3000
10

touchdown phase. Table 2 defines the phase-indicating subscripts for variables. For
example, ẋlo is the forward velocity of the CM at lift-off, θ̇td+ is the angular velocityof the leg just before touchdown, θtd is the leg angle of just before or just after
touchdown, etc. Table 3 shows the physical parameters, together with the values
used in later simulations. Note that Jl is an ‘equivalent leg inertia’ composed of the
moment of inertia of the leg mass of the leg, and mass of the body. Let us define themass of the body as Mb , and suppose the leg has mass Ml and moment of inertia Ju .
The mass of the leg is located below the hip joint, with the distance d. Then, ‘the362

S.-H. Hyon and T. Emuraequivalent moment of inertia of leg’ Jl is calculated as
Jl = Ju +Mb Ml 2d .
Mb + Ml

Note that if d = 0, then Jl coincides with the leg inertia Ju . For simplicity,
we assume total mass M = Mb + Ml is located on the hip joint without loss ofgenerality.
2.2. Equation of motion
At the stance phase, the leg compresses and extends, and the angular momentum of
the robot around the contact point evolutes under gravity. The dynamics is describedas:M r̈ + Kl (r − r0 ) − Mr θ̇ 2 = Mg(1 − cos θ) + f



d
(1)
Jl θ̈ + Jb φ̈ + (Mr 2 θ̇) = rMg sin θdt J φ̈ + K (θ − φ) = τ ,bhswhere f is the control force of the leg, while τs is the control torque of the hip joint,
which are applied during stance. The spring is initially loaded with the same valueof gravity force (Mg).
At the flight phase, the CM of the robot moves along a ballistic flight path and theangular momentum of the robot around the CM is preserved. Then, the dynamics isgiven by:ẍ = 0 z̈ = −g(2)Jl θ̈ + Jb φ̈ = 0
Jb φ̈ + Kh (θ − φ) = τv ,where τv represents the control torque of the hip joint, which is applied during flight.
By assumption (iii) in Section 2.1, the velocities of the generalized coordinates
change instantaneously at the touchdown phase, according to:Jl cos θtd

µtd−
 ẋtd+ = ẋt d− −Jl + Mr02Jl sin θtd żtd+ = żt d− − Jl + Mr 2 µtd−
0
(3)
Mr
0


=
θ̇
−
µ
θ̇

td+
td−
td−


Jl + Mr02φ̇td+ = φ̇td−ṙtd+ = żtd+ cos θtd − ẋtd+ sin θtd ,Energy-preserving control of a running robot363µtd− := ẋtd− cos θtd + żtd− sin θtd + r0 θ̇td− .

(4)

where

See Appendix A for derivation of these equations.
At lift-off phase, there are no discontinuous changes except for ṙlo = 0 due toassumption (iii).

3. ANALYSIS ON ENERGY DISSIPATION AND PHASE SWITCHING

If the total mechanical energy is completely preserved and normal phase switchingof the system is ensured, it is expected that energy-efficient gaits are autonomously
generated. Therefore, preserving energy will be our most important control goal
for this purpose and we call this idea the energy-preserving control strategy.
However, since energy loss is unavoidable in the real world, we should carefully
analyze why and how much energy can be lost. As for the model described in
the previous section, energy dissipation occurs for two reasons. One is the control
input. Whenever if we apply control input to achieve some control goals, there isalways energy consumption. The other reason is the impulsive force imposed at thetouchdown phase. In our model, the leg spring cannot absorb the impulse normal to
the longitudinal axis of the leg, while it can do along the leg axis.
The energy change between just before touchdown and just after touchdown is
uniquely determined by kinetic energy changes. This is because the configuration
variables remain constant during the impulse phase and, hence, the potential energy
does not change.
Etd+ − Etd−

td+
1
1
1
M(ẋ 2 + ż2 ) + Jb φ̇ 2 + Jl θ̇ 2
=
2
2
2
td−


1
1
2
2
2
2
M(ẋtd+ + żtd+ ) − M(ẋtd− + żtd− ) +
=
2
2
1 2
1 2
− Jl θ̇td−.
+ Jl θ̇td+
2
2

1
1
2
2
Jb φ̇td+− Jb φ̇td−
2
2

Substituting (3) into this equation yields:
Etd+ − Etd− = −

MJl
µ2 .
2(Jl + Mr02 ) td−(5)

Therefore, we call µtd− the energy dissipation coefficient, because if the condition:µtd− = 0,(6)

holds at touchdown and no control inputs are applied to the robot, then the totalmechanical energy of the system is preserved during running. Therefore, we cansay that (6) is a necessary condition for ‘complete passive running’.

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S.-H. Hyon and T. Emura

Figure 3. Two kinds of stubbing.

The next important thing we should consider is the condition for sustaining
running without falling to the groud. This condition can be defined by the following
statements.
(I) Normal phase switching is ensured(II) Angles of the body and leg at stance phase are bounded within certain region
Condition (I) is very important because our model is a hybrid system, which ismainly composed of two phases — stance phase and flight phase. If the transition
between two phases fails, the system cannot evolve anymore. A necessary condition
is found to be:ṙtd+ < 0.

(7)This condition comes from the first equation of (1), which describe the dynamics ofthe leg length r at the stance phase. In (1), at the moment of touchdown, Mr θ̇ 2 andMg(1−cos θ) are positive, and Kl (rtd+ −r0 ) = 0 holds. Therefore, if ṙtd+ > 0, there
are no attractive vector fields that constrain the foot to the ground. This prevents therobot from entering to stance phase from flight phase.
Note that (7) is not always satisfied. Figure 3 shows such two unfortunate
situations. Figure 3a indicates stubbing just after lift-off, while Fig. 3b shows
stubbing just before touchdown. The situation Fig. 3b cannot be recovered andshould be avoided in advance, while the situation in Fig. 3a can be avoided byretracting the leg during the flight phase (since the foot is mass-less, the leg length
is arbitrarily controlled ‘without influence on the other dynamics’).
For the same reason, the axial velocity of the leg just before lift-off must be
positive:ṙlo− > 0.

(8)

Fortunately, this condition is automatically satisfied if (7) holds because at the
stance phase, only repelling force is applied to the direction of the leg axis [16].
Therefore, (7) is sufficient.
Condition (II) is obvious, but it is difficult to derive the sufficient condition
mathematically because the dynamics at the stance phase is non-integrable. We
will pose this as a future task. Instead, we will limit ourselves to say that thiscondition could be almost neglected if we suppose no slip occurs between the footEnergy-preserving control of a running robot

365

and ground, as in Section 2.1. Actually, it was never observed in the simulation thatcondition (II) is violated before the robot lifts off the ground.

4. NON-DISSIPATIVE TOUCHDOWN CONTROL AND QUASI-PERIODICRUNNING GAITS

4.1. Control goal
If no energy is dissipated during running, the solution curve of this hybrid non-linear
system lies on the energy-invariant manifold. If this invariant hybrid flow forms aphysically meaningful periodic orbit, it will be the ‘gait’ we are looking for. For thispurpose, we derive a controller to ensure both equations (6) and (7) in this section,and call this new controller ‘non-dissipative touchdown control’.
We can do this by applying control inputs only at the flight phase and allowing
the robot to move freely at the stance phase with zero inputs. The reason why weare doing so is that if no energy dissipation occurs at touchdown (µtd− = 0), nointeraction between the robot and environment takes place, even if control inputs
are applied at the flight phase. Therefore, even if internal energy is not preserved
due to the control efforts, the total mechanical energy of the robot ‘including thepower source of the actuators’ is exactly preserved as:Ėsys := Ė − τ ψ̇ = 0,(9)where τ means control inputs defied by:
f = τs = 0,
τv =: τ.

(10)
(11)

Our control problem is to find the control input τ that makes the robot landat the time Tv , with (θtd , θ̇td− ) satisfying (6) and (7), for any given lift-off states(ẋlo , θlo , θ̇lo , φlo , φ̇lo ). Although this is a dead-beat control to bring Tv , θtd and θ̇td− tothe desired values, we only have to choose Tv and θtd , because θ̇td− is automaticallycalculated by (6).
There are, however, a large number of such pairs (Tv , θtd ) satisfying (7). Therefore, we determined it by the most simplest way described below. First, we choose
Tv so that the vertical velocity at touchdown and lift-off is the same in magnitude
and opposite in direction:
Tv =

2żlo
.
g

(12)

Next, we set the desired touchdown angle θ̄ to be symmetric to the lift-off angle
about the vertical axis:θ̄ = −θlo .

(13)

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S.-H. Hyon and T. Emura

Then, from (6) we obtain the desired touchdown angular velocity θ̇¯ as:
1(ẋtd− cos θtd + żtd− sin θtd )r0
1= − (ẋlo cos θlo + żlo sin θlo )r0¯= θ̇ =: θ̇.

(14)żtd+ = żtd− = żlo − gTv = −żlo ,ẋtd+ = ẋtd− = ẋlo ,

(15)
(16)

ṙtd+ = żtd cos θtd − ẋtd sin θtd= −żlo cos θlo + ẋlo sin θlo= −ṙlo .

(17)

θ̇td = −lo

Additionally, considering:

we obtain:

From this, we can see (7) is satisfied as long as we use (12) and (13), and initialconditions are appropriately chosen (described later).
Before closing this section, note again that (12) and (13) are no more than one
option. Moreover, this choice does not always results in one-periodic gait. Actually,
the variable set that falls into a one-periodic gait (or neutral orbit [8]) is unique fora given energy level and physical parameters.
4.2. Derivation of the controller
Using new variables:ψ = θ − φ,σ = Jb φ + Jl θ,

(18)
(19)

the dynamics of flight phase can be rewritten as:ẍ = 0 z̈ = −g
2h
2

ψ
=
−
τ
ψ̈
+


h

Kh



σ̈ = 0.

(20)

The last equation represents the conservation of the angular momentum of the robot.
Since these are independent second-order linear ODEs, we can easily dead-beat ψand ψ̇, by once-switching of the constant inputs. The reasons why we use dead-beatcontrol are (i) it is optimal from the viewpoint of input magnitude and (ii) it makes
it easy to minimize control inputs via some adaptive laws as shown in Section 5.2.

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367

Figure 4. Control input.

Discretizing (20) using piecewise constant inputs (Fig. 4) defied by:τ1 , if 0  t < Tv /2τ=τ2 , if Tv /2  t < Tv ,(21)

(where t is the time after the lift-off) and (20), the control inputs can be calculated
as:

 
ψ̄
ψ
τ1
lo
= B(Tv )−1− A(Tv )
,
(22)
τ2
ψ̇lo
ψ̇¯
where:
ψ̄
ψ̇¯

1
=
Jb

θ̄
(Jb + Jl ) ¯
θ̇



σlo
+ C(Tv )
σ̇lo


,

(23)

and:

1
cos(h Tv )sin(h Tv )hA(Tv ) = −h sin(h Tv ) cos(h Tv )h Tvcos(h Tv ) − cos1 2B(Tv ) =h Tv
Khh sin(h Tv ) − h sin
2


1 Tv
.
C(Tv ) =
0 1



h Tv

2


h Tv
h sin2−1 + cos

Since det(B(Tv )) = (h /Kh2 ){2 sin(h Tv /2) − sin(h Tv )}, control inputs (22)always exist, except when Tv = 0 or Tv = 2π /h .

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S.-H. Hyon and T. Emura

Figure 5. Simulation result of non-dissipative touchdown control. The top two graphs show time
evolutions of energy level (E) and control inputs (τ1 and τ2 ). The lower four graphs represent the
selected images of the Poincaré map, where notation ‘D’ represents the time derivatives of preceding
variables. Note that the position x is manually reset at each iterative crossing of the section forvisibility. Although all the points are bounded in some regions, no fixed points appeared. From this,we can see the gait obtained is a quasi-periodic one.

Energy-preserving control of a running robot

369

4.3. Simulation results and discussion
We have simulated the action of the non-dissipative touchdown control (21)–(23),together with (12)–(14), for wide range of initial conditions. The initial conditionmeans the states of just after touchdown hereinafter. Instead of finding themfor complete passive running via numerical methods such as Newton–Raphson’s
method, we just approximate them for the sake of simplicity (see Appendix B).
Figure 5 depicts one of the simulation results. Here, the free parameters of initialvalues are chosen to be θ0 = 0.25 rad and ẋ0 = 2 m/s. This corresponds to the
middle speed of running for the robot of Table 3. We can see from the images of the
Ponincaré map that the obtained gaits are quasi-periodic ones, which can be seen
in some Hamiltonian systems [17]. Their periods depend on the initial conditions.
Since the initial conditions are parameterized in θ0 and ẋ0 , we can easily get oneperiodic gaits. Figure 6 shows such a pair of parameters. Despite week restrictions
to the initial conditions (Appendix B), all solutions starting from them were found to
be stable running gaits; thus, the controller allows a wide region of initial conditions.
This is also the case for different physical parameter sets. Figure 7 shows stick
pictures of high-speed steady running at 5 m/s.
Looking at the time evolution of the energy, the period of the energy level is foundto be identical to those of state variables. Since the gaits obtained are non-linear
flows on the energy invariant manifold that meets (9):Ėsys := Ė − τ ψ̇ = 0,

Figure 6. Parameter pairs of complete passive running gaits (ẋ0 versus θ0 ).

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S.-H. Hyon and T. Emura

Figure 7. Stick picture of the subsequent two steps of steady running: running speed 5 m/s. The robot
moves from left to right.

it is quite natural to ask what happens when there are external disturbances. Figure 8
shows one of the simulation results in the case that we create an external disturbanceduring running. In this example, a large constant force f = 200 N is applied to therobot along to the leg axis at the stance phase (at 30th step). As a result, the energy
level is increased by about 25 J. Nevertheless, the robot continues stable running.
Note that according to the energy change, the gait has jumped to new quasi-periodic
orbits. This robustness against disturbances comes from the fact that our controller
does not depend on the energy level itself.
We have also tested a disturbance ‘at the flight phase’. However, in many cases,
the robot cannot recover the touchdown error and eventually falls down. This be
improved when we combine some feedback controllers together with the dead-beatcontroller.

5. ORBITAL STABILIZATION

Since a one-legged running robot is essentially unstable, stabilization is the mostimportant control goal and we have achieved it in the previous section. However,
the obtained gaits are quasi-periodic ones, not limit cycles. Nor are they passive
running gaits because the control input has non-zero value. Moreover, they requirevery large control inputs whenever inappropriate initial conditions are given. In thissection, we propose an adaptive orbital stabilization controller and adaptive input
minimization controllers.
5.1. Stabilization to a desired periodic orbit by adaptation of the touchdown angle
What should we do to change the period? Remember Section 4.1, where we notedthe choice of the desired touchdown angle is not unique. Considering the period of
orbits is identical to that of states on the Poincaré section, we invented the following
stepwise adaptation law:
 − 1 (θ (k) + θ (k − p)), if k > p
lo
lo
(24)
θ(k) =
2−θlo (k), else,

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371

Figure 8. Simulation results of the disturbance test. The top two graphs show time evolutions ofenergy level (E) and control inputs (τ1 and τ2 ). The lower four graphs represent the selected images of
the Poincaré map, where notation ‘D’ represents the time derivatives of preceding variables. Note thatthe position x is manually reset at each iterative crossing of the section for visibility. The disturbanceis applied at the 30th step, then a new gait appears, which can specified if we compare this with Fig. 5.

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S.-H. Hyon and T. Emura

where k  1 is the iteration step and p  1 is the desired period. The meaning of
this adaptation law is clear when considering the case of p = 1. In this case, thedesired touchdown angle is the negative one half of the summation of the current liftoff angle and the previous lift-off angle. This corresponds to the bisection methodin numerical analysis.
We note that this method is incidentally similar to the delayed feedback controller,
originally proposed by Pyragas, to stabilize some chaotic orbits to the desired
periodic orbit, if we consider the touchdown angle as a control input to the discrete
dynamical system (Poincaré map) [18, 19].
5.2. Input minimization via adaptive stiffness control and adaptive leg force control
To minimize control inputs, we focus on the magnitude of the control inputs.
Since we are using piecewise control input shown as Fig. 4, input magnitude isrepresented by a stepwise value:τ (k) := τ2 (k) − τ1 (k).

(25)

Considering the robot does not require any control inputs and internal energy ispreserved during complete passive running, it can be said that ‘if τ is increasing,the internal energy is insufficient and vice versa’. Therefore, if we can control theinternal energy level appropriately, τ could be controlled to zero! We propose twokinds of method below.
The first one is to control spring stiffness adaptively. For example, the hip stiffness
adaptation law is given by: (k) + γ · τ (k), if k > 2

 h
(26)
h (k + 1) =1
1
+, if k = 1,
 Kh
Jb Jl
where γ > 0 is the adaptation gain (note that h can be controlled by stiffness ofthe hip spring Kh ). If we remember animals can change their joint stiffness via pairsof antagonistic muscles, joint stiffness adaptation (or natural frequency adaptation)
may be a quite natural control. In the same manner, the leg spring adaptation law
also can be derived.
The second one is to control the internal energy by the external force during the
stance phase. For example, (piecewise constant) leg force can be used as:
f (k) = γf · sign(τ (k − 1)) · τ (k − 1),(27)

where γf > 0 is the adaptation gain. In the same manner, the hip torque also can beused to produce external force during the stance phase.
Note that the second method changes the total mechanical energy, while the first
method does not; it only transfers energy between the controller (motor) and theoscillator of hip spring.

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373

5.3. Simulation results and discussion
First, we will show the stabilization of a two-periodic orbit using (24), under the
initial conditions identical to those of Section 4.3. Figure 9 shows the simulation
results that actually asymptotically orbitally stabilized to a two-periodic orbit.
Interestingly, the energy level is bounded by the range of the original quasi-periodic
one.
Next, we will show the result of the stabilization to one-periodic with two kinds of
input minimization. Figure 10 shows the simulation, where input minimization via
spring stiffness adaptation law (26) is activated from the 60th step. The adaptaion
gain γ was set to 0.1; Figure 11 shows the same plots of input minimization via
the leg force adaptation law (27). The adaptaion gain γf was set to 40.
We can see from Figs 10 and 11 that both the adaptation laws work well andcontrol inputs eventually converges to zero! Thus, we have succeeded in complete
passive running. If we look at the energy level, it converges to 105.3 J in Fig. 10,
while in Fig. 11 it jumps from 105.3 J to 111.3 J.
We also tested stabilization to arbitrary periodic gaits and confirmed that for somedesired periods, especially for ‘the periods of odd number’, numerical solutions
did not converge. This interesting results will have a relationship with the issue of‘limitation of delayed feedback control’ discussed in the literature [19]. Complete
passive running was not found for multi-periodic gaits. It was also shown that there
are significant differences in convergence speed, depending on the initial conditions
or target period. Although we have not yet performed mathematical analysis, we
expect the reason lies in the structure of the model. Related to this problem, we
refer the reader to Poulakakis et al., interesting result [20], where they shows a
simplified planar quadruped model has two kind of passive bounding gaits.

6. CONCLUSIONSWe presented a new simple controller for a planar one-legged passive running robot
having a telescopic leg and springs.
First, we proposed an energy-preserving control strategy for energy-efficient andautonomous gait generation. This strategy is implemented as a new touchdown
controller at the flight phase. Specifically, based on the energy dissipation analysis,non-dissipative touchdown control to ensure energy preservation and normal phaseswitching was derived. Simulation results shows that the robot can hop from awide set of initial conditions and the generated running gaits are found to be quasiperiodic orbits, which can be seen in Hamiltonian systems. Since controlled runninggaits exist for every admissible energy level, they have some robustness against
disturbances.
Next, it was shown that simple adaptation laws, similar to delayed feedback controllers for chaotic systems, can asymptotically stabilize these quasi-periodic gaits
to the periodic ones of a desired period, with some limitations. In particular, espe-

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Figure 9. Simulation results of the orbital stabilization to a two-periodic gait. The top two graphs
show time evolutions of energy level (E) and control inputs (τ1 and τ2 ). The lower four graphs
represent the selected images of the Poincaré map, where notation ‘D’ represents the time derivativesof preceding variables. Note that the position x is manually reset at each iterative crossing of thesection for visibility. The images of the Poincaré map asymptotically converge to two fixed points.
This means we have a stable two-periodic orbit.

Energy-preserving control of a running robot

375

Figure 10. Simulation results of the orbital stabilization controller and the spring stiffness adaptation.
The top two graphs show time evolutions of energy level (E) and control inputs (τ1 and τ2 ). The lower
four graphs represent the selected images of the Poincaré map, where notation ‘D’ represents the time
derivatives of preceding variables. Note that the position x is manually reset at each iterative crossingof the section for visibility. The input minimization via the spring stiffness adaptation law (26) is
activated from the 60th step. The images of Poincaré map asymptotically converge to one fixed point
and the control inputs eventually converge to zero!

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S.-H. Hyon and T. Emura

Figure 11. Simulation results of the orbital stabilization controller and the leg force adaptation. The
top two graphs show time evolutions of energy level (E) and control inputs (τ1 and τ2 ). The lower
four graphs represent the selected images of the Poincaré map, where notation ‘D’ represents the time
derivatives of preceding variables. Note that the position x is manually reset at each iterative crossingof the section for visibility. The input minimization via the leg force adaptation law (27) is activatedfrom the 60th step. Different from Fig. 10, the fixed point is slightly shifted to another point due tothe energy change. Nevertheless, control inputs eventually converge to zero!

Energy-preserving control of a running robot

377

cially for the one-periodic gait, by applying adaptation laws, the robot eventuallyhops without any control inputs.
Since our energy-preserving strategy is clear and implementation of the controller
is straightforward, we believe it can be easily applied to a wide class of legged
mechanisms [21]. We are now constructing experimental models of one-legged andbiped robots to verify the controller [22].
Acknowledgement
S.-H. H. thanks to Claude Samson for discussion about literature [13].

REFERENCES
1. K. Masuoka, A mechanical model of repetitive hopping movements, Biomechanics 5, 251–258(1980) (in Japanese).
2. M. Raibert, Legged robots that balance. MIT Press, Cambridge, MA (1985).
3. J. Hodgins and M. Raibert, Biped gymnastics, Int. J. Robotics Res. 9 (2), 115–132 (1990).
4. G. Zeglin and B. Brown, Control of a bow leg hopping robot, in: Proc. Int. Conf. on Robotics
and Automation, Leuven, pp. 793–798 (1998).
5. S. H. Hyon, T. Emura and T. Mita, Dynamics-based control of one legged hopping robot, J. Syst.
Control Engng 217 (2), 83–98 (2003).
6. D. Koditschek and M. Buehler, Analysis of a simplified hopping robot, Int. J. Robotics Res. 10(6), 587–605 (1991).
7. R. M’Closkey and J. Burdick, Periodic motions of a hopping robot with vertical and forward
motion, Int. J. Robotics Res. 12 (3), 197–218 (1993).
8. W. Schwind and D. Koditschek, Control of forward velocity for a simplified planar hopping
robot, in: Proc. Int. Conf. on Robotics and Automation, Nagoya, pp. 691–696 (1995).
9. R. Alexander, Three uses for springs in legged locomotion, Int. J. Robotics Res. 9 (2), 53–61(1990).
10. C. Thompson and M. Raibert, Passive dynamic running, in: Experimental Robotics I, V. Hayward and O. Khatib (Eds), pp. 74–83. Springer, Berlin (1989).
11. M. Ahmadi and M. Buehler, Stable control of a simulated one-legged running robot with hip andleg compliance, Trans. Robotics Automat. 13 (1), 96–104 (1997).
12. M. Ahmadi and M. Buehler, The ARL Monopod II running robot: control and energetics, in:Proc. Int. Conf. on Robotics and Automation, Antwerpen, pp. 1689–1694 (1999).
13. C. François and C. Samson, A new approach to the control of the planar one-legged hopper, Int.
J. Robotics Res. 17 (11), 1150–1166 (1998).
14. T. McGeer, Passive bipedal running, in: Proc. Royal Society of London: Biological Sciences(1990).
15. S. Hyon and T. Emura, Quasi-periodic gaits of passive one-legged hopper, in: Proc. Int. Conf.
on Intelligent Robotics and Systems, Lausanne, pp. 2625–2630 (2002).
16. H. Goldstein, C. Poole and J. Safko, Classical Mechanics, 3rd edn. Addison-Wesley, Reading,MA (2002).
17. V. I. Arnold, Mathematical Methods of Classical Mechanics. Springer, Berlin (1978).
18. K. Pyragas, Continuous control of chaos by self-controlling feedback, Phys. Lett. A170, 421–428(1992).
19. T. Ushio, Limitation of delayed feedback control in nonlinear discrete-time systems, Trans. Circ.
Sys. I 43, 815–816 (1996).

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20. I. Poulakakis, E. Papadopoulos and M. Buehler, On the stable passive dynamics of quadruped
running, in: Proc. Int. Conf. on Robotics and Automation, Taipei, pp. 1368–1373 (2003).
21. S. Hyon and T. Emura, Aerial posture control for 3D biped running using compensator aroundyaw axis, in: Proc. Int. Conf. on Robotics and Automation, Taipei, pp. 57–62 (2003).
22. T. Emura, S. Hyon, Y. Kuroda and M. Suzuki, A biped hopper controlled around yaw axis bybody-twisting motion, in: Proc. Int. Conf. on Mechatronics, Longhborough, pp. 505–510 (2003).
23. T. McGeer, Passive dynamic walking, Int. J. Robotics Res. 9 (2), 62–82 (1990).
24. A. Goswami, B. Espiau and A. Keramame, Limit cycles and their stability in a passive biped
gait, in: Proc. Int. Conf. on Robotics and Automation, Minneapolis, MN, pp. 246–251 (1996).

APPENDIX A: DERIVATION OF IMPULSE EQUATION (3)Here, we will derive impulse equation (3). Let us begin with the constrained
equation of motion. At the flight phase, equations of motion are described asfollows:M ẍ = 0 M z̈ = −Mg(A.1)
Jl θ̈ + Jb φ̈ = 0
Jb φ̈ + Kh (θ − φ) = 0,
where we removed the control input for simplicity.
At the stance phase, the robot has a constraint as:
x + r sin θ = 0
z − r cos θ = 0.

(A.2)

Using this constraint, we can obtain constrained equations of motion at the stancephase:
M ẍ = λx M z̈ = −Mg + λz
(A.3)
Jl θ̈ + Jb φ̈ = λx r cos θ + λz r sin θJb φ̈ + Kh (θ − φ) = 0,
where λx and λz are Lagrange multipliers, which correspond to the ground reaction
forces of the horizontal and vertical direction respectively. The coefficient terms ofthese multipliers are each component of the Jacobian matrix of (A.2). Combining(A.2) and (A.3), we can obtain the same equations (1) as in Section 2.2.
Now, let us derive the impulse equation. The following calculations are basedon Lagrange’s impulse equation of motion. At inelastic impact, phase constraint
appears suddenly. Due to that, the velocities change instantaneously, while theconfiguration remains unchanged. In our case, constraint means equation (A.2),
i.e. the foot sticks on the ground. This constraint yields an impulsive force(called the constraint impulse). The impulse occurs during a short period of timeEnergy-preserving control of a running robot379

t := ttd+ − ttd− , i.e. during just before touchdown and just after touchdown. The
velocity changes due to the impulse during the time interval t can be calculated
from (3) to be:Mẋ = λ̂x Mż = λ̂
z
(A.4)

 Jl θ̇ = λ̂x r cos θ + λ̂z r sin θJb φ̇ = 0,
touchdown. ż, θ̇where ẋ := ẋtd+ − ẋtd− is the velocity change before/after
 ttd+ttd+
λx dt and λ̂z = ttd−λz dt
and φ̇ are also defined in the same manner. λ̂x = ttd−represent the constraint impulse. These equations represent the instantaneouschange of kinetic energy due to the impulsive forces.
Remembering impulse is applied only to the perpendicular direction of thelongitudinal axis of the leg and there is no impulse to the tangential axis of theleg (see the discussion about our assumption in Section 2.1), we haveλ̂x = p̂ cos θ(A.5)
λ̂z = p̂ sin θ,
where p̂ is the impulse normal to the leg axis. Then, (A.4) is re-written as:Mẋ = p̂ cos θ Mż = p̂ sin θJl θ̇ = r p̂
Jb φ̇ = 0.

(A.6)

On the other hand, from the time derivative of (A.2), the following velocity
constraint holds just after touchdown.
ẋtd+ + ṙtd+ sin θtd + r0 θ̇td+ cos θtd = 0(A.7)
żtd+ − ṙtd+ cos θtd + r0 θ̇td+ sin θtd = 0,
where we used θtd = θtd− = θtd+ , since the configuration variables do not change
during a short period of impact. From this equation, we obtain:ẋtd+ cos θtd + żtd+ sin θtd + r0 θ̇td+ = 0.

(A.8)

Substituting (A.6) into (A.8), the constraint impulse normal to the longitudinal
axis of the leg is obtained as follows:

380

S.-H. Hyon and T. Emuraẋtd+ +cos θtdsin θtdr0
cos θtd + żtd+ +sin θtd + r0 θ̇td+ + p̂ = 0
MMJl
MJl (−ẋtd− cos θtd − żtd− sin θtd − r0 θ̇td− ). (A.9)→ p̂ =
Mr02 + Jl

From (A.6) and (A.9), we can finally obtain (3).

APPENDIX B: APPROXIMATION OF THE ONE-PERIODIC SOLUTION

The initial conditions for one-periodic passive running could be easily approximatedif we suppose synchronization between the two oscillatory motions of leg swinging
and compression (see Ref. [11] for details).
Initial conditions to be determined are (ẋ0 , ż0 , θ0 , θ̇0 , φ0 , φ̇0 , ṙ0 ), where we choose(ẋ0 , θ0 ) as free parameters and let (ż0 , θ̇0 , φ0 , φ̇0 , ṙ0 ) be calculated. Supposing thatthe leg is swung at the natural frequency of the hip spring both at the stance phaseand the flight phase yields:h (Ts + Tv ) = 2π,

√

(B.1)

where h := Kh ((1/Jb ) + (1/Jl )) is the natural frequency of leg swinging.
Although the stance duration Ts depends on the initial conditions at the stance phase,near the vertical running (θ ≈ 0), we get:π
= π/ .
(B.2)
Ts ≈ √
Kl /M
Then, flight duration Tv becomes:
Tv = 2π h − Ts = π

2
1
.
−
h 

(B.3)

From these equations, the initial vertical velocity of the CM is calculated to be:ż0 = −2g/Tv = −

2gπ
.
1
2
−
h 

(B.4)

Next, from (6), θ̇0 becomes:θ̇0 = −

1(ẋ0 cos θ0 + ż0 sin θ0 ).
r0

(B.5)

Since we assume the natural oscillation of the hip spring, we obtain:
φ0 = −

Jl
θ0 ,
Jb

(B.6)

Energy-preserving control of a running robot

φ̇0 = −

Jl
θ̇0 .
Jb

381

(B.7)

Finally, from (3), ṙ0 is shown to be:ṙ0 = ż0 cos θ0 − ẋ0 sin θ0 .

(B.8)

Note that the solutions starting from these approximated initial conditions are notstable except for ‘trivial’ vertical hopping, i.e. even if we can obtain the exact initialcondition for passive running, the robot eventually falls to the ground, because
passive running is unstable (the Poincaré map has eigenvalues greater than 1). Thisis the point that differs most from passive walking robots because they have somebasins of attraction, where the robot can walk down a slope without control [23, 24].

ABOUT THE AUTHORS
Sang-Ho Hyon received the ME degree from Waseda Univerisity in 1998 and
PhD degree from Tokyo Institute of Technology in 2002. He is currently a
Research Associate in the Department of Bioengineerign and Robotics of Tohoku
University. His research interests include legged locomotion and non-linear
dynamical systems. He is a member of the RSJ, JSME, SICE and IEEE.

Takashi Emura received the PhD degree from Tohoku University in 1969. He
is currently a Professor in the Department of Bio-Robotics of Tohoku University.
He succeeded in walking a biped dynamically and found that the semicircular
canal should be defined as an angular velocity sensor from 1972 through 1973.
His research interests include legged robots, automatic navigation vehicles, highprecision servomechanism for NC machines, etc. He is a member of the RSJ,
JSME, SICE, JSPE and IEEE.