Proceedings of the 2004 IEEE
International Conference on Robotics & Automation
New Orleans, LA • April 2004

Running Control of a Planar Biped Robot based on
Energy-Preserving Strategy
Sang-Ho Hyon

Takashi Emura

Department of Bioengineering and Robotics
Graduate School of Engineering, Tohoku University
Aramaki-Aza-Aoba 01, Sendai 980-8579, Japan
Email: sangho@ieee.org

Department of Bioengineering and Robotics
Graduate School of Engineering, Tohoku University
Aramaki-Aza-Aoba 01, Sendai 980-8579, Japan
Email: emura@emura.mech.tohoku.ac.jp

Abstract— In this paper, we extend running controller ofpassive one-legged hopper to a planar biped robot with torso, andevaluate the controller on simulations. The controller is derived
based on energy-preserving strategy and it actually preserves
mechanical energy at touchdown. Interestingly, zero dynamics
of decoupling controller (dynamics about pair of controlled leg)is found to be stable. Combining simple attitude controller atstance phase generates stable periodic running gaits of arbitral
period. The control performance is shown to be better than asimple PD-feedback control of leg placement.

hip spring Kh
BODYφJb (x,z)MτLEG JuθI. I NTRODUCTION

leg spring
Kl

A. Background
After the Raibert’s excellent works [1], running robots havebeen widely studied both experimentally [2][3][4][5][6] andtheoretically [7][8]. On the other hand, recently there are manystudies on biped humanoid robots with the aim of practicalapplication. Therefore, enhancing the mobility of them isimportant target to be reached. Running control of bipedhumanoid robots is included in such targets. Energy-efficient
control of fast running is especially crucial for autonomous
humanoid robots because it directly extends operation time.
In this connection, there are some remarkable researches onenergy-efficient running control. Tompson and Raibert showedthat spring-driven one-legged hopping robot can hop without
any inputs, provided if the initial conditions were appropriately
chosen [9]. Ahmadi and Buehler applied Raibert’s algorithm tothis robot and realized energy-efficient hopping in simulationand experiment [10]. François and Samson derived a rather
systematic controller based on the general control method usedin nonlinear oscillatory system [11].
B. Energy-preserving controller for one-legged hopper
Motivated from their works, we proposed, in [12], alternative controller based on its energy analysis for a planar onelegged robot shown in Fig. 1.
The underlining principle is energy-preserving control strategy. This means the controller preserves system energy as
much as possible. The most important reason why we use thisstrategy is; if the system energy is preserved, it is expectedthat the system autonomously generates natural periodic gaits,
just as some class of Hamiltonian system exhibit natural periodic orbit [13]. Instead of depending on some pre-calculated
0-7803-8232-3/04/$17.00 ©2004 IEEE

Z
X

Fig. 1.

r

f
FOOT(no-mass)

Passive one-legged hopper

periodic solutions, or target (desired) dynamics, analysis onenergy change of the original nonlinear hybrid system were
utilized as shown below.
First, we choose desired touchdown angle θ d and angularvelocity θ̇d at the moment of lift-off to meet the following
energy non-dissipation condition:µtd− := ẋtd− cos θtd + żtd− sin θtd + r0 = θ̇td− = 0(1)

Subscript “td” means just the moment of touchdown and “td-”or “td+” mean just before / after touchdown hereinafter. Since
the energy change between just before and after touchdown iscalculated as
M Jl
µ2 ,
(2)
Etd+ − Etd− = −2(Jl + M r02 ) td−
condition (1) means there is no energy exchange between the
robot and the ground, provided if no control input applied
during stance phase. Having determined θ d and θ̇d , finally wecan apply simple linear dead-beat controller because the flight
dynamics is integrable.
As a result, interesting quasi-periodic gaits, which can beseen in some Hamiltonian system, were found, and both period

3791

4

107

2
τ [Nm]

E [J]

108

106
105

0

−2

104

−4

103
0

−6
0

20
40
time [s]

τ
1
τ
2

50
steps

100

−1.85

C. Paper organization−1.9

The purpose of this paper it to extend the controller ofpassive one-legged hopper to biped robots. This was partiallydone on a 3D biped model in [17], where a rotor rotatingaround yaw-axis of torso was introduced. In this paper, weconsider a highly nonlinear planar biped model having massive
legs and torso and try to achieve stable periodic running gaits
of it. Specifically, we derive dead-beat controller at flightphase based on the energy-preserving strategy to preserve
mechanical energy at touchdown. Then, we combine somestance-phase controllers to get stable running gaits.
The paper is organized as follows: Section II introduces
our new biped running robot and the equations of motion of
simplified model are given. Section III extends the controller
described in Section I to a planar biped robot with torso, by
introducing nonlinear decoupling control and target dynamics.
Section IV shows simulation results of biped running. Section
V concludes this paper.

zD(k)

xD(k)

2.2
2.1

−1.95
−2

2
1.9
−0.18

−2.05
−0.16
x(k)

−0.14

−2.4

0.5

−2.5

0

0.47

0.475
z(k)

0.48

−0.2

0
0.2
φ(k)

0.4

φD(k)

θD(k)

−2.6

−0.5

−2.7
−2.8

−1

−2.9
0.25 0.3 0.35 0.4
θ(k)stabilization and one-periodic passive running were achieved
in simulation. It was also found that an adaptive control oftouchdown angle, which is similar to the delayed feedback
controller for chaotic system [15], can asymptotically stabilize
these quasi-periodic gaits to desired periodic ones. Especiallyfor 1-periodic gait, by using some additional adaptive controllers, the robot eventually hops without any control inputs,that is, complete passive running is obtained. Fig. 2 is an
example of simulation results, which shows an adaptation
control law achieves complete passive running. Fig. 3 isthe stick pictures of high-speed passive running at 5 [m/s].
Complete description and results can be found in the literature
[14].

−1.5

II. A PLANAR BIPED ROBOT

Vertical height [m]Fig. 2. Simulation results of the orbital stabilization controller and the spring
stiffness adaptation. The top two graph shows 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 thetime derivatives of preceding variables. Please note that the position x is
manually reset at each iterative crossing of the section, for visibility. The
input minimization via spring stiffness adaptation law is activated from the60th step. You can see the images of Poincaré Map asymptotically converge
to one fixed point and the control inputs eventually converge to zero!

1
0.5
0
5

6

7

8
Horizontal distance [m]

9

10

11

Fig. 3. Subsequent two steps of 1-periodic passive running gait. Runningspeed is 5 [m/s]. The robot moves from left to the right

Figure 4 shows a CAD model of newly developed planar
biped robot, SkipperII. The robot has two springy telescopic
legs swinging around hip joints. Leg actuators are mounted
parallel to the leg spring. Overall height of the robot is 0.75[m] and the total weight is about 7 [kg]. The hardware designand experiments will be presented elsewhere.
Figure 5 shows the definition of mathematical model. The
generalized coordinates are defined as the position of centerof gravity (CoG), x = (x g , zg )T ∈ R2 , the attitude of thetorso, φ ∈ R1 , and joint angles, ψ = (ψ 1 , ψ2 )T ∈ R2 . Mb
and m are the mass of torso and leg respectively. I and J
are the moment of inertia about CoG of the torso and CoG ofthe leg respectively. All principal axes of each rigid part arecoincident with their center axes. Table I shows the physicalparameters, together with the values used in later simulations.
This model is highly nonlinear because it has massive legs
and torso, whose CoG are located away the hip joint.
Additionally, the following assumptions are imposed on themodel:(A) Mass of the foot (unsprung mass) is negligible(B) The foot does not bounce back, nor slip the ground(inelastic impulsive impact)

3792

φ

Lb

Torso
Mb, I

(xg,zg)Leg2

m, J

Fig. 4.

Z

CAD model of a new planar biped robot, SkipperII

ψ2 −ψ
1
θ2

L

Leg1

−θ1

m, J
r0

X
(C) The springs are mass-less and non-dissipating
A running motion is composed of successive phase transitions; stance → flight → touchdown → stance → · · ·. Below
we show the equations of motion at each phase in abbreviatedfrom.

Lateral viewFig. 5.

Definition of mathematical model

A. Equations of motion at stance / flight phase
The equations of motion at stance phase (right leg support)
can be derived by Lagrangian formulation.




M 0
0
2×3
 ẍ
 0 M
  φ̈ 
N11 N12 
03×2
ψ̈N21 M22




0


f
 −M g 
λ



=
(3)
 H1  + 03×1 + 0τH2Here, M = 2m + M b is the total mass and g is gravity
T∈ R1×2 , and N22 ∈acceleration. N 11 ∈ R1 , N21 = N12
2×2
1R
are inertia matrix. H 1 ∈ R and H2 ∈ R2×1 arenonlinear terms including centrifugal force, Coriolis force,gravity force and the leg spring forces.
The generalized foces are f = (f 1 , f2 )T ∈ R2 , where f1
and f2 are the forces of leg actuators, and τ = (τ 1 , τ2 )T ∈ R2 ,where τ1 and τ2 are the torques of hip actuators, and λ =(λx , λz )T ∈ R2 represents ground reaction forces. Thanks
TABLE IR OBOT PARAMETERS
Variables
L, Lb , r0
Mb , m
I, J
Kl

Unit
m
kg
kgm2
N/m

Values0.16, 0.2, 0.08
4.11, 2.13
0.05 , 0.015
3000

to assumption (C), we assume f = 0 throughout this paper.
Dynamics of flight phase can be obtained by removing λ andf from (3).
B. Equations of motion at touchdown phase
From assumption (B), touchdown equation can be expressedas Lagrange’s impulsive equation.




M 0
0
2×3
 ∆ẋ
 0 M
  ∆φ̇  = E(q)T λ̂p , (4)N11 N12 
03×2∆ψ̇N21 M22
where ∆ẋ := ẋtd+ − ẋtd− means instantaneous velocitychanges, so does ∆ė and ∆ṗ. λ̂p ∈ R1 is the constraint impulse perpendicular to leg axis, associated with instantaneous
velocity constraint at touchdown:


ẋ0
(5)
[ cos θ1 sin θ1 0 −(L + r0 ) 0 ]  φ̇  = 0,ψ̇where x0 ∈ R2 is the hip position, and r 0 is the nominal leg
length. Note that the constraint impulse along leg axis is completely absorbed by the leg spring, due to the assumption (A)and (C), and hence does not appears in (4). After coordinatetransformation from x 0 to x, (5) becomes toẋE(q)  φ̇  = 0,
(6)
ψ̇

3793

where E(q) ∈ R1×5 is jacobian of velocity constraint, whichappears in (4).

Next, we define a target dynamics. Conservation law of theangular momentum around CoG can be expressed as
J0 φ̇ + J1 ψ̇1 + J2 ψ̇2 = P0 ,III. C ONTROLLER IMPLEMENTATION VIA DECOUPLING
A. Stance phase controller
It is well known that a torso mounted below the hip joint
has passive stability [3]. In most of biped robots, however, thetorso is located above the hip joint like Fig. 5. In contrast to
one-legged robot, in which no control inputs are applied, thebiped robot cannot hold its torso upright posture without pitch
control, because of reaction forces from hip joints.
Here we are temporarily using simple pitch control, regardless of expense of actuator energy:
τ1 = −K1p φ − K1d φ̇,(7)where K1p ≥ 0 and K 1d ≥ 0. The supporting leg is assumed
to be Leg1 indicated in Fig. 5.
For swinging leg (Leg2), we are also temporarily controllingit byτ2 = −K2p (ψ2 + ψ1 ) − K2d (ψ̇2 + ψ̇1 ).

(8)

Under this controller, counter oscillation of each leg is expected during running.
Remarks:
Since we are not satisfied with this somewhat“factitious” controller that can deteriorate some useful intrinsic
nonlinear dynamics at stance phase. Specifically, as long as
we use that controller, internal energy cannot be preserved.
The solution will be introducing hip springs that can preserve
energy and produce oscillatory motion, as in the case ofpassive one-legged model.
B. Flight phase controller
To apply the controller of one-legged model to the biped,
decoupling control and target dynamics are introduced. Here
we suppose the next touchdown leg is Leg1.
First, the equation of motion (3) is decoupled using a newcontrol input. Rewriting the lower part of (3), we obtains
N11 φ̈ + N12 ψ̈ + H1 = 0
N21 φ̈ + N22 ψ̈ + H2 = τ .

(9)
(10)

Equation (9) represents time differential of conservation law ofangular velocity around CoG of the robot. Rearranging theseequations, we obtain
−1

φ̈
−H1
N11 N12
.
(11)
=
N
N
21
22
−H
ψ̈
2+τ
Then we decouple the lower part of this equation by newcontrol inputs u 1 and u2 defined later (partial feedback lenearization):

 

ψ¨1
u1
=
(12)
u2
ψ¨2
This equation means we can control ψ̈ arbitrarily by newcontrol inputs u 1 and u2 .

(13)

where Ji (i = 0, 1, 2) is nonlinear inertia terms and P 0 means
initial angular momentum of flight phase. If we choseψ̇2 =1(P0 − J0 φ̇0 − J1 ψ̇1 )
J2

(14)

(φ̇0 is the initial value of φ̇ at flight phase), we get thefollowing target dynamics.
= φ̇0

φ̇

(15)

This means “another first integral of motion is created by a
feedback control”. Note that it suits normal biped running gait
because this constraint holds approximately, when the bothlegs are swung symmetrically ( ψ˙1 + ψ˙2 = 0). The control
input u2 that achieve (14) can be easily calculated by the time
derivative of (14)
u2 =

1
J2

d
dt

P0 −

J0
J2

d
dt

φ̇0 −

d
dt

J1
J2

ψ̇1 −

J1
u1 (16)J2

Then, the remained task is to determine control input, u 1 in
(16). The control objective is to dead-beat the absolute angleof the swing leg θ 1 (Fig. 5) and its velocity θ˙1 to some desired
values at given fixed time T v (flight time). As we explainedin Section I, desired values are chosen to preserve energy
at touchdown. For the biped robot case, condition of energy
preservation is given by:
λ̂p = 0,(17)

where λ̂p can be calculated explicitly from (4) and (6). For
a given flight time T v , however, there are many pairs of θ dand θ̇d , as in the case of one-legged model. For example, toachieve orbital stabilization, θ d can be chosen as follows.
θd (k) =− 21 (θlo (k) + θlo (k − p)),−θlo (k),if k > p
else,

(18)

where k > 1 is the iteration step and p > 1 is a desired period.
This is the same adaptive control law for passive one-legged
hopper in [12]. Note that the desired angular velocity θ̇d (k) is
automatically determined by (17) accordingly.
Since (12) is a trivial second order linear ODE, we caneasily dead-beat ψ 1 and ψ˙1 , by only once-switching of theconstant inputs. Note that θ 1 = ψ1 + φ and θ˙1 = ψ˙1 + φ̇. If
we define new variables




φ
θ1
Φ :=
(19)
, Θ :=φ̇θ˙1and descretize the lower part of (12) using the piecewise
constant inputs
u1

=

u1a ,
u1b ,

if 0 ≤ t < Tv /2
if Tv /2 ≤ t < Tv ,(20)

the dead-beat control inputs can be calculated as follows:

3794



u1a
u1b



0.8

= B(Tv )−1 {Ψ(Tv ) − A(Tv )Ψ(0)}

3
x [m]
g
z [m]

0.6

(21)

g

where


A(Tv ) =B(Tv ) =xgD [m/s]z D [m/s]

1 Tv
0 1

 3

2
8 Tv
1
2 Tv

g

1



0.2

,

(22)

1
8 Tv
1
2 Tv

0

0


2

Ψ(Tv ) = Θd − A(Tv )Φ(0)

−0.2
1

(23)
(24)

1.5

2
2.5
Time [s]1and Θd is the desired value of Θ.

34

ψ D [rad/s]
1
ψ D [rad/s]

2

φD [rad/s]

2

φ [rad]

0
−2

−0.5
−4
−1
1

1.5

2
2.5
Time [s]

10

−6
1

3

5

1.5

2
2.5
Time [s]

3

30

τ
1
τ2

P [W]
1
P [W]

20

2

10
0

0
−10

−5
−20
−10
1

1.5

2
2.5
Time [s]

−30
1

3

1.5

2
2.5
Time [s]

3

2

0.1

1

0.05
φD [rad/s]

zgD [m/s]

Fig. 6. Time evolutions of stable 1-periodic running gait. The top fourgraphs shows time evolutions of each state variable, where notation “D”
represents the time derivatives of preceding variables. xg is manually resetat each iterative crossing of the section, for visibility. The left bottom graphis the hip torques, while the right bottom graph shows power consumption ateach hip actuator.

0
−1

(25)

0.5
zg [m]

0.6

−0.1
−2

10

10

5

5

ψ2D [rad/s]

ψ1D [rad/s]

It is worth comparing the proposed controller with a simple
PD-feedback leg placement at flight phase:

0
−0.05

−2
0.4

B. Comparison with a simple PD-feedback leg placement

0−5

The main difference between this and the proposed controller
is whether the angular velocity at touchdown is controlled
or not. Fig. 10 shows the phase portraits of the simulation,
where control parameter was chosen as K sp1 = 100, Ksd1 =10, Ksp2 = 100, Ksd2 = 10. Comparing to Fig. 7, we cansee the symmetricity is damaged large. Although the gaits arestill stable owing to (18), energy dissipation always occurs at
touchdown, and it makes control inputs quite large at stance2
2.5
Time [s]60We have simulated the action of the controller proposed in
the previous section. The simulation starts from stance phaseof Leg1 and initial height z g0 , or initial vertical velocity ż g0
was chosen large so that the swinging leg does not stub against
ground. Fig. 6 shows time evolutions of each state variable andFig. 7 represents phase portraits. Fig. 8 shows corresponding
animation. They show stable 1-periodic running gaits. Control
parameters are determined as K p1 = 200, Kd1 = 50, Kp2 =100, Kd2 = 10 in (7). Interestingly, dynamics about Leg2
(zero dynamics of decoupling controller) is found to be stable.
Actually, the motion of Leg2 indicates counter oscillation ofLeg1. Although the running gait seems to have symmetricity,
the motion of torso is slightly asymmetric, as recognized fromthe right top graph of Fig. 7. This asymmetricity become
more significant if the feedback gains of (7) become smaller.
Without attitude control, the robot falls down after a few steps.
The bottom two graphs of Fig. 6 indicate that the attitudecontrol at stance phase requires more power than a dead-beatcontrol at flight phase.
We also simulated stabilization to (unknown) multi-periodic
gaits. For example, if we set p = 2 in (18), we obtain a stable
2-periodic running gait, as shown in Fig. 9.
The controller is also applied to a more precise model using
DADS, commercial simulation software (www.cybernet.co.jp),
and we obtained almost same results. Animation video will be
included in the conference proceedings.

1.5

2

IV. S IMULATION RESULTS−Ksp1(θ1 − θd ) − Ksd1 (θ̇1 )−Ksp2(ψ2 + ψ1 ) − Ksd2 (ψ̇2 )

−1
1

3

ψ [rad]1
ψ [rad]0.5

A. Stable periodic running

τ=

2

0.4

−10
−1

0

2
φ [rad]40ψ [rad]

0.5

6
−3

x 10

0
−5

−0.5

0
ψ [rad]
1

0.5

1

−10
−1

−0.5

1

2

Fig. 7.
Phase portraits of stable 1-periodic running gait. Notation “D”
represents the time derivatives of preceding variables. Note that the scaleof the abscissa of right top is 10−3 and small horizontal curve running form
φ = 0 [rad] to 2 × 10−3 [rad] does not mean an instantaneous jump, but
indicates the result of high-gain PD-feedback about body pitch.

3795

0.5

1

0

0.6
0.4

−0.5

4

4.5
5
5.5
Horizontal distance [m]

6

−2
0.4

6.5

0.1

1

0.05
φD [rad/s]

2

0
−1

ψ1D [rad/s]

Fig. 8. Stic animation of subsequent three steps of 1-periodic biped running,corresponding Fig. 6. The robot moves from left to the right.

−1
0.5
zg [m]

0.6

−1.5
−0.03

10

10

5

5

ψ2D [rad/s]

3.5

zgD [m/s]

0
−1

0.2
0

0
−5

−10
−0.5

−0.01
φ [rad]00ψ [rad]

0.5

0
−5

0
ψ [rad]

0.5

−10
−0.5

1

0

−0.02

2

−0.05

−2
0.4

0.5
zg [m]

0.6

−0.1
−2

10

10

5

5

ψ2D [rad/s]

ψ1D [rad/s]

φD [rad/s]

2

zgD [m/s]

Vertical height [m]

1
0.8

0
−5

−5
−10
−1

0
ψ [rad]0.5

1

2
φ [rad]4

Fig. 10. Phase portraits of stable running gait using simple PD-feedbackleg placement. Notation “D” represents the time derivatives of preceding
variables. In the bottom two figures, small vertical lines running form ψ̇1,2 =0 [rad/s] to 5 [rad/s] implies instantaneous velocity changes associated withtouchdown impulse equation. This shows simple PD-feedback controller does
not preserve energy at touchdown.

6
−3

x 10

0

−10
−1

−0.5

0

−0.5

1

0
ψ [rad]

0.5

DOF at roll axis and applying the same controller. Although
our controller cannot be applied to humanoid robot directly,
we believe there is some embedding transformation.

1

2

R EFERENCES
Fig. 9.
Phase portraits of stable 2-periodic running gait. Notation “D”
represents the time derivatives of preceding variables.

phase. In this simulation, the power consumption at hip joints
exceeded 100 [W].
V. C ONCLUSION
In this paper, we extended running controller of passive
one-legged hopper to a planar biped robot with torso, andevaluated the controller on simulations. The controller wasderived based on energy-preserving strategy and it actuallypreserved mechanical energy at touchdown phase, as in thecase of one-legged model. Interestingly, zero dynamics of
decoupling controller (dynamics about pair of controlled leg)
was found to be stable. Combining simple attitude controller atstance phase generated stable periodic running gaits of arbitral
period. The control performance was better than a simple PDfeedback control about leg placement.
The next task is to realize complete passive running gaits.
To do so, we should introduce hip springs to make legs
swung passively both at flight phase and stance phase. Thisimplies replacing stance phase controller shown in Section IIIB with another stabilizing controller, where the control input
eventually converges to zero. If this goal is achieved, the phase
portraits of complete passive biped running gaits will exhibit
strong symmetricity, as explained in [16]Extending planar controller to 3D biped model is also
important work. This was partially done by introducing one
DOF at yaw-axis (rotor) in [17] and we are now adding more

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