Robot Phase Entrainment on Quadruped CPG Controller

Vítor Matos and Cristina P. Santos

Industrial Electronics Department, School of Engineering, University of Minho, Guimarães, Portugal

Keywords:

Central Pattern Generator, Quadruped locomotion, Locomotion Entrainment.

Abstract:

Central Pattern Generators are used in several kinds of robot locomotion, from swimming and ﬂying, to bipeds,

quadrupeds and hexapods. It is thought that this approach can yield better results in dynamical and natural

environments. In this work we expand a previous quadruped locomotion controller and propose a method

to couple the step cycle phase onto the locomotor CPG of a quadruped robot, creating a feedback pathway

to coordinate the phases of each leg to the phase of the step cycle. This approach is tested in a simulated

quadruped robot and the performed locomotion is evaluated. Results demonstrate that the proposed phase

coupling synchronizes the swing step phase of ipsilateral legs to the respective step phase of the cycle and

show an improvement in stability of the performed walk gait.

1 INTRODUCTION

Motor control is a complex problem in robotics,

highly relevant in tasks such as manipulation and

grasping, environment exploration and navigation,

human-robot interaction and locomotion, all in de-

manding dynamical environments. Several processes

must be addressed to achieve an acceptable level of

performance and autonomy in motor control, as per-

ception, planning, execution, feedback and mechan-

ics.

We endeavor to achieve general and autonomous

quadruped locomotion on natural environments. In

this work we explore further contributions in the

feedback process, ubiquitous for short and long-term

adaptation of any kind of legged locomotion.

Legged locomotion can be achieved using typi-

cal model based planning algorithms (Buchli et al.,

2009), or limit-cycle based control, such as Central

Pattern Generators (CPGs) (Fukuoka et al., 2003).

CPG controllers are well suited for locomotion be-

cause: 1) produce stable rhythmic patterns, provid-

ing robustness against transient perturbations; 2) en-

able the design of distributed implementations; 3) are

generally easier to modulate through control param-

eters that may be used as higher level command sig-

nals; and 4) due to its stability, CPGs are appropri-

ate for the integration of sensory feedback signals

and to achieve entrainment with the mechanical body

(Ijspeert, 2008).

CPG systems can be designed starting from dif-

ferent conceptual approaches. They can be designed

ﬁrstly from the CPG model as a feedforward gen-

erator, and only then the effect of feedback signals

is included and the loop is closed; endorsing what

kind of information should be considered and how

will affect the ﬁnal behaviour (Righetti and Ijspeert,

2008; Fukuoka et al., 2003). Or can be designed

from the beginning with the closed-loop goal in mind,

using feedback signals to tightly generate trajecto-

ries (Maufroy et al., 2010; Buchli and Ijspeert, 2008).

Systems of coupled oscillators are widely used

for modeling CPGs, and while there exists exten-

sive work and methods for analyzing these dynamical

systems, less work has been carried out on methods

and frameworks for synthesizing oscillators that have

to exhibit a speciﬁc desired behavior. For instance,

in (Buchli et al., 2006) it is presented a framework

for characterizing and designing oscillators, as well

as deﬁning desired perturbations in order to achieve

frequency-locking, phase-locking or any speciﬁc out-

put signal shape.

Step phase feedback plays an important role in lo-

comotion, allowing the adaptation of the onset of the

swing and stance phases (Pearson, 2008). These were

explored in legged robots, whether through phase re-

setting (Aoi and Tsuchiya, 2006) or phase transitions

depending on load/unloading of the legs (Righetti and

Ijspeert, 2008; Maufroy et al., 2010)

In this paper, we explore a different approach for

phase feedback. We devise a phase feedback inspired

in (Fukuoka et al., 2003) and try to entrain the CPG

260

Matos V. and P. Santos C..

Robot Phase Entrainment on Quadruped CPG Controller.

DOI: 10.5220/0004037802600265

In Proceedings of the 9th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2012), pages 260-265

ISBN: 978-989-8565-22-8

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

oscillators with the phase of the robot’s periodic dy-

namics, coordinating the phase of the step cycle of

each leg with the sensed body motion.

The goal of this contribution is to include step

phase feedback onto a previous developed locomotor

system and study its inﬂuence on locomotion’s per-

formance. It is based on past work for modeling a

limb-CPG (Matos and Santos, 2010). This previous

work only considered the CPG network as a rhythmic

open-loop controller. In this paper, we discuss the ef-

fects of the addition of feedback on the rhythmogenic

ability of the CPGs, and propose a methodology to

explore the possibilities of physical entrainment with

the system.

The proposed feedback couples the CPG system

to the pendulum rolling motion of the projected Cen-

ter of Gravity (pCOG). Robot’s sensory information

regarding body angle, joint position and foot touch

sensors are used to calculate pCOG, which modu-

lates the frequency of the leg oscillator through the

feedback mechanism. The goal of this feedback is to

avoid the swinging of a leg before the robot pCOG is

transferred to the opposite supportpolygon, and this is

achieved by synchronizing the oscillator’s phase with

the performed step phase.

The inclusion of feedback is expected to improve

robot performance, herein measured by the Support

Stability Margin (SSM), an adequate measure for a

static stable. Besides, the proposed feedback should

not affect the required duty factor and phase relation-

ships of a crawl gait.

Simulations are conducted on a simulated en-

vironment with the model of an AIBO quadruped

robot. We study the robot’s performance regarding

velocity, SSM and the correct execution of the step

phases. We also explore the inﬂuence of integrat-

ing this phase feedback along with the coupling net-

work of CPGs (Matos and Santos, 2010). Simulation

results show an improvement on robot performance

regarding SSM, maintaining the desired general fea-

tures of the crawl gait such as duty factor and inter-

limb phase relationships.

2 LOCOMOTOR SYSTEM

The locomotor system generates the trajectories for

the leg joints, producing the locomotor motions of the

robot. Similarly to previous work, the locomotor sys-

tem is composed by a network of four Central Pat-

tern Generators (CPGs) modeled as nonlinear oscil-

lators (Matos and Santos, 2010). Each CPG is able

to endogenously output the rhythmic signals that will

control each joint on a single leg.

2.1 Central Pattern Generator

The concept of biological locomotor CPG includes

the idea of hierarchical organized unitary oscillators,

the unit-CPG. A single unit-CPG controls and acti-

vates the antagonistic muscle pairs, controlling the

movements of a single joint.

The hip unit-CPG is modeled using a nonlinear

oscillator:

˙x

= α



µ−r



−O

) −ω

(1)

˙z

= α



µ−r



+ ω

−O

) (2)

1−β

−az

+ 1

, (3)

with r

−O

)

+ z

x(t)

solution is the angle of the hip joint of leg

i at instant t. This rhythmic solution has an ampli-

tude of

√

µ, an offset O

and an angular frequency

ω. Frequency is modulated according to the current

phase of the oscillator, enabling the generation of a

trajectory with stance and swing phase of different

durations (Righetti and Ijspeert, 2008). We want to

achieve a 0.75 duty factor for the walk gait, setting

β = 0.75. Parameter α controls the relaxation of the

solution to the stable orbit and a the toggling speed of

ω.

The modulation of the generated trajectories with

respect to their amplitude, frequency and offset, is

carried out explicitly and smoothly through the speci-

ﬁcation of a set of parameters, and allows for a dis-

tributed organization due to its entrainment proper-

ties.

Knee joints are controlled according to the corre-

sponding hip swing joint in a simple fashion. When

the leg performs the swing phase, the knee ﬂexes to

a ﬁxed angle θ

. During the stance phase, the knee

extends to θ

This motion is generated by applying the follow-

ing second order system:

˙y

= v

, ˙v

= −

−g

) −bv

(4)

−az

(5)

whose stable solution y converges to a goal ﬁxed

point g, changing between θ

and θ

depending on

the step phase. Relaxation is controlled by parameter

2.2 Interlimb Coordination

In order to generate the desired quadruped gait, we

couple the four CPGs in a network with variable phase

RobotPhaseEntrainmentonQuadrupedCPGController

261

Table 1: Phase relationships for walking gait.

i LF LF LF RF RF LH

j RF LH RH LH RH RH

−π −

3π

−

relationships. Coupling between two oscillators is

achieved by applying a rotation matrix R





, where

is the desired relative phase between oscillator i

and j. Eqs (1,2) are extended with the rotation matrix

members:

˙x

= . . . + k

osc

∑

j6=i



cosφ

−z

sinφ



, (6)

˙z

= . . . + k

osc

∑

j6=i



sinφ

+ z

cosφ



, (7)

where i and j speciﬁes the CPG of leg ∈

{LF, RF, LH, RH}. Parameter k

osc

determines the

strength of the interlimb coupling in the network,

where every connection has equal weight. Phase re-

lationships for the walk gait are presented in table 1,

note that φ

= −φ

The achievedcoordination among the CPGs is sta-

ble and ﬂexible through the modulation of the desired

phase relationships. It is also stable and robust to

perturbations of phase, making the system return to

the desired phase relationships, dependingon the cou-

pling strength k

osc

and assuming a limited range in the

perturbation magnitude.

2.3 Robot Phase Coupling

The goal of using phase coupling in our CPG ap-

proach is to synchronize the phases of the CPG net-

work to the dynamics’ phase of the robot. The act

of walking exhibits periodic motions, from which we

extract the phase of the robot’s locomotion or robot

phase. We use the periodic motion of the projected

Center of Gravity to calculate the robot’s phase, con-

sidering the body angle, joint positions and touch sen-

sors. The proposed coupling tries to synchronize the

generated swing phase of the CPGs with the measured

point in the step cycle in which the robot has its pro-

jected Center of Gravity (pCOG) over the contralat-

eral support polygon during the walk gait (ﬁg. 1).

If this phase coupling is achieved correctly, the

swing phase of each leg will happen when the pCOG

is over the contralateral support side, ensuring that the

weight is not over the swinging leg, and thus the robot

does not fall over it. This feedback mechanism by co-

ordinating the swing phases with the correct support

polygon presents the potential of improving the walk,

by increasing the stability, forcing the pCOG to be

over the side with most legs supporting the body. This

A B C D E

Figure 1: Oscillatory movement exhibited by the pro-

jected Center of Gravity during the walk gait. The pCOG

moves between the contralateral triangular support poly-

gons. Open circles denote feet ground contact.

improvement on stability is specially desired when

the body weight goes from one side to the other (A

to B in ﬁg. 1). The normal tendency is to fall over

the swing leg. The proposed feedback is expected to

improve this.

To achieve this kind of entrainment between our

oscillators and the robot dynamics we follow the gen-

eral guidelines from (Buchli et al., 2006). We spec-

ify the desired perturbation effect on the oscillator’s

phase using its polar representation and transform it

back to its cartesian representation.

Consider the oscillator from eq.(1,2) in polar co-

ordinates and consider the movement of pCOG in the

frontal plane due to the robot’s rolling motion as a

simple oscillatory motion with its phase described by

= arctan2



pCOG



. We couple the robot’s phase

(φ

) with the oscillator’s phase (φ

) with a desired

phase difference of φ

and coupling constant k

, as fol-

lows:

= ω

+ k

sin



−φ



, (8)

˙r

= α



µ−r



. (9)

In cartesian coordinates this phase coupling be-

comes

˙x

= α



µ−r



−O

) −z

, (10)

˙z

= α



µ−r



−(x

−O

)ω

, (11)

= ω

−

cos



+ φ



−x

sin



+ φ

i

. (12)

We choose the desired phase differences to respect

the followingrules (ﬁg. 2): i) when the robot leans left

with its pCOG over the left support polygons, φ

= 0,

the right legs should perform the swing phase, −π <

< 0; ii) when the robot leans right and the pCOG

is to the right side, φ

= π, the left legs should swing

while the right legs perform the stance phase, −π <

< 0; and iii) the oscillators should reﬂect the phase

relationship and sequence of the walk gait and exhibit

a relative phase difference of

among themselves.

Phase relationships are: φ

, φ

−3π

, φ

3π

, φ

−π

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262

Swing Stance

=0 Á

=¼

pCOGy

pCOG over left pCOG over right

LH LF

RF RH

RH RF

LF LH

Figure 2: Limit cycle of the pCOG oscillation on the

frontal plane (pCOG

). When the pCOG is on the left side

(pCOG

> 0) the robot’s phase is φ

= 0 , and φ

= π when

is on the right side pCOG

< 0. Each CPG performs the

swing phase when −π < φ

< 0.

3 SIMULATIONS

A series of simulations were performed in a sim-

ulated environment with the model of Sony AIBO

quadruped robot in Webots

First we study the coupling of interlimb coordi-

nation and robot phase by changing the values of the

coupling weights k

osc

, k

. We make a systematic pa-

rameter exploration on the parameter tuple hk

osc

, k

iin

the range [0, 9.5], in steps of 0.5. In each run the robot

locomotes with a desired nominal gait, a statically sta-

ble walk gait (β = 0.75) for 10 s and the required in-

formation is recorded. We then compare and discuss

the obtained average Support Stability Margin (SSM)

and the achieved velocities. SSM is the smallest dis-

tance of the pCOG to the edge of the polygon deﬁned

by the supporting feet projection onto the plane with

the gravitational acceleration as its normal. SSM is

an indicator that tells if the pCOG is inside the sup-

port polygon at all times, when considering statically

stable gaits as the walking gait.

We then choose and use the hk

osc

, k

i values that

result in the best walk in terms of trade-off between

the average SSM and the achieved velocity, to com-

pare and quantify the improvement of the walk with-

out and with robot phase coupling. We analyze the

robot’s performance regarding velocity, SSM and dis-

cuss improvements over the execution of the step

phases.

Parameter Exploration. Interlimb coupling, k

osc

and robot phase coupling, k

inﬂuence the walk in

different ways. While interlimb coupling simply co-

ordinates the phase relationships between the CPGs,

robot phase coupling tries to coordinate the phase of

each CPG to the phase of the robot.

From ﬁg. 3(a) we can see that velocity does not

Webots mobile robot simulator: http://

www.cyberbotics.com/

change when we increase or decrease the strength of

interlimb coupling, k

osc

. However, when changing

the strength of phase coupling, k

, the achieved ve-

locity is inﬂuenced, decreasing when the coupling in-

creases. This is a possible indicator that the oscillators

are being adapted to respect the current step phase of

the walk, being slowed down to match the robot dy-

namics. For k

> 4.5 the velocity decreases greatly,

suggesting that beyond this point the inﬂuence and

strength of this phase coupling is no longer adequate

and tries to stop the robot.

Similarly, SSM shows no major variation for a

changing interlimb coupling strength, k

osc

(ﬁg. 3(b)).

The major determinant of the achieved SSM is the

phase coupling strength, k

. There is a range of k

where the SSM shows higher values, [1;3.5]. It sug-

gests that the CPGs are being coordinated according

to the robot dynamics, correcting the execution of the

step phases. However, above 3.5 the SSM decreases

to low values, similarly to the velocity.

The velocity achieved without phase coupling was

0.134 m.s

−1

osc

= 1, k

= 0) and the obtained SSM

was 6.14 mm. The highest obtained SSM was

12.97 mm, when using k

osc

= 2.5, k

= 2.5, with

achieved velocity 0.098 m.s

−1

. We consider this re-

sult to be a fair trade-off between the achieved SSM

and velocity for a walk gait.

Locomotion Comparison. With phase coupling

osc

= 2.5, k

= 2.5) it is expected that the left legs’

swing phases are performed when the pCOG is over

to the right side of the support polygon. We can ver-

ify this is true in ﬁg. 4 (right) since the swing phase of

both left legs (ascending trajectories) are performed

when pCOG

< 0. These results show that the pro-

posed phase coupling synchronizes the swing step

phase of ipsilateral legs to the respective step phase

of the cycle. The nominal step period is 0.8 s, from a

swing period of 0.2 s and a duty factor of 0.75. When

we employphase coupling the interaction of the CPGs

with the robot’s phase changes slightly the achieved

average step period, from 0.8 s to 1.2 s, while main-

taining the chosen duty factor, adapting the swing pe-

riod to 0.3 s. This adaptation did not change the rela-

tive phases among the CPGs, maintaining the desired

interlimb coordination of the nominal walk gait.

Let us verify the overall effect on the achieved

step cycle duration and compare the achieved SSM.

Fig. 5 showsthe achievedSSM overthe two runs. The

dotted (solid) lines show the achieved SSM without

(with) phase coupling. Positive values denote that the

pCOG lies inside the support polygon, while negative

values denote a position outside the support polygon,

with a distance to the nearest edge correspondent to

RobotPhaseEntrainmentonQuadrupedCPGController

263

0 2 4 6 8

osc

0.02

0.04

0.06

0.08

0.1

0.12

(a) Velocity achieved on each run.

0 2 4 6 8

osc

−4

−2

(b) Mean Support Stability Margin (SSM) on each run.

Figure 3: k

osc

, k

parameter exploration, and its effects on velocity (m.s

−1

) and phase coupling. Velocity and SSM are mostly

inﬂuenced by the value of k

8 8.2 8.4 8.6 8.8 9

−15

−10

−5

8 8.2 8.4 8.6 8.8 9

−15

−10

−5

pCOG

, Left joints

Time (s)

left rightright leftright

Without feedback With feedback

Figure 4: pCOG position in the frontal plane (solid blue)

and left hip joints’ trajectories (dashed red:fore and dotted

yellow:hind). Swing phases correspond to the ascending

parts of the trajectories. Without phase coupling the hind

leg (dotted yellow) swing onset happens while the pCOG

is in the ipsilateral side (left panel), meaning pCOG

> 0.

With phase coupling (right panel) the swing phases on both

left legs happen when the pCOG is in the contralateral side,

pCOG

< 0.

the absolute value.

We can verify in ﬁg. 5 that the performed SSM in-

creases when phase coupling is employed. Negative

values of SSM indicate the robot may fall over the

swinging leg. The moments of the step where pCOG

falls outside the support polygon (negativevalues) are

reduced from 66% of the step phase without feedback

(dotted), to 29% of step phase when feedback is em-

ployed (solid). The average value of SSM also in-

creases due to the maintenance of the pCOG inside

the support polygon.

The walk gait sequences from the two simulations

are shown in ﬁg. 6. We can see that without phase

coupling (top) the pCOG is generally closer to the

edge of the support polygon than with phase coupling

(bottom). pCOG also is in the same side in the onset

of some swing phases when phase coupling is not em-

ployed, at 9.10 s and 9.50 s, which was solved when

phase coupling is employed (bottom, at 9.10 s and

9.60 s). The concerning point of contralateral swing

0 0.2 0.4 0.6 0.8 1

−20

Step cycle

SSM (mm)

Figure 5: SMM without (dotted) and with (solid) robot

phase coupling. The average SSM is 6.14 mm without

phase coupling and 12.97 mm with phase coupling. Us-

ing phase coupling, increases the chance of brief four feet

support (solid).

onset was dealt with by the proposed feedback mech-

anism and the result was to achieve brief four feet

support between these contralateral phases (bottom,

at 9.00 s and 9.50 s).

Results also show that front feet touching the

ground incorrectly during the swing phase decreased

from 12.60% to 11.90% of the swing phase period,

and hind feet from 1.30% to 0.07%. The unwanted

lifting of the feet occurring during stance phase de-

creased in the fore legs from 23.40% to 5.09% of the

stance period.

4 CONCLUSIONS

In this contribution we try to take advantage of the

properties of oscillators, typically used to model

CPGs on legged robots. We investigate and propose

a method to couple the CPG rhythmic activity to the

step phase of a quadruped robot, trying to create a link

between the robot dynamics and the walking motion

of the locomotor controller.The goal of the proposed

feedback is to maintain the pCOG over the contralat-

eral side of the current swinging leg, in order to im-

prove stability and avoid the swinging of a leg which

ICINCO2012-9thInternationalConferenceonInformaticsinControl,AutomationandRobotics

264

9.00 s

9.10 s

9.20 s

9.30 s 9.40 s

9.50 s

9.60 s

9.70 s

9.80 s

9.00 s 9.10 s 9.20 s 9.30 s 9.40 s 9.50 s 9.60 s 9.70 s 9.80 s

Figure 6: Performed gait sequence of the walk gait without (top) and with (bottom) phase coupling. Black dot is the position

of the projected Center of Mass. Colored ﬁlled dots represent stance trajectories and empty dots swing trajectories. (red:left

fore, blue: right fore, yellow:left hind, green: right hind) In this ﬁgure, the reference frame is centered on the robot.

is supporting the body weight.

Results show that the phase coupling adapts the

generated trajectories and performs what is proposed.

A systematic exploration of coupling parameters was

conducted to study the inﬂuence of interlimb coupling

and the proposed phase coupling onto the CPG and

ﬁnal walk gait, and the best parameters were used to

conduct further comparisons. The obtained measure-

ments indicate an improvement of the walk gait, dou-

bling the SSM.

However,further work should be performed in

evaluating its effect on different walking conditions,

such as in inclined planes, irregular terrains and small

perturbations. We will also study the integration with

other kinds of feedback within the same framework,

such as phase transition (Righetti and Ijspeert, 2008)

and postural control (Sousa et al., 2010), as well as on

other legged robots, including robots with compliant

actuators.

ACKNOWLEDGEMENTS

This work is funded by FEDER Funding supported

by the Operational Program Competitive Factors -

COMPETE and National Funding supported by the

FCT - Portuguese Science Foundation throughproject

PTDC/EEACRO/100655/2008. Vítor Matos is sup-

ported by PhD Grant SFRH/BD/62047/2009, granted

by the Portuguese Science Foundation.

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