Low-dimensional Feature Extraction for Humanoid Locomotion using
Kernel Dimension Reduction
Jun Morimoto, Sang-Ho Hyon, Christopher G. Atkeson, and Gordon Cheng
Abstract— We propose using the kernel dimension reduction(KDR) to extract a low-dimensional feature space for humanoid
locomotion tasks. Although humanoids have many degrees of
freedom, task relevant feature spaces can be much smaller
than the number of dimension of the original state space. We
consider an application of the proposed approach to improve
the locomotive performance of humanoid robots using an extracted low-dimensional state space. To improve the locomotive
performance, we use a reinforcement learning (RL) framework.
While RL is a useful non-linear optimizer, it is usually difficult
to apply RL to real robotic systems – due to the large numberof iterations required to acquire suitable policies. In this study,
we use the extracted low-dimensional feature space for RL sothat the learning system can improve task performance quickly.
The kernel dimension reduction method allows us to extract thefeature space even if the task relevant mapping is non-linear.
This is an essential property to improve humanoid locomotive
performance since stepping or walking dynamics involves highly
nonlinear dynamics. We show that we can improve stepping andwalking policies by using a RL method on an extracted feature
space by using KDR.

I. I NTRODUCTION
Reinforcement learning (RL), which does not require a
precise environmental model, can be a useful technique toimprove task performance of real robots. However, one drawback of utilizing RL is that it usually requires a large numberof iterations to improve policies in a high-dimensional space.
Thus, applications of RL have been limited to robots withsmall numbers of degrees of freedom [1]–[4].
In our approach, we propose using kernel dimension
reduction (KDR) [5], [6] to extract a low-dimensional feature
space so that the learning system can improve task performance even when a target robot model has many degrees of
freedom including the complexity of humanoid robots.
In this study, we focus on improving the locomotive performance of humanoid robots as an application of our learning framework. The dynamics of biped robots characteristically includes contact and collision with the ground. Modeling the interaction with the ground can be very cumbersome.
Using RL methods can be a suitable approach to improve
J. Morimoto is with the Japan Science and Technology Agency, ICORP,
Computational Brain Project, and with ATR Computational Neuroscience
Laboratories. xmorimo@atr.jp
S. Hyon is with the Japan Science and Technology Agency, ICORP,
Computational Brain Project, and with ATR Computational Neuroscience
Laboratories. sangho@atr.jp
C. G. Atkeson is with the Robotics Institute, Carnegie Mellon University.

cga@cs.cmu.edu
G. Cheng is with the Japan Science and Technology Agency, ICORP,
Computational Brain Project, and with ATR Computational Neuroscience
Laboratories. gordon@atr.jp

Fig. 1. (Left) Our human sized hydraulic humanoid robot CB developedby SARCOS. height: 1.59 m, total weight: 95 kg. (Right) Simplified 3D
biped simulation model of our humanoid robot.

biped walking. We use the extracted low-dimensional feature
space for RL.
We extract the low-dimensional feature space from stepping and walking dynamics without explicitly identifying
rigid body parameters and without the use of a groundcontact model. KDR allows us to extract the feature spaceeven if the task relevant mapping is non-linear. This isessential property to improve humanoid locomotive performance since stepping or walking dynamics involve highly
nonlinear dynamics. We show that we can improve stepping
and walking policies by using a RL method applied to theextracted feature space.
We apply our learning framework to a biped simulation
model (see Fig. 1(Right)) of our humanoid robot CB (seeFig. 1(Left)) [7].
In our approach, we first construct a stepping and a walking controller based on our previous study [8]. The previous
study proposed using the center of pressure to detect thephase of the robot dynamics for both stepping and walking(Fig. 2). We used simple periodic functions (sinusoids) as
desired joint trajectories (see Appendix). We showed thatsynchronization of the desired trajectories at each joint withthe detected phase from the center of pressure could generate
stepping and walking movements. In this study, we modulate
the amplitude of the sinusoids according to the current stateof the robot to improve locomotive performance.
In Section II, our learning framework is introduced. In
Section III-A, we explain how we applied the kernel dimension reduction (KDR) method to the tasks of stepping andwalking. In Section III-B, we describe our implementation ofa RL method on a low-dimensional feature space extracted

(A)

(B)

z
y

z

y

x

COM

COM

roll

0

(C)

z

y

COP

COP

0

pitch

COM

y
COP

Fig. 2. Inverted pendulum model represented by the center of pressure(COP) and the center of mass (COM). ψroll denotes roll angle of thependulum. ψpitch denotes pitch angle of the pendulum.

using KDR. In Section IV, we show our simulation results.
Our biped controller as proposed in [8] is introduced inappendix.
II. L EARNING FRAMEWORKIn this study, we consider extracting a low-dimensionalfeature space to improve task performance by using KDR.
We assume that nominal stepping and walking controllers areprovided (see Appendix), and our learning system improves
the performance of these controllers. Since the nominal controller can generate periodic movements, we only consider
the robot state at a Poincaré section.
For example, we consider the dynamics ξ̇ = g(ξ) of astate vector ξ ∈ Rn . The Poincaré map is a mapping froman n − 1 dimensional surface S defined in the state space
to itself [9]. If ξ(k) ∈ S is the k-th intersection, then the
Poincaré map h is defined by ξ(k + 1) = h(ξ(k)). In ourstudy, we defined the section which satisfies the roll angle
defined by the COM and COP equaling zero ψ̇ roll = 0 (seeFig. 2).
Since we consider an application of feature extraction
to an RL framework, we are interested in figuring out theappropriate feature vector which can predict state variables
xr ∈ Rr used to represent a reward function r(x r ). We thenformulate the problem to find the state vector x ∈ R n as:p(xr (k + 1)|xf (k), u(k))  p(xr (k + 1)|x(k), u(k)), (1)where x ∈ R denotes the state vector in the originalstate space. We consider a projection of the state to the lowdimensional feature vector x = Bx f such that n < m, where
B is a projection matrix. u ∈ R l is the output of a policy.
Then, in our learning framework, we consider three sets ofstate variables: 1) the state variables used to represent reward
function in the original state space x r , 2) the original statevariables xf , and 3) the state variables in the feature space
after projection x.
The relationship in (1) implies that the low-dimensionalfeature space x(k) tries to keep the Markov property for thestate xr . We use KDR to derive the projection matrix B.
fm

The policy of the learning system is updated and outputs
the next action only at this Poincaré section.
To improve task performance, we stochastically modulate
the amplitude of the sinusoidal patterns according to thecurrent policy π w :πw (x(k), u(k)) = p(u(k)|x(k); w),(2)where w is the parameter vector of the policy π w . In thefollowing sections, we explain how we extract the lowdimensional feature space and how we acquire the control
policy πw .
III. F EATURE EXTRACTION AND POLICY IMPROVEMENT
A. Kernel dimension reduction
We use kernel dimension reduction (KDR) [5], [6] toextract a low-dimensional feature space. Here we considera regression problem in which we try to explain a variable
Y ∈ Y by using a variable X ∈ X . The target of KDR is
to find a low-dimensional subspace Z of the original input
space X such that we can keep information of the variable
Y even in the subspace Z.
More concretely, KDR tries to find a projection matrix B,
which specifies the subspace of the original high-dimensional
state space, to make the conditional distribution p(Y |Z) closeto p(Y |X):
p(Y |X)  p(Y |Z),(3)
where Z = B T X. In our study, Y corresponds to x r (k + 1),
X includes xf and u, and Z includes x and u in (1).
This method does not impose assumptions on either the
distributions of X or the conditional distribution P (Y |X).
The idea of KDR is to map random variables X and Y toreproducing kernel Hilbert spaces (RKHS) [10], [11] andevaluate conditional independence using cross-covariance
operators.
Let HX be an RKHS of functions on X induced by the
kernel function k X (·, X) for X ∈ X . We also define thespace HY and the kernel function k Y (·, Y ). Then, we define
the cross-covariance between a pair of functions f ∈ H X
and g ∈ HY as follows:< g, ΣY X f >HY= EXY [(f (X) − EX [f (X)])(g(Y ) − EY [g(Y )])] (4)for all functions f and g, where Σ Y X is a cross-covariance
operator. Similarly, we define covariance operators Σ XX
and ΣY Y . Now, we can use these operators to define a
conditional cross-covariance operators as:
ΣY Y |X = ΣY Y − ΣY X Σ−1
XX ΣXY(5)

This definition assumes that Σ XX is invertible.
To the projection matrix B, we try to minimize
 deriveTr Σ̂Y Y |Z , where Σ̂Y Y |Z is the empirical conditional covariance operator which corresponds to (5).
Let {xi , yi }N
i=1 denote N samples from the joint distribution p(X, Y ), and let K Y ∈ RN ×N and KZ ∈ RN ×N
denote the Gram matrices computed over {y i } and {zi =B T xi }. Then, this minimization problem can be formulated
in terms of KY and KZ , so that the optimal projection matrix
B ∗ is derived as:B ∗ = arg min Tr KcY (KcZ + N IN )−1(6)
B

where IN is the N × N identity matrix, and  is a regularization coefficient [5], [6]. The matrix K c denotes the centeredkernel matrices

 

1
1
Kc = IN − 1N 1TN K IN − 1N 1TN , (7)NN
where 1N = (1, . . . , 1) is the vector with all elements equal
to 1. We used a Gaussian kernel:
||xi − xj ||2,(8)
Kij = k(xi , xj ) = exp −2s2
where Kij is the ij element of the Gram matrix, and s
denotes the parameter of the kernel.
B. Policy improvement by using a reinforcement learning
methodWe use a policy gradient method proposed by [12] toimprove the the stepping and walking policies.
The basic goal is to find a policy π w (x, u) = p(u|x; w)that maximizes the expectation of the discounted accumulated reward:∞Eπw {V (k)} = Eπw

γ i−k r(i) ,

(9)

i=k

where r denotes reward, V (k) is the actual return, w is theparameter vector of the policy π w , and γ, 0 ≤ γ < 1, is thediscount factor.
In this study, x represents an extracted feature vector andu denotes the output of a policy. We modulate amplitudes ofsinusoidal patterns as the output of our stepping and walking
policies (see Section IV). Both the approximated valuefunctions and the policies are represented by a normalized
Gaussian network [13].
IV. SIMULATION
We applied our proposed method to a simplified simulation
model of our humanoid robot CB [7] (Fig. 1(Right)).
A. Improvement of biped stepping performance
We applied our proposed method to improve stepping inplace.
We modulate the amplitude A step in (16) to improve thestepping performance. We defined the target of the steppingtask to keep the desired state at ψ roll . We use a reward
function:(10)r = −0.1(ψdroll − ψ roll )2
for this stepping task, where ψ droll = 2.0◦ . The learning
system also receives a negative reward r = −1 if the bipedmodel falls over.
Since the reward function is defined as a function of thevariable ψ roll , we try to find a low-dimensional feature spacethat can predict ψ roll at the next step by using KDR. We use

200 samples to find the projection matrix B. The parameterof the kernel in (8) is s = 0.75.
The simplified humanoid model has 10 joints, and thebase link has 6 degrees of freedom. We did not considertranslational degrees of freedom of the base link since ourhumanoid robot CB does not have a sensor to detect the 3D
position of the base link. Therefore, we need to consider 13
degrees of freedom. As a consequence, we need to consider
a 26 dimensional state space as the original state space that
includes the time derivative of each degree of freedom. We
applied KDR for the original 26 dimensional state space
to find the proper projection to a 1-dimensional feature
space. The number of dimensions of the feature space ispredetermined.
Figure 3 shows learning performance of the stepping task
on the extracted feature space. This results showed that theextracted feature can be used to learn the stepping task.
Here we compared with the learning performance using
randomly extracted feature space. We randomly selected theelements of the projection matrix B such that B T B = 1 forthis comparison. Figure 3 shows that the learning system
could not acquire a good stepping policy and had large
variance in stepping performance when we used a randomly
selected feature space.
Figure 4 shows the relationship between the extracted
1-dimensional feature and the roll angle ψ roll . This result
showed that KDR extracted a 1-dimensional feature, which
has high correlation to the roll angle (correlation coefficient
was 0.93). This is interesting because the learning system
automatically finds a feature space which corresponds to theroll angle of the center of mass while a number of bipedwalking studies have emphasized that humanoid robots haveinverted pendulum dynamics (see Fig. 2), with the top of thependulum at the center of mass and the base at the center ofpressure [14]–[17].
Note that the reward function is a function of the rollangle. However, it is not obvious whether the proper variable
to predict the roll angle at the next time step (k + 1) is theroll angle at current time (k) or not.
Figure 5 shows a comparison between control performanceof an acquired policy and that of the initial policy. The rollangle is kept around desired state by using the acquired
policy.
An acquired stepping movement is shown in Fig. 6.
B. Improvement of biped walking performance
We also applied our proposed method to improve walking
performance.
We modulate the amplitude A walk in (17) to generate
forward movement for the biped walking task. The targetof the walking task is to increase the angular velocity of thependulum ψ̇ pitch (see Fig 2(C)) at the Poincaré section. We
use the reward function:r = 0.1(ψ̇ pitch )(11)

for this biped walking task. The learning system also receivesa negative reward r = −1 if the biped model falls over.

0.2

10

KDR
Random0

Acquired
InitialDesired−0.2−0.4

Ψroll [°]

Average reward

8

−0.6

6

4

−0.8
2
−1
0

5

10
Trial

15

20

Fig. 3. Learning performance of the stepping controller in the simulated
environment. The blue solid line represents the learning performance using afeature space extracted by KDR. The red dashed line represents the learning
performance using a randomly selected feature space. The learning system
could not acquire a good stepping policy and had large variance in stepping
performance when we used a randomly selected feature space. Means andstandard deviations of the learning performances were derived from 10
simulation runs.

0
0

5

10
Steps

15

20

Fig. 5. Comparison between control performance of an acquired policy
and that of the initial policy. The roll angle ψroll at the Poincaré sectionψ̇roll = 0 (solid line). The dotted line represents the desired angle for this
stepping task.

Extracted state x

20

15

10

5

0
0

1

2

3
4
roll
ψ [°]

5

6

7

Fig. 4. Relationship between the COM roll angle ψroll and the extracted
feature x. The extracted feature is highly correlated to the COM roll angle
ψroll . The correlation coefficient was 0.93. The extracted feature capturesphysical property of the robot model.

Since the reward function is defined as a function of thevariable ψ̇ pitch , we try to find the low-dimensional feature
space that can predict ψ̇ pitch at next step by using KDR.
In this case, we also use 200 samples to find the projection
matrix B. The parameter of the kernel in (8) is s = 0.75.
We applied KDR for the original 26 dimensional state
space to find the proper projection to a 3-dimensional feature
space. The number of dimensions of the feature space ispredetermined.
Figure 7 shows learning performance of the walking task
on the feature extracted by using KDR. This results showedthat the extracted feature can be used to learn the walking
task.
We again compared with the learning performance using
randomly extracted feature space. Since the amount of the
reward could be mostly explained only by the output ofpolicies Awalk , the learning system could acquire walking
policies even using a randomly selected feature space. However, learning performance kept increasing only when weused the feature space extracted by KDR possibly because
the extracted feature space has a larger emphasis on the
reward.
Figure 8 shows a comparison between control performanceof an acquired policy and that of the initial policy for the

Fig. 6. Acquired stepping movement. The red thin line represents desired
angle. the pendulum state represented by the light gray line behind the redline came close to the desired state at the Poincaré section. The light graysphere represents the center of mass.

walking task. The pitch angular velocity ψ̇ pitch had a larger
value using the acquired policy compared to using the initial
policy.
Figure 9 shows the acquired walking performance.
V. DISCUSSION
We proposed using KDR to extract a low-dimensionalfeature space for a humanoid locomotion task. In this study,
we used the extracted low-dimensional state space for RL sothat the learning system could improve task performance. We
showed that we could improve stepping and walking policies
by using a RL method on the extracted feature space by using
KDR. In our future work, we will consider application of the
proposed method to our new humanoid robot CB [7].
So far, we empirically determined the sufficient dimensionality of the feature space. Automatic selection of thenumber of dimensions for the humanoid locomotion tasks isalso included as part of our future work.
ACKNOWLEDGMENT
This material is based upon work supported in part by
the National Science Foundation under grants CNS-0224419,
DGE-0333420, ECS-0325383, and EEC-0540865.
A PPENDIX
Our biped controller uses a coupled phase oscillator model
to modulate the phase of the sinusoidal patterns. The aim

(a) t=0.0 s(b) t=3.2 s(c) t=6.5 s(d) t=9.7 s(e) t=13.0 s

Fig. 9. Acquired walking pattern. Average walking speed is 0.16 m/s for initial 20 seconds. The red line represents the starting position. We showed one
snap in three walking steps.

1

Parameters used in the controllers are summarized in Table
I.

Average reward

0.8
0.6

A. Phase detection of the robot dynamics

0.4

As shown by our previous study [8], we can use the centerof pressure y cop and the velocity of the center of pressureẏcop to detect the phase of the robot dynamics:


ẏcop
φ(ycop ) = − arctan
,
(12)
ycop

0.2
KDR
Random

0
0

20

40

60

80

100

Trial

Fig. 7.
Learning performance for the walking task in the simulated
environment. The blue solid line represents the learning performance using afeature space extracted by KDR. The red dashed line represents the learning
performance using a randomly selected feature space. Since the amount of
the reward could be mostly explained only by the output of policies Awalk ,the learning system could acquire walking policies even using a randomly
selected feature space. However, learning performance kept increasing only
when we used the feature space extracted by KDR possibly because theextracted feature space has a larger amount of information on the reward.
Means and standard deviations of the learning performances were derived
from 10 simulation runs.
20

Acquired
Initiald/dt ψpitch [°/s]

15
10
5
0
−5
−10
0

5

10
Steps

15

20

Fig. 8. Comparison between control performance of an acquired policy andthat of the initial policy. The pitch angular velocity ψ̇pitch at the Poincaré
section ψ̇roll = 0 (solid line).

of using the coupled phase oscillator model is to synchronize periodic patterns generated by the controller with the
dynamics of the robot. To use the coupled phase oscillator
model, detection of the phase of the robot is needed. We
introduce a method to detect the robot phase in Section
A. We briefly explain phase coordination for biped walking
in Section B. As in our previous study [8], we use simple
sinusoidal patterns as nominal trajectories for each joint. We
describe the design of the nominal trajectories for stepping
movement in Section C, and walking movement in Section D.

where ycop = (ycop , ẏcop ) (see Fig. 2). We use a simplified
COP detection method introduced in [8].
B. Phase coordination
In this study, we use four oscillators with phases φ ic ,where i = 1, 2, 3, 4. We introduce couplings between theoscillators and the phase of the robot dynamics φ(y cop ) in(12) to regulate the desired phase relationship between theoscillators:φ̇ic = ωc + Kc sin(φ(ycop ) − φic + αi ),(13)where αi is the desired phase difference, K c is a coupling
constant, and ω c is natural angular frequency of oscillators.
We
use
four
different
phase
differences,{α1 , α2 , α3 , α4 } = {− 12 π, 0.0, 12 π, π}, to make symmetricpatterns for a stepping movement with the left and rightlimbs (see Section C.2), and also to make symmetricpatterns for a forward movement with the left and rightlimbs (see Section D).
C. Stepping controller for lateral movement
1) Side-to-side controller for lateral movement: First, we
introduce a controller to generate side-to-side movement. We
control the hip joints θ h roll and the ankle joints θ a roll (Fig.
10(A)) for this movement. Desired joint angles for each jointare:θhd roll (φc ) = Ah roll sin(φc ),dθa roll (φc ) = −Aa roll sin(φc ),

(14)
(15)

where Ah roll and Aa roll are the amplitudes of a sinusoidal
function for side-to-side movements at the hip and the ankle
joints, and we use an oscillator with the phase φ c = φ1c .

(B)

(A)
z

TABLE Iz

PARAMETERS OF THE BIPED STEPPING AND WALKING MODEL FOR EACHyh_rollh_rollh_pitchk_pitcha_pitcha_rollEXPERIMENTx

a_rollFig. 10. Stepping controller: (A) Controller for side-to-side movement.
(B) Controller for foot clearance.

2) Vertical foot movement to make clearance: To achievefoot clearance, we generate vertical movement of the feet(Fig. 10(B)) by using simple sinusoidal trajectories:
θhd pitch (φc ) = (Apitch + Astep ) sin(φc ) + θhrespitch ,θkd pitch (φc ) = −2(Apitch + Astep ) sin(φc ) + θkrespitch ,θad pitch (φc ) = −(Apitch + Astep ) sin(φc ) + θarespitch , (16)where Apitch is the amplitude of a sinusoidal function toachieve foot clearance, θ hrespitch , θkrespitch , θarespitch represent
the rest posture of the hip, knee, and ankle joints respectively.
We use the oscillator with phase φ c = φ1c for right limb
movement and use the oscillator with phase φ c = φ3c , whichhas phase difference of φ 3c = φ1c +π, for left limb movement.
We modulate the amplitude of the sinusoidal patterns bychanging Astep according to the current pendulum state forthe stepping task (see Sections IV-A).
D. Biped walking controller
To walk forward, we use an additional sinusoidal trajectory. Thus, the desired nominal trajectories for right hip andankle pitch joints become:θhd pitch = Apitch sin(φ1c ) + Awalk sin(φ2c ) + θhrespitch ,θad pitch = −Apitch sin(φ1c ) − Awalk sin(φ2c ) + θarespitch , (17)where the phase φc = φ2c has 12 π phase difference of φ 1c .
We use φ3c and φ4c for left limb instead of φ 1c and φ2c ,where the phase φc = φ4c has π phase difference of φ 2c .
We then modulate the amplitude of the sinusoidal patterns
by changing A walk according to the current pendulum stateof the biped walking task (see Section IV-B).
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