IS Technical Report UEC-IS-2006-1

Motor Planning and Sparse Motor Command Representation∗
Yutaka Sakaguchi† and Shiro Ikeda‡†

University of Electro-Communications, Graduate School of Information Systems.
‡The Institute of Statistical Mathematics.

Abstract
The present article proposes a novel computational approach to the motor planning. In this approach, each
motor command is represented as a linear combinationof prefixed basis patterns, and the command for a giventask is designed by minimizing a two-termed criterion
consisting of a task optimization term and a parameter
preference (i.e., sparseness) term. The result of a computer simulation with a single-joint reaching task confirmed that our “representation-based” criterion for motor planning appropriately worked, together with showing that the resultant trajectory qualitatively replicatedFitts’ law.
keywords: motor planning, sparse representation, parametric command representation
1 Introduction
Although information representation in the brain is oneof the central problems in neuroscience, most studies on
motor control have not paid enough attention to this problem. The present article studies motor planning from aviewpoint of command representation.
Motor planning is a problem to design a command
pattern for a given motor task, and many computational
studies have been conducted to solve this problem. For
the upper-limb reaching movement, for example, various optimality criteria have been proposed, including
jerk-minimum principle[5] and torque-change minimum
principle[11].
Whilst these theories succeeded in explaining thecharacteristic features of empirical data, they focused
mainly on the kinematic/dynamic properties of movement trajectories, and did not try to design the command
pattern directly. On the other hand, Harris & Wolpert[7]proposed a novel concept that the neural system designs a motor command whose resultant endpoint variance is minimized with a presence of signal dependent
noise (SDN). This theory links the motor command andtask requirement (i.e., endpoint accuracy) via SDN, andpresents a method to plan motor commands based not on
the kinematic space but on the command space.
In the present article, we aim to take another step toward studying the relationship between motor planningand command representation. In concrete, we propose
a simple parametric model for command representation
and a two-termed optimality criterion based on “sparseness” preference, referring to a computational theory∗ This article is submitted to Neurocomputing.

on visual information representation. We formalize theproblem and show how to solve it. Finally, we show thevalidity of the present framework by a computer simulation.
2Parametric motor command representation

In our model, each motor command is represented as alinear combination of a fixed set of patterns, called “basis” patterns. This framework shares the idea with thesynergy decomposition model[1, 2], where each motor
command pattern to multiple muscles u(t) is represented
as a linear sum of synergy patterns {w j (t)} with weights{a j } and time shifts {d j } as
u(t) =
a j w j (t − d j ).
(1)
jIn the present study, we regard those patterns with diﬀerent time shifts as diﬀerent basis patterns, which resultsinu(t) =
a j w j (t). (a j ≥ 0)(2)
j

In the following, we assume the basis is already given
and we focus our discussion on how to find the optimal
parameters {a j } for a given task, that is, motor planning1.
3

Motor planning criterion dependent on commandrepresentation

One natural definition of the criterion for motor planningis the following:
E = [Task Error] + λ[Additional Cost].

(λ > 0) (3)The first term corresponds to the task constraints/requirements, such as terminal condition (i.e.,the hand must stay around the target for certain time),
and is common to most computational models of motor
planning. Because it is an “ill-posed problem” to find asolution satisfying simply the task requirement, variousforms of additional costs have been proposed, such as
jerk[5], torque-change[11], and endpoint variance[7], todetermine the unique solution.
Here, we assume the preference on the parameters,
instead of kinematics/dynamics cost, and use it as thesecond term. To this end, we refer to the computational
1 In the synergy decomposition model, the problem was to find optimal parameters and basis patterns which minimized the error between
the observed muscle activities and reconstructed command patterns,
which is diﬀerent from motor planning.

model of visual information representation developed byOlshausen and Field[8].
In their scheme, a visual image I(x, y) (x and y represents position in the image) was given by a generativemodel
a j φ j (x, y) + ε(x, y),(4)I(x, y) =asminimize E {a j }I; {φ j }2I(x, y) −=a φ (x, y) + λ
S (a )j jjj= [Approximation Error] + λ[Parameter Preference],(5)
where S (a j ) corresponds to the logarithm of the priorprobability density function of a j .
Notice that the second term of this cost function
played an essential role in reproducing the receptive field
properties of V1 neurons, and at the same time, gave rise
to rich theoretical discussion, such as independent component analysis (ICA)[3] and sparse coding[9].
Comparing Eq.(3) with Eq.(5), we see that these costfunctions well correspond to each other: The first terms
represent the requirement of the system and the second
terms represent the parameter preference. Considering
this correspondence, we define a criterion for motor planning as follows

S (a j).
(6)
E = [Task Error] + λIf we adopt|θ(t) − θT |2 + λ|a j | p ,(7)jwhere θ(t) is the joint angle at time t, θT is the target. We
consider t as discrete time step, and t f and t p are movement time and pausing time, respectively. We usually
set p = 1 (i.e., L1 constraint) for sparseness perference,but below we also think of the case p = 2 (i.e., L2 constraint), which prefers the minimum “energy” solution,
for comparison.
Because θ(t) can be written as a linear combination of{a j }, this problem can be solved using the quadratic programming (QP) for p = 1 and 2. Note that this problem
is similar to the “Lasso” model[10] when p = 1.
On the other hand, we can define le (·) as
0|x| ≤ e, (e ≥ 0),(8)le (x) =∞|x| > eand define the [Task Error] term ast
f +t ple (θ(t) − θT ),t=t f

which forces θ(t) to stay between θT ±e during [t f , t f +t p ]and [Task Error] = 0. In this case, the problem can berewritten as
minimize|a j | p s.t. |θ(t) − θT | ≤ e (t f ≤ t ≤ t f + t p ).
j(9)This can be solved with the linear programming (LP) andQP when p = 1 and p = 2, respectively.
Application of these algorithms dramatically reduces
the computational cost, and therefore, the motor planningcan be solved quite eﬃciently.
5jt
f +t pt=t fwhere {φ j (x, y)} is the basis and ε(x, y) is Gaussian noise.
This framework has the same form as our parametricmodel in Eq. (2). They assumed that our visual systemtried to minimize the following two-termed cost function,
which is the logarithm of the posterior distribution,

j|a j | p , then the problem becomes

jjx,y

Simulated experiment

5.1 Method
In order to examine the validity of above frameworkand the eﬀect of diﬀerent optimization methods, we ran
a computer simulated experiment, taking a single-joint
reaching task as an example.
Figure 1 (a) shows the structure of the model, which isalmost the same as one-dimensional arm model in Harris& Wolpert[7]. The motor command u(t) was transferred
into the joint torque τ(t) through two first-order low-passfilters, whose time constants (T 1 and T 2 ) were 30 and 40
ms, respectively:|a j | as the second term, that is, penal-j

ize larger weights, the system will tend to choose motor
commands consisting of a smaller number of basis patterns. This means that the motor command is designed
based on sparse representation.
4 Motor planning and optimization algorithms
In the proposed framework, the motor planning problem is solved by minimizing the cost function in Eq.(6).
When the motor system is linear, the problem can besolved by well known optimization techniques by setting
the [Task Error] term properly.
Let us imagine a reaching movement of a single-jointarm (See a detailed example in Sec. 5). If we define thet
f +t p|θ(t) − θT |2 , and the second term
[Task Error] term as(T 1 T 2 )d2 τdτ(t) + (T 1 + T 2 ) (t) + τ(t) = u(t).
2dt
dt(10)Dynamics of the arm was given byI

t=t f2d2 θdθ(t) + b (t) + kl20 θ(t) = τ(t),2dt
dt(11)

a.

b.
Basis Patternsu(t)MotorCommand

11+T1s EMGMuscle

11+T2sτ(t)Joint Angle

Torqueθ(t)Relative Pattern Level

Musculo-Skeltal Model

0.04 s10.08 s0.12 s0.5

00Elbow

0.16 s

0.04

0.08

0.12

0.16

0.2

Time (sec)Figure 1. Model structure and pre-defined basis patterns. a. Motor command is imposed to the single-joint skeletal
system through two first-order linear filters. b. Four pre-defined basis patterns are prepared.
where θ(t) is the joint angle, I(= 0.25) is the inertia, b(=0.2) is the viscosity, k(= 0.1) is the elasticity, l0 (= 0.35)
is the arm length. In total, therefore, the motor system
was a fourth order linear system.
On the other hand, the motor command u(t) was given
by Eq.(2). In this experiment, the pre-defined basis patterns w j (t) were four rectangular pulses with diﬀerent
time-width but with the same energy (i.e.,w2j (t)∆t =5.2 ResultFigure 2 (a) shows the command patterns and joint trajectories obtained by four diﬀerent optimization methods.
Parameter values were chosen so that the resultant movement times were similar for all methods. Typical threephase command patterns and smooth movement trajectories were generated, and angular velocity (data notshown) showed an almost-symmetric bell-shaped profile,common to all conditions. This confirms that the proposed method worked appropriately as a motor planningalgorithm.
Two other major features are pointed out here. First,
while the movement trajectories were similar among different optimization formalizations, the command patterns showed diﬀerent features: Command patterns for
p = 2 (B and D in Fig. 2(a)) were more continuousand minute (i.e., containing a larger number of basis patterns), compared to those for p = 1 (A and C). Thismeans that sparser representation were obtained with L1
constraints on the weights.
Second, movement time t f depended on θT . Figure
2 (b) shows this dependency in more detail2 . The upper
panel indicates the joint trajectories for diﬀerent valuesof η when we solve Eq.(7) with p = 1, and the lower
panel shows the relationship between the log movement
distance and movement time for diﬀerent η values. The
data for four targets are nicely located on the regression
line. This means that the present result reproduced Fitts’law[4], saying that the movement time can be approximated by a + b (log(l) - log(w)), where l is movement
distance and w is target size (a and b are constants).

t

1, where ∆t is the width of discrete time step. See Figure
1 (b)). This assumption is based on the discussion thatsingle joint movement can be achieved by controlling thewidths and heights of rectangular motor commands [6].
To be more specific, we prepared positive and negative
rectangular pulses starting at 30 diﬀerent times (from 0 s
to 0.8 s with an interval of 0.04 s).
Task performance was evaluated by the sum of theendpoint error during the pausing time t p (= 0.4 s) andmovement time t f with a weight η (i.e., faster movement
was preferable). Resultantly, our objective function was|a j | p .
E {a j }, t f θT ; {w j } = η t f + [Task Error] + λj(12)Note that movement time was also determined by optimizing this function.
We solved this problem using LP and QP by rewriting
it as in Eqs.(7) and (9). In concrete, we solved the problem using these algorithms for each t f value chosen fromthe region [0.1 s, 0.4 s] with an interval of 0.02 s and thensearched the optimal value of t f which gave the minimal
cost.
Other experimental conditions were as follows: The
initial joint angle was 0 rad, and the target angle (θT ) waschosen from 0.2, 0.4, 0.8 and 1.6 rad. ( π/2). Optimization problem was solved with IMSL C Math Library (Visual Numerics, Inc.). We ran the simulation for diﬀerent
values of parameters λ, η and e, and found their appropriate values producing meaningful results.

6Concluding remarks

In the present article, we introduced a parametric framework of motor command representation, and showed thepossibility of the parameter preference to work as a criterion for motor planning.
2 This dependency changed according to η when λ and e were fixed.
Meaningful results were obtained for appropriate values of η: t f wasstuck to the upper or lower limit for too small or too large η, respectively.

3

a.

b.
30

10Eq.(7), p = 1
λ = 0.15
η = 0.002

2

θT = 0.2
0.4
0.8
1.620

A

θT = 1.6

1.5

2

0.5

230
20

1.5
10

Joint Angle θ (rad)Eq.(9), p = 1e = 0.002
η = 100

Joint Angle θ (rad)10-300Command uC

1.5

0.4
0.2-10-20

Eq.(7), p = 2
λ = 0.15
η = 0.002

Eq.(7), p = 1
λ = 0.30, η = 0.002

0.8

10B

Joint Trajectory for
different η values

Joint Trajectory

Command Pattern-10-20-30
30
20
100

1
0.5

0.5
000.2

0.40.6

λ = 0.45, η = 0.002
2
1.5

012

0.5

1.5

0

1

00.2

0.4

0.6

Time (sec)

-10
0.5
-20

Eq.(9), p = 2
e = 0.002
η = 100

030

2

Distance vs. MT
0.4

20
1.5

MT (sec)

D-30

10
10-100.5-20-30

00.2

0.4

0.6

000.2

0.4

0.6

0.30.2

η = 0.15
η = 0.3
η = 0.45

0.1
00.1

0.2

0.5

1.0

2.0

Distance log θTTime (sec)Figure 2. Results of computer simulated experiments. a. Planned command patterns and resultant trajectories forfour diﬀerent optimization methods: (A) QP solution to Eq.(7) with p = 1, (B) QP solution to Eq.(7) with p = 2, (C)LP solution to Eq.(9) with p = 1, and (D) QP solution to Eq.(9) with p = 2. b. Relationship between log movement
distance and movement time for diﬀerent η values in case A.
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A simulated experiment shows that Fitts’ law canbe qualitatively replicated with a single-joint lineararm model. We are now examining whether ourrepresentation-based criterion also works for the movement with non-linear and/or redundant motor systems.
Creating optimal basis for a given task environment is afuture research topic.
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