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Neural Networks 14 (2001) 381±393

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Contributed article

Quantitative examinations for multi joint arm trajectory planningÐusing
a robust calculation algorithm of the minimum commanded torquechange trajectory
Yasuhiro Wada a,*, Yuichi Kaneko a, Eri Nakano b, Rieko Osu c, Mitsuo Kawato c,daNagaoka University of Technology, 1603-1 Kamitomioka, Nagaoka-shi, Niigata, 940-2188, Japanb
Advanced Telecommunication Research Laboratories, Kyoto, 619-0288, Japanc
ERATO Dynamic Brain Project, Kyoto, 619-0288, Japand
ATR Human Information Processing Research Laboratories, Kyoto, 619-0288, Japan
Received 23 March 2000; revised 26 December 2000; accepted 26 December 2000

Abstract
In previous research, criteria based on optimal theories were examined to explain trajectory features in time and space in multi joint arm
movement. Four criteria have been proposed. They were the minimum hand jerk criterion (by which a trajectory is planned in an extrinsickinematic space), the minimum angle jerk criterion (which is planned in an intrinsic-kinematic space), the minimum torque change criterion(where control objects are joint links; it is planned in an intrinsic-dynamic-mechanical space), and the minimum commanded torque change
criterion (which is planned in an intrinsic space considering the arm and muscle dynamics). Which of these is proper as a criterion for
trajectory planning in the central nervous system has been investigated by comparing predicted trajectories based on these criteria withpreviously measured trajectories. Optimal trajectories based on the two former criteria can be calculated analytically. In contrast, optimal
trajectories based on the minimum commanded torque change criterion are dif®cult to be calculated, even with numerical methods. In somecases, they can be computed by a Newton-like method or a steepest descent method combined with a penalty method. However, for a realistic
physical parameter range, the former becomes unstable quite often and the latter is unreliable about the optimality of the obtained solution.
In this paper, we propose a new method to stably calculate optimal trajectories based on the minimum commanded torque change criterion.
The method can obtain trajectories satisfying Euler±Poisson equations with a suf®ciently high accuracy. In the method, a joint angle
trajectory, which satis®es the boundary conditions strictly, is expressed by using orthogonal polynomials. The coef®cients of the orthogonal
polynomials are estimated by using a linear iterative calculation so as to satisfy the Euler±Poisson equations with a suf®ciently highaccuracy. In numerical experiments, we show that the optimal solution can be computed in a wide work space and can also be obtainedin a short time compared with the previous methods.
Finally, we perform supplementary examinations of the experiments by Nakano, Imamizu, Osu, Uno, Gomi, Yoshioka et al. (1999).
Estimation of dynamic joint torques and trajectory formation from surface electromyography signals using a neural network model.
Biological Cybernetics, 73, 291±300. Their experiments showed that the measured trajectory is the closest to the minimum commanded
torque change trajectory by statistical examination of many point-to-point trajectories over a wide range in a horizontal and sagittal work
space. We recalculated the minimum commanded torque change trajectory using the proposed method, and performed the same examinations as previous investigations. As a result, it could be recon®rmed that the measured trajectory is closest to the minimum commanded
torque change trajectory previously reported. q 2001 Elsevier Science Ltd. All rights reserved.
Keywords: Minimum commanded torque change criterion; Euler±Poisson equation; Orthogonal polynomial; Trajectory planning; Optimization

1. Introduction
The following two characteristics have been well demonstrated about the feature of a point-to-point human arm movement on a plane (Abend, Bizzi, & Morasso, 1982). (1) The path
is a roughly straight line, but is slightly curved. (2) The velo* Corresponding author. Fax: 181-258-47-9500.
E-mail address: ywada@nagaokaut.ac.jp (Y. Wada).

city pro®le is bell shaped with a single peak. Several models
have been proposed to explain these features (Bizzi, Accornero, Chapple, & Hogan, 1984; Flash & Hogan, 1985;
Nakano, Imamizu, Osu, Uno, Gomi, Yoshioka et al., 1999;
Rosenbaum, Loukopoulos, Meulenbroek, Vaughan, & Engelbrecht, 1995; Uno, Kawato, & Suzuki, 1989). In particular, thefollowing criteria for trajectory planning based on optimal
principles have been proposed.
The minimum hand jerk criterion (Flash & Hogan, 1985)

0893-6080/01/$ - see front matter q 2001 Elsevier Science Ltd. All rights reserved.
PII: S 0893-608 0(01)00026-0

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Y. Wada et al. / Neural Networks 14 (2001) 381±393

is planned in an extrinsic-kinematic space, the minimum
angle jerk criterion is planned in an intrinsic-kinematic
space (Rosenbaum et al., 1995), the minimum torquechange criterion (Uno et al., 1989), where control objects
are joint links, is planned in an intrinsic-dynamic-mechanical space, and the minimum commanded torque change
criterion (Nakano et al., 1999) is planned in an intrinsic
space considering the arm and muscle dynamics and using
representations for motor commands controlling the muscle
tensions. The former two criteria are planned in a kinematic
space, and the latter two criteria are models that depend on
the dynamics of the arm. The ®nal target in this paper is to
discuss which of these trajectory planning criteria based on
optimal principles is appropriate as a criterion for trajectory
planning in the central nervous system.
Incidentally, analytical solutions for the minimum handjerk criterion and the minimum angle jerk criterion can becalculated easily. However, the latter two models, whichsmoothen the change of the torque or the commanded
torque, must be solved under several constraints with thenonlinear dynamics and boundary conditions at the startingand ®nal points. That is, we have to solve a nonlinear, optimization problem.
In general, it is quite dif®cult to obtain an optimal solution. Nakano et al. (1999); Sakuraba, Osu, Nakano, Wada,
and Kawato (2000) reported that the trajectory predicted by
the minimum commanded torque change criterion is the
trajectory closest to a human trajectory among trajectories
predicted by the above criteria. However, it has been very
dif®cult to establish a robust method for obtaining an optimal trajectory based on the minimum commanded torquechange criterion at various starting points, ®nal points, anddynamic parameters. The optimal trajectory can be obtainedby the Newton-like method (Uno et al., 1989), and the steepest descent method (Nakano et al., 1999). The Newton-like
method is an algorithm to guarantee the local optimality of
the converged trajectory mathematically; however, the algorithm often becomes unstable according to the parametervalue of the dynamics such as the viscosity, and then,often diverges. The algorithm also occasionally diverges
according to the positions of the starting and ®nal points.
On the other hand, the optimal solution can be stable andcomputed comparatively more easily using the steepest
descent method. However, the optimality of the solution
cannot be guaranteed. Because it was dif®cult for Nakano
et al. (1999) to calculate an optimal trajectory of the minimum commanded torque change, comparisons were madeon quasi-optimal trajectories based on the minimum
commanded torque change obtained using the steepest
descent method and measured trajectories.
Calculating the optimal trajectory is necessary andindispensable for explaining human arm movements bya mathematical model. Therefore, the need is high foran algorithm that can calculate the optimal trajectory
accurately and without fail based on the minimum
commanded torque change criterion, regardless of thepositions of the starting and ®nal points or the value ofthe dynamic parameter.
In this paper, ®rst of all, we propose a trajectory calculation method based on the minimum commanded torquechange criterion, which satis®es Euler±Poisson equations
with a suf®ciently high accuracy. These Euler±Poisson
equations, which are derived from the functional optimal
problem, provide the necessary condition for obtaining the
optimal trajectory. In the proposed method, the joint angle
trajectory is expressed by a system of orthogonal polynomials for time, and each coef®cient of the orthogonal polynomials is estimated by using a linear, iterative operation tosatisfy the Euler±Poisson equations as much as possible.
We show the numerical experiment results of the minimum
commanded torque change trajectory generation for a twojoint arm on two planes (the horizontal and sagittal planes)
and also point out that the optimal trajectory, which satis®es
the Euler±Poisson equations with a suf®ciently high accuracy, can be obtained in a short time, regardless of thepositions of the starting and ®nal points or the value ofthe dynamic parameter. Finally, we discuss the adequacyof the trajectory planning criterion. We examine four trajectory planning criteria in multi joint reaching movements,
which can explain the universal features in arm trajectories
well, as mentioned above, quantitatively and statisticallyusing many movement trajectories measured on various
and not local spaces (Nakano et al.,1999).

2. Optimal criteria for trajectory planning andcalculation algorithm
2.1. Minimum hand jerk criterionWith the minimum hand jerk criterion (Flash & Hogan,1985) a trajectory is planned according to the arm position(x, y) in an extrinsic-kinematic space given from visionindependently of the musculoskeletal dynamics. The trajectory is planned so as to minimize the time integral of jerk
(three times differentiation of the hand position with respect
to time). The objective function is given as follows:
1 Ztf
CJ 
2 0

(

d3 x
dt3

!2

d3 y
1
dt3

!2 )
dt

1

Here, tf shows the duration.
In a point-to-point movement, the trajectory predicted by
the criterion is a straight line trajectory in the Cartesian
coordinate. The tangential velocity is a bell shape with asingle peak. Flash and Hogan (1985) claimed that the minimum hand jerk criterion can reproduce a trajectory corresponding well with movement data from human subjects.
An attractive point is that the analytical optimal trajectory,
which satis®es the minimum hand jerk trajectory, can beobtained easily.

Y. Wada et al. / Neural Networks 14 (2001) 381±393t
x t  x 1 x 2 x  210
tf
J

s

s

f

!3

t
115
tf

!4

t
26
tf

!5 !
2

t
y t  y 1 y 2 y  210
tf
J

s

s

f

!3

t
115
tf

!4

t
26
tf

!5 !
3

s

s

f

f

Here, x , y and x , y correspond to the position of thestarting point and the position of the ®nal point, respectively.

383

reproduce the gradually curved trajectory unable to be
explained by the minimum hand jerk criterion. Moreover,movements that pass via points and movements to which an
external force is applied can be reproduced. The torque iscalculated from an equation of two link manipulators (Eq.
(7)).

t1  {I1 1 I2 1 2M2 L1 S2 cosu2 1 M2 L1 2 }u 1 1 I2
1 M2 L1 S2 cosu2 u 2 2 M2 L1 S2 2u_ 1 1 u_ 2 u_ 2 sinu2
1 B11 u_ 1 1 B12 u_ 2 1 g{ M1 S1 1 M2 L1 sinu1
1 M2 S2 sin u1 1 u2 }

2.2. Minimum angle jerk criterion
A trajectory based on the minimum angle jerk criterion isplanned so as to minimize the change of the angle acceleration. That is, the object function of the minimum angle jerk
criterion is expressed by minimizing the integration of thejerk over the motion duration, as shown in Eq. (4):!2
2
1 Ztf Xd 3 uidt
4
CAJ 
2 0 i1 dt3

u i shows the joint angle of joint i here. An analytical solution of the criterion can be easily obtained in the samemanner as the minimum hand jerk model. The minimum
angle jerk trajectory is expressed as the following equations,which are the ®fth spline functions.
!3
!4
!5 !
t
t
t
f
AJ
s
s
ui t  ui 1 ui 2 ui  210
115
26
tf
tf
tf
0 # t # tf 5Here, usi and ufi show the joint angle at the starting point andthe ®nal point of joint i, respectively. The trajectory
predicted by the criterion is independent of the in¯uenceof the arm dynamics and the motion duration, except thearm length. The point-to-point movement trajectory plannedby the criterion is a straight path in the joint space. The
trajectory transformed in the Cartesian coordinate is gradually curved according to the arm position.
2.3. Minimum torque change criterion
The minimum torque change trajectory is planned so thatthe change of torque t generated by each joint becomes thesmallest. The object function is given by the following
equation:
2 
1 Ztf X
dt i 2dt
6
CTC 
2 0 i1 dt

t i shows the torque generated by joint i. The minimum
torque change criterion (Uno et al., 1989) depends on thearm dynamics. The minimum torque change criterion cant2  I2 1 M2 L1 S2 cosu2 u 1 1 I2 u 2 1 M2 L1 S2 u_ 1 2 sinu2
1 B22 u_ 2 1 B21 u_ 1 1 gM2 S2 sin u1 1 u2 7Here, t 1 and u 1 show the torque and the joint angle ofthe shoulder, and t 2 and u 2 show the torque and joint
angle of the elbow, respectively. Ii, Mi, Li, and Si, show
the inertia moment surroundings of link i (i  1, 2),mass, arm length, and the distance from the positionof the joint to the position of the center of the gravity.
The three-dimensional shape of a male's arm wasmeasured by a Cyberware Laser Range Scanner. We
calculated the arm as a homogeneous material with a
speci®c gravity of 1.0 and computed its mass, center ofmass, and moment of inertia from its volume. The arm
parameters for each subject were calculated using theratio of the arm length based on the measured data. Bij
shows the viscosity coef®cient expressing the in¯uenceof the angular velocity of joint j on the torque of joint i.
Links 1 and 2 correspond to the upper arm and theforearm. Moreover, g is the acceleration of gravity.
The term concerning g shows the force supporting thearm. Therefore, g is set to 0 for movement in the horizontal plane. Incidentally, in the minimum torquechange model proposed by Uno et al. (1989), only thelink dynamics is regarded as the controlled object.
The minimum torque change trajectory is extremely
sensitive to the value of the viscosity. Most of the viscosity
measured around a joint is ascribed to a biochemical andmechanical reaction process within the muscle when it
receives impulses and generates tension, which is not
ascribed to a passive property of the joint (Akazawa,
1994). Considering the viscosity in calculating the torquemeans that both the link dynamics and muscle are regarded
as controlled objects. It is not appropriate to use a non-zero
viscous value to calculate the torque because the literalminimum torque change model takes the actual torquearound the joint as an object for optimization (Flash,1990). Therefore, the actual torque with zero viscosity
Bij  0 is calculated from the dynamics equation of a twolink manipulation.

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Y. Wada et al. / Neural Networks 14 (2001) 381±393

2.4. Minimum commanded torque change criterion
Even if the hand position is determined in the extrinsic
coordinates, the joint angles or the muscle lengths cannot
be uniquely determined because of redundant degrees of
freedom. When a desired trajectory is determined in the
joint angle coordinate, the actual torques around the joints
can be calculated by an inverse dynamics equation.
However, there are also an in®nite number of possiblecombinations of agonist and antagonist muscle tensions
that can generate the same torques. The degrees of freedom of a -motoneurons, which innervate each muscle, arehigher than those of the muscles, and cortical motor
neurons may have higher degrees of freedom than a motoneurons. Even if the time pro®les of muscle tensions
are speci®ed, the ®ring rates of the cortex or spinal cord
neurons cannot be uniquely determined. Regardless ofthese indeterminacies, the actual hand trajectories showcommon invariant characteristics, and electromyographic(EMG) signals appear in typical triphasic patterns. Theseobservations suggest that the brain solves these ill-posed
problems based on some principles.
The minimum motor command change model (Kawato,
1992, 1996) has been proposed for trajectory planning in an
intrinsic-dynamic-neural space. The minimum angle jerk
criterion and the minimum torque change criterion can becalculated from a comparatively easy physical parameter.
Because the minimum motor command change model canconceptualize signals at the a -motoneuron or cortical motoneuron level, all indeterminacies can be constrained at eachlevel. Although an attempt has been made to estimate themotor commands at the muscle level (Koike & Kawato,1995), it is extremely dif®cult to estimate the motor
commands of the spinal cord or cortex by modeling theinformation processing from a central system to a peripheral
system.
A quantitative model, not a conceptual model, isneeded to actually compute an optimal trajectory. Therefore, ®rst, a minimum commanded torque change modelthat approximates the minimum motor command change
model and has computability is proposed, while positively appreciating the assumption of non-zero viscosity
by Uno et al. (1989). In the literal minimum torquechange model, only the link dynamics is regarded as
the controlled object, whereas the minimum commanded
torque change model, both the link dynamics andmuscles are regarded as controlled objects. We employ
motor commands at the peripheral level, in other words,
we use signals controlling muscle tensions to model theminimum commanded torque change criterion. In termsof indeterminacy, however, the minimum commanded
torque change model solves problems at the samelevel, that is, the torque level as the minimum torquechange model.
With the minimum commanded torque change criterion,
the link dynamics and the muscles are controlled, and the

trajectory is planned so as to minimize the time integral of
the square of the commanded torque change rate. The object
function is conceptually different though is given by an
expression identical to the minimum torque change criterion
as Eq. (8) shows,N 
1 Ztf X
dtci 2CCTC 
dt8
2 0 i1 dt
Here, tf indicates the movement time, and N shows thenumber of joints. t c shows the commanded torque. Incidentally, the commanded torque does not correspond to themechanical output torque. It is an approximated torque forthe motor commands. A point-to-point movement based on
the criterion is planned in consideration of the dynamics
parameters of the mass of the arm and the inertia movement,
etc., shown in Eq. (7). The feature of the trajectory generated by the criterion changes according to the start postureand ®nal posture.
In this paper, Eq. (7) for a two-link, two-joint arm wasused for the calculation of the commanded torque. It was
extremely dif®cult to predict a trajectory satisfying thecriterion as well as the minimum torque change criterion.
In particular, because the value of the viscosity was not set
to 0 for the minimum commanded torque change criterion, it
became more dif®cult to obtain the solution by the Newtonlike method.
In our study, we use the following formula, which wasestimated from the actual torque and viscosity during staticforce control (Gomi & Osu, 1998), to acquire viscous valuesof diagonal components (B11, B22) and off-diagonal components (B21, B12) for each trajectory. Here, for simplicity,mamean absolute torques (shoulder:tma
1 , elbow:t2 ) during
movement are used as the actual torques.
B11 ; B12  B21 ; B22   0:63 1 0:095tma
1 ;ma
0:175 1 0:0375tma
2 ; 0:76 1 0:185t2 

Nm= rad=s

9

We propose a new method to calculate the minimum
commanded torque change trajectory, which is basedon the Euler±Poisson equation and can predict an optimal trajectory robust for the change of the value of the
dynamics parameter such as a viscosity. Naturally, themethod can also be applied to the minimum torquechange criterion.
3. A prediction algorithm for trajectories based on theminimum commanded torque change criterion using theEuler±Poisson equation
3.1. Euler±Poisson equation
In this section, a calculation algorithm for trajectories
based on the minimum commanded torque change criterionusing the Euler±Poisson equation is described. Basically,
the algorithm calculates a trajectory that satis®es the

Y. Wada et al. / Neural Networks 14 (2001) 381±393

Euler±Poisson equations with a high accuracy and theboundary conditions strictly. The object function of theminimum commanded torque change criterion (Eq. (8)) isexpressed as follows.
1 Ztf
F t; u1 ; u2 ; u_ 1 ; u_ 2 ; u 1 ; u 2 ; u13 ; u23 dt ! Min
CCTC 
2 0
10
P2 ÿ c 2 3
3
Here, F  i1 dti =dt ; u1 and u2 show the differential
coef®cient of the third order with respect to the time of u 1
and u 2, respectively. By the boundary conditions of thestarting point and the ®nal point (position, velocity, andacceleration), the following Euler±Poisson equations forthe above optimal problem can be derived as a necessary
condition for an extreme value.
E1 

E2 

2

3

We use the Jacobian polynomial (Szego, 1975) as the set oforthogonal polynomials to express Dui t (see Appendix A).
The Jacobian polynomial can be represented as Q~ k t 
64t3 1 2 t3 P~ k6;6 t: We de®ne the difference trajectory
Dui t as the following equations, which are normalized to
0 # t # 1.
X
64aik t3 1 2 t3 P~ k6;6 t
14
Dui t k

Dui t is composed of the summation of Jacobian orthogonal
polynomials; the boundary conditions of the position, velocity, and acceleration at the starting point and ®nal point all
satisfy 0. From the above, the minimum commanded torquechange trajectory ui t is given by Eq. (15).

ui t  usi 1 usi 2 ufi  210t3 1 15t4 2 6t5  1 64t3 1

2
d 2
d
2
d
2
F2
F1 2
F2 3
F0
_

2u1
dt 2u 1dt 2u 1dt 2u13
2
d 2
d2 2
d3 2
F2
F1 2
F2 3
F0
2u2
dt 2u_ 2
dt 2u 2
dt 2u23

11

Therefore, u 1 and u 2, which satisfy these two Euler±Poisson
equations for every movement time, are the minimum
commanded torque change trajectory.
3.2. A system of orthogonal polynomials for expression of
the minimum commanded torque change trajectory
The minimum angle jerk trajectory is one of the good
approximations of the minimum commanded torque change
trajectory. Here, let us assume that the minimum
commanded torque change trajectory ui t (i  1, 2) canbe expressed by the following equation.

ui t  uAJ
i t 1 Dui t

12Here, uAJ
i t corresponds to the minimum angle jerk trajec-

tory, and Dui t is denoted as the difference between theminimum commanded torque change trajectory and theminimum angle jerk trajectory. The trajectory of Eq. (12)must satisfy the following boundary conditions of the starting position and ®nal position.

ui 0  usi ;u_ i 0  u i 0  0;

ut tf   ufi ;u_ i tf   u i tf   0

i  1; 2

13

385

2 t3

K
X
k0

aik P~ k6;6 t

15

6;6
x obtained by a recurrence
Jacobian polynomials Pk11
formula from the ®rst to the (k 1 1)th order are shown inAppendix A.

3.3. An estimation algorithm of parameter aik
To estimate aik satisfying the Euler±Poisson equations
with a high accuracy, we propose the following two
innovations. (1) The left side of the Euler±Poisson
equation is expressed as a linear summation concerningaik, and (2) the coef®cient aik is estimated using an
iterative calculation by the discretization of the movement time.
3.3.1. A linear summation concerning parameter aik
Eq. (15) is substituted for Eq. (11), and this is arranged to
become a linear expression of aik such as Eq. (16).
E1 t; u1 ; u2 ; u_ 1 ; u_ 2 ; ¼; u13 ; u23 


2 X
K
X
i1 k0

_

_





1 hik t; u1 ; u2 ; u 1 ; u 2 ; u 1 ; u 2 aik

1 J1 t; u1 ; u2 ; u_ 1 ; u_ 2 ; ¼; u13 ; u23 
E2 t; u1 ; u2 ; u_ 1 ; u_ 2 ; ¼; u13 ; u23 
2 X
K
X

_

_





In the next section, we show concrete equations of the minimum angle jerk trajectory and the second term on the rightside of Eq. (12).



3.2.1. A difference trajectory Dui t
uAJi t in Eq. (12) is the minimum angle jerk trajectory,
which satis®es the boundary conditions (Eq. (13)). Therefore, Dui t should be a function that makes the boundary
conditions at the starting point and ®nal point all become 0.

Eq. (16) is obtained according to the standard of the following (A) and (B).

i1 k1

2 hik t; u1 ; u2 ; u 1 ; u 2 ; u 1 ; u 2 aik

1 J2 t; u1 ; u2 ; u_ 1 ; u_ 2 ; ¼; u13 ; u23 16(A) Terms 1; A1 ; A2 ´ui nHere, [ ] ui n indicate product terms of the inside

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Y. Wada et al. / Neural Networks 14 (2001) 381±393

Table 1
Terms in 1hik, 2hik, J1, and J2
1hik, 2hik

Sum of A1 term a
Sum of A2, B1, A2 ; A3 ´B1 terms

J1 , J 2

Sum of B2, A1 ; A4 ´B2 terms a
Sum of B3, A2 ; A3 ´B3 terms

A1

cos u1 ; cos u1 1 u2 ; cos2 u1 ; cos2 u1 1 u2 ; cos u1 cos u2 ; cos u1 cos u1 1 u2 ; cos u2 cos u1 1 u2 A2cos(u 2), cos2(u 2)A3sin(u 2), sin2(u 2), sin(u 2)cos(u 2)A4

sin u1 ; sin u1 1 u2 ; sin u1 cos u1 ; sin u1 cos u1 1 u2 ; cos u1 sin u1 1 u2 ; sin u1 sin u2 ; sin u1 cos u2 ; cos u1 sin u2 ; sin u2 sin u1 1 u2 ;
sin u2 cos u1 1 u2 ; cos u2 sin u1 1 u2 ; sin u1 1 u2 cos u1 1 u2 

B1

ui 1 ; u21 ; u11 2 ; u21 2 ; u11 u21

B2

u11 2 ; u21 2 ; u11 u21 ; u11 3 ; u21 3 ; u11 2 u21 ; u11 u21 2 ; u11 4 ; u21 4 ; u11 3 u21 ; u11 u21 3 ; u11 2 u21 2 ; u12 2 ; u22 2 ; u12 u22 ; u11 2 u12 ;u12 u21 2 ; u11 u12 ; u21 u12 ; u11 u22 ; u21 u22 ; u11 u13 ; u11 u23 ; u21 u13 ; u21 u23 ; u12 2 u22 ; u12 u22 2 ; u11 u21 u12 ; u11 u21 u22 ; uiAJ  2 ; ui AJ  3 ;ui AJ  4 i  1; 2u21 5 ; u11 u21 4 ; u11 2 u21 3 ; u12 u21 3 ; u21 3 u22 ; u12 2 u21 ; u11 u22 2 ; u21 u22 2 ; u11 2 u13 ; u21 2 u13 ; u11 2 u23 ; u21 2 u23 ; u12 u13 ; u22 u13 ;B3u12 u23 ; u22 u23 ; u11 u12 u22 ; u21 u12 u22 ; u11 u21 u13 ; u11 u21 u23 ; u11 u21 2 u12 ; u11 2 u21 u22 ; u11 u21 2 u22 ; u21 6 ; u11 u21 5 ; u11 4 u21 2 ;u11 2 u21 4 ; u11 3 u21 3 ; u12 u21 4 ; u11 4 u22 ; u21 4 u22 ; u11 2 u12 2 ; u21 2 u12 2 ; u11 2 u22 2 ; u21 2 u22 2 ; u12 3 ; u22 3 ; u12 2 u22 ;u12 u22 2 ; u11 3 u13 ; u21 3 u13 ; u11 3 u23 ; u21 3 u23 ; u13 2 ; u23 2 ; u13 u23 ; u11 3 u12 u21 ; u11 u12 u21 3 ; u11 3 u21 u22 ; u11 u21 3 u22 ;u11 2 u12 u21 2 ; u11 2 u21 2 u22 ; u11 2 u12 u22 ; u21 2 u12 u22 ; u11 u21 u12 2 ; u11 u21 u22 2 ; u11 2 u21 u13 ; u11 u21 2 u13 ; u11 2 u21 u23 ; u11 u21 2 u23 ;u11 u12 u13 ; u12 u13 u21 ; u11 u13 u22 ; u11 u12 u23 ; u13 u21 u22 ; u11 u22 u23 ; u12 u21 u23 ; u21 u22 u23 ; u11 u21 u12 u22 ; ui AJ  4 ; ui AJ  5 ; ui AJ  6 ; ui AJ  4 u11 ;ui AJ  4 u21 ; ui AJ  4 ui 1  2 ; ui AJ  4 u21 2 ; ui AJ  4 u11 u11 ; ui AJ  5 u11 ; ui AJ  5 u21 ; i  1; 2

aFor the sagittal plane.

terms of [ ] and ui n . Refer to Table 1 for A1 and A2.
(B) 1; A2 ; A3 ´ui 4 ´u_ j ; ui 4 ´ u_ j 2 ; ui 4 ´u_ 1 ´u_ 2 ; ui 4 ´u j ; ui 5 ´u_ j i; j  1; 2
However, [ ].[ ] shows the product with each term of[ ]. Refer to Table 1 for A2 and A3.
Here, J1 and J2 are composed of terms; Eq. (15) is notn(n  2,
substituted according to the standard, and uAJi ¼, 6). The composition terms of 1hik, 2hik, J1 and J2 areshown in Table 1.
3.3.2. A discretization of the movement time
Time t is dispersed to M divisions with equal intervals forthe normalized movement time t0 ; t1 ; t2 ; ¼; tm ; ¼; tM  t0 
0; tM  1: Here, if a parameter aik expressing an optimal
trajectory exists, the trajectory of Eq. (16) satis®es Eq. (11).
That is, Eq. (16) satis®es the following equation.
2 X
K
Xi1 k0

_

_





l hik tm ; u1m ; u2m ; u 1m ; u 2m ; u 1m ; u 2m aik33
1Jl tm ; u1m ; u2m ; u_ 1m ; u_ 2m ; ¼; u1m
; u2m
0

l  1; 2;

0 # m # M17Here, the above equation is rearranged using l hikm lhik tm ; ¼ and Jlm  Jl tm ; ¼ as follows.
2 X
K
Xi1 k0l hikm aik 1 Jlm  0

18

Eq. (18) holds for all discretization time values. Then,
2(M 1 1) linear equations of aik are derived. Theseequations are expressed as follows, in matrix form.
332
2¼ 1 h2K0J10
1 h100
7
7
6
6
6 .
6 . 7
.. 7
7
6 .
6 . 7
]
. 72
6 .
6 . 7
7 a 3 6
7
6
7 10
7
6
6
6 1 h10m ¼ 1 h2Km 76
6J
7 6 1m 7767
6
7
76 a11 7 6
7
6
76
6 .
6
.
.
7 6 . 7
7
6 ..
.. 7
7
]
.
76
7
6
6
.. 7 6
76
7
6
7
6
.
7
6
6
7 6J 77
6 1 h10M ¼ 1 h2KM 76
7
6
1M
76
7
6
6
1
76 a1K 7
70
6
6
7 6
76
7
6 h
¼
7
h
J
6 2 100
6 20 7
2 2Ko 767
76 a20 7 6
7
6
7
6 .
7 6
.. 7
76
6 .
6 .. 77
6
76 . 7 6 . 7
6 .
].
76 .. 7 6
7
6
74
7
6
5 6
7
7
6 h
6
¼
2 h2Km 7
6 2 10m
6 J2m 7
7 a2K
7
6
6
6 .
6 . 7
.. 7
7
6 .
6 . 7
6 .
6 . 7
]
. 7
5
5
4
4
¼
J
h
h
2M
2 10M
2 2KM
Ha 1 J  0

Y. Wada et al. / Neural Networks 14 (2001) 381±393

Fig. 1. Relationship between the number of parameters K and uEumax . A, B,
C, D: trajectories in Fig. 3.

Therefore,a  2H# J

19

#Here, H indicates a pseudo inverse matrix of H. aik isestimated by an iterative calculation of the following
method.
1. The initial value of parameter aik is set to 0, andjoint angle trajectory u i is calculated from Eq. (15).
2. 1 hikm ;2 hikm ; J1m ; and J2m are calculated at every timetm (m  0, 1, ¼, M).
3. aik is obtained from Eq. (19).
4. A new joint angle trajectory u i is calculated from Eq.
(15) using aik computed in step 3.
Steps 2±4 are repeated until aik converges. In otherwords, in Eq. (19), let d (d  0, 1, 2, ¼) denote anindex of the iterative calculation; the iterative calculation can be shown by the following equation.
ad11  2H ad # J ad 20

4. Numerical experiments of point-to-point movementon the horizontal and sagittal planes
The formation of minimum commanded torque change
trajectories with the same starting point, ®nal point, andmotion duration as measured trajectories by Nakano et al.
(1999) was carried out. The error of the Euler±Poisson
equation was de®ned by the maximum value of the absolutevalue of the error, such as Eq. (21), to evaluate eachpredicted trajectory.
max

0#tm #1

2
Xi1

uEip tm  2 Ei tm u  uEumax

21

Here, Eip tm  shows the optimal value of Ei tm . It is clear

387

Fig. 2. Convergence of uEumax . A, B, C, D: trajectories in Fig. 3.

that Eip t  0 (0 # t # 1) holds from Eq. (11). Whether the
trajectory satis®es the Euler±Poisson equation can be
judged according to the error de®nition. One thousand two
hundred minimum commanded torque change trajectories
were generated and measured by Nakano et al. (1999). First,
the relationship between the number of parameters K andthe error uEumax was examined in the proposed method. The
value of K was changed (i.e. 30, 32, 34, ¼, 70). All of theinitial values of aik were set to 0.
Fig. 1 shows changes in error uEumax according to thenumber of parameters K. The horizontal axis shows thenumber of parameters K, and the vertical axis is uEumax
calculated by Eq. (21). A, B, C, and D correspond to thefour trajectories shown in Fig. 3. Though the error uEumax
decreases as K increases, it converges near K  54. The
number of optimal parameters K differs according to the
trajectory. Therefore, in the following, we use the numberof parameters K that minimizes the error of uEumax for eachtrajectory.
Fig. 2. indicates the convergence of error uEumax in 100
times iterative calculations. The horizontal axis shows theiterative number of estimations of parameter aik and thevertical axis shows uEumax calculated by Eq. (21). uEumax
for all trajectories converges to a small value below about
10 29. It can therefore be said that the Euler±Poisson equations are satis®ed with a high accuracy at every movement
time of the trajectory. Moreover, the error clearly converges
in a small number of iterative calculations, that is, about 20
times. Consequently, optimal trajectories can be obtained by
the proposed method regardless of whether the trajectories
can be solved by the Newton-like method or not.
The minimum commanded torque change trajectories on
the horizontal plane obtained by the proposed method areshown in Fig. 3. The Y-axis of (a) is forward of the body, andthe origin is the position of a shoulder. The optimal trajectories shown by the thick line cannot be obtained by the
Newton-like method. Moreover, it can be understood thatthe features of human arm movements, that is hand

388

Y. Wada et al. / Neural Networks 14 (2001) 381±393

Fig. 3. Minimum commanded torque change trajectories calculated using the proposed method in the horizontal plane.

trajectories, are not straight lines, but gradually curved, andthe velocity pro®les each have a single peak and are reproduced well.
Table 2 indicates values of the object function (integration of the second power of the commanded torquechange) for two trajectories solved by each method on
the horizontal plane and the sagittal plane. It can beunderstood that the object function of the proposed
method is smaller than or equal to the other methods
for any case. Moreover, it can be con®rmed that the
proposed method can also obtain the converged trajectory
solution, which is unable to be calculated by the Newtonlike method.
To con®rm how the proposed method and the Newtonlike method satisfy the Euler±Poisson equations, E1 and E2
at every movement time were calculated.
Fig. 4 shows the absolute values of E1 and E2 at every
time point for the proposed method and the Newton-like
method. The horizontal axis indicates the normalized
time, and the vertical axis is the absolute values of E1 andE2. It can be understood that the proposed method satis®es
the Euler±Poisson equations better than the Newton-like
method during the whole movement time.

5. Consideration
In experiments, we examined the validity of the minimum
hand jerk criterion, minimum angle jerk criterion, minimum
torque change criterion, and minimum commanded torquechange criterion by comparing actual trajectories withtrajectories predicted by these optimization criteria. For
movements within the horizontal and sagittal planes, theminimum commanded torque change model was able toreproduce the spatial characteristics of measured trajectories. In this case, the magnitudes and directions of curvatures were better than those of the other three models. The
minimum torque change model could reproduce neither themagnitudes nor the directions of curvatures. The minimum
hand jerk model, which always predicts straight paths,
showed a lack of correlation with the measured trajectories
regarding the whole deviations. 1 Even though the minimum
1
We investigated the trajectory curvatures quanti®ed as an area bounded
by a start-to-goal straight line and the hand path. This area was named thewhole deviation. The whole deviation concerns the direction in which a
trajectory had curved. If a trajectory had curved right relative to the vector
from the start to the target, the area was designated positive; on the other
hand, a trajectory that curved left was given a negative sign.

Y. Wada et al. / Neural Networks 14 (2001) 381±393

389

Fig. 4. Absolute values of E1 and E2 (trajectory A in Fig. 3. O: Newton-like method; *: the proposed method.

angle jerk model could explain the direction of the curvature, it predicted trajectories with obviously excessive
curvatures (see Appendix B).
In previous studies, Hollerbach (1990); Osu, Uno, Koike,
and Kawato (1997) suggested that planning in the joint
space cannot explain a gently curved hand trajectory. As
shown in Table 3, the slopes relating whole deviations of
the minimum hand jerk and actual trajectories were zero(the mean error the slope was 1 in both planes). For minimum angle jerk trajectories, slopes were from 1.4 to 2.6 in
the horizontal plane and from 1.3 to 1.5 in the sagittal plane(the mean error of the slope was 1 and 0.34, respectively).
Hence, it can be considered that the minimum angle jerkmodel is quantitatively three times better than the minimum
hand jerk model in the sagittal plane. Although the minimum angle jerk model has a qualitative weak point inpredicting trajectories that are too curved, it is quantitatively
a better model compared with the minimum hand jerkmodel. Accordingly, we demonstrated that it is impossible
to completely reproduce actual data with trajectory planningin the kinematic space. As Flash (1990) pointed out, wecon®rmed that literal minimum torque change trajectories,

computed without consideration of the viscosity, cannot
reproduce any actual trajectory. Moreover, by the analysisof the trajectory, the velocity, and the acceleration and thetorque, it was shown that the minimum commanded torquechange criterion was able to explain the temporal features ofactual hand trajectories best among the criteria. It can therefore be said that the minimum commanded torque change
criterion is a better model than the other three criteria
because the movement of the entire workspace can be
explained in time and space. Moreover, the minimum
commanded torque change criterion most similarly
reproduced the feature of the measured trajectory in pointto-point movements on the sagittal plane. The wholedeviations in minimum commanded torque change trajectories on the horizontal plane and the sagittal plane shown in
Table 3 predict that the curvatures of the trajectories on bothplanes have almost the same tendency. Therefore, results
such as a linear relation between the whole deviation of a
trajectory measured in a horizontal plane and the wholedeviation of a trajectory measured in a sagittal plane seemappropriate for the minimum commanded torque change
criterion.

390

Y. Wada et al. / Neural Networks 14 (2001) 381±393

the minimum commanded torque change trajectory using
the proposed method; the same examinations as Nakano etal. (1999) were performed. As a result, it was recon®rmedthat the measured trajectory was closest to the minimum
commanded torque change trajectory, like the result by
Nakano et al. (1999).

Table 2
Values of the performance index (the time integral of the square of thecommanded torque change rate)

Horizontal plane
The proposed method
Newton-like method
Steepest descent method
Sagittal plane
The proposed method
Newton-like method
Steepest descent method
a

1

2a

33.4432
33.4432
33.461992.1030±
92.1660

93.784493.7845
93.8458

40.3807±40.3932

Acknowledgements
This study was performed through Special Coordination
Funds for the promotion of Science and Technology of the
Science and Technology Agency.

±Shows that the method cannot calculate the optimal trajectory.

6. Summary

Appendix A

It was shown that the proposed method has the following
advantages by numerical experiments. The trajectory
obtained by the proposed method is similar to the trajectoryof the mathematically guaranteed Newton-like method. An
optimal trajectory, which cannot be obtained by the
Newton-like method, is able to be calculated. Even if thevalue of the viscosity, which is one reason that a converged
solution is unable to be obtained by the Newton-like
method, is greatly changed, it can be con®rmed that the
proposed method is able to obtain an optimal solution.
This was shown to obtain the converged solution in a very
short period of time compared with the previous method.
Though we searched for the most appropriate number ofparameters K in the paper, a value of around K  60 issuf®cient as shown in Fig. 1. An optimal solution can beobtained by an iterative calculation of several ten times. The
proposed method is a general method for a nonlinear, optimal problem. Therefore, the method can be applied tovarious optimization problems. However, an important
remaining problem is to clarify the convergence conditionsof the method theoretically.
Finally, we performed supplementary examinations of
the experiments by Nakano et al. (1999). We recalculated

A.1. Jacobian polynomial
A Jacobian polynomial (Szego, 1975) is as follows. In
general, there is family Rk uk  0, 1, 2, ¼) of polynomialsde®ned by sections a and b.
Zb
however; k ± lRk ; Rl w  Rk xRl xw xdx  0a22
Rk and Rl to which Eq. (22) is satis®ed are called an orthogonalization polynomial system concerning weight w in
section a and in section b. This orthogonalization polynomial is obtained by the third recurrence formula (Szego,1975). In particular, the following weight w(x) is de®nedby section (21, 1).
w x  1 1 xa 1 2 xba; b . 21

23

Function Pka;b x k  0; 1; ¼; which is orthogonal
concerning Eq. (23), is a Jacobian polynomial. Here,Z1
Pka;b x w x1=2 Pl a;b x w x1=2 dx
Pka;b ; Pl a;b w 
21

24Table 3Results of a t-test for measured whole deviations and simulated whole deviations. (r: correlation coef®cient; t: t values; df: degrees of freedom; *p , 0.05;**p , 0.01; HJ: minimum hand jerk criterion; AJ: minimum angle jerk criterion; TC: minimum torque change criterion; CTC: minimum commanded torquechange criterion)
Subjects

MM

Modelsr

A Horizontal plane
HJ
0
AJ
0.727
TC
2 0.365
CTC
0.798
B Sagittal plane
HJ
0
AJ
0.751
TC
2 0.400
CTC
0.777

YS

TT

t

df

Slope

r

t

df

Slope

r

t

df

Slope

±
14.56**
5.39**
18.18**

±
189
189
189

0
1.933
2 0.943
0.819

0
0.680
2 0.298
0.691±
12.62**
4.25**
12.99**

±
185
185
185

0
2.645
2 0.993
1.302

0
0.835
2 0.3630.654±
20.17**
5.18**
11.50**

±
177
177
177

0
1.440
2 0.6180.485±
15.12**
5.81**
16.44**

±
177
177
177

0
1.487
2 0.485
0.894

0
0.733
2 0.3220.829±
14.20**
4.49**
19.54**

±
174
174
174

0
1.278
2 0.302
0.900

0
0.788
2 0.358
0.771±
17.04**
5.11**
16.12**

±
177
177
177

0
1.265
2 0.375
0.700

Y. Wada et al. / Neural Networks 14 (2001) 381±393

391

Table 4
Results of Bonferroni's t-test for the mean squared errors of measured trajectories and simulated trajectories. Pairwise comparisons were carried out for fourmodels. t: t values; df: degrees of freedom; *p , 0.05; **p , 0.01; HJ: minimum hand jerk criterion; AJ: minimum angle jerk criterion; TC: minimum torquechange criterion; CTC: minimum commanded torque change criterion. The model of which the amount of mean squared errors was signi®cantly small is shown
to the right of the level of signi®cance)
Subjects

A Horizontal plane
HJ±AJ
Trajectory
Velocity
Acceleration
Torque
HJ±TC
Trajectory
Velocity
Acceleration
Torque
HJ±CTC
Trajectory
Velocity
Acceleration
TorqueAJ±TC
Trajectory
Velocity
Acceleration
TorqueAJ±CTC
Trajectory
Velocity
Acceleration
TorqueTC±CTC
Trajectory
Velocity
Acceleration
TorqueB Sagittal plane
HJ±AJ
Trajectory
Velocity
Acceleration
Torque
HJ±TC
Trajectory
Velocity
Acceleration
Torque
HJ±CTC
Trajectory
Velocity
Acceleration
TorqueAJ±TC
Trajectory
Velocity
Acceleration
TorqueAJ±CTC
Trajectory
Velocity
Acceleration
TorqueTC±CTC
Trajectory
Velocity
Acceleration
Torque

MM

YS

t

df

9.35**
12.02**
13.32**
2.78**
11.80**
12.60**
13.27**
1.09
4.07**
4.00**
0.59
5.16**
7.24**
6.06**
4.81**
5.08**
12.42**
15.27**
14.67**
8.45**
14.00**
14.84**
14.63**
12.80**

190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190
190

4.38**
5.79**
6.28**
1.71
6.71**
6.80**
6.94**
1.16
2.79**
0.69
2.59*
1.68
3.46**
2.41*
1.68
0.68
6.87**
6.90**
5.26**
4.09**
7.89**
6.93**
5.44**
5.43**

178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178

HJ
HJ
HJ
HJ
HJ
HJ
HJ
CTC
CTC
CTC
AJ
AJ
AJ
TC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
HJ
HJ
HJ
HJ
HJ
HJ
CTC
HJ
AJ
AJ
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC

TT

t

df

13.54**
17.00**
17.50**
5.60**
7.66**
8.35**
8.83**
2.87**
0.17
0.93
0.71
5.04**
0.75
1.50
3.57**
10.66**
14.22**
17.60**
16.79**
12.53**
7.77**
8.96**
9.13**
6.82**

186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186
186

2.67**
3.80**
4.55**
2.65**
6.60**
6.75**
6.93**
1.55
3.60**
2.19*
0.57
1.34
3.57**
2.94**
2.52*
1.63
5.83**
5.98**
4.60**
3.88**
7.28**
6.74**
5.51**
5.19**

175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175
175

HJ
HJ
HJ
HJ
HJ
HJ
HJ
HJ

CTC
TC
TC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
HJ
HJ
HJ
HJ
HJ
HJ
HJ
CTC
CTC
AJ
AJ
AJ
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC

t

df

3.25**
6.42**
8.76**
0.83
10.28**
10.67**
10.55**
1.18
4.58**
5.66**
3.99**
5.09**
8.30**
7.11**
5.79**
2.78**
6.02**
9.51**
10.45**
7.15**
13.12**
13.80**
12.92**
12.54**

178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178

2.47**
3.53**
3.98**
0.52
6.38**
6.60**
6.79**
0.35
3.88**
3.00**
0.66
3.47**
3.78**
3.39**
3.33**
0.20
5.87**
6.03**
4.50**
3.89**
7.89**
7.43**
6.53**
6.31**

178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178
178

HJ
HJ
HJ
HJ
HJ
HJ
CTC
CTC
CTC
CTC
AJ
AJ
AJ
TC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC
HJ
HJ
HJ
HJ
HJ
HJ
CTC
CTC
CTC
AJ
AJ
AJ
CTC
CTC
CTC
CTC
CTC
CTC
CTC
CTC

392

Y. Wada et al. / Neural Networks 14 (2001) 381±393

Consider the following equation.
w x  1 1 x6 1 2 x6no1=2 no1=2
 1 1 x6 1 2 x6
1 1 x6 1 2 x6 1 1 x3 1 2 x3 1 1 x3 1 2 x3325

1 2 x Pk6;6 x is as follows at both ends

Qk x  1 1 xof section 1 and 21.

3

Qk 21  Qk 1  0;

dQk x
dQk x
 0;dx x21dx x1d2 Qk xd2 Qk x0
dx2 x21
dx2 x126
Here, by the coordinate transformation of t  (x 1 1)/2,
Q~ k t  64t3 1 2 t3 P~ k6;6 t is obtained.
A.2. Jacobian polynomial of k 1 1
The following Jacobian polynomial of k 1 1 is given as
follows, where the coef®cient of the highest degree isassumed as 1,
P06;6 x  1

and predicted trajectories. Table 3 summarizes the correlation coef®cients, the results of a test on the correlations, andthe slopes of the regression lines of all of the subjects in thehorizontal and sagittal planes. All of the minimum hand jerk
trajectories were straight; this indicated that they were notcorrelated with those of actual trajectories. The minimum
angle jerk trajectories were curved more than the actual
trajectories. Most of the whole deviations of the minimum
torque change trajectories had negative correlations to thoseof actual trajectories, and the correlation coef®cients were
low. The whole deviations of the minimum commanded
torque change trajectories were smaller than, or approximately the same as, those of actual trajectories, and thecorrelation coef®cients were high.
As a result of the second analysis, we carried out a ttest for the MSEs of the position, velocity, acceleration,and torque of all of the trajectories measured in bothplanes. In 72 total comparisons between the minimum
commanded torque change model and the other threemodels, and with the four characteristics, three subjects,
and two planes (3 £ 4 £ 3 £ 2), the MSEs of the minimum commanded torque change model for 62 comparisons were smaller (signi®ciantly smaller, P , 0.05)(Table 4). We roughly summarized that the minimum
commanded torque change model was the best one for
the MSEs.

P16;6  x
P26;6 x  x2 2

References

1
15

.. ..
. .
6;6
6;6
x  x 2 ak Pk6;6 x 2 bk Pk21
x
Pk11

ak 

bk 

xPk6;6 x; Pk6;6 xw

k  0; 1; 2; ¼

Pk6;6 x; Pk6;6 xw

k  1; 2; ¼

Pk6;6 x; Pk6;6 xw

6;6
6;6
Pk21
x; Pk21xw

6;6
x  0:However, P21

Appendix B. Quantitative examination of the spatial andtime features of trajectories
We recalculated the minimum commanded torque change
trajectory using the proposed method, and performed thesame examinations as Nakano et al. (1999). Comparedwith the actual trajectories, many of the minimum angle
jerk trajectories and minimum torque change trajectories
are largely curved towards the outside and inside of thebody, respectively. As a result of the analysis, we plotted
the correlations between the whole deviations of measured

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