1ATR Human Information Processing Research
Laboratories,
Nakano, Eri,
Hiroshi Imamizu,
Rieko Osu,
Yoji Uno,
Hiroaki Gomi,
Toshinori Yoshioka, and
Mitsuo Kawato.
Quantitative examinations of internal representations for arm
trajectory planning: minimum commanded torque change model. A
number of invariant features of multijoint planar reaching movements have been observed in measured hand trajectories. These features include roughly straight hand paths and bell-shaped speed profiles where the trajectory curvatures between transverse and radial movements
have been found to be different. For quantitative and statistical
investigations, we obtained a large amount of trajectory data within a
wide range of the workspace in the horizontal and sagittal planes (400 trajectories for each subject). A pair of movements within the
horizontal and sagittal planes was set to be equivalent in the elbow
and shoulder flexion/extension. The trajectory curvatures of the
corresponding pair in these planes were almost the same. Moreover,
these curvatures can be accurately reproduced with a linear regression
from the summation of rotations in the elbow and shoulder joints. This
means that trajectory curvatures systematically depend on the movement
location and direction represented in the intrinsic body coordinates.
We then examined the following four candidates as planning spaces and
the four corresponding computational models for trajectory planning.
The candidates were as follows: the minimum hand jerk model in an
extrinsic-kinematic space, the minimum angle jerk model in an
intrinsic-kinematic space, the minimum torque change model in an
intrinsic-dynamic-mechanical space, and the minimum commanded torque
change model in an intrinsic-dynamic-neural space. The minimum
commanded torque change model, which is proposed here as a computable
version of the minimum motor command change model, reproduced actual
trajectories best for curvature, position, velocity, acceleration, and
torque. The model's prediction that the longer the duration of the
movement the larger the trajectory curvature was also confirmed.
Movements passing through via-points in the horizontal plane were also
measured, and they converged to those predicted by the minimum
commanded torque change model with training. Our results indicated that
the brain may plan, and learn to plan, the optimal trajectory in the
intrinsic coordinates considering arm and muscle dynamics and using
representations for motor commands controlling muscle tensions.
Hand trajectories measured for planar reaching
movements are known to have common invariant spatiotemporal features,
namely, a roughly straight hand path, a bell-shaped tangential velocity profile (Abend et al. 1982 Three "indeterminacy problems" are involved in planning and
executing reaching tasks with a visually guided arm (Kawato
1992 Many approaches have been proposed to explain how the brain resolves
such problems (Bernstein 1967 Because most targets for movements are provided in external visual
coordinates and the achieved trajectory is roughly straight, it seems
natural at first sight that the necessary constraints to solve the
indeterminacy problem are in the extrinsic coordinates (Flash
and Hogan 1985 Neural recording data consistent with the four different planning
spaces have been obtained. Examples are Georgopoulos et al.
(1982 Recent studies have investigated performance changes or adaptation
processes in artificially altered environments to objectively discuss
the trajectory planning space. If a hand trajectory is planned in a
kinematic space, it will be changed under a kinematic transformation in
the visual space, but it will not be affected by any altered dynamics,
e.g., the force field where the viscosity was altered. However, the
dynamic model makes the opposite prediction. Results supporting
kinematic planning have been reported by both Wolpert et al.
(1995) One way to investigate the space in which trajectories are planned is
to compare actual trajectories with trajectories predicted by the
optimization criterion defined in each space. We used the minimum hand
jerk (Flash and Hogan 1985
ABSTRACT
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES
INTRODUCTION
TOP
ABSTRACT
INTRODUCTION
METHODS
RESULTS
DISCUSSION
APPENDIX A
APPENDIX B
REFERENCES
; Flash and Hogan
1985
; Kelso et al. 1979
; Morasso
1981
; Uno et al. 1989a
), and smooth acceleration (Koike and Kawato 1995
). These invariant features can be
observed in rapidly executed movements without on-line correction. The hand trajectory seems to be planned for the execution of such a
feed-forward controlled movement. The experiments by Bizzi et al. (1984)
suggest that deafferented monkeys can reach a target with their hands by feed-forward control alone, and the whole trajectory from the initial position to the final position is preplanned.
). There are an infinite number of spatiotemporally
possible routes leading to the target, but it is necessary to select a
final unique trajectory (trajectory determination problem). 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 (a problem of coordinates transformation). When a desired trajectory is determined in the joint angle coordinate, actual torques around the joints can be calculated by an inverse dynamics equation. However, there are also an infinite number of
possible combinations of the agonist and antagonist muscle tensions
that can generate the same torques. The degrees of freedom of
-motoneurons, which innervate each muscle, are higher than those of
the muscles, and cortical motor neurons may have higher degrees of
freedom than
-motoneurons. Even if the time profiles of muscle
tensions are specified, the firing rates of the cortex or spinal cord
neurons cannot be uniquely determined (a problem of motor command
generation). Regardless of these indeterminacies, the actual hand
trajectories show common invariant characteristics, and
electromyographic (EMG) signals appear in typical triphasic patterns.
These observations suggest that the brain solves these ill-posed
problems based on some principles (for further detail of these
problems, see Kawato 1996
).
; Bizzi et al.
1984
; Saltzman and Kelso 1987
). Optimization
models have been proposed for single joint movements (Nelson
1983
) and for multijoint movements (Dornay et al.
1996
; Flash and Hogan 1985
; Kawato
1992
, 1996
; Uno et al. 1989a
,b
). These models are objectively and
experimentally examinable because of their quantitative predictions. It
has already been confirmed that most of the criteria proposed for
single joint movements are unable to reproduce the smoothness of the
velocity or acceleration in multijoint movements (Kawato
1996
). In addition, the combination of the virtual trajectory
and minimum jerk models proposed by Flash (1987)
is also
a quantitatively examinable model. The claims or advantages of this
model are high arm stiffness, invariance of virtual trajectory, and
simple desired trajectory (straight virtual trajectory). However, the
arm stiffness measured during movement is not high (Gomi and
Kawato 1996
,
1997
).1 The
invariance of virtual trajectory does not hold well because not only
the amplitude but also the temporal patterns of EMG signals are
different between straight and natural movements (Osu et al. 1997
). A quite different trajectory from the actual one is
predicted with actually measured low stiffness and a straight virtual
trajectory (Katayama and Kawato 1993
). Due to these
reasons, this model does not seem to be an attractive candidate for now.
). However, the possibility that trajectories are
planned in intrinsic joint torque or motor command space has been
pointed out (Kawato 1992
; Uno et al.
1989a
). Different spaces where optimization principles are
applied predict different trajectory properties. Hence, in this study,
we discuss the problem of the planning space by experimentally
investigating the properties of executed trajectories. As a candidate
of the coordinates frame for trajectory planning, we first considered
the extrinsic coordinates represented by factors such as the position
within the task space, and the intrinsic coordinates based on inherent
expressions in the motor system such as the joint angle, muscle length,
muscle tension, torque, and motor command. Second, following on the
work of Osu et al. (1997)
, we classified spaces that
either solely rely on kinematic aspects, e.g., the hand position, joint
angle, and muscle length, or on both kinematic and dynamic aspects,
e.g., torque and muscle tension. Finally, the dynamic parameters are divided into mechanical variables, e.g., torque and muscle tension, and
neural representations that depend on nervous system processing, e.g.,
the motor command controlling muscle tension. These three methods of
classification allow us to consider the following four plausible
candidates as planning spaces: an extrinsic-kinematic space (e.g., the
Cartesian coordinates of the hand position, hand movement direction, or
the polar coordinates), an intrinsic-kinematic space (e.g., joint angle
or muscle length), an intrinsic-dynamic-mechanical space (e.g., torque
or muscle tension), and an intrinsic-dynamic-neural space (e.g., the
motor command controlling muscle tensions, the firing rates of motor
neurons, or Purkinje cells in the cerebellum).
, 1986
) for an extrinsic-kinematic space,
Lacquaniti et al. (1995)
for an intrinsic-kinematic
space, Scott and Kalaska (1995
, 1997
),
Sergio and Kalaska (1998)
for an
intrinsic-dynamic-mechanical space, and Keller (1973)
,
Shidara et al. (1993)
, and Gomi et al. (1998)
for an intrinsic-dynamic-neural space. Especially, for eye movements, it has been reported that firing frequencies of motor
neurons (Fuchs et al. 1988
; Keller 1973
)
and cerebellar Purkinje cells (Gomi et al. 1998
)
represent dynamic motor commands specifying necessary muscle forces and
torques. Sergio and Kalaska (1998)
showed in arm control
that each firing pattern of the primary motor cortex neurons obtained
in different movement tasks is similar to the corresponding temporal
profile of force necessary in each task.
, and Flanagan and Rao (1995)
using
kinematic transformation, and by Flash and Gurevich
(1991)
, Shadmehr and Mussa-Ivaldi (1994)
,
Lackner and Dizio (1994)
, and Conditt et al.
(1997)
using dynamic transformation. On the contrary, the results of Uno et al. (1989a)
, Uno et al.
(1995)
, Osu et al. (1997)
, Gomi and
Gottlieb (1997)
, and Uno, Imamizu, and Kawato (unpublished observations) support dynamic planning. The differences in these results can be ascribed to whether or not the internal model for the
alternation has been sufficiently learned or the transformation is
strong enough to change the optimal hand trajectory (Kawato 1996
). This controversial problem will continue, because the
results depend on the settings of delicate task conditions, the number of learning trials, and the instructions given in the transformation experiments. In this paper, we adopted the tasks under ordinary space
without using the transformation to discuss trajectory planning spaces.
), minimum angle jerk, minimum
torque change (Uno et al. 1989a
), and minimum commanded torque change models, whose objective functions are defined in the
above spaces, respectively. In the next section, we describe four
optimal trajectory formation models and specifically propose the
minimum commanded torque change model. The diagram in Fig. 1 shows the spaces for trajectory
planning and the corresponding models. In a comparison of actual
trajectories with predicted trajectories, the fundamental assumption is
that actual trajectories are close to planned trajectories. This
assumption is confirmed by Osu et al. (1997)
, and by
partially referring to Katayama and Kawato (1993)
and
Gomi and Kawato (1996)
.
View larger version (109K):
[in a new window]
Fig. 1.
Conceptual schema of trajectory planning spaces. Motor cortex conveys
motor commands MCc to -motoneuron of the spinal cord, and the motor
command MC
derived from the
-motoneuron activates muscles it
innervates. Muscle activation is controlled by these impulses, muscle
tensions then arise, and finally actuated torques are generated to
realize trajectory.
Utilization of data obtained in limited locations makes it possible for the conclusion to be dominated by the selected trajectories. There is also the possibility that the differences between model predictions are small for some specific movements. Hence generalized studies should utilize a large number of movements executed at many possible locations within the workspace.
In the first experiment, a large amount of data on point-to-point movements was obtained in the horizontal plane at shoulder level and in the sagittal plane passing through the shoulder. First, the relationship of trajectory curvatures between movements executed in the horizontal plane and those executed in the sagittal plane was investigated. Second, the trajectory curvatures were modeled from the locations and directions where the movements were accomplished. Third, we used the data to compute optimal trajectories to explore the space where the trajectories were planned.
In the second experiment, we examined the influence of the duration of movements on hand trajectories because dynamic planning models, but not kinematic planning models, predict changes in the trajectory with movement duration.
The subjects were requested to train for via-point movements in the
third experiment. We compared the characteristics of measured trajectories and the characteristics of optimal trajectories for each
model. Note that trajectory planning models have previously been
examined for via-point movements (Flash and Hogan 1985;
Okadome and Honda 1992
; Uno et al.
1989a
). However, studies have yet to examine each model by
observing changes in trajectory properties due to training.
Minimum commanded torque change model
The minimum hand jerk model (Flash and Hogan 1985),
defined in an extrinsic-kinematic space, is an attractive candidate as a trajectory planning model for humans. Here, the jerk is defined by
differentiating the hand position (x, y) three times by time t in Cartesian coordinates. In this model, the objective
function
![]() |
Soechting and Lacquaniti (1981) discussed trajectory
planning in a kinematic joint space based on the observation that the shoulder and elbow move while maintaining a linear relationship between
the joint angles near the edge of the workspace. Rosenbaum et
al. (1995)
have discussed the coordinated movements of arm and
trunk using optimization criteria defined in the joint space. This
model is named here the minimum angle jerk model as one possible exemplification of the optimization theory. This model always predicts
straight paths in the joint space. The criterion function to be
minimized is expressed as
![]() |
Trajectory planning in an intrinsic-dynamic space is another candidate
capable of accounting for actual data. The minimum torque change model
(Uno et al. 1989a) is classified as
"intrinsic-dynamic-mechanical." This model was able to reproduce
the properties of hand trajectory influenced by arm dynamics, arm
posture, external forces, and movement duration (Uno et al.
1989a
; Uno and Kawato 1996
). However, as is
detailed later, it has been pointed out that the viscous values should
be set to zero in the literal minimum torque change model (Flash
1990
), and this literal model is discriminated from our new
one, the minimum commanded torque change model.
The formulas of the original minimum torque change model (Uno et
al. 1989a) and the minimum commanded torque change model proposed here are the same and both include the viscous values. However, the original paper by Uno et al. (1989a)
misnamed the model and used incorrect values for inertia and viscosity.
The actual trajectories are not at all similar to the trajectories predicted by using our viscous values and an inertia value by Uno et al. (1989a)
that is too large or those predicted
by using our inertia value and the viscous values of Uno et al.
(1989a)
that are too small (but not zero). The combination of
wrong parameters (large inertia and small viscous values) contingently
leads the prediction of trajectories very similar to the actual ones.
Therefore our minimum commanded torque change trajectory is at first
sight similar to the original minimum torque change trajectory.
However, our paper provided appropriate parameters, accurate
interpretation, and proper designation to the original model.
The minimum motor command change model (Kawato 1992,
1996
) has been proposed for trajectory planning in an
intrinsic-dynamic-neural space. The bursting of motor neurons or the
cerebellar Purkinje cells observed in rapid movements, e.g., saccadic
eye movements or ocular following responses, apparently seems
to go against the principle of smoothness. However, a bell-shaped
velocity profile and smooth acceleration were observed even in saccades
as well as in arm movements (Harris and Wolpert 1998
).
It has been demonstrated that the temporal profile of the firing
frequency of motor neurons (Fuchs et al. 1988
;
Keller 1973
) or the cerebellar Purkinje cells (Gomi et al. 1998
; Shidara et al. 1993
)
changes according to such smooth temporal profiles of velocity and
acceleration of eye movements. Gomi et al. (1998)
reported that the firing patterns of cerebellar Purkinje cells were
represented by a linear summation of position, velocity, and
acceleration and smoothly changed over time correlating with the
dynamic component of the necessary torque. Consequently, smoothness of
central motor commands has already been observed in neuronal recording data.
It is reasonable to consider that the principle of smoothness should be
applied in the motor command space because the degrees of freedom are
higher at the CNS level as mentioned in the INTRODUCTION. Because the minimum motor command change model can conceptualize the
signal at the -motoneuron or cortical motoneuron level, the indeterminacy can be constrained at each level. Although an attempt has
been made to estimate the motor commands at the muscle level (Koike and Kawato 1995
), it is extremely difficult to
estimate the motor commands of the spinal cord or cortex by modeling
the information processing from a central system to a peripheral
system. A quantitative model, not a conceptual model, is needed to
actually compute an optimal trajectory. Therefore we are the first to
fully propose a minimum commanded torque change model that approximates the minimum motor command change model and has computability, while
positively appreciating the assumption of nonzero viscosity by
Uno et al. (1989a)
. In the literal minimum torque change
model, only the link dynamics are regarded as the controlled object, whereas in the minimum commanded torque change model, both link dynamics and muscles are regarded as controlled objects (Fig. 2). We employ motor commands at the
peripheral level, in other words, we use signals controlling muscle
tensions to model a minimum commanded torque change criterion. In terms
of indeterminacy, however, the minimum commanded torque change model
solves problems at the same level, that is, the torque level as the
minimum torque change model.
|
In the minimum torque change model and minimum commanded torque change
model, the objective function to be minimized is expressed by
![]() |
![]() |
![]() |
![]() |
![]() |
![]() |
(1) |
|
Most of the viscosity measured around a joint is ascribed to a
biochemical and mechanical 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 viscosity in calculating torque means that both link
dynamics and muscles are regarded as controlled objects (Fig. 2). It is not appropriate to use the nonzero viscous value to calculate torques
because the literal minimum torque change model takes the actual torque
around the joint as an object for optimization (Flash
1990
).
The commanded torque is calculated in consideration of the muscle
viscosity and is intended to conceptually approximate the command of
the -motoneuron MC
(see Fig. 1). The term commanded torque
indicates the torques ascribed to a muscle elastic property, which
reflects the motor command mainly controlling the rest-length of a
muscle. In other words, the motor command controlling muscle tensions
and consequently muscle torques designate this commanded torque. The
commanded torque includes a component for compensating damping by
muscle viscous properties (viscous torque), namely, the commanded
torque is torque before being damped by such viscous properties (Fig.
2). To generate actual torque, the motor command controlling muscle
tensions must compensate for the influence of muscle viscosity. By
considering link dynamics as well as muscle properties, the commanded
torque reflects a representation of a motor command more closely than
the actual torque.
This conception is illustrated by the following expressions. The
equations conceptually interpret a model approximation of the minimum
motor command change model. We expressed in a vector form in
Eqs. 2-5. The terms depending on position, velocity, and acceleration in Eq. 1 were rearranged using different
symbols for concise explanation. The predicted trajectories were
actually calculated by Eq. 1 but not by Eqs.
2-5. The minimum commanded torque change model does not model the
cortical or spinal motor command itself. Here, we use the term motor
command as the command already conveyed to the muscle and as that which
controls tensions at the muscle level. That is to say, the influence of
reflexes at the spinal or cortical level has been already involved in
this motor command. Because there is no neural delay included in this final torque generating process, we did not consider any neural delay
in the following equations.
Equation 2 defines the actual torque a. The
first and second terms in Eq. 1 correspond to the term
R in Eq. 2. The third and sixth terms are
expressed by the term H in Eq. 2. We assume that
there is no passive joint viscosity. The actual torque is determined as
Eq. 3a according to the Kelvin-Voight model
(Özkaya and Norbin 1991
). R, H, K, and
Bm denote the inertia matrix, the centrifugal
force and the Coriolis force, the stiffness matrix, and the muscle
viscosity matrix, respectively. We defined the component ascribed to
the muscle out of Bij in Eq. 1 as
Bm in Eq. 3a.
and
r represent the current position and the equilibrium position of the joint in the vector form. The elements of the matrix
K are coefficients of joint stiffness due to each muscle elasticity. This first elastic term in Eq. 3a is defined as
the commanded torque
c (Eq. 3b). The negative
designated Bm indicates that
c is
reduced by viscous force relating to the shortening velocity of the
muscle. The commanded torque can then be rewritten as Eq. 4,
which explicitly shows that the commanded torque considers viscous
force. The actual torque does not consider it (Eq. 2)
![]() |
(2) |
![]() |
(3a) |
![]() |
(3b) |
![]() |
(4) |
![]() |
(5a) |
![]() |
![]() |
(5b) |
![]() |
(5c) |
The coefficient of viscosity is known to vary with joint angle, angular
velocity, or stiffness during postural control and movement
(Bennet 1993; Bennet et al. 1992
;
Gomi and Osu 1998
). No study has ever reported the
viscous value during multijoint movements. In our study, we use the
following formula, which was estimated from the actual torque and
viscosity during static force control (Gomi and Osu
1998
), to acquire viscous values of diagonal components
(B11, B22) and
off-diagonal components (B21,
B12) for each trajectory. Here, for simplicity,
mean absolute torques (shoulder:
1ma,
elbow:
2ma) during movement are used as the
actual torques
![]() |
![]() |
(6) |
![]() |
METHODS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Experimental setup
POINT-TO-POINT MOVEMENTS. In the first experiment, the subjects were three right-handed males, 22-26 yr old (MM, YS, and TT). They sat in a chair, and their shoulders were fixed to the back of the chair with a harness. For reaching movements in the horizontal plane, the table was adjusted to lift the subjects' arms to shoulder level; the subjects' wrists were braced so that movement was constrained to the two degrees of freedom for the elbow and shoulder (Fig. 3A). Since the movements within the horizontal plane were performed at a slightly lower level than the shoulder level, a part of the hand and elbow was in contact with the surface. A semitransparent low-friction Teflon sheet covered the table. For movements in the sagittal plane, the subjects performed the experiment on a transparent acrylic board suspended along the sagittal plane passing through their right shoulders (Fig. 3C). The subject's elbow was constrained to be in the plane of hand movement because the abduction/adduction of shoulder joint was restricted by the flat board. Hand movements were executed in this plane.
|
VIA-POINT MOVEMENT. The subjects were two right-handed males (KH and AH) and one right-handed female (NH), 20-23 yr old. The setup of the measuring and experimental apparatus was the same as in the first horizontal experiment. A cathode ray tube (CRT) screen was placed in front of the subjects. Three circles and a cross were projected on this screen, which indicated the initial position, the via-point, and the final position (with radii of 1, 2, and 2.5 cm, respectively), and the current position of the hand measured by the OPTOTRAK (Fig. 3E).
If the via-point was set on the horizontal plane, the subjects would have frequently disturbed their visual field with their arms. To prevent occlusion of the via-point, we used a CRT screen. The subjects' task was to move their arms from the initial position to the final position by passing through a via-point within a time limit (KH, 559-675; AM, 570-690; NH, 525-635 ms). The time limit of movement duration was decided according to the distance between the initial position (shoulder angle at 59°, elbow angle at 99°) and the final position (shoulder angle at 14°, elbow angle at 91°). The different initial and final positions and time limit are due to the different arm lengths of each subject. The via-point was extinguished at the onset of movement, which was indicated by a beep sound to avoid on-line correction of movements. Eleven via-points were selected and equally arranged on the perpendicular bisector of the start-goal straight line (Fig. 3F). The combination of the initial position (open circle), final position (filled circle), and via point (asterisk) was regarded as a set. One of the 11 sets was randomly presented for each trial. First, the subjects performed 10 trials for each of the 11 via-points shown as asterisks in Fig. 3F for a total of 110 trials (early stage of training). In the second task, the subjects were trained for five via-points shown as open circles in Fig. 3F (training). Thirty trials were performed for each via-point, amounting to 150 trials. Finally, we carried out the third task to test the effects of training (late stage of training), where the number of trials and via-points were the same as in the early stage of training. The subjects obtained feedback concerning their individual hand paths and movement durations after each trial. The subjects took a brief rest between tasks.Filtering
The position data were digitally filtered by a sixth-order
Butterworth filter with an upper cutoff frequency of 10 Hz. Derivatives of the position data were computed by applying a three-point local polynomial approximation. The actual beginning and end positions of
each movement were determined using a two-dimensional curvature with a
3-mm1 threshold (Imamizu et al. 1995
).
Hence the movement duration, which was calculated from these start and
end positions, was longer than the duration first required as a task condition.
For point-to-point movements, if the subject made a corrective movement, or the velocity at the end position was >5% of the maximum velocity, the data were rejected as a failure. A trajectory with a velocity profile that deviated from an average two times larger than the standard deviation was taken out of the analysis as an outlier (see APPENDIX A for details).
Analysis
TRAJECTORY CURVATURES WITHIN THE HORIZONTAL AND SAGITTAL PLANES.
Using data from the first experiment, we first investigated the
correlation of trajectory curvatures between the horizontal and
sagittal planes. The curvature of each trajectory was quantified as an
area bounded by a start-to-goal straight line and the hand path (Fig.
4A). This area was named
whole deviation; W (Osu et al. 1997). The
whole deviation concerned the direction in which the trajectory curved.
If the trajectory curved right relative to the vector from start to
target, the area was designated positive; on the other hand, a
trajectory that curved left was given a negative sign. We examined the
relationship between the curvatures of the paired trajectories within
the horizontal and sagittal planes by calculating the correlation
coefficients between horizontal and sagittal whole deviations. Here,
the data utilized were from common trials in both planes adopted by the
criteria mentioned above.
|
CONTRIBUTION OF THE COORDINATED ROTATION OF JOINTS. Next, the trajectory curvatures were linearly regressed from the sum of rotations of the elbow and shoulder joints from the initial position to the target (coordinated rotation of joints).
The coordinated rotation of joints
![]() |
(7a) |
![]() |
(7b) |
![]() |
(7c) |
![]() |
(8) |
EXAMINATION OF MODELS.
We compared measured trajectories with those predicted using each
trajectory planning model. As constraints to simulate the trajectories
for each model, we used the initial and final positions and movement
durations determined from actual data. The velocity and acceleration
were assumed to be zero at the initial and final positions. The minimum
torque change trajectories and the minimum commanded torque change
trajectories were computed by the steepest descent method
(APPENDIX B). We compared measured and predicted
trajectories for spatiotemporal properties. First, the correlations of
whole deviations between both measured and predicted trajectories were
compared. If the whole deviations of the measured trajectories
completely corresponded to the predicted trajectories, the slope of the
regression line fitted by the least-squares method would be 1.0, and
the correlation coefficient (r) would be 1.0. Second, the
mean squared errors (MSE) were obtained for the position, velocity,
acceleration, and torque between those measured and predicted as
follows
![]() |
(9) |
CONTRIBUTION OF MOVEMENT DURATION.
In the second experiment, whole deviations were linearly regressed from
the coordinated rotation of joints and the signed movement duration
(T) as follows
![]() |
(10) |
CHANGE IN TRAJECTORY PROPERTY WITH TRAINING.
For via-point movements, we focused on the transverse movement passing
through a via-point in the front of the body. We analyzed trials where
the hand entered the target circle within the time limit. For example,
the hand did not always pass through a given via-point. Hence we
divided the area among the nearest and furthest via-points into eight
equal parts, and the hand trajectories that passed through each part of
the area were averaged (Fig. 11, A and B, area
divided by dotted lines). The trajectory data were normalized by
Atkeson and Hollerbach's (1985) method before averaging.
![]() |
RESULTS |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Trajectory curvatures within the horizontal and sagittal planes
The averaged entire movement durations were 593 ± 56 (SD) ms, 561 ± 51 ms, and 550 ± 57 ms in the horizontal plane and 620 ± 55 ms, 644 ± 53 ms, and 634 ± 60 ms in the sagittal plane using subjects in the order MM, YS, and TT. The 191, 187, and 179 trajectories in the horizontal plane and 179, 176, and 179 trajectories in the sagittal plane, in that order, passed the criteria explained in METHODS (Filtering) and were used for further analysis.
In Fig. 5 we show samples of each of the
five trajectories with large positive (B and F),
small absolute (C and G), and large negative
(D and H) whole deviations. As previous research
has demonstrated (Atkeson and Hollerbach 1985;
Haggard and Richardson 1996
; Uno et al.
1989a
), hand trajectories are gently curved in some specific
regions of the workspace, as seen in this figure. On the horizontal
plane, the hand paths of transverse movements appear to be convex
toward the outside, whereas those of radial movements seem to be
relatively straight. In the sagittal plane passing through the
shoulder, paths for up-and-down movements are outwardly convex, and
those for back-and-forth movements are relatively straight.
|
Significant positive correlations were obtained between the whole deviations of the hand trajectories in the horizontal and sagittal planes as shown in Fig. 6 (with subjects in the order MM, YS, and TT: r = 0.75, 0.65, 0.86; slope = 0.93, 1.47, 0.97; t (169, 164, 159) = 14.59, 21.50, 11.11; P < 0.001, 0.001, 0.001). The slopes of the regression lines fitted by the least-squares method were almost 1 except for YS. From these results, it was found that a pair of trajectory curvatures are similar when they correspond to each other within the horizontal and sagittal planes. However, a pair of trajectories that are different in the task space are identical with regards to the flexion/extension of the shoulder and elbow joints (see Fig. 3, B and D). Therefore the trajectory curvatures can be determined depending on the change of arm posture in the intrinsic body space. Furthermore, similar relationships were found between the horizontal and sagittal planes for whole deviations of the minimum commanded torque change trajectories (Fig. 7). Significant positive correlations with slope almost 1 were obtained between whole deviations predicted in the horizontal plane and those predicted in the sagittal plane for all of the subjects (r = 0.79, 0.81, 0.63; slope = 1.12, 1.02, 0.87; t (169, 164, 159) = 16.52, 17.54, 10.22; P < 0.001, 0.001, 0.001). The slopes for actual data and predicted data indicated the same trend for the two subjects MM and TT (Figs. 6 and 7) but not for subject YS.
|
|
Contribution of the coordinated rotation of joints
Table 2 summarizes the results of the linear regression of the whole deviation from the coordinated rotation of joints for all subjects. The regressions were significant, and squared correlation coefficients were high for all subjects (the mean r2 in the order of the horizontal and sagittal planes: 0.75 ± 0.14, 0.73 ± 0.06). In Fig. 5, the shaded area below each trajectory represents the amount of reconstructed whole deviations by the coordinated rotation of joints (data from TT with the highest r2). Note that this linear model only predicts the whole deviation and not any trajectory or path. The reconstructed whole deviations closely simulate the actual whole deviations concerning magnitudes and directions. The positive coefficient values in a of Eq. 8 statistically confirm that trajectories having large positive whole deviations appeared in right-to-left and down-to-up movements (Fig. 5, B and F), and those having large negative whole deviations were mainly observed in left-to-right and up-to-down movements (Fig. 5, D and H). This also suggests that trajectories having small whole deviations were found in radial and fore-and-aft movements in the polar coordinates of the shoulder (Fig. 5, C and G). These results indicated that trajectory curvatures can be reproduced well from the coordinated rotation of joints related to dynamics, as discussed later.
|
Examination of models
Figure 8A shows the trajectories measured and predicted by each criterion in the horizontal plane. In comparison with actual trajectories, many of the minimum angle jerk trajectories and minimum torque change trajectories were largely curved toward the outside and inside of the body, respectively. The same tendencies were shown for data measured in the sagittal plane. The velocity (B and C), acceleration (D and E), and torque profiles (F and G) of the sample movement denoted by the arrows are shown in Fig. 8. The predictions of the minimum commanded torque change model were best fitted to data relating to all properties, namely, velocity, acceleration, and torque.
|
The correlations between the whole deviations of measured and predicted
trajectories, and regression lines are indicated in Fig.
9. Table 3
summarizes the correlation coefficients, the results of a test on the
correlations, and the slopes of the regression lines of all subjects in
the horizontal and sagittal planes. The correlation coefficients for
the three models were significant, except for the minimum hand jerk
model. All of the minimum hand jerk trajectories were straight, so they
were not correlated with those of actual trajectories (the mean
correlation coefficient: 0, mean slope: 0). It was quantitatively
indicated that the minimum angle jerk trajectories were curved larger
than actual trajectories (in horizontal and sagittal planes, the mean
correlation coefficients, 0.75 and 0.76; mean slopes, 2.0 and 1.35).
Most of the whole deviations of the minimum torque change trajectories
had negative correlations to those of actual trajectories, and the
correlation coefficients were low (the mean correlation coefficients,
0.38 and
0.36; mean slopes,
1.09 and
0.39). The whole
deviations of the minimum commanded torque change trajectories were
smaller than, or approximately the same as, those of actual
trajectories, and the correlation coefficients were high (the mean
correlation coefficients, 0.70 and 0.80; mean slopes, 0.80 and 0.84).
In subject TT, the minimum angle jerk model has a better
correlation than the minimum commanded torque change model in the
horizontal plane, however, the scattered whole deviations to right and
left as shown in Fig. 9, C and D, imply that the
minimum angle jerk model cannot predict small whole deviations observed
in actual trajectories. This tendency is common in all subjects.
|
|
We carried out t-tests for the MSEs of the position, velocity, acceleration, and torque of the trajectories measured in both planes. The histograms of the MSEs of torques are shown in Fig. 10. In 72 total comparisons between the minimum commanded torque change model and other three models, four characteristics, three subjects, and two planes (3 × 4 × 3 × 2), the MSEs of the minimum commanded torque change model for 71 (66) comparisons were smaller (significantly smaller, P < 0.05). We roughly summarized for the MSEs that the minimum commanded torque change model was the best, the minimum hand jerk model the second best, the minimum angle jerk model the third best, and the minimum torque change model the worst. In the experiments in the horizontal and sagittal planes, we found that the minimum commanded torque change model quantitatively and statistically predicted trajectories the best.
|
Contribution of movement duration
A total of 190 trajectories was used in the analysis. The average
duration of movement was 688 ± 131 ms. Table
4 summarizes the results of linear
regression calculated using Eq. 10. The regression was
significant and the squared correlation coefficient was high. These
results quantitatively indicated that the trajectory curvatures were
dependent on the duration of movement and the locations of the initial
and final points quantified by the coordinated rotation of joints.
Moreover, these results support the prediction of Uno and Kawato
(1996) that longer movement duration causes larger trajectory
curvature. We will explain this prediction later in the paper.
|
Change in trajectory property with training
The number of trajectories adopted for analysis were 52, 61, and 48 for the early stage of training, and 68, 73, and 62 for the late stage of training for subjects in the order KH, AM, and NH. The average duration of movement was 951 ± 96 ms, 901 ± 140 ms, and 847 ± 59 ms for the early stage of training, and 878 ± 80 ms, 883 ± 89 ms, and 863 ± 58 ms for the late stage of training in the same order.
Figure 11 shows the mean paths and tangential velocity profiles measured (A, B, D, and E; data from subjects AM and NH) and those predicted for each model of subject AM (C and F). The trajectories were predicted using the experimentally specified initial position, final position, via-point, and averaged movement duration of the measured data from subject AM. To clearly show the variety of tangential velocity profiles depending on the via-points locations, we changed the amplitudes of velocity profiles corresponding to the deviations between the start-goal straight line and via-points in Fig. 11 (D-F). When a via-point was set near the body with respect to start-goal line, the deviation was designated negative. For example, a bottom path (A, left) and a bottom profile (D, left) were derived from common data.
|
The mean paths changed to asymmetric shapes on the right and left in the late stage of training in comparison with the relatively symmetric shapes in the early stage of training (Fig. 11A). Note that the minimum hand jerk model predicts symmetric shapes of the path, whereas all other models predict asymmetric shapes.
In the data from subject AM, the tangential velocity profiles had double peaks in the early stage of training, both in trials with a via-point far from and near to the body. However, in the late stage of training, a double peak appeared only when a via-point was near to the body, and the time at maximum velocity changed from earlier to later as the via-point shifted toward the body. According to the data from subject NH, the double peaks of the tangential velocity profiles, which were not clearly observed in the early stage of training, were solely visible in near via-points in the late stage of training. For almost all data, the single peaks observed for far via-points after learning are also observed at the single trial level (33/36, 38/38, and 33/37; number of single-peak trials/number of adopted trials in the order AM, NH, and KH). The time at maximum velocity in the late stage of training was different from the early stage of training. This time changed from the middle of the movement duration to earlier in the far via-points. For all subjects, tangential velocity profiles appeared to be more asymmetric in shape on the right and left in the late stage of training than in the early stage of training when via-points were set further from the body. In the late stage of training, the characteristics of the actual tangential velocity profiles for all subjects stated above were very similar to those of the minimum torque change trajectories and the minimum commanded torque change trajectories (Fig. 11, D-F).
We do not show the data of subject KH, because we could not observe any training effects. However, the minimum torque change and minimum commanded torque change models reproduced the paths and tangential velocity profiles of actual data well. It should be noted that subject KH had experience with via-point movements in a different experimental design, so he already possessed some skills.
![]() |
DISCUSSION |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
We first discuss the relationship between the minimum commanded
torque change model and the minimum jerk model of equilibrium trajectory (Flash 1987) rather than the actual
trajectory. The two models are close in the sense that smoothness is
imposed on representations related to muscle elastic characteristics.
The space of trajectory planning for the minimum jerk model of
equilibrium trajectory is considered to be more centrally located in
the brain from the visual input side than that of the original minimum
hand jerk model because the former is related to a form of motor
command (equilibrium trajectory), whereas the latter is defined in a
purely sensory visual space. Similarly, the minimum commanded torque change model is more central from the motor output side than is the
literal minimum torque change model. Thus the two approaches starting
from the input and output extremes seem converge to
intrinsic-dynamic-neural representations for trajectory planning.
However, conceptual differences still exist between the two approaches.
The planned trajectory itself takes into account the dynamics of the
muscle, arm, and the environment in the minimum commanded torque change
model, but not in the minimum jerk model of equilibrium trajectory. A recent model by Harris and Wolpert (1998)
can be
classified as a version of the minimum motor command change model.
In the present experiments, we adopted initial position, final
position, and movement duration as the independent variables to examine
trajectory curvature. Movement distance is an additional plausible
independent variable. Because an enormous number of trajectories are
needed to be measured if all of four parameters were to be set as
independent variables, the number of parameters should be limited. The
most influential parameters in trajectory curvatures were found the
initial and final positions. As shown in the experiment by Uno
et al. (1989a) and our experiments, the movement duration only
modestly affects the trajectory curvature. Furthermore, in preliminary
experiments, we found that variations in movement amplitude lead to
little change in curvature. Because we examined the three most
influential variables in trajectory curvatures out of four variables,
we may claim that our experiments add considerable generality to the
previous work.
Trajectory curvatures within the horizontal and sagittal planes
We first explored whether the trajectory is planned in extrinsic or intrinsic coordinates by comparing trajectory curvatures between the horizontal and sagittal planes. Movements in the horizontal and sagittal planes can be executed by the same extension/flexion of elbow and shoulder joints. In this respect, each pair of movements in both planes dynamically corresponds to each other. The coordinated rotations of joints among the corresponding movements were the same. For example, the left-to-right transverse movement in the horizontal plane was equivalent to the downward up-and-down movement in the sagittal plane.
We examined the correlation between whole deviations of hand trajectories in the horizontal plane and in the sagittal plane. There was a variety of trajectory curvatures measured at various locations, and the trajectories of corresponding pairs tended to be curved with the same direction and amplitude. This suggests a strong relationship between the trajectory curvatures and variables in the body space such as flexion/extension.
Furthermore, the correspondence of curvatures in both planes suggest a weak gravitational influence on movements, and thus common properties of hand trajectories achieved with the same coordinated rotations of joints. Such common features were found in the predicted trajectories by the minimum commanded torque change model. Moreover, similar slopes of actual data (Fig. 6) and model predictions (Fig. 7) obtained from two of three subjects suggest that this model can predict a trajectory that has almost the same curvature as the actual trajectory. The movements within both planes have the same kinematic variables (coordinated rotation of joint) in the intrinsic coordinates; however, kinematic variables in the extrinsic coordinates are not the same between these planes. Hence it seems that this equivalence supports planning in intrinsic space but not extrinsic space. However, we cannot discuss if the trajectory is planned in dynamic space or kinematic space from these results.
The dynamic model can predict different trajectories for the different
planes because it depends on gravity. On the other hand, the kinematic
model is not influenced by gravity. In the present experiments, we
assumed that gravity had little influence on movements within the
horizontal plane, because the subjects' arms were supported by a
table. For movements within the sagittal plane, gravity should be
considered because of the lack of support. Even in this case, the hand
trajectory was found to be quite insensitive to weights held during
movements both in the real data and the theoretical calculation
(Atkeson and Hollerbach 1985; Uno et al. 1989a
). That is, a small effect of weights was observed on the optimal trajectories predicted by the original minimum torque change
model (Uno et al. 1989a
). Even if that trajectory is
generated based on dynamic criteria, the hand trajectory is apparently
not much affected by gravity because its influence is almost spatially uniform and thus not large enough to change the optimal trajectory. This is the theoretical background why the dynamic model such as the
minimum commanded torque change model can reproduce similar curvatures
shown in the trajectories measured within the horizontal and sagittal planes.
Wolpert et al. (1994) proposed the effect of visual
perceptual distortion on trajectory planning to explain the gently
curved hand trajectories observed in transverse movements. Conversely, Osu et al. (1997)
demonstrated that hand trajectories
are also curved within the frontoparallel plane for which the visual
perceptual distortion is negligible (Foley 1980
;
Indow and Watanabe 1988
). We assume that the visually
distorted lines within the horizontal and sagittal planes are different
from each other. If the idea of Wolpert et al. (1994)
was appropriate, we would find different curvatures between the actual
trajectories measured in these planes. The experimental results here
showed almost the same trajectory curvatures measured in these planes
as mentioned above for two of three subjects. Our outcome may be
considered as further counterevidence against the effect of visual
perceptual distortion on trajectory planning.
Contribution of the coordinated rotation of joints
We were able to quantitatively demonstrate that trajectory curvatures are dependent on the coordinated rotation of joints. In a large coordinated rotation of joints, namely a transverse movement when the shoulder and elbow rotate toward the same direction, the interactional torques such as off-diagonal components of inertial matrices and centrifugal forces increase with the same signs; therefore the torque around the joint is enlarged. In contrast, when the coordinated rotation of joints is small, the force components related to the shoulder and elbow joints have the inverse signs of each other, and torque is resultingly small. Therefore the coordinated rotation of joints is closely connected to the torque generated around the joint. We can say that the coordinated rotation of joints is a kinematic variable, but it is closely related to the arm dynamics.
Examination of models
For movements within the horizontal and sagittal planes, the minimum commanded torque change model was able to reproduce the spatial characteristics of measured trajectories, in this case, the magnitudes and directions of curvatures, better than the other three models. The minimum torque change model could neither reproduce the magnitudes 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. Even though the minimum angle jerk model could explain the direction of the curvature, it predicted trajectories with obviously excessive curvatures.
In previous studies, Hollerbach (1990) and Osu et
al. (1997)
suggested that planning in the joint space cannot
explain a gently curved hand trajectory. As shown in Fig. 9, the slopes
relating whole deviations of the minimum hand jerk and actual
trajectories were zero (the mean error of the slope was 1 in both
planes). For minimum angle jerk trajectories, slopes were from 1.4 to
2.7 in the horizontal plane and from 1.3 to 1.5 in the sagittal plane (the mean error of the slope was 1 and 0.35, respectively). Hence it
can be considered that the minimum angle jerk model 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 in predicting trajectories that are too curved, it is
quantitatively a better model compared with the minimum hand jerk
model. Accordingly, we demonstrated that it is impossible to completely
reproduce actual data with trajectory planning in the kinematic space.
As Flash (1990)
pointed out, we confirmed that literal
minimum torque change trajectories, computed without consideration of
the viscosity, cannot reproduce any actual trajectory.
Contribution of movement duration
Uno and Kawato (1996) predicted that longer
movement duration causes a viscous term to be relatively larger than
the other terms, namely, inertia, centrifugal force, and the Coriolis
force. Uno and Kawato (1996)
reported that a
trajectory is curved toward the outside (right side) with high viscous
values, and low viscous values lead the trajectory to curve inside
(left side). Mathematically, the original minimum commanded torque
change trajectory, which is computed by multiplying the movement
duration by
, is the same as that computed by multiplying viscosity
coefficients by
without changing the movement duration (Uno
and Kawato 1996
). Therefore a longer movement duration can be
predicted to cause a more curved trajectory. This was confirmed in
behavioral experiments (Uno and Kawato 1996
). The
results of our second experiment quantitatively supported these
previous studies with an enormous amount of data.
Movements performed in various places within a plane may have different
levels of difficulty. If a difficult movement leads to a highly curved
trajectory, we consider the possibility of the movement duration being
long due to this difficulty. In this case, we cannot discuss the
contribution of the movement duration to the trajectory curvature
because the movement duration and the coordinated rotation of joints
are correlated variables. Therefore we performed the following analysis
solely to investigate the influence of the movement duration.
![]() |
(11) |
![]() |
(12) |
Change in trajectory property with training
For via-point movements, we averaged paths and tangential velocities using measured data and compared them with the paths and tangential velocities predicted by each model. The measured trajectory properties were closer to those predicted by the minimum torque change model, the minimum commanded torque change model, and the minimum angle jerk model with training. However, the minimum torque change model and minimum angle jerk model could not reproduce the characteristics of actual trajectories in the first experiment. Because the minimum commanded torque change model was able to accurately explain data measured for both point-to-point movements and via-point movements, this model seemed to be the best of the four models. The results of the present experiments suggested that this model is applicable to skilled rather than to novel movements.
Minimizing the commanded torque change or the motor command change is the implicit constraint, and the constraints for the target, movement duration, and via-point are explicit. The latter constraints are given from the external world, whereas the former constraints are not. Even if the hand path, velocity, and acceleration of movements are provided as the explicit constraints, the CNS will have to finally compute torques or motor commands capable of meeting these constraints. The subjects can meet the implicit constraints such as minimizing the commanded torque change while meeting these explicit constraints in the process of training. There is also the possibility of adopting a different criterion to execute skilled or unskilled movements. In either case, the CNS seems to have the tendency to finally learn to perform optimization in the higher space, e.g., the commanded torque or motor command space.
The actual properties of paths and tangential velocities in the late stage of training converged with those predicted by the minimum commanded torque change model. This model could explain the properties of hand trajectories for both point-to-point movements and via-point movements. Taking all of these results together obtained in the present experiments, we were able to suggest that hand trajectory is planned in the intrinsic coordinates considering arm and muscle dynamics and using representations for motor commands controlling muscle tensions.
![]() |
APPENDIX A |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Dissimilarity index of velocity profiles
We quantified the tangential velocities of all of the data by
normalizing the trajectories for the duration and distance of the
movement (Atkeson and Hollerbach 1985). The mean of
these normalized tangential velocity profiles was regarded as the
reference profile. We calculated the maximum common area among areas
surrounded by a given normalized tangential velocity profile and the
x-axis, and areas bounded by the reference profile and the
x-axis. The whole area surrounded by these two profiles was
divided into the common part (Fig. A1,
light shaded area) and noncommon part, that is, the summation of the
differences between a normalized profile and the reference profile at
each time instance (dark shaded area). Then the ratio of the noncommon
area to the whole area was determined for each trajectory. We defined
this ratio as a dissimilarity index of the reference profile and actual
profile as shown below.
|
![]() |
APPENDIX B |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|
Calculation of optimal trajectories
The minimum torque change and minimum commanded torque change
trajectories were computed by combining the steepest descent method and
penalty method. The objective function is defined as
![]() |
For instance, we set the initial to 500 and performed 10,000 repetitive computations for each of 9 sets, namely, a total of 90,000 times to compute the minimum commanded torque change trajectories in
the horizontal task of the first experiment. The value of i + 1-th
at the Nth set was calculated by the following equation
![]() |
![]() |
![]() |
ACKNOWLEDGMENTS |
---|
This study was partially supported by Special Coordination Funds for promoting Science and Technology from the Science and Technology Agency of Japan, and by a Human Frontier Science Program grant to M. Kawato.
![]() |
FOOTNOTES |
---|
Address for reprint requests: M. Kawato, ATR Human Information Processing Research Laboratories, 2-2 Hikari-dai, Seika-cho, Soraku-gun, Kyoto 619-0288, Japan.
1
Gribble et al. (1998)
presented a different interpretation of data of Gomi and Kawato
(1996)
using a complicated nonlinear muscle model. Although
their muscle model is complicated and looks biologically plausible, it
is not a quantitatively validated model. For example, their model is an
overdamped system that is against the previous observations that the
musculoskeletal system is underdamped. Another trivial mistake of their
approach is that they simulated free movements while using parameter
values derived from constrained movements. Subjects in Gomi and
Kawato (1996)
generated ~2.5 times larger torques compared
with free movements because of the manipulandum load. Thus the observed
stiffness in the experiment is expected several times larger than in
movements without any load. If the same load with the experiments were
applied in the simulation of Gribble et al. (1998)
, our
experimental results (movements and stiffness) may not be reproduced by
constant change of equilibrium position.
2 The three-dimensional shape of a male's arm was measured by a Cyberware Laser Range Scanner. We calculated the arm as a homogeneous material with a specific gravity of 1.0 and computed its mass, center of mass, and moment of inertia from its volume. The arm parameters for each subject were calculated using the ratio of the arm length based on the measured data.
The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked "advertisement"n accordance with 18 U.S.C. Section 1734 solely to indicate this fact.
Received 14 April 1998; accepted in final form 7 January 1999.
![]() |
REFERENCES |
---|
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
---|