INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

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; 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.

Three "indeterminacy problems" are involved in planning and executing reaching tasks with a visually guided arm (Kawato 1992). 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).

Many approaches have been proposed to explain how the brain resolves such problems (Bernstein 1967; 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.

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). 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).

Neural recording data consistent with the four different planning spaces have been obtained. Examples are Georgopoulos et al. (1982, 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.

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), 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.

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), 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 this window] [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
is minimized, where t_f is the movement duration. This model always predicts straight paths regardless of the influence of arm dynamics, arm posture, external forces, and movement duration.

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
where _i is the ith joint out of n joints. Because a straight trajectory in the joint space was thought to be much more curved than the actual trajectory in the extrinsic hand space, this class of model has been considered to be inappropriate (Hollerbach 1990; Osu et al. 1997). However, we will quantitatively demonstrate that this model can predict the actual trajectory curvature better than the minimum hand jerk model.

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.

View larger version (17K): [in this window] [in a new window]   Fig. 2. Conceptual block diagram for intrinsic-dynamic models. Motor command u is transmitted to muscle and commanded torque c is generated through muscle elastic property. c is dumped by muscle viscous property, and actual torque a is obtained. u designates c capable of producing necessary a to complete a given movement considering torque deducted through viscous property. Minimum torque change model only regards link dynamics as controlled object (- · - · -), minimum commanded torque change model regards both link dynamics and muscles as controlled objects (- - -).

In the minimum torque change model and minimum commanded torque change model, the objective function to be minimized is expressed by
where _i is either the actual torque in the minimum torque change model, or the commanded torque in the minimum commanded torque change model, generated around the ith joint out of n joints. The shoulder is 1, and the elbow is 2. The actual torque with zero viscosity or commanded torque with nonzero viscosity is computed from the following dynamics equation of a two-link manipulator

(1)

I_i, M_i, L_i, S_i, and g represent inertia, mass, arm length, center of mass of link i, and acceleration due to gravity (9.8 [m/s²]). Links 1 and 2 correspond to the upper arm and the forearm. The values of I_i, M_i, and S_i are estimated from the measured link length L_i for each subject.² B_ij is the viscosity coefficient expressing the influence of the angular velocity of joint j on the torque of joint i. Table 1 summarizes the physical parameters of the arm for each subject. Value g is set to zero for experiments in the horizontal plane because a table supports the arm. In the sagittal plane, torques are computed by considering the force supporting the arm against the acceleration of gravity g.

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 B_m 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 B_ij in Eq. 1 as B_m 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 B_m 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)

The actual torque can be further expressed as Eq. 5a assuming that K(u), _r(u), and B_m(u) have the linear dependence with respect to motor command u (Katayama and Kawato 1993)

(5a)

(5b)

(5c)

In this formula, k₁ and b₁^m represent the coefficients of u that express elasticity and viscosity. The intrinsic elasticity and viscosity independent of the motor command in the musculoskeletal system are denoted by k₀ and b₀^m. ₀ expresses the equilibrium point when the motor command is zero. r is a coefficient of u determining the equilibrium point. The term p in Eq. 5c corresponds to the first term of Eq. 5b. The second and third terms of Eq. 5b are approximated to the term qu, which is dependent on the motor command assuming that change in  is slower than change in u, and the higher order term of u is negligible. According to the minimum commanded torque change model, the change over time of the second term in Eq. 5c, namely the motor command change, is thought to be approximately minimized.

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: ₁^ma, elbow: ₂^ma) during movement are used as the actual torques

(6)

In this study, we compute the literal minimum torque change trajectories with zero viscosity, and the minimum commanded torque change trajectories with viscous values estimated by using Eq. 6. We note that the minimum torque change model we test in this paper uses appropriate inertial and viscous parameters and minimizes torque change assuming zero viscosity. No parameter fitting was performed for any model.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

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.

View larger version (48K): [in this window] [in a new window]   Fig. 3. Experimental setup, degrees of freedom for arm, and workspace. Subjects executed point-to-point movements on the horizontal board (A) or sagittal board (C). Movement within horizontal plane is performed by flexion/extension (fl/ex) of elbow and horizontal fl/ex of shoulder after abduction (abduction/adduction: ab/ad) with 90° (B). Movement within sagittal plane is performed by fl/ex of elbow and shoulder (D). Shaded area in top left figures included in B and D shows workspace where initial and final positions were located in horizontal and sagittal planes. Gray crosses on upper arm indicate humeral rotation is the same between horizontal and sagittal planes (B and D). E: subjects performed via-point movements while watching a cathode ray tube (CRT) screen. F: via-points indicated by asterisks and rounded by circles were used in early and late stages of training, and in training, respectively.

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).

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.

View larger version (17K): [in this window] [in a new window]   Fig. 4. Parameters that explain trajectory curvatures. A: filled black area is a "whole deviation" as index of trajectory curvature. The sign of a right curved trajectory for a vector from initial to final position is defined as positive, and the sign of a left curved trajectory is defined as negative. B: coordinated rotation of joints is summation of quantity 1 that subtracts initial shoulder angle from final shoulder angle, and quantity 2 that subtracts initial elbow angle from final elbow angle.

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).

(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)

where n, (x_n^raw, y_n^raw), (x_n^model, y_n^model), and N denote the number of 5-ms sampling points, the coordinates of actual data, those of predicted data, and the total sampling number, respectively. The MSEs allow trajectory comparisons including temporal variations. Bonferroni's t-test was used to test the differences among the models. If a certain model almost always gave the smallest MSEs, we considered that model to be statistically best for predicting actual trajectories.

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)

The movement duration was provided with the same sign as the whole deviation of each trajectory. Because similar results were obtained for all subjects to those in the first experiment, we combined the data of all of the subjects. If trajectory planning is based on the dynamic criteria, the movement duration was predicted to affect the hand trajectory (Uno and Kawato 1996).

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

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION APPENDIX A APPENDIX B REFERENCES

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.

View larger version (36K): [in this window] [in a new window]   Fig. 5. A and E indicate axes settings within horizontal and sagittal planes for B-D and F-H that show hand trajectories and reconstructed areas for model represented by the coordinated rotation of joints. B and F indicate 5 trajectories for positive whole deviations. C and G indicate 5 trajectories for small absolute whole deviations (straight line). D and H show 5 trajectories for negative whole deviations. Solid line is a measured hand trajectory, and shaded area below each trajectory is amount of reconstructed whole deviations by coordinated rotation of joints. For shaded area, actual path changes its shape by expanding or contracting related to perpendicular of start-goal straight line, so that its whole deviation becomes the same as that reconstructed by coordinated rotation of joints. Shaded area shows ability of coordinated rotation of joints to model the actual whole deviation. Linear model only predicts the whole deviation not any trajectory or path. ×, , and (0, 0), the initial, final, and shoulder positions.

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.

View larger version (18K): [in this window] [in a new window]   Fig. 6. Correlations of whole deviations measured in the horizontal and sagittal planes of all subjects. A: subject MM. B: subject YS. C: subject TT.

View larger version (19K): [in this window] [in a new window]   Fig. 7. Correlations of whole deviations predicted by minimum commanded torque change model in horizontal and sagittal planes of all subjects. A: subject MM. B: subject YS. C: subject TT.

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.

View larger version (28K): [in this window] [in a new window]   Fig. 8. Examples of trajectory properties measured and predicted by minimum hand jerk, minimum angle jerk, minimum torque change, and minimum commanded torque change models are shown by solid gray lines, dotted lines, dash dot lines, broken lines, and solid lines, respectively. A: paths. B and C: velocities. D and E: accelerations. F and G: torques. ×, , and (0, 0), the initial, final, and shoulder positions. Profiles shown in B-G are derived from trajectories whose initial and final positions are indicated by arrows in A.

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.