Motor learning in reaching tasks leads to homogenization of task space error distribution

A human arm, up to the wrist, is often modelled as a redundant 7 degree-of-freedom serial robot. Despite its inherent nonlinearity, we can perform point-to-point reaching tasks reasonably fast and with reasonable accuracy in the presence of external disturbances and noise. In this work, we take a closer look at the task space error during point-to-point reaching tasks and learning during an external force-field perturbation. From experiments and quantitative data, we confirm a directional dependence of the peak task space error with certain directions showing larger errors than others at the start of a force-field perturbation, and the larger errors are reduced with repeated trials implying learning. The analysis of the experimental data further shows that a) the distribution of the peak error is made more uniform across directions with trials and the error magnitude and distribution approaches the value when no perturbation is applied, b) the redundancy present in the human arm is used more in the direction of the larger error, and c) homogenization of the error distribution is not seen when the reaching task is performed with the non-dominant hand. The results support the hypothesis that not only magnitude of task space error, but the directional dependence is reduced during motor learning and the workspace is homogenized possibly to increase the control efficiency and accuracy in point-to-point reaching tasks. The results also imply that redundancy in the arm is used to homogenize the workspace, and additionally since the bio-mechanically similar dominant and non-dominant arms show different behaviours, the homogenizing is actively done in the central nervous system. Significance The human arm is capable of executing point-to-point reaching tasks reasonably accurately and quickly everywhere in its workspace. This is despite the inherent nonlinearities in the mechanics and the sensorimotor system. In this work, we show that motor learning enables homogenization of the task space error thus overcoming the nonlinearities and leading to simpler internal models and control of the arm movement. It is shown, across subjects, that the redundancy present in the arm is used to homogenize the task space. It is further shown, across subjects, that the homogenization is not an artifact of the biomechanics of the arm and is actively performed in the central nervous system since homogenization is not seen in the non-dominant hand.

while planning and executing movements (Mussa-Ivaldi et al. 1985). One 4 3 such signature of such inhomegenieties is the observeddirection dependence of error in 4 4 reaching tasks was done by (Gordon et al. 1994). who showd that the variability in the and N(J) is defined as where n is the number of trials. One of the simplest models of learning is that of a first-order process where the error is 1 8 4 reduced exponentially. The variation of peak error e(t) as a function of time t, can be written where β is a parameter that describes the rate of change of error and is independent of the 1 8 7 current error. The evolution of error from the above can be written as.
where α is the initial error and β is the learning rate. In our case, the error is the maximum deviation (perpendicular distance) of the hand from the 1 9 0 straight line connecting the fixation box and the target box in each of the 8 directions. Additionally, instead of a continuous function of time, the error is for each trial, and we have obtained using MCMC for the subject above along direction 90 degree (column 4 in Table   2  0  8 above) is shown in Supplemental Table 2. The values of α and β obtained using the well- are in good agreement. It may be mentioned that the R language was used to implement the 2 1 2 MCMC and Levenberg-Marquardt algorithms. The Supplemental Table 2 shows the values of gives a much better confidence to the obtained numbers. We have used MCMC for all 22 2 1 7 subjects and for all 8 directions. Experiments were divided into three epochs --a pre-adaptation, an adaptation to an external Arm model and tracker positions to measure joint rotation angles. We trained 12 subjects to learn point-to-point reaching movements using their dominant 2 3 2 hand, along 8 directions, in a force-field which was set using the force-field perturbation 2 3 3 equation (1). In this experiment the perturbation was proportional to the velocity of the hand 2 3 4 but perpendicular to the hand movement direction. We used an experimental setup shown in 2 3 5 figure 1 A.The data for the three phases -a pre-adaptation baseline period, followed by a 2 3 6 force field perturbation (dynamic perturbation) and finally a post-adaptation phase when the the three phases. the learned force field is turned off in the post adaptation period. This washout error 2 5 0 converges to baseline levels. learning is a first-order process where the error is reduced 2 5 2 exponentially, and the variation of peak error is as shown in equation (4). To compute the 2 5 3 learning in force--field perturbation trials, the errors were fitted with an exponential fit using 2 5 4  Figure 1D).

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To obtain a more geometric view, we plot the mean error and variation in the ellipse is fitted (motivated by the velocity distribution at a point seen in a robot --(see text 2 6 0 after equation (3)) with the mean error in the 8 directions for each of the three data sets. The 2 6 1 mean error is denoted by a "circle" and the "line" through the "circle" denote the variation last 5 trials --the errors are large when the lateral force is applied and as the trials progress, can be observed that the error ellipse in the baseline (when no external force is applied) is the 2 6 7 smallest. area of the ellipse for all the subjects. The ratio is large in the first 5 trials and the ratio in the 8.25e-06, t (11) = 7.80). There was good difference in the mean ellipse area between the 1.82e-06, t (11) = 9.13). However, the area of the ellipse is larger in the last 5 trials as  Overall, the main findings are that the maximum error due to the external force-field 2 8 5 is very large when it is applied and due to learning the error decreases with trials. This can be Differences in the ratio of the major to minor axis may reflect a difference in the intrinsic 3 0 7 biomechanics of the human arm. In contrast, differences in the ratio may also reflect the 3 0 8 effect of neural control that assists in homogenizing the workspace of the human arm. To assess this, we tested and compared the ratio of the major to minor axis of the ellipse across  field perturbation in each of the 8 directions with the non-dominant hand for a typical subject.

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Again, the mean error is denoted by a "circle" and the "line" through the "circle" denote the 3 1 6 variation with trials. We fit ellipses through the mean error along the 8 directions. This is 3 1 7 shown in figure 3 A for the baseline, the first 5 and the last 5 trials. It can be seen that as trials  The mean ellipse area in baseline period (mean = 7.19 േ 2.66) was significantly less D; p = 5.83e-06, t (9) = 9.43). There was significant difference in the mean direction ellipse 37.07 േ 27.90; p = 1.39e-04, t (9) = 6.31). The ellipse area gradually become smaller with 1 9 | P a g e practice over the course of about two hundred trials which implies some learning is taking Taken together these findings indicates that the difference in ratio of the major to the hand and non-dominant hand --most likely due to active neural control. Comparison of directional error ellipse eccentricity in baseline (black), starting of perturbation.

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Maximum error, learning rate and redundancy along directions: As mentioned earlier, the learning rate β and the maximum error α see equation (4)  200 trials (along all directions taken together) for a particular subject was 6.40 and 0.006.

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Similar results were obtained for all the 12 subjects. To investigate the variation of α and β 3 4 8 along each direction, the force-field perturbation trail data are sorted along the 8 directions. This is shown for a typical representative subject is shown in Supplemental Table 1 = 5.54, p = 3.24e-5, see figure 4 C) indicative that some directions have higher initial errors.

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Furthermore, there is a clear statistical difference in learning rate along 90° and 225° (F As mentioned earlier, the human hand is known to be redundant. In the model shown in performing planar point-to-point reaching motions and hence, the human hand is assumed to homogenization of error distribution due to a force-field perturbation, we computed the null hand for all subjects in the baseline period. Consistent with our hypothesis, N(J) was lesser 3 7 7 along the minor axis (mean = 0.06 ± 0.05) compared to the major axis (mean = 0.11 ± 0.07, p 3 7 8 = 0.006, t (9) = 3.49,) for the dominant hand. For the non-dominant hand, there was no 3 7 9 difference in N(J) values along the major and minor axis.
We have earlier observed that during learning homogenization of the error takes trials. Taken together we can suggest that redundancy may also play a role in making error and minor axis (blue).

Learning of a visuo-motor perturbation:
3 9 1 To test whether homogenization is a property related specifically to learning of the newly 3 9 2 imposed biomechanics or is a more general feature of motor learning we next examined 10 3 9 3 subjects while they learnt point-to-point reaching movements with the visuo-motor 3 9 4 perturbation, equation (2), along 8 directions, where in each case, the cursor was rotated by typical subject. Again, the mean error is denoted by a "circle" and the "line" through the 3 9 9 "circle" denote the variation with trials. We fit ellipses through the mean error along the 8 directions. This is shown in figure 6 A for the baseline, the first 5 and the last 5 trials. Unlike The mean ellipse area in baseline period (mean = 6.06 In this study, we have presented variation in task space error along directions in point-to-4 2 7 point reaching tasks. We demonstrated that the motor learning is homogenizing the 4 2 8 workspace of the human arm possibly to increase the efficiency and accuracy. In addition, 4 2 9 our results also showed a significantly larger use of redundancy along the directions with What is more significant is the observation that this anisotropy in errors distribution is In control theory, it is well known that it is significantly easier to design controllers for a 4 4 2 linear system to achieve a desired accuracy. More specifically, for a robot to follow a desired Our results indicate that some directions appear to be easier to learn and some are more shoulder) and the effort required is largest when the arm is moving in the direction of the 4 6 6 major axis and likewise the inertia seen, and the effort required is smaller when the arm is 4 6 7 moving along the minor axis and a mechanistic view of more/less error along direction of 4 6 8 more/less effort is consistent with this observation.

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The task space positioning error for a robot, in the presence of external disturbances and noise, is also related to the stiffness (or impedance) of the robotic arm. The impedance of  all directions show the same error distribution (see figure 4). This is consistent with our study 4 9 6 as learning a visuo-motor rotation involves learning of an internal kinematic model and unlikely requires the homogenization since the inhomogeneities are relate to dynamics.
Homogenization by feedback control internal models: As mentioned previously, movements associated with larger inertia will have greater muscle 5 0 0 loads and as a consequence of signal dependent noise will generated larger errors. In this 5 0 1 context, it is interesting that the distribution of errors in the baseline condition is low and well as internal models that ensure homogeneity even during the early feedforward driven inhomogeneous biomechanics. Although, we do not propose a mechanism on how feedback 5 0 8 gains and internal models are relearnt following the perturbation, our results suggest that a 5 0 9 form of learning that is sensitive not just to the magnitude of the errors as a consequence first 5 1 0 order learning, but of the learning rates themselves that are sensitive to the direction of 5 1 1 movement that helps homogenise errors. We suggest that such directional specific learning conducted during reaching tasks, subjected to a force-field disturbance, that the 5 1 7 generalization is bi-modal, perhaps reflecting basis elements that encode direction bi- modally. This work also does not discuss the direction dependence of positioning error in 5 1 9 reaching tasks. In this work, our results indicate that the generalization is not bi-modal --the In previous work we showed through first and second order correlations of null space variability--a proxy for joint redundancy--a possible role for joint redundancy in motor 5 2 5 learning of dynamics and kinematics. Here, we extended this correlation to study the 5 2 6 directional dependency of redundancy and its possible impact on motor learning. In 5 2 7 congruence and extension with our previous study, we observed that the redundancy 5 2 8 exhibited a directional axis that aligned with the learning axis, which could also explain the 5 2 9 observed homogenization. Furthermore, this redundancy axis was only observed for the speculate that a greater redundancy may allow better learning by increasing available options, 5 3 4 contributing to homogenising errors across directions. trajectory, when the arm is perturbed by a lateral force, decreases with practice and 5 4 0 approaches the situation when no lateral force is applied. In this paper, we investigate the trials where no lateral force is applied. Moreover, it is observed that the eccentricity of the