Figure 1.
Proof of concept: illustration that the hand jerk and torque change costs are more discernible during reaching to a bar than to a point.
A. Simulated hand paths for point-to-point movements in the horizontal plane. Targets (T1 to T6) were located approximately as in [11]. B. Simulated hand paths for the point-to-bar case. The starting points are the same as in panel A, but we replaced the target points by target lines/bars. The shaded areas emphasize the amount of difference between these two cost functions.
Figure 2.
A. Illustration of the experimental paradigm.
The reachable region from the sitting position is emphasized on the bar. The 5 initial postures under consideration are also shown (P1 to P5). B. Experimental trajectories for a representative subject. Dotted lines depict the initial arm posture of the subject (upper arm and forearm). The average fingertip path is shown in thick black line for each initial posture, from P1 to P5. The 20 trials are depicted in thin gray lines for every initial postures. C. Experimental angular displacements and finger velocity profiles for the most typical subject. First column: joint displacements at the shoulder and elbow joints; Second column: Finger velocity profiles with shaded areas indicating the standard deviation. Time is normalized, not amplitude.
Figure 3.
Model of the arm and definition of the parameters.
The extrinsic and intrinsic coordinates are denoted by and
, respectively.
is the total arm length, while
and
are the upper arm and forearm lengths. The subscript 1 denotes the shoulder joint. These segments have mass
, inertia
and distance to the center of mass
, with
. The Cartesian bar equation is given by
. The solid and dotted lines are the measured and simulated paths, respectively. The parameters RP, MV and sIPC are the reached point, movement vector angle and the signed index of path curvature.
Table 1.
Classical cost functions already proposed in the literature.
Table 2.
General movement features.
Figure 4.
Quantitative experimental results.
A. Reached point (final finger position) on the bar for each initial posture from P1 to P5 (RP parameter). The unit on the vertical bar is normalized by the arm length (percentage). The horizontal zero baseline is the level of the shoulder joint. Each point indicates the average location of the pointing movement, and error bars indicate the variability (standard deviation) across subjects. B. Movement vector angle (MV). The graph gives the angle between the movement vector and the horizontal line. Negative and positive values correspond to downward and upward movements, respectively. C. signed Index of Path Curvature: The graph depicts sIPC values for every initial posture. Positive and negative values correspond to globally concave and convex paths, respectively. D. Joint coupling. values are reported. Low values indicate low level of correlation between the shoulder and elbow angular displacements. E. Amplitudes of angular displacements. The graphs correspond to the shoulder (left) and elbow (right) joints, respectively. The magnitude of joint displacements (in degrees) is given for all initial postures.
Figure 5.
Inverse optimal control results: details for the most typical subject.
A. Weighting coefficients, i.e., elements of the vector (normalized by the maximum value). B. Contribution of each cost ingredient with respect to the total cost, for each simulation. The contribution of the
cost is computed as
. It is visible that mainly the energy and the angle acceleration are involved in general, with low contributions of the hand and angle jerks and a residual contribution of the geodesic cost. Torque, torque change, and effort costs do not contribute at all. C. Finger paths obtained from the best cost combination found by the inverse optimal procedure. Errors between the measured paths and the simulated ones (
and
parameters) are reported, for each initial posture. Note that this is the best criterion, and that any other cost combination would replicate the data less accurately with respect to metric 1.
Table 3.
Reconstruction errors after inverse optimal control.
Figure 6.
Inverse optimal control results for the 20 subjects using metric 1.
A. Weighting coefficients, i.e. elements of the vector (normalized such that the sum equals 1). Each bar corresponds to one subject. B. Contribution of each cost ingredient to the total cost, for each subject. The energy and angle acceleration costs, which are predominant in the total movement cost, are highlighted with shaded areas. This result is not evident when looking only at the weighting vector.
Figure 7.
Predictions of the different tested models.
A. Typical experimental data in order to facilitate comparisons (already depicted in Figure 2). B–H. Predicted hand paths for each model. I. Hybrid model, maximizing smoothness and minimizing energy (with a ratio 10∶1 for the energy component). Black and white bars are reported to show the regions on the bar for which the cost is relatively close to the optimal one (here, black areas correspond to movement costs below the 10% threshold relative to the minimum cost value).
Figure 8.
Areas between simulated and recorded finger paths.
This parameter qualifies as a general error measure. Values were first averaged across initial postures for each participant, and then, the mean and standard deviation were finally reported across participants. It is apparent that the energy and angle jerk/acceleration models performed quite well (with a lower standard deviation for the energy model), while the geodesic and hand jerk models performed moderately. The worst models were the torque change, effort and torque models, given in decreasing order of performance. The best model was the hybrid model, in agreement with the results provided by the inverse optimal control approach.
Figure 9.
Comparisons between models and real data, for relevant parameters.
A and B depict the reached point (RP) and movement vector (MV) parameters, which are the relevant parameters for the finger path. An analysis confirms that energy and angle jerk models, as well as the hybrid model, are quite efficient in predicting the terminal point on the bar and the movement direction (upward or downward). C and D depict the signed index of path curvature (sIPC) and joint coupling (), and are reported for the sake of completeness. However, they are not relevant when the final point is poorly predicted by a model. It is apparent that only the hybrid model is able to predict successfully these additional parameters (sIPC and joint coupling
). Parameters reported on the graphics: parameter
is the cumulative error across all starting positions
:
, with
being one of the following parameters: RP, MV, sIPC, or joint coupling; parameter
is the correlation coefficient between the simulated and measured data.
Figure 10.
Simulated angular displacements and finger velocity profiles.
A. Angular displacements at the shoulder and elbow joints. B. Finger velocity profiles. In both graphs, solid lines correspond to the experimental data, which are recalled from Figure 10 to facilitate comparisons. Dashed lines correspond to the simulated data (averaged across subjects), for the hybrid model, mixing the minimization of the mechanical energy expenditure and the angle acceleration energy. Shaded areas indicate the standard deviation. Time is normalized, but not amplitude.