Low-Dimensional Synergistic Representation of Bilateral Reaching Movements

Abstract

Kinematic and neuromuscular synergies have been found in numerous aspects of human motion. This study aims to determine how effectively kinematic synergies in bilateral upper arm movements can be used to replicate complex activities of daily living (ADL) tasks using a sparse optimization algorithm. Ten right-handed subjects executed 18 rapid and 11 natural-paced ADL tasks requiring bimanual coordination while sitting at a table. A position tracking system was used to track the subjects' arms in space, and angular velocities over time for shoulder abduction, shoulder flexion, shoulder internal rotation, and elbow flexion for each arm were computed. Principal component analysis (PCA) was used to generate kinematic synergies from the rapid-paced task set for each subject. The first three synergies accounted for 80.3 ± 3.8% of variance, while the first eight accounted for 94.8 ± 0.85%. The first and second synergies appeared to encode symmetric reaching motions which were highly correlated across subjects. The first three synergies were correlated between left and right arms within subjects, whereas synergies four through eight were not, indicating asymmetries between left and right arms in only the higher order synergies. The synergies were then used to reconstruct each natural-paced task using the l₁-norm minimization algorithm. Temporal dilations of the synergies were introduced in order to model the temporal scaling of movement patterns achieved by the cerebellum and basal ganglia as reported previously in the literature. Reconstruction error was reduced by introducing synergy dilations, and cumulative recruitment of several synergies was significantly reduced in the first 10% of training task time by introducing temporal dilations. The outcomes of this work could open new scenarios for the applications of postural synergies to the control of robotic systems, with potential applications in rehabilitation. These synergies not only help in providing near-natural control but also provide simplified strategies for design and control of artificial limbs. Potential applications of these bilateral synergies were discussed and future directions were proposed.

Keywords: activities of daily living; bilateral upper limb movements; kinematic synergies; motor control; principal component analysis.

Figure 1

(A) Sensor placement on body. S₁ and S₂ are positioned at the lateral head of the clavicles, S3 is placed on the right side on the body’s midaxillary line, S₄ and S₆ are placed on the outer side of the elbow, and S₅ and S₇ are placed on the wrists between the distal heads of the radius and ulna. P₁ indicates the coronal plane, P₂ indicates the sagittal plane, P₃ indicates the plane normal to V _ae, and P₄ indicates the plane containing V _ae and V _aw. P₃ and P₄ exist for both the left and right arms. (B) Tasks executed during study. Each panel is labeled with a number corresponding to the task in Table 2 which is shown. Subjects start each task with their hands in the rest positions marked by the visible red boxes.

Figure 2

Fraction of variance accounted for by each synergy (bars) and total from 1 to n synergies (line). The first synergy accounts for 57.98 ± 6.35% of variance, the first six synergies account for 91.05 ± 1.69%, and the first eight synergies account for 94.82 ± 0.85%. Dotted line shows 0.95 threshold.

Figure 3

Correlation analysis of the first three synergies. (A) Pearson coefficient of determination, r², averaged across the 45 unique combinations between the specified synergies of each of 10 subjects. Statistical differences found using one-way ANOVA tables with α = 0.05 with Tukey post hoc tests. Comparison of synergy 1 to synergy 1 and synergy 1 to synergies 2 and 3, among all pairings of synergy 2, and among all pairings of synergy 3 yielded significant differences. (B) Correlation coefficient, r, between each subject for synergies 1, 2, and 3. Only unique pairings of subjects are shown using the upper triangle matrices. Synergy 1 appears highly positively correlated except for subject 6, who is highly negatively correlated.

Figure 4

Angular velocity profile for first eight synergies of subject 6. Vertical axis is unitless velocity since synergy is unscaled. Rows are labeled with letters indicating the side (“R” for right or “L” for left), the joint (“S” for shoulder or “E” for elbow), and the rotation (“A” for abduction in positive direction, “F” for flexion in positive direction, “I” for internal rotation in positive direction) for each DoF. Synergy 1 and 2 involve flexion and internal rotation of shoulder along with extension of elbow, implying a reaching motion, whereas synergy 3 involves extensions at the shoulder and flexion at the elbow.

Figure 5

Posture visualization for first eight synergies of subject 6 (columns) over six normalized time instances (rows). Position at T = 0% is the subject’s position averaged across the first 50 samples of all tasks. Synergies were integrated up to each time point and multiplied by a gain such that normal joint range of motion is not violated. Mirror symmetry between left/right arms can be seen in the first three synergies whereas synergies 4–8 have asymmetric motions. Subject’s non-dominant hand tends to go to a single position and hold steady while the dominant hand appears to move continuously.

Figure 6

Pearson’s coefficient of determination comparing right arm joints to left arm joints of the each synergy. One-way ANOVA and Tukey post hoc show that synergy 1 and synergy 2 each had significantly higher coefficients of determination than synergies 4–8, while the coefficient for synergy 3 was significantly higher than synergy 4 and 8. Statistical difference found in first three synergies implies asymmetric motion between left and right arms in higher order synergies.

Figure 7

Normalized reconstruction error when recruiting from synergies 1–8 with and without dilations. The reconstruction error for each synergy was averaged over degrees of freedom, subjects, and tasks. Dilated synergies were longer than undilated synergies by 25, 50, and 75% of the difference between minimum reconstruction task length and the synergy length. SDs are across subjects and tasks.

Figure 8

Three examples of reconstruction progression for left shoulder abduction. (A) Repetition 2 of task 27 for subject 8, the best-performing reconstruction with error across all joints of 2.22%. (B) Repetition 2 of task 24 for subject 5, error of 2.4%. (C) Repetition 1 of task 22 for subject 10, error of 7.3%. The early reaching phases of tasks were typically not as accurately reconstructed as the later manipulation phases.