Alternating Minimization for Time-Shifted Synergy Extraction in Human Hand Coordination

Identifying motor synergies – coordinated hand joint patterns activated at task-dependent time shifts – from kinematic data is central to motor control and robotics. Existing two-stage methods first extract candidate waveforms (via SVD) and then select shifted templates using sparse optimization, requiring at least two datasets and complicating data collection. We introduce an optimization-based framework that jointly learns a small set of synergies and their sparse activation coefficients. The formulation enforces group sparsity for synergy selection and element-wise sparsity for activation timing. We develop an alternating minimization method in which coefficient updates decouple across tasks and synergy updates reduce to regularized least-squares problems. Our approach requires only a single data set, and simulations show accurate velocity reconstruction with compact, interpretable synergies.

💡 Research Summary

The paper addresses the problem of extracting motor synergies—coordinated patterns of hand joint velocities that may be activated at different time offsets—from kinematic recordings. Traditional approaches first obtain a set of candidate waveforms using linear dimensionality reduction (e.g., SVD) and then, in a separate stage, select time‑shifted templates through sparse optimization. This two‑stage pipeline requires at least two distinct datasets (one for template extraction, another for coefficient estimation), which doubles the experimental burden and does not jointly optimize the waveforms and their activations.

To overcome these limitations, the authors propose a unified optimization framework that simultaneously learns a compact set of synergies and the sparse, time‑shifted activation coefficients for each grasping task. The core of the model is a convolution‑mixture representation: each synergy (s_j) (a vector of length (T_s) for all joints) is embedded into the observation window of length (T) by a Toeplitz‑like shift matrix (D_{jk}). For a given task (g), the observed velocity matrix (v_g) is expressed as
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