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Table 1 Mathematical description of the HALS algorithm

From: A muscle synergies-based movements detection approach for recognition of the wrist movements

Definition Description References
\( {M}^k=M-\sum \limits_{p\ne k}{C}_p{W}_p^T=M-{CW}^T+{C}_k{W}_k^T \) Time-varying synergy model [25]
\( \frac{\partial {D}_F^k\left({M}^k\left\Vert {c}_k{w}_k\right.\right)}{\partial {c}_k}={c}_k{w}_k^T{c}_k-{M}^k{w}_k \)
\( \frac{\partial {D}_F^k\left({M}^k\left\Vert {c}_k{w}_k\right.\right)}{\partial {w}_k}={c}_k^T{c}_k{w}_k-{M}^k{c}_k \)
Computing the local gradient of the cost function [25, 26]
\( {w}_k\leftarrow \frac{1}{c_k^T{c}_k}{\left[{M}_k^T{a}_k\right]}_{+}=\frac{1}{c_k^T{c}_k}\max \left\{\varepsilon, {M}^k{w}_k\right\}{c}_k\leftarrow \frac{1}{w_k{w}_k^T}{\left[{M}_k^T{a}_k\right]}_{+}=\frac{1}{w_k^T{w}_k}\max \left\{\varepsilon, {M}^k{c}_k\right\} \) The updating rules [ξ]+ = max {ε, ξ} is a positive small integer (usually10−10). [25, 26]