Robot Hand Synergy Mapping Using Multi-factor Model and EMG Signal

Abstract

In this paper, it is investigated how a robot hand can be controlled from a human motion and an EMG signal in a tele-operation system. The proposed method uses a tensor to represent a multi-factor model relevant to different individuals and motions in multiple dimensions. Therefore, the synergies extracted by the proposed algorithm can account for not only various grasping motions but also the different characteristics of different people. Moreover, a synergy-level controller which generates motion and force of the robot is developed with postural synergies and an EMG signal. The effectiveness of the proposed new mapping algorithm is verified through experiments, which demonstrate better representation of hand motions with synergies and greater performance on grasping tasks than those of conventional synergy-based algorithms.

Appendix: Tensor Representation

This section is a condensation of [11, 12]. For details of tensor, please read the references.

A tensor is a higher-order generalization of the vector (first-order tensor) and matrix (second-order tensor). When we used a matrix to represent motion data set, the rows usually contained the channels for the joint angles, and the columns for motion samples. However, when we considered the multiple factors of human and used a tensor framework, the motions were grouped by each factor so that they constituted an Nth-order tensor.

The order of tensor Y $\in {\mathbb {R}{^{I_{1}\times I_{2}\times \ldots \times I_{N}}}}$ is N. The mode-n vectors of an Nth-order tensor $\underline{\mathrm{Y}}$ are defined as the $I_{n}$-dimensional vectors obtained by varying index $I_{n}$ while keeping the other indices fixed. All mode-n vectors can be arranged together as column vectors to compose a mode-n flattening matrix $\mathbf {Y_\mathrm {n}}$ $\in {\mathbb {R}{^{^{I_{n} \times (I_{1}I_{2},\ldots ,I_{n-1}I_{n+1},\ldots ,I_{n})}}}}$. The $I_{n}$-dimensional vectors of $\mathbf{{Y}}_\mathrm {n}$ are obtained from tensor $\underline{\mathrm{Y}}$ by varying index $I_{n}$ while keeping other indices fixed.

The multiplication of a high-order tensor Y $\in {\mathbb {R}{^{I_{1}\times I_{2}\times \ldots \times I_{N}}}}$ by a matrix $\mathbf{{A}}\in {\mathbb {R}{^{J_{n} \times I_{n}}}}$ is a mode-n product of tensor $\underline{\mathrm{Y}}$ by $\mathbf {A}$, which is denoted as $\underline{\mathrm{Y}} \times _{n} \mathbf {A}$. It can also be expressed in terms of flattened matrices. The entries of the product are given as

$$\begin{aligned} (\underline{\mathrm{Y}}\times _{n} \mathbf{{A}})_{i_{1}\ldots i_{n-1}j_{n}i_{n+1}\ldots i_{n}} = \sum _{i_n}d_{i_{1}\ldots i_{n-1}i_{n}i_{n+1}\ldots i_{n}}a_{j_{n}i_{n}} \end{aligned}$$

(11)

The tensor decomposition of $\underline{\mathrm{Y}}$ seeks for N orthonormal mode matrices as Eq. (12), which is obtained by HOSVD.

$$\begin{aligned} \underline{\mathrm{Y}}=\underline{\mathrm{G}} \times _{1}\mathbf{{U}}_\mathrm {1} \times _{2}\mathbf{{U}}_\mathrm {2} \ldots \times _{n}\mathbf{{U}}_\mathrm {n} \end{aligned}$$

(12)

The column vectors of $\mathbf{{A}}_\mathrm {n}$ are the orthonormal basis vectors of the mode-n unfolding matrix $\mathbf{{Y}}_\mathrm {n}$. Core tensor $\underline{\mathrm{G}}$ governs the relationship among mode matrices $\mathbf{{U}}_\mathrm {n}$.

Consequently, tensor representation is helpful to treat multi-factorization problem, as matrix is able to decompose factor using non-negative matrix factorization such as PCA and NMF.