Wrist autonomy based on upper-limb synergy: a pilot study

Abstract

Incorporating an electrically powered wrist can largely improve the dexterity of a prosthetic hand when grasping various objects; however, it also intensifies the difficulty of the hand’s operation due to the introduction of extra degrees of freedom (DOFs). The mechanism of multi-joint synergy in human body movements provides a new sight to solve this problem. In this paper, focusing on four typical manipulation activities of daily life (ADLs), 10 upper-limb joint angles were collected and analyzed first to verify the existence of synergy. Then, a linear regression model was established to predict the wrist rotation angle from the shoulder and elbow joints, which can be directly used as a control reference for achieving wrist autonomy. For both healthy and amputee subjects, experimental platforms were established and control tests were conducted, wherein the task completion time and compensatory movement during the four ADLs were evaluated. The results show that our synergy-based wrist autonomy method can effectively improve the completion efficiency of multiple ADLs without increasing the control complexity. Also, it can significantly reduce the compensatory movements of multiple joints compared to traditional prostheses using an idle wrist.

Graphical Abstract

Appendices

Appendix 1

The position of the IMU is considered a dynamic coordinate system. From the obtained data, the attitude matrix of the dynamic coordinate system of the joint can be calculated, that is, the rotation transformation matrix with respect to the stationary global coordinate system.

$$R=R\left(z,\varphi \right)R\left(y,\theta \right)R\left(x,f\right)=\left[\begin{array}{ccc}c\varphi c\theta & c\varphi s\theta sf-s\varphi cf& c\varphi s\theta cf-s\varphi sf\\ s\varphi c\theta & s\varphi s\theta sf+c\varphi cf& s\varphi s\theta cf+c\varphi sf\\ -s\theta & c\theta sf& c\theta cf\end{array}\right]$$

(2)

where φ,θ,f are the three Euler angles of yaw, pitch, and roll of IMU output, respectively.

The elements of the column vector of the rotation matrix R are the projections of the XYZ axes of the dynamic coordinate system onto each of the axes in the stationary coordinate system, so the vector representation of each axis of the dynamic coordinate system in the stationary coordinate system can be obtained as:

$$\overrightarrow{{\varvec{O}}{\varvec{X}}}=\left[c\varphi c\theta ,s\varphi c\theta ,-s\theta \right]$$

(3)

$$\overrightarrow{{\varvec{O}}{\varvec{Y}}}=\left[{\text{c}}\varphi s\theta sf-s\varphi cf,s\varphi s\theta sf+c\varphi cf,c\theta sf\right]$$

(4)

$$\overrightarrow{{\varvec{O}}{\varvec{Z}}}=\left[c\varphi s\theta cf-s\varphi sf,s\varphi s\theta cf+c\varphi sf,c\theta cf\right]$$

(5)

When solving the three angles A, B, and C of the shoulder, simply project the Y-axis vector of the elbow coordinate system ${\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}$ in the corresponding plane of the shoulder coordinate system, and the angle between the vector obtained by the projection and the corresponding shoulder coordinate axis is the desired angle. The formula for calculating the projection of a vector in the plane is introduced here:

$$pro{j}_{V}\left(\overrightarrow{X}\right)=T{({T}^{T}T)}^{-1}{T}^{T}\overrightarrow{{\varvec{X}}}$$

(6)

Where $\overrightarrow{{\varvec{X}}}$ is the vector being projected and T is the basis vector matrix of the projection plane V. Then the angles A, B, and C can be found by the following equations:

$$A={\text{acos}}\left(\frac{pro{j}_{XOY\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\cdot {\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{shoulder}}{\left|pro{j}_{XOY\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\right|\cdot \left|{\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{shoulder}\right|}\right)$$

(7)

$$B={\text{acos}}\left(\frac{pro{j}_{YOZ\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\cdot {\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{shoulder}}{\left|pro{j}_{YOZ\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\right|\cdot \left|{\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{shoulder}\right|}\right)$$

(8)

$$C={\text{acos}}\left(\frac{pro{j}_{XOZ\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\cdot {\overrightarrow{{\varvec{O}}{\varvec{X}}}}_{shoulder}}{\left|pro{j}_{XOZ\_shoulder}\left({\overrightarrow{{\varvec{O}}{\varvec{Y}}}}_{elbow}\right)\right|\cdot \left|{\overrightarrow{{\varvec{O}}{\varvec{X}}}}_{shoulder}\right|}\right)$$

(9)

D and E can be derived directly from the equation for calculating the angle between the axes in the IMU coordinate system:

$$D={\text{acos}}\left(\frac{{\overrightarrow{OY}}_{elbow}\cdot {\overrightarrow{OY}}_{shoulder}}{\left|{\overrightarrow{OY}}_{elbow}\right|\cdot \left|{\overrightarrow{OY}}_{shoulder}\right|}\right)$$

(10)

$$E={180}^{\circ }\text{-acos}\left(\frac{{\overrightarrow{OY}}_{wrist}\cdot {\overrightarrow{OY}}_{elbow}}{\left|{\overrightarrow{OY}}_{wrist}\right|\cdot \left|{\overrightarrow{OY}}_{elbow}\right|}\right)$$

(11)

According to the principle of the uniqueness of the relative transformation matrix between two rotational matrices, F, G, H, I, and J can be solved. Taking G as an example, the transformation matrix of the elbow coordinate system with respect to the shoulder coordinate system can be expressed as:

$${{}^{shoulder}R}_{elbow}={R}_{shoulder}^{-1}{R}_{elbow}$$

(12)

At the same time, this transformation matrix can also be expressed by the existing angle as:

$$\begin{array}{c}{{}^{shoulder}R}_{elbow}=R\left(y,-C\right)R\left(z,90-D\right)R\left(x,G\right)R\left(z,90\right)R\left(x,180\right)=\\ \left[\begin{array}{ccc}-c(D)c(C)c(G)-s(C)s(G)& -s(D)c(C)& s(C)c(G)-c(D)c(C)s(G)\\ -s(D)c(G)& c(D)& -s(D)s(G)\\ -c(D)s(C)c(G)+c(C)s(G)& -s(D)s(C)& -c(C)c(G)-c(D)s(C)s(G)\end{array}\right]\end{array}$$

(13)

The result of G can be obtained by using the equality of the corresponding elements of the matrix.

$$G=\mathrm{atan}\left({{}^{shoulder}R}_{elbow}(\mathrm{2,3})/{{}^{shoulder}R}_{elbow}(\mathrm{2,1})\right)$$

(14)

The transformation matrix of the wrist coordinate system with respect to the elbow coordinate system can be expressed as:

$${{}^{elbow}R}_{wrist}={R}_{elbow}^{-1}{R}_{wrist}$$

(15)

At the same time, this transformation matrix can again be expressed by the existing angle as:

$${{}^{elbow}R}_{wrist}=R(x,E-180)R(y,F)$$

(16)

A result of F can be obtained.

$$F=\mathrm{atan}\left({{}^{elbow}R}_{wrist}(\mathrm{1,3})/{{}^{elbow}R}_{wrist}(\mathrm{1,1})\right)=\left[\begin{array}{ccc}c(F)& 0& s(F)\\ -s(E)s(F)& c(E)& c(F)s(E)\\ -c(E)s(F)& -s(E)& c(E)c(F)\end{array}\right]$$

(17)

H, I, and J are the same, and no further derivation is made here.

Appendix 2

$${F}^{^{\prime}}={K}_{0}+A*{K}_{A}+B*{K}_{B}+C*{K}_{C}+D*{K}_{D}+E*{K}_{E}+G*{K}_{G}$$

Table 2

Table 2 Summary of synergistic analysis of upper limb joints in healthy individuals