SCIENCE CHINA Information Sciences, Volume 64 , Issue 5 : 152209(2021) https://doi.org/10.1007/s11432-020-3059-9

## In-hand manipulation of a circular dynamic object by soft fingertips without angle measurement

• AcceptedSep 1, 2020
• PublishedApr 8, 2021
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### Supplement

Appendix

According to 24, the matrix $\boldsymbol{A}~=~\left[\boldsymbol{A}_{1},~\boldsymbol{A}_{2}\right]~\in~\mathbb{R}^{11~\times~6}$ is defined as \begin{align*}\boldsymbol{A}_{1} = \begin{bmatrix} \boldsymbol{J}_{1}^{\rm T}\begin{bmatrix}c_1 \\ - s_1\end{bmatrix} & \boldsymbol{0}_{4\times 1} & - \frac{\partial}{\partial\boldsymbol{q}_{1}}\varphi_{r_{1}} & \boldsymbol{0}_{4\times 1} \\ \boldsymbol{0}_{4\times 1} & \boldsymbol{J}_{2}^{\rm T}\begin{bmatrix} - c_2 \\ s_2\end{bmatrix} & \boldsymbol{0}_{4\times 1} & - \frac{\partial}{\partial\boldsymbol{q}_{2}}\varphi_{r_{2}} \\ - c_1 & c_2 & -J_{x_1} & -J_{x_2} \\ s_1 & - s_2 & -J_{y_1} & -J_{y_2} \\ 0 & 0 & R & -R\end{bmatrix}, \boldsymbol{A}_{2} = \begin{bmatrix} \bar{a}_1\begin{bmatrix}s_1 \\ c_1\end{bmatrix} & \boldsymbol{0}_{4\times 1} & \dfrac{\beta_{\theta}}{D_L}\boldsymbol{J}_{1}^{\rm T}\begin{bmatrix}y_1-y_2 \\ x_2-x_1\end{bmatrix} \\ \bar{a}_2 \begin{bmatrix}s_2 \\ c_2\end{bmatrix} & \boldsymbol{0}_{4\times 1} & -\dfrac{\beta_{\theta}}{D_L}\boldsymbol{J}_{2}^{\rm T}\begin{bmatrix}y_1-y_2 \\ x_2-x_1\end{bmatrix} \\ J_{x_1} + J_{x_2} & 0 & 0 \\ J_{y_1} + J_{y_2} & 0 & 0 \\ 0 & -\frac{1}{2}Rf_dD_{12} & 0 \end{bmatrix} \end{align*} for $\bar{a}_i=r_iD_i^{-1}~\boldsymbol{J}_{i}^{\rm~T}$, $J_{x_1}~=~\frac{\partial}{\partial\boldsymbol{x}}\varphi_{r_{1}}$, $J_{y_1}~=~\frac{\partial}{\partial\boldsymbol{y}}\varphi_{r_{1}}$, $J_{x_2}~=~\frac{\partial}{\partial\boldsymbol{x}}\varphi_{r_{2}}$, $J_{y_2}=~\frac{\partial}{\partial\boldsymbol{y}}\varphi_{r_{2}}$.

### References

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• Figure 1

(Color online) Grasp and orientation of a circular object by pads of the thumb and index fingers.

• Figure 2

Kinematics of hemispherical soft-fingertip hand grasping a circular object.

• Figure 3

Kinematic relationships at contact, where the normal force relative angle $\beta(\bar{\boldsymbol{\nu}})~=~\alpha_{1}-~\alpha_{2}+~\pi~\in~\mathbb{R}$ is depicted. All angles are measured according to the right-hand rule and being consistent with the inertial frame, i.e., in (a) $\alpha_1<0$, $\alpha_2>0$, $\phi_1<0$, $\phi_2<0$, and for (b) $\alpha_1>0$, $\alpha_2<0$, $\phi_1>0$, $\phi_2>0$.

• Figure 4

(Color online) The optimal grasping of a circular object [44].

• Figure 5

(Color online) Proposed optimal grasping and orientation control of a circular object with $\theta_d=0$ rad.

• Figure 6

Optimal grasping with $\beta_d=\pi$.

• Figure 7

(Color online) Optimal grasping: (a) ${\rm~CoM}_o$ and the object angle $\theta$, (b) internal angles $\alpha_i$, for $i=1,2$.

• Figure 8

Optimal grasping and orientation control with $\beta_d=\pi-\theta_d$.

• Figure 9

(Color online) Optimal grasping and orientation control: (a) ${\rm~CoM}_o$, (b) internal angles $\alpha_i$, (c) the object angle $\theta$, protectłinebreak (d) orientation angle error $\overline{\Delta\theta}$.

• Figure 10

(Color online) Comparative study between the proposed approach and [43]: object coordinates, internal angles and error angles. (a) Without object angle measurement; (b) with object angle measurement.

• Figure 11

Optimal grasping and orientation control with $\beta_d=\pi-\theta_d$ and $\theta_d=-0.2$ rad.

• Figure 12

(Color online) Optimal grasping and orientation control with $\theta_d=-0.2$ rad: (a) ${\rm~CoM}_o$, (b) $\hat{\theta}$, (c) the object angle $\theta$, (d) orientation angle error $\overline{\Delta\theta}$.

• Figure 13

(Color online) Optimal grasping under gravity effect: performance of the $\Delta~f_i$ and the object coordinates, $i=1,2$, where $\bar{p}(0)~=~[0.15,~0.2]^{\rm~T}$ m and $\theta~(0)$ = 0.0 rad with $f_d~=~2.5$ N and $\theta_d~=~0$ rad. (a) Contact force error; (b) object coordinates.

• Figure 14

(Color online) Optimal grasping under gravity effect: performance of the $\Delta~f_i$ and the object coordinates, $i=1,2$, where $\bar{p}(0)~=~[0.075,~0.155]^{\rm~T}$ m and $\theta(0)~=~0.0$ rad with $f_d~=~2.5$ N and $\theta_d~=~0.2$ rad. (a) Contact force error; (b) object coordinates.

• Figure 15

(Color online) Performance of the stable grasping and orientation of a circular object avoiding any object information: (a) $t=0$ s, (b) $t=5$ s, (c) $t=10$ s, (d) $t=15$ s, (e) $t=20$ s. (f) $t=25$ s.

• Table 1

Table 1Physical parameters of the robotic fingers

 Parameter Value (m) Parameter Value $L_{i1}$ $0.1$ $M_{i1}$ $0.5$ kg $L_{i2}$ $0.08$ $M_{i2}$ $0.3$ kg $L_{i3}$ $0.06$ $M_{i3}$ $0.15$ kg $L_{i4}$ $0.03$ $M_{i4}$ $0.1$ kg $l_{cm_{i1}}$ $0.045$ $I_{i1}$ $6.5\times~10^{-4}$ kg$\cdot$m$^{2}$ $l_{cm_{i2}}$ $0.035$ $I_{i2}$ $2.5\times10^{-4}$ kg$\cdot$m$^{2}$ $l_{cm_{i3}}$ $0.025$ $I_{i3}$ $5\times~10^{-5}$ kg$\cdot$m$^{2}$ $l_{cm_{i4}}$ $0.01$ $I_{i4}$ $2\times~10^{-6}$ kg$\cdot$m$^{2}$

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