Modeling and Control of an Octopus Inspired Soft Arm under Prescribed Spatial Motion Constraints

Abstract

Precise control of soft robots remains challenging due to their highly compliant nature. Existing kinematic models may not enable accurate control performance as they do not account for actuation forces and dynamics. This paper tackles the problem of precise motion control for a soft robotic arm with longitudinal muscle actuators. We develop an integrated modeling and control framework that incorporates dynamics and actuation forces for improved accuracy. A key contribution is deriving and implementing a mathematical model of the soft muscle actuators using minimum norm optimization. Among, actuator saturation is addressed through a tension limiting function. Based on the whole model, we develop a dynamic surface controller with performance constraint to precisely control the soft arm. This controller makes soft robot subsequent interactions more secure. To assess the approach, numerical simulations and physical experiments are designed to verify the feasibility and rationality.

Appendix A

The rotation matrix R is converted into a unit quaternion Q, which can be described by the following formula:

$$\begin{aligned} R_{0}=\left[ \begin{array}{ccc} r_{11} &{} r_{12} &{} r_{13} \\ r_{21} &{} r_{22} &{} r_{23} \\ r_{31} &{} r_{32} &{} r_{33} \end{array}\right] \Rightarrow \left\{ \begin{array}{l} q_{1}=\frac{\sqrt{1+r_{11}+r_{22}+r_{33}}}{2} \\ q_{2}=\frac{r_{32}-r_{23}}{4 q_{1}} \\ q_{3}=\frac{r_{13}-r_{31}}{4 q_{1}} \\ q_{4}=\frac{r_{21}-r_{12}}{4 q_{1}} \end{array}\right. \end{aligned}$$

(1)

However, Eq. A1 must meet $q_1\ne 0,r_{11}+r_{22}+r_{33}+1>0$, briefly, $1+tr(R)>0$. If $q_1=0,tr(R)=-1$, then, we need to discuss $\max \{r_{11},r_{22},r_{33}\}$ when converting to quaternion. The form of $q_2,q_3,q_4$ is determined by the value in $r_{11},r_{22},r_{33}$. In addition, C(Q) represents the process of converting a unit quaternion into a coordinate rotation matrix. It can be written as $C(Q)=$

$$\begin{aligned} \begin{aligned} \begin{bmatrix} q_{1}^{2}\!-\!q_{2}^{2}\!-\!q_{3}^{2}\!+\!q_{4}^{2} &{} 2 \left( q_{1} q_{2}\!+\!q_{3} q_{4}\right) &{} \!2\left( q_{1} q_{3}\!-\!q_{2} q_{4}\right) \\ 2\left( q_{1} q_{2}\!-\!q_{3} q_{4}\right) &{} \!-q_{1}^{2}\!+\!q_{2}^{2}\!-\!q_{3}^{2}\!+\!q_{4}^{2} &{} 2\left( q_{2} q_{3}\!+\!q_{1} q_{4}\right) \\ 2\left( q_{1} q_{3}\!+\!q_{2} q_{4}\right) &{} \!2\left( q_{2} q_{3}\!-\!q_{1} q_{4}\right) &{} \!-\!q_{1}^{2}-q_{2}^{2}\!+\!q_{3}^{2}\!+\!q_{4}^{2} \end{bmatrix} \end{aligned} \end{aligned}$$

(2)

For operator $A(\cdot )$, it can be expressed as

$A(\lambda )=\left( \begin{array}{cccc}0 &{} -\lambda _{1} &{} -\lambda _{2} &{} -\lambda _{3} \\ \lambda _{1} &{} 0 &{} -\lambda _{3} &{} \lambda _{2} \\ \lambda _{2} &{} \lambda _{3} &{} 0 &{} -\lambda _{1} \\ \lambda _{3} &{} -\lambda _{2} &{} \lambda _{1} &{} 0\end{array}\right)$. where $\lambda \in \mathbb {R}^3$.