Forward Kinematics of Object Transporting by a Multi-Robot System with a Deformable Sheet

TL;DR

A computational algorithm based on the FK method is presented to obtain all possible solutions with the given initial sheet shape and the robot team formation to demonstrate the effectiveness, completeness, and efficiency of the FK algorithm.

Abstract

We present object handling and transporting by a multi-robot team with a deformable sheet as a carrier. Due to the deformability of the sheet and the high dimension of the whole system, it is challenging to clearly describe all the possible positions of the object on the sheet for a given formation of the multi-robot system. A complete forward kinematics (FK) method is proposed for object handling by an $N$-mobile robot team with a deformable sheet. Based on the virtual variable cables model (VVCM), a constrained quadratic problem (CQP) is formulated by combining the geometric constraints and minimum potential energy conditions of the system. Analytical solutions to the CQP are presented and then further verified with the force closure condition. We present an FK algorithm based on the FK method to obtain all possible solutions with the given initial sheet shape and the robot team formation. We demonstrate the effectiveness, completeness, and efficiency of the FK algorithm with experimental results and case study examples.

Figures (6)

• Figure 1: The system configuration and experimental setup for a four-robot team. (a) Experimental setup and system configuration. Positions of the robot $\bm{r}_i$, the holding point $\bm{p}_i$ and the object $\bm{p}_o$ in $\mathcal{W}$. (b) The initial shape of the sheet in $\mathcal{W}_S$. Virtual cables $l_i$ is determined by the contact point $\bm{v}_o$.
• Figure 2: Three possible static equilibrium conditions for a five-robot formation. (a) $\mathcal{I}_t = \{ 1,2,3,4,5\}$ and $k=|\mathcal{I}_t|=5$. (b) $\mathcal{I}_t = \{ 1,2,3,4\}$ and $k=|\mathcal{I}_t|=4$. (c) $\mathcal{I}_t = \{ 1,2,4\}$ and $k=|\mathcal{I}_t|=3$.
• Figure 3: A robot system that both the initial sheet shape and robot formation are square. The taut cable number is four, i.e., $k=4$, but the number of maximum linearly independent equations for \ref{['eq_A11']} is $k_1=2$, that is, we only need to use the constraints of three independent taut cables ($k_1+1=3$). (a) Initial sheet shape and the contact point. (b) Robot formation. (c) System configuration.
• Figure 4: Example 1: Experimental and computational results for three FK solutions of the four-robot formation to handle an object with a deformable sheet.$\mathcal{V}_N^0=[-0.32 \; -0.42; \; 0.80 \; -0.38; \; 0.75 \; 0.71; \; -0.37 \; 0.66]^T$ m, and $\mathcal{R}_N=[0.21 \; 0.12; \; 0.80 \; 0.04; \; 0.90 \; 0.55; \; 0.44 \; 0.72]^T$ m. A straight line segment (taut cable) is formed when the plastic sheet is subject to tension and marked by a blue solid line. (a) The contact point $\bm{v}_o$ between the object and the sheet in $\mathcal{W}_S$. (b) The object position $\bm{p}_o$ and robot configuration $\mathcal{R}_N$ in $\mathcal{W}$. (c) The snapshots of experimental system configuration.
• Figure 5: Example 2: A case study for an eight-robot formation. Each sub-figure contains the robot formation configuration $\mathcal{R}_N$ and the object position $\bm{p}_o$ in $\mathcal{W}$ (left), and the contact point with the sheet $\bm{v}_o$ in $\mathcal{W}_S$ (right). The solid blue line represents the taut cable and the red solid points represents positions $\bm{p}_o$ and $\bm{v}_o$. The sheet shape is a regular octagon, and the radius of its circumscribed circle is $0.9$ m. The blue dotted line indicates the sub-formation of the robots involved in the handling the object. $\mathcal{R}_N=[0.5 \; 0;\; 0.35 \; 0.35; \; -0.05 \; 0.5; \; -0.35 \; 0.35; \; -0.50 \; 0; \; -0.30 \; -0.35; \; 0 \; -0.50; \; 0.35 \; -0.40]^T$ m. Taut cable sets $\mathcal{I}_t$ are (a) $\{4,5,8\}$. (b) $\{1,5,7,8\}$. (c) $\{3, 4,5, 8\}$. (d) $\{3,5,7,8\}$. (e) $\{4,5,6,8\}$. (f) $\{1,3,5,7,8\}$.

Theorems & Definitions (4)

• Lemma 1
• Proof 1
• Lemma 2
• Proof 2