# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 2 Model Establishment of a Multi-Terminal Clamping System
### Simplification and Transformation of the Internal Force Space  
Next, we discuss the stable clamping internal force of the multi-point-contact collaborative handling system proposed in this paper. When more than two contact points conduct a collaborative clamp, it is difficult to determine the null space basis V proposed in Section 2.1 because V does not represent the intuitive meaning of the internal force/moment. Hence, we write Equation (6) as follows:  
\[ \begin{split}{}^{w}h_{o}={}^{w}h_{I1}+{}^{w}h_{I2}+\ \dots\ +{}^{w}h_{I8}\,\\ {}^{w}h_{r1}=c_{1,1}{}^{w}h_{I1}+c_{1,2}{}^{w}h_{I2}+\ \dots\ +c_{1,n}{}^{w}h_{I8}\,\\ {}^{w}h_{r2}=c_{2,1}{}^{w}h_{I1}+c_{2,2}{}^{w}h_{I2}+\ \dots\ +c_{2,n}{}^{w}h_{I8}\,\\ \vdots\\ {}^{w}h_{r7}=c_{7,1}{}^{w}h_{I1}+c_{7,2}{}^{w}h_{I2}+\ \dots\ +c_{7,8}{}^{w}h_{I8}.\end{split}  \tag{7}\]  
Thus, Equation (6) becomes  
\[{}^{w}h= \begin{bmatrix}{}^{w}h_{o}\\ {}^{w}h_{m}\end{bmatrix} = \begin{bmatrix}G\\ \text{C}\end{bmatrix} \ {}^{w}q_{l}=\text{U}^{-1}\ {}^{w}q_{l} \tag{8}\]Once the seven internal force vectors \({}^{w}h_{r1}\dots\,{}^{w}h_{r,7}\) are given, matrix \(C\) is determined automatically. However, under the same net force requirement, the internal force vector \({}^{w}h_{r1}\dots\,{}^{w}h_{r,7}\) has an infinite number of values. This paper determines the internal force direction of stable clamping according to the terminal model. Thus, the desired internal force for stable clamping is adaptively tracked under this set of internal force directions. Therefore, we extract the internal force vector Equation (9) from Equation (8):  
\[{}^{w}h_{m}=C^{w}q_{l} \tag{9}\]  
In fact, the workspace formula represented by Equations (1)-(9) can merge and decompose the force vectors at the contact points based on the task and model degrees of freedom. Therefore, in order to reduce the computational complexity and simplify the system, we can simplify the internal force workspace based on the degree of freedom of the terminal model and the characteristics of the task (this article does not involve tasks involving local or individual \(\Sigma_{hi}(i=1,\dots,n)\) contact objects). According to the method we want to introduce in this paper, without loss of generality, we make the following reasonable assumptions:  
1. No torque vector is formed at a single contact point. 2. No internal force is generated between two parallel contact points on a terminal. 3. No internal torque is generated between all contact points, so only internal forces are involved in the control based on reference trajectories. Based on the above assumptions, the initial internal force workspace becomes a four-contact terminal force vector workspace. According to the system model depicted in Figure 2, the coordinate systems at the contact points \(\Sigma_{h1}\) and \(\Sigma_{h2}\) are merged into \(\Sigma_{hi}\). The new coordinate direction remains unchanged, while the origin changes to \(O_{ha}\) at the midpoint of the connected line between \(O_{h1}\) and \(O_{h2}\). Furthermore, as \(\Sigma_{ha}\) is a new cooperation terminal, its force vector is determined by the force vector combination at the contact point of \(\Sigma_{h1}\) and \(\Sigma_{h2}\). We merge \(\Sigma_{h3}\) and \(\Sigma_{h4}\) to form \(\Sigma_{hb}\); \(\Sigma_{h5}\) and \(\Sigma_{h6}\) to form \(\Sigma_{h6}\); and \(\Sigma_{h7}\) and \(\Sigma_{h8}\) to form \(\Sigma_{hd}\) in the same way. As presented in Figure 2, the force vector is simplified similarly.