# 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
Therefore, the force vector applied to the object by the eight terminal contact points \({}^{w}q_{h}\equiv\left[{}^{w}h_{h1}^{T}{}^{w}h_{h2}^{T}{}^{w}h_{h3}^{T}\dots \,{}^{w}h_{h8}^{T}\right]^{T}\) described in Section 2.1 becomes \({}^{w}q_{h}\equiv\left[{}^{w}h_{ha}^{T}{}^{w}h_{hb}^{T}{}^{w}h_{hb}^{T}{}^{w}h _{hd}^{T}\right]^{T}\), and accordingly, the force/moment vector at the tip of the virtual rod \({}^{w}q_{l}\equiv\left[{}^{w}h_{l1}^{T}{}^{w}h_{l2}^{T}{}^{w}h_{l3}^{T}\dots \,{}^{w}h_{l8}^{T}\right]^{T}\) becomes \({}^{w}q_{l}\equiv\left[{}^{w}h_{la}^{T}{}^{w}h_{lb}^{T}{}^{w}h_{lc}^{T}{}^{w}h _{ld}^{T}\right]^{T}\). Thus, Equation (7) becomes  
\[{}^{w}h_{o}={}^{w}h_{la}+{}^{w}h_{lb}+{}^{w}h_{lc}+{}^{w}h_{ld} \tag{10}\] \[{}^{w}h_{r1}={}_{c1,1}{}^{w}h_{la}+{}_{c1,2}{}^{w}h_{lb}+{}_{c1,3}{ }^{w}h_{lc}+{}_{c1,4}{}^{w}h_{ld}\] \[{}^{w}h_{r2}={}_{c2,1}{}^{w}h_{la}+{}_{c2,2}{}^{w}h_{lb}+{}_{c2,3}{ }^{w}h_{lc}+{}_{c2,4}{}^{w}h_{ld}\] \[{}^{w}h_{r3}={}_{c3,1}{}^{w}h_{la}+{}_{c3,2}{}^{w}h_{lb}+{}_{c3,3}{ }^{w}h_{lc}+{}_{c3,4}{}^{w}h_{ld}\]  
Defining the direction of the intuitive force while handling an object with a human hand is trivial. Therefore, the following internal force group is used as an intuitive force given to the expected internal force group:  
\[{}^{w}h_{r1}=\frac{1}{2}({}^{w}h_{la}-{}^{w}h_{lb}) \tag{11}\] \[{}^{w}h_{r2}=\frac{1}{2}({}^{w}h_{lc}-{}^{w}h_{ld})\] \[{}^{w}h_{r3}=\frac{1}{2}({}^{w}h_{la}+{}^{w}h_{lb})\ -\frac{1}{2}({}^{w}h_{lc}+{}^{w}h_{ld}\ )\]  
Thus, matrix \(C\) in Equation (11) is determined as the intuitive expectation value matrix \(C_{d}\) as follows:  
\[C_{d}= \begin{bmatrix}0.5I_{6}&-0.5I_{6}&0_{6}&0_{6}\\ 0_{6}&0_{6}&0.5I_{6}&-0.5I_{6}\\ 0.5I_{6}&0.5I_{6}&-0.5I_{6}&-0.5I_{6}\\ \end{bmatrix}  \tag{12}\]The simplified generalized internal force vector \({}^{w}q_{md}\equiv\left[{}^{w}h_{r1}^{T}{}^{w}h_{r2}^{T}{}^{w}h_{r3}^{T}\right]^{T}\) is the expected generalized internal force vector. Thus, Equation (11) becomes  
\[{}^{w}q_{md}\equiv C_{d}*^{w}q_{l} \tag{13}\]  
Although the expected internal force vector \({}^{w}q_{md}\) determined in Equation (13) is intuitive and in line with conventional thinking from the perspective of understanding, it significantly increases the difficulty of setting the parameters. It even makes it impossible to achieve favorable control from a control perspective. Therefore, we perform a favorable transformation of the internal force workspace appropriate for the control method proposed in this paper. Considering the robustness of the multi-terminal adaptive scheme introduced in this paper, it is necessary to determine one of the terminals and control its absolute motion to enable the other terminals to control the force tracking of the relative motion of this terminal. Note that the internal force vector does not contribute to the object's motion but represents the mechanical stress exerted on the object. There are infinite representations of internal force vectors as null space vectors. The time-varying complex internal force space is difficult to determine, and it is difficult to accurately control the collaborative task of manipulating unknown objects using multiple terminals. One of this paper's contributions is modelling the multiple terminals as an accurate controllable space. Based on the model in Equation (10), the selected internal force group is used as the control variables and converted to meet the control needs. In this paper, the desired internal force group in Equation (13) and the internal force group from the slave terminal position to the master terminal are converted in the following equation to achieve the control goal. The internal force group from the slave terminal to the master terminal is defined as follows:  
\[ \begin{array}{l}{}^{w}h_{rb}=\frac{1}{2}({}^{w}h_{la}-{}^{w}h_{Ib})\,,\\ {}^{w}h_{rc}=\frac{1}{2}({}^{w}h_{la}-{}^{w}h_{lc})\,,\\ {}^{w}h_{rd}=\frac{1}{2}({}^{w}h_{ld}+{}^{w}h_{ld})\,\,.\end{array}  \tag{14}\]  
Figure 2: Simplified system workspace.