# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 2 Model Establishment of a Multi-Terminal Clamping System
### Formalizing the Workspace
The force vector generated at the top of \({}^{w}I_{hi}\) is defined as  
\[{}^{w}h_{li}\equiv\left[{}^{w}F_{li}^{\mathrm{T}}\ \ wN_{li}^{\mathrm{T}} \right]^{\mathrm{T}} \tag{1}\]  
where \({}^{o}F_{li}\) and \({}^{o}N_{li}\) are the force and moment applied at the tip of the virtual rod \(i\), respectively. The prefix \(w\) indicates that the vector is defined in the robot's coordinate system \(\Sigma_{w}\). The vector \({}^{w}I_{hi}\) is calculated from the force and moment applied to the object by the contact terminal \(i\). The sum of the combined forces on an object \({}^{w}F_{o}\) and moment \({}^{w}N_{o}\) is calculated from the sum of \({}^{w}I_{hi}\). \[{}^{w}h_{o}\equiv\left[{}^{w}F_{o}^{\mathrm{T}}\ \ wN_{o}^{\mathrm{T}} \right]^{\mathrm{T}}=\ G^{w}q_{l} \tag{2}\]  
where \(G\equiv[\mathrm{I}_{6}\ \mathrm{I}_{6}\ \mathrm{I}_{6}\ \ldots]\), \({}^{w}q_{l}\equiv\left[{}^{w}h_{I1}^{\mathrm{T}}\ \ wI_{I2}^{\mathrm{T}}\ \ wI_{I3}^{\mathrm{T}}\ \ldots\ ^{w}h_{ln}^{\mathrm{T}}\right]^{\mathrm{T}}\), and \(\mathrm{I}_{6}\) is a \(6*6\) identity matrix with a rank of \(6\) because the matrix \(G\) maps \(n*6\) dimensional vectors to six-dimensional  
Figure 1: Four terminals equipped with two optical bionic force sensor contact points cooperate to clamp and stably handle heavy objects of any shape. vectors. Therefore, its null space range is \((6*n-6)\)-dimensional, with \(n=8\) in this paper. We define V as the null space basis of \(G\). The general interpretation of Equation (2) is  
\[{}^{w}q_{l}=G^{-}\ {}^{w}h_{o}+\text{V}\ {}^{w}h_{m} \tag{3}\]  
where \({}^{w}h_{m}\) is an arbitrary \((6*n-6)\)-dimensional vector corresponding to V and \(G^{-}\) is the generalized inverse matrix of \(G\). Note that \(\text{V}^{w}h_{m}\) belongs to the null space of \(G\). Thus, \({}^{w}h_{m}\) corresponds to the internal force/moment that can be controlled separately without affecting the net force. In addition, V and \({}^{w}h_{m}\) are not fixed, which is important for the follow-up force-tracking control. Equation (3) can be rewritten as  
\[ \begin{split}{}^{w}q_{l}=[G^{-}\ \text{V}]\genfrac{[}{]}{0.0 pt}{}{w}{w}h_{m} \end{split} =\text{U}^{w}h\\ \text{U}\equiv\ [G^{-}\ \text{V}]\ \in\ \mathbb{R}^{6n+6n}, {}^{w}h\equiv\ \genfrac{[}{]}{0.0 pt}{}{w}{w}h_{m}\end{split}\ \in\ \mathbb{R}^{6n}. \tag{4}\]  
where \({}^{w}h\) is a generalized force vector. The internal forces \({}^{w}h_{m}\) can be expressed as a \((n-1)6\) dimensional force vector set:  
\[{}^{w}h_{m}\equiv \begin{bmatrix}{}^{w}h_{r1}\\ {}^{w}h_{r2}\\ \vdots\\ {}^{w}h_{r(n-1)}\end{bmatrix}  \tag{5}\]  
For a given \({}^{w}q_{l}\), the force/moment vector \({}^{w}h\) is obtained by solving Equation (4). \[{}^{w}h=\text{U}^{-1}\ {}^{w}q_{l} \tag{6}\]  
Next, we introduce stable clamping and handling with multi-contact points in inner and outer working spaces, utilizing the controlling vectors and the required coordinate systems both simultaneously and independently. The physical meaning of this setup is summarized as follows. Matrix \(G\) maps the forces exerted by the multiple terminals and contact points to the external forces of the object. Matrix V is the null space basis of \(G\), and matrix U maps a set of external and internal forces to the forces of multi-terminal contact points.