# 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
Matrix \(C\) determined by Equation (14) will be named \(C_{e}\), the internal command force parameter. \[C_{e}= \begin{bmatrix}0.5I_{6}&-0.5I_{6}&0_{6}&0_{6}\\ 0.5I_{6}&0_{6}&-0.5I_{6}&0_{6}\\ 0.5I_{6}&0_{6}&0_{6}&-0.5I_{6}\end{bmatrix}  \tag{15}\]  
When \({}^{w}q_{me}\equiv \begin{bmatrix}{}^{w}h_{rb}^{T}{}^{w}h_{rc}^{T}{}^{w}h_{rd}^ {T}\end{bmatrix} ^{\mathsf{T}}\) is determined as a generalized command internal force vector, then Equation (14) becomes:  
\[{}^{w}q_{me}\equiv C_{e}*^{w}q_{l} \tag{16}\]  
If we determine the expected internal force set, the command internal force group is uniquely determined since we can perform the following conversion between Equations (13) and (16). \[ \begin{array}{l}C_{d}^{-}{}{}^{w}q_{md}={}^{w}q_{l}\\ {}^{w}q_{me}=C_{e}C_{d}^{-}{}^{w}q_{md}\end{array}  \tag{17}\]  
where \(C_{d}^{-}\) is the generalized inverse of \(C_{d}\). The transformation in Equation (17) is critical for introducing force-tracking adaptive control. The relationship between the expected and the command internal force in Equation (17) is represented in Figure 3, where the force applied by each terminal on the object is transferred to the origin of the object coordinate system through a virtual rod. \({}^{w}F_{r1},{}^{w}F_{r2},{}^{w}F_{r3}\) represents the intuitive expected internal force extracted from Equation (11) in \({}^{w}h_{r1},{}^{w}h_{r2},{}^{w}h_{r3}\), and \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) is the internal instruction force--the key control quantity in this article. Due to the four terminals in zero space, the combined external force \({}^{w}F_{o}\) applied to the object is controlled near a fixed value according to the internal force in changing the demand of the object's movement. It is easy for operators to regularly observe the internal force \({}^{w}F_{r1},{}^{w}F_{r2},{}^{w}F_{r3}\). Through observation, we conclude that it meets the demand for stable clamping of objects. However, as a control variable, achieving internal force-tracking tasks is difficult because a tracked end is a tracking end itself, which poses significant difficulties in the controller design. Due to the four terminals in zero space, the combined external force F applied to the object is controlled near a fixed value according to the internal force changing the demand of the object's movement. In order to solve such difficulties, this article proposes the instruction of the internal force \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) (extracted from \({}^{w}h_{r1},{}^{w}h_{r2},{}^{w}h_{r3}\) in Equation (14)) through Equations (11)-(17). As shown in Figure 3, \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) is the internal force between one terminal and the other terminals, which is crucial for force-tracking control. If this end is determined as the absolute-motion control object, it will solve the problem of the tracked end itself being a tracking end. This greatly reduces the uncertainty of control. As shown in Equation (17), \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) is uniquely determined when \({}^{w}F_{r1},{}^{w}F_{r2},{}^{w}F_{r3}\) is determined. In Section 3, we will track a fixed \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) on a one-dimensional holding line while determining \({}^{w}F_{r1},{}^{w}F_{r2},{}^{w}F_{r3}\).