# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 2 Model Establishment of a Multi-Terminal Clamping System
### Absolute Relative Motion Control  
Human hands can hold and handle unknown large objects of any shape in any way stably and flexibly. It is worth highlighting how complex time-varying internal forces and motion are regulated through perceptual intelligence. In this paper, we simulate the ability of humans to develop a multi-terminal robot control algorithm. In the above human process, a person perceives that any part of their palm is in contact with the object and can be consciously determined as the master terminal, while other parts of their palm in contact with the object can be determined as the slave terminal. Moreover, the slave terminal clamp (increase the internal force) to the master terminal is determined by consciousness to stabilize the clamping. Furthermore, it is important to keep the relative movement between the master terminal and the slave terminal within a small stable range (time-varying force and motion tracking). Similarly, the task space of the robot can be broadly divided into the combined motion of the absolute motion of the master terminal and the relative motion of the slave terminal. The absolute motion is performed by the master terminal and is designed to achieve accurate and robust tracking of the target reference trajectory. On the other hand, relative motion is conducted by the slave terminal, designed to accept the movement of the master terminal. Thus, the so-called closed kinematic chain constraints can be satisfied. The overall framework is shown in Figure 4; in this paper, \(\Sigma_{ha}\) presented in Figure 2 is identified as the master terminal, Terminal-a, responsible for absolute motion control. The other \(\Sigma_{hi}\), \((j=b,c,d)\) are slave terminals, named Terminal-j \((j=b,c,d)\), and are responsible for relative motion control. Figure 4: The overall framework. Figure 3: Relationship between the expected internal force and the commanded internal force. Figure 2 illustrates the relationship between each terminal and object motion in the robot coordinate system, which can be explained with the following equation:  
\[\left\{ \begin{array}{l}{{}^{w}P_{hi}}={{}^{w}P_{o}}-{{}^{w}R_{hi}}\ {{}^{hi}l_{i}}\\ {{}^{w}\dot{p}_{hi}}={{}^{w}p_{o}}-{{}^{\left({w}R_{hi}}\ {{}^{hi}l_{i}}\right)^{ \times w}\omega_{o}}\\ {{}^{w}\omega_{hi}}={{}^{w}\omega_{o}}\end{array} \right. \tag{18}\]  
where \({{}^{w}p_{hi}}(i=a,b,c,d)\) is the position vector of the simplified four terminals in the robot coordinate system, \((\cdot)^{\times}\) is the cross product executed by the oblique symmetry matrix operator, \({{}^{w}R_{hi}}(i=a,b,c,d)\) represents the transformation matrix of each terminal coordinate system relative to the robot coordinate system, \({{}^{w}p_{o}}\) and \({{}^{w}\omega_{o}}\) are the translation velocity and angular velocity of the object in the robot coordinate system, respectively, and \({{}^{hi}l_{i}}\) is the virtual rod represented by the terminal coordinate system. The relative position \({{}^{hi}p_{rj}}\) and orientation \({{}^{hi}Q_{rj}}\) of the three slave terminals, Terminal-j \((j=b,c,d)\), and the one master terminal, Terminal-a, in Equation (18) are expressed in the slave terminal coordinate system as follows:  
\[{{}^{hi}p_{rj}}={{}^{w}R_{hj}}{{}^{T}}\left({{}^{w}p_{ha}}-{{}^{w}p_{hj}}\right) \tag{19}\]  
\[{{}^{hi}Q_{rj}}={{}^{w}R_{hj}^{T}}{{}^{w}Q_{ha}} \tag{20}\]  
In an ideal state, there is no relative motion between each terminal, so the relative position \({{}^{hi}p_{rj}}\) and orientation \({{}^{hi}Q_{rj}}\) should remain constant. The robot coordinate system is the default and will be omitted in subsequent equations. At time t, the expected translation velocity and angular velocity corresponding to the object are expressed by \({{}^{d}_{o}}(t)\) and \({\omega_{o}^{d}}(t)\), respectively.