# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 3 Multi-Terminal Adaptive Motion
### Problem Description  
Since the exact position and motion of the object during the time-varying motion are uncertain, and the stiffness \(k\) of the object is unknown, completing time-varying internal force tracking directly from the terminal relative motion control is impossible. Therefore, in this paper, we propose an adaptive scheme for tracking the expected internal forces of multiple terminals.  
The internal command force is defined in Section 2.2. In the following derivation, the slave Terminal-i \((j=b,c,d)\) will track the relative position \({}^{hj}P_{rj}\) between its coordinate system and the master terminal Terminal-a, i.e., the reference position trajectory of the direction \(\overrightarrow{O_{hj}O_{ha}}\) to achieve the desired command internal force. The time-varying force tracking for this specified direction can be transformed into a virtual spring two-stage motion, as depicted in Figure 5\(p_{rje}\)\((j=b,c,d)\) is defined as the measurement position of the slave terminal in the \(\overrightarrow{O_{hj}O_{ha}}\) direction, and \(p_{rjo}\) is defined as the position of the initial contact point between the slave terminal and the object and the initial \(\overrightarrow{O_{hj}O_{ha}}\), \(p_{rjo}=p^{hje}\), \((j=b,c,d)\). Extracting the expected command internal force \(F_{rj}^{d}\), \((j=b,c,d)\) from the generalized command internal force vector \({}^{w}q_{me}\) in Equation (17), the internal force error can be expressed as \(\Delta F_{rj}=F_{rj}-F_{rj}^{d}\). In position control mode, assume \(p_{rje}^{d}=p_{rje}\), and the stiffness of an unknown object is \(k_{\rm e}\). Then, the internal command force can be expressed by  
\[F_{rj}=k_{\rm e}\big{(}p_{rjo}-p_{rje}\big{)} \tag{29}\]Equation (29) shows that if it is known that the reference position \(p_{rj^{\prime}}^{r}\) (\(j=b,c,d\)) of the direction \(\overrightarrow{O_{hj}O_{ha}}\) depends on the exact position of the object \(p_{rj^{\prime}}\), (\(j=b,c,d\)) and the exact stiffness of the object \(k\), then it can be calculated from the required and desired internal force \(p_{rj}^{r}\) (\(j=b,c,d\)).  
\[p_{rj}^{r}=p_{rjo}-\frac{F_{rj}^{d}}{k_{e}} \tag{30}\]  
However, the object's exact position during the handling process is uncertain in actual control. In Figure 5\(p_{rjo}\), (\(j=b,c,d\)) is the initial contact point position, and in the subsequent control process, it completely depends on the dynamic environment which is composed of the master terminal and the objects to calculate. The time-varying internal force is kept near the fixed value in this dynamic environment and thus fully considers the impact of the master terminal and objects' motions on the internal force. As a result of the stiffness of unknown objects \(k\), the error of the model parameters, the control accuracy, and the existence of uncertain factors (such as noise), the trajectory \(p_{rjo}\) (\(j=b,c,d\)) is uncertain, so it is impossible to directly calculate the relative motion reference position trajectory of the slave terminal.  
Note: The internal force workspace transformation in Section 2.2 proposed the direction of the internal command force, which meets the control requirements. In the same sense, each slave terminal coordinate system origin \(O_{hj}(j=b,c,d)\) tracks the desired position at the holding line \(\overrightarrow{O_{hj}O_{ha}}\) pointing to the origin of the master terminal coordinate system \(O_{ha}\). Therefore, next, we will focus on the one-dimensional variables at the holding line \(\overrightarrow{O_{hj}O_{ha}}\) of \(p_{hjo}\), (\(j=b,c,d\)) and \(p_{hjo}\), (\(j=b,c,d\)) in dynamic and uncertain environments.