# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 4 Experiments
### Manipulating Containers  
In practice, the object stiffness varies significantly depending on the materials. To prove that the proposed method is suitable for anthropomorphic clamping and handling of objects of any stiffness and shape, a container weighing up to 3 kilograms containing debris was manipulated. Unlike the soft package, the container's surface was hard, flat, and conical in shape. The high stiffness of the container meant that adaptive force tracking became more difficult.  
To demonstrate the environmental adaptability and user-friendliness of the proposed method, all control parameters were the same as in the previous experiments. Since the container was loaded with debris and the container's surface was smooth, the model contact point was not enough to meet the friction required by rotation. Note that only the translational operation of the previous experiments' position trajectory was executed in this experiment. If there are enough contact points, the arbitrary rotation task can be completed. The key poses of this experiment are depicted in Figure 12.  
The corresponding expected internal force was \([^{w}F_{r1}{}^{w}F_{r2}{}^{w}F_{r3}]=[-13N,-5N,-36N]\), and the expected internal command force group was \([^{w}F_{rb},^{w}F_{rc},^{w}F_{rd}]=[-13N,-22N,-27N]\), with historical records illustrated in Figure 11b. These records infer that the adaptive process was longer than the specified internal force-tracking error and the time amplitude was larger. The tracking command internal force error of each slave was less than 2 N (8%).  
The absolute position trajectory is depicted in Figure 7a. Similarly, the relative position trajectory and position tracking error are presented in Figures 13 and 14, respectively. Note that due to the translational manipulation, there was no significant deviation from the relative position after the internal force stabilized, as shown in Figure 10b. However, due to the high stiffness and smooth surface, the position fluctuated greatly and tracking the expected command internal force in the initial stage required more time.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Methods**} & **Steady-State** & \multirow{2}{*}{**Overshoot**} & \multirow{2}{*}{**Setting Time**} & **Requirements for** & **Applicability to** \\  & & & & & **Stability Parameters** & **Multi-Terminal Collaboration** \\ \hline Adaptive impedance control & & & & & \\ based on a reference & \textless{}1\text{N}\) & 15\% & 5 s & Low & Strong \\ trajectory & \textless{} & 76\% & 36 s & Non-convergence & \textless{} \\ Constant impedance control & \textless{}8 N & 43\% & 16 s & Non-convergence & \textless{} \\ Variable impedance control & \textless{}3 N & 32\% & 13 s & High & Weak \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison of force-tracking performance in dynamic and uncertain environments.  
Figure 12: Key poses of translational high-stiffness container.  
So far, the effectiveness and accuracy of the proposed adaptive reference trajectory method have been proven. This method allows for manipulating unknown objects with arbitrary shapes through multi-terminal collaboration, which tracks the expected command internal force and relative motion control algorithm in the specified internal force direction. Compared with pure motion control and constant impedance control, ineffectiveness in manipulating unknown objects has been demonstrated in [13], and thus no comparative experiments will be conducted in this paper. Thus, we conclude that:  
1. The proposed method effectively extracts and regulates holding lines for complex internal force spaces during multi-terminal force closure collaborations and maintains contact forces.  
Figure 13: The xz plane and yz plane projection of the container translational relative motion trajectory.  
2. The motion estimation in the proposed multi-terminal collaboration method is important for dealing with multi-terminal time-varying motion.