# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 4 Experiments
### Soft Package Manipulation Experiment
#### 4.2.2 Internal Force Tracking Analysis  
The following experiments aimed to prove the effectiveness and superiority of the proposed method for arbitrary trajectory handling of large objects with unknown information in an arbitrary way using multi-terminal actuators. These experiments focused on simplification, transformation, and adaptive tracking methods for complex internal force spaces.  
Here, the operator observed the desired internal force \([^{w}F_{r1},^{w}F_{r2},^{w}F_{r3}]=[-7N,-5N,-24N]\), which satisfied the clamping motion requirement in a given intuitive sense and converted it into the desired commanded internal force \([^{w}F_{rb},^{w}F_{rc},^{w}F_{rd}]=[-7N,-13N,-18N]\) employing Equation (17), as presented in Section 4.2.1. The above operation is essential for multi-terminal arbitrary clamping and handling of unknown objects of arbitrary shape due to the reasonableness of the desired  
Figure 10: Relative motion error: (**a**) relative position trajectory error of each slave; and (**b**) relative direction trajectory error of each slave.  
internal force \([{}^{w}F_{r1},{}^{w}F_{r2},{}^{w}F_{r3}]=[-7N,-5N,-24N]\) enhancing the possibility of tracking the desired command internal force on the holding line \(\overrightarrow{O_{hj}O_{ha}}\), \((j=b,c,d)\). Failing the task because of slippage or insufficient internal force between the slave ends was also avoided. Figure 3 fully illustrates this point. In addition, the internal forces exerted on the object from the three slave terminals are shown in Figure 11a. When the reference position began to be adaptively updated in the direction of the hold line, the initial overshoot of each slave terminal was about 25%. The internal command force stabilized at the desired command internal force group \([-7N,-13N,-18N]\) in about 7 s. During the subsequent operation, the internal force was always controlled near the expected command internal force with the help of the reference trajectory. Note that the initial trajectory time was calculated from the beginning of the second stage of the entire experiment, and the internal force history was recorded from the first stage. The direction of \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) was along the holding line \(\overrightarrow{O_{hj}O_{ha}}\), \((j=b,c,d)\), and Figure 3 highlights that the force \({}^{w}F_{la}\) at the master end was time-varying. In this case, adaptively tracking the direction of \({}^{w}F_{rb},{}^{w}F_{rc},{}^{w}F_{rd}\) to the holding line \(\overrightarrow{O_{hj}O_{ha}}\), \((j=b,c,d)\) and the desired value under the constraint that the relative motion direction of the multiple slave to the master end remains constant is a unique contribution of this paper.  
Figure 11a illustrates the error response. Despite the uncertainty of the object stiffness and position, the tracking command internal force error of each slave was less than 1 N (6%). When adaptive control was disabled, the system's overshoot increased. One of the main goals of these controllers is to avoid force overshoot during the contact phase while maintaining internal force-tracking errors. In the following experiments, this phenomenon will become even more critical. Meanwhile, in this paper, we also compare the effectiveness and applicability of our proposed method with other types of control algorithms in force tracking of uncertain and unknown moving targets, especially in multi-terminal environments. With support from references [18; 19], as well as numerous similar studies, we constructed a comparative table (Table 2). From the table, it can be observed that the adaptive impedance control method based on reference trajectories established in this paper is more competitive in terms of the error response, parameter settings requirements, and computational complexity of the multi-terminal model.  
Figure 11: Internal force-tracking error: (**a**) history of the internal force group of the soft manipulation package; and (**b**) history of the internal force group of the manipulation container.