# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 4 Experiments
### Soft Package Manipulation Experiment  
#### 4.2.1 Motion Control Analysis  
The soft package was an elastomer with an uneven and irregular surface which could be manipulated to any shape. The mechanical response to the applied force was nonlinear. The soft-body wrapping could quickly absorb and regenerate the force during the translation and rotation of multi-terminal gripping objects. The key poses for the entire experimental process are presented in Figure 6. In the first stage, the operator visually judged the shape of the object through remote operation commands and decided the initial clamping posture of the four terminals, ensuring that each contact point (3D biomimetic optical force sensor) had a contact force of about 1N with the object. Based on the first stage, the second stage triggered the proposed command internal force adaptive tracking algorithm. Each slave terminal actuator from the end effector was close to the master terminal along the direction of its hold line \(\overset{\wedge}{O_{hj}O_{hh}}\), \((j=b,c,d)\), until it adaptively obtained the desired command internal force. In the third stage, the object's specified position trajectory and direction trajectory were manipulated under the joint operation of absolute, relative, and reference trajectory control of unknown adaptive motion. The absolute motion position trajectory is depicted in Figure 7a, and the direction trajectory is illustrated in Figure 7b, revealing that the main end actuator trajectory was smoother, suggesting that the master terminal actuator tracked the desired trajectory well. The absolute position and direction errors are shown in Figure 8a,b. Due to the existence of  
Figure 7: Absolute motion control: (**a**) absolute position trajectory; and (**b**) absolute direction trajectory. unknown uncertainties, such as coordinate calibration error, numerical error and Kinematics uncertainty, the absolute error has a small and acceptable oscillation near zero. The projection of each slave relative position trajectory on the xz and yz planes of the robot's coordinate system is illustrated in Figure 9. The slave trajectory highlights that after triggering the adaptive tracking algorithm of the internal command force in the second stage, each reference trajectory was generated by adaptively estimating unknown parameters from the terminal; each slave terminal was close to the master terminal along the holding line, tracking the trajectory of the internal command force. At first, the holding line \(\overrightarrow{O_{hj}O_{ht}}\), \((j=b,c,d)\), which is the vector \({}^{hb}{}_{Prb}\), \({}^{hc}{}_{Prc}\), \({}^{hd}{}_{Prd}\) in Figure 5, varied and oscillated in a certain range of directions within its coordinate system. Thus, special attention should be paid to the fact that the internal forces traced on the holding line do not necessarily satisfy the desired internal forces, assuming an artificially given initial relative position between each terminal, including frictional requirements. Therefore, the expected value of the commanded internal force \({}^{w}F_{rb}\), \({}^{w}F_{rc}\), \({}^{w}F_{rd}\) directly given without considering the expected internal force \({}^{w}F_{r1}\), \({}^{w}F_{r2}\), \({}^{w}F_{r3}\) in Figure 3 was insufficient to meet the system stability requirements. If a reasonable expected internal force \({}^{w}F_{r1}\), \({}^{w}F_{r2}\), \({}^{w}F_{r3}\) is given according to Equation (17), it is converted to a reasonable expected value of \({}^{w}F_{rb}\), \({}^{w}F_{rc}\), \({}^{w}F_{rd}\), avoiding the above instability. \(\overset{\wedge}{k}\) and the unknown parameters introduced in Section 3 were estimated in a stable range around the holding line under reasonable desired commanded internal force conditions, with each slave end being under the control constraint of a constant relative position and the feedback force controller. Thus, the relative trajectory was well maintained with the help of the reference trajectory. The relative position and direction trajectory errors are depicted in Figure 10a,b.