# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 5 Conclusions  
In this paper, a multi-terminal actuator adaptive cooperation method is proposed to manipulate unknown objects for a system with closed-chain motion and dynamic uncertainty of the multi-arm, multi-terminal actuator and multi-contact sensing model. Drawing on the intelligent behavior of human clamping and handling of unknown objects, simplification of and conversion to effective control is established for the complex time-varying internal force working space during multi-terminal collaborative clamping and handling of unknown objects. Based on Lyapunov theory, an algorithm for multiples slaves to estimate the object stiffness and motion uncertainty in the direction of the internal command force is proposed. Moreover, impedance control is used to generate a reference trajectory for multi-slave ends, maintaining the desired command internal force on a reasonable holding line to track the motion of the master terminal. In this way, the stability and accuracy of the entire control system are realized. The proposed method is simple, stable, and robust to an object's shape, position, and stiffness changes. Figure 14: (**a**) Position trajectory error of b slave terminal in container translation. (**b**) Position trajectory error of c slave terminal in container translation. (**c**) Position trajectory error of d slave terminal in container translation. Experiments on a two-arm robot system with multi-terminal, multi-contact sensing points verified that the proposed control method shows good position and force-tracking performance when the multi-terminal actuator clamps manipulate unknown objects of any shape or stiffness in any way. The specific results, such as experimental data, show that the relative position oscillation amplitude of each slave position \(p_{hb},p_{hc},p_{hd}\) under the reference trajectory of motion estimation after tracking the expected command internal force to a stable position is less than 1 mm (except for in the case of a violent change in the direction of motion of the main end) and the directional oscillation amplitude is less than 1 degree. For objects with a low stiffness, the internal force error of each command from the terminal is less than 1 N (6%). Additionally, the internal force error of the tracking command for objects with a high stiffness is less than 2 N (8%). In addition, the method proposed in this article focuses on the collaborative force closure collaboration of multiple terminals (rather than the internal force closure collaboration of fixtures) and is limited to the model being formulated with eight sensing contact points and four terminals. However, the proposed algorithm can be extended to the collaborative force closure collaboration of \(n\) multiple sensing contact points (tactile research) and \(n\) multiple terminal actuators. Future work will focus on developing anthropomorphic robots with multiple terminals, utilizing self-adjusting expected internal forces based on sliding perception and more functional and nuanced models, and manipulating objects with complex and variable external forces. Accordingly, the new algorithms for these purposes are expected to have more requirements and lead to more difficulties. Conceptualization, Z.W. ; methodology, Z.W. and J.D. ; software, Z.W. ; validation, F.S. ; writing--original draft preparation, Z.W. and F.S. ; writing--review and editing, J.D. ; supervision, J.D. ; funding acquisition, J.D. All authors have read and agreed to the published version of the manuscript. This research was funded by the National Natural Science Foundation of China, grant number 62275066. Not applicable. The authors would like to gratefully acknowledge the National Natural Science Foundation of China and would like to thank the editors and reviewers for their valuable comments and constructive suggestions.