# Model-Based Adaptive Collaboration of Multi-Terminal Internal Force Tracking
## 1 Introduction  
Collaborative force closed collaboration can perform more complex and uncertain tasks in a wide range of application scenarios, such as rapid and stable handling or large object rescue in nonstructural hazardous environments [1, 2, 3, 4]. These cooperative systems form a closed operation chain and form a group of motion constraints through contact forces. Thus, the degree of freedom of the closed operation chain system is significantly reduced to form a task chain. During this process, the complex time-varying internal forces generated by the dynamic system formed by various force and motion transmission components (referred to as terminals in this article) and objects are crucial for the stability of the task. Effective control of the workspace is crucial. Due to the existence of motion uncertainty and force uncertainty, the controller design of closed cooperative systems with multiple terminals cooperating in force becomes more complex and unstable. At present, in order to solve the kinematics uncertainty when manipulating unknown objects in the range of dual terminals, researchers have proposed several solutions using advanced control theory. One example of this is the use of sliding mode control to improve the trajectory-tracking accuracy [5, 6]. An adaptive control scheme [7, 8] was proposed for the cooperative control of multi-arm robots in the case of unknown object information. The concept of absolute relative motion was proposed for the application scenario of grasping (force closure within the fixture) to move large objects [9, 10]. A time-varying internal force control scheme includes impedance control and hybrid position/force control. However, whether this is executed by adaptive control, proposed in [11; 12] to improve the time-varying force-tracking performance of objects with unknown stiffness, or by hybrid position/force control, proposed in [13; 14; 15; 16] with the decoupling task space as the position and force subspaces, to a certain extent, the scheme depends on the known environment information and kinematics model. A method was proposed in [17] to solve the time optimal path tracking problem of two robots executing collaborative grasping tasks. The time optimal path was determined by dynamic programming, and a task space admittance control scheme was used to generate a contact model. This method is suitable for grasping general objects in surface contact with robots. However, the problem of time-varying internal force tracking for uncertain targets has not been resolved [18]. After analyzing the end effector and environment of the robot, the contact force was used as the feedback force of a position-based impedance controller to actively track the desired dynamic uncertain environmental force. In order to reduce the force-tracking error caused by environmental position uncertainty, variable impedance control based on online adjustment of impedance parameters was proposed to compensate for the unknown environment and dynamic expected force. The introduction of variable-stiffness adaptive impedance control to adjust internal forces, which was proposed in an adaptive hybrid impedance control method for dual-arm robots [19], is the most valuable reference related to this work. However, variable-stiffness adaptive impedance control for robot terminals requires high control parameter settings, which can easily lead to system instability. These methods, which to some extent adapt to uncertain environments, improve the robustness of robot systems to uncertain dynamic environmental forces in collaborative force closed collaboration. However, such research is still in the exploration and improvement stage, and more experiments and methods are needed to improve its performance and applicability for numerous application scenarios. In both static- and weak-dynamic-force environments, the force-tracking performance of robots has been improved, enabling better tracking of target forces [20; 21; 22].