# Multi-Robot Trajectory Planning and Position/Force Coordination Control in Complex Welding Tasks
## 1 Introduction
The master and the slave arm should've meet the certain constraints. The master arm is controlled by the position control mode, and the slave arm follows the motion trend of the master arm detected by the force/torque sensor mounted at the wrist of the slave arm. The master/slave force control approach for the coordination of two arms which carries a common object cooperatively was proposed in [8; 9], and the necessity of force control for cooperative multiple robots is also pointed out. However, the slave arm needs to have a fast following response [10], otherwise it will lead to system instability. Force/torque sensor and position controller are difficult to achieve high-speed response in actual control systems. Therefore, the master-slave control strategy is only suitable for low-speed applications; (2) Hybrid motion/force control. The basic idea is that two arms work equally and coordinated by the centralized control. The position control is used in the free space, and the force control is used in the constrained space, such as [11; 12; 13; 14]. A difficulty in implementing the hybrid control law in rigid environment is knowing the form of the constraints active at any time. And it also needs to sacrifice some performance by choosing low feedback gains, which makes the motion controller \"soft\" and the force controller more tolerant of force error; (3) Synchronization control. The basic idea is to track the desired trajectory which is generated by the desired force based on the dynamic model of the manipulators. The control problem is formulated in terms of suitably defined errors accounting for the motion synchronization between the manipulators involved in the cooperative task. The concept of motion synchronization was used in [15; 16]. An adaptive control strategy was adopted to track the desired trajectory, ensuring the synchronization position errors converged to zero. In addition, intelligent control strategies were also used in the coordination control of nonlinear cooperativemanipulator systems [17; 18; 19]. However, the synchronization control is based on the dynamic model. Although the synchronization error at the end effector of the manipulator was considered, due to the difficulties in dynamic modeling, over-complexity control model, strong coupling and nonlinearity, it has not been applied in most actual control systems; (4) Impedance control. The basic idea is to achieve the adjustment of the position based on the force error. Impedance control is a stable and effective method widely used in many fields including coordination. The coordination strategy of the object based on impedance control was studied in [20; 21; 22; 23; 24]. The force acting on the object was decomposed into the external force that contributes to the object's motion and the internal force by the end-effector of both arms. Following the guidelines in above references, the external impedance and the internal impedance were combined in a unique control framework [25]. Compared with the first three control strategies, impedance control overcomes the shortcomings of the above control methods, and can effectively control the internal and external force. In contrast, impedance control has been widely used in multi-robot position/force coordination control. However, a closed-chain system is often subject to external dynamic disturbances and calibration errors in the actual industrial system. These factors can cause time-varying trajectory deviations at the end-effector, and time-varying trajectory deviations can cause unknowns and dynamically changing external forces and internal forces between multi-robot and the operated object. Most of the current impedance control methods that used in the above studies are constant, and the presented control strategies are not feasible for the unknown and dynamically changing trajectory errors. To the best of our knowledge, no research has been reported to solve the trajectory deviation problem during the multi-robot coordination with actual industrial robotic systems.