# Multi-Robot Trajectory Planning and Position/Force Coordination Control in Complex Welding Tasks
## 4 Symmetrical Adaptive Variable Impedance Control for Multi-Robot Coordination
### Symmetrical Adaptive Variable Impedance Control for Coordination Control  
A symmetrical adaptive variable impedance position/force coordination strategy is proposed to solve the influence of the unknown trajectory deviation which is caused by the external disturbance forces. The basic idea is to first decompose the resultant forces into the internal and external force which are exerting on the contact points of the operated object. And ideally, the desired internal and external force can be obtained. Consider the disturbance force which is caused by the trajectory deviation is dynamically changing, so the adaptive variable impedance is proposed to track the desired position and force. The first step is to decompose the internal and external force. The desired resultant force of the transfer robots need to exert on the contact point of the operated object is given as shown Equation (17). In order to achieve tracking the desired external force and internal force of the operated object by the transfer robots, Equation (17) can be expressed as a form of force exerting on the contact point of the operated object with transfer robots, it is shown as Equation (20). \[\mathbf{h}=\mathbf{h}_{E}+\mathbf{h}_{I}. \tag{20}\]  
where \(\mathbf{h}_{E}\) and \(\mathbf{h}_{I}\) represent the external force and internal force exerting on the contact point of the operated object with the transfer robots, respectively. They are meet the following equation. \[\left\{ \begin{array}{l}\mathbf{h}_{E}=\mathbf{W}^{\dagger}\mathbf{h}_{m}\\ \mathbf{h}_{I}=\mathbf{V}\mathbf{h}_{i}\end{array} \right.. \tag{21}\]  
According to the Equation (21), the proposed internal and external force can be known. In actual control, the wrench of the transfer robots exerting on the contact point of the operated object can be detected by a six-dimensional force/torque sensor which is installed at the end-effector of the robot, and the wrench can be denoted by \(\mathbf{h}_{r}\). It is further possible to decompose \(\mathbf{h}_{r}\) into the external forces and internal forces as shown in Equation (22). \[\left\{ \begin{array}{l}\mathbf{h}_{Er}=\mathbf{W}^{\dagger}\mathbf{W}\mathbf{h}_{r}\\ \mathbf{h}_{Ir}=(\mathbf{I}-\mathbf{W}^{\dagger}\mathbf{W})\mathbf{h}_{r}\end{array} \right., \tag{22}\]  
where \(\mathbf{h}_{Er}\) and \(\mathbf{h}_{Ir}\) represent the actual values of the external force and the internal force which are resolved based on the measured values by the six-dimensional force/torque sensor. In the second step, position/force coordination control is based on symmetrical adaptive variable impedance. The schematic diagram of multi-robot position/force coordination control based on a symmetrical adaptive variable impedance is shown in Figure 6.  
From Figure 6, a symmetric internal and external impedance coordination strategy is used for the transfer robots, and the position control strategy is used for the welding robot to just follow the object's motion. For the transfer robots, the inner impedance controller is composed of transfer robots and the operated object, and the outer impedance controller is composed of the operated object and the environment (external disturbance or external force caused by trajectory deviations). In the actual control, a symmetrical coordination method is adopted to convert the internal and the external force which is exerted on the contact point of the operated object at the end-effector of the single robot. The purpose of the inner impedance controller is to track the desired internal force and modify the movement trajectory of the end of the transfer robot based on the force deviation. The purpose of the outer impedance controller is to track the desired external force and modify the movement trajectory of the operated object according to the force deviation. Then the transfer robots and welding robot update the respective movement trajectories according to the corrected trajectory of the operated object.