# 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
Considering that the desired internal forces, external forces, and the uncertain wrench which is caused by external disturbances are time-varying functions, an adaptive variable impedance control strategy is adopted. Therefore, the system is divided into an adaptive variable impedance inner loopcontroller and an adaptive variable impedance outer loop controller. The diagram of symmetrical internal and external adaptive impedance control is shown in Figure 7.  
As shown in Figure 7, the desired movement trajectory of the operated object which inputs by the system is \(\mathbf{T}_{md}(t)\). In the ideal conditions, it exists \(\mathbf{T}_{md}(t)=\mathbf{T}_{mr}(t)\). According to the closed-chain constraint conditions, the trajectories (\(\mathbf{T}_{mr1}(t)\), \(\mathbf{T}_{mr2}(t)\), \(\mathbf{T}_{mr3}(t)\)) of multi-robot can be obtained, where \(\mathbf{T}_{mr1}(t)=\mathbf{T}_{r1}(t)\), \(\mathbf{T}_{mr2}(t)=\mathbf{T}_{r2}(t)\), \(\mathbf{T}_{mr3}(t)=\mathbf{T}_{r3}(t)\). When uncertainty forces caused by the trajectory deviations exist, they can be obtained by the two six-dimensional force/torque sensor which are denoted by \(\mathbf{h}_{r1}\) and \(\mathbf{h}_{r2}\). According to Equation (22), the measurement of the resultant forces can be decomposed into external forces(\(\mathbf{h}_{E1}\), \(\mathbf{h}_{E2}\)) and internal forces(\(\mathbf{h}_{r1  
Figure 6: The schematic diagram of multi-robot position/force coordination control. Figure 7: The diagram of symmetrical internal and external adaptive impedance control. When there is a force deviation between the measured external force and the desired external force \(\mathbf{h}_{Ed}\), the system will obtains an error trajectory \(\mathbf{\sigma}_{m}(t)\) through the adaptive variable impedance external loop controller, then the corrected trajectory \(\mathbf{T}_{mr}(t)\) of the operated object is obtained, and the trajectory of each robot is further corrected. When there is a force deviation between the measured internal force and the desired internal force (\(\mathbf{h}_{Id1}\) and \(\mathbf{h}_{Id2}\)), the system obtains error trajectories (\(\mathbf{\sigma}_{1}(t)\) and \(\mathbf{\sigma}_{2}(t)\)) through the adaptive variable impedance inner loop controller, and then transfer robots's correction trajectories (\(\mathbf{T}_{r1}(t)\) and \(\mathbf{T}_{r1}(t)\)) can be obtained. The adaptive variable external impedance law is proposed as Equation (23), as shown in Figure 7.  
\[\left\{ \begin{array}{l}\mathbf{M}[\mathbf{T}_{mr}(t)-\mathbf{T}_{md}(t)]+[\mathbf{B}+\Delta \mathbf{B}(t)][\mathbf{T}_{mr}(t)-\mathbf{T}_{md}(t)]=\mathbf{h}_{Er1}(t)+\mathbf{h}_{Er2}(t)-\mathbf{h }_{Ed}(t)\\ \\ \Delta\mathbf{B}(t)=\frac{\mathbf{B}}{\mathbf{T}_{mr}(t)-\mathbf{T}_{md}(t)}\Phi(t)\\ \\ \Phi(t)=\Phi(t-\lambda)+\sigma\frac{\mathbf{h}_{Idi}(t)-\mathbf{h}_{Iri}(t)}{\mathbf{B}_{I }}\end{array} \right. \tag{23}\]  
\[\left\{ \begin{array}{l}\mathbf{M}_{I}[\hat{\mathbf{T}}_{ri}(t)-\hat{\mathbf{T}}_{mri}(t) ]+[\mathbf{B}_{I}+\Delta\mathbf{B}_{I}(t)][\hat{\mathbf{T}}_{ri}(t)-\hat{\mathbf{T}}_{mri}(t)] =\mathbf{h}_{Iri}(t)-\mathbf{h}_{Idi}(t)\\ \\ \Delta\mathbf{B}_{I}(t)=\frac{\mathbf{B}_{I}}{\mathbf{T}_{ri}(t)-\mathbf{T}_{mri}(t)}\Phi(t)\\ \\ \Phi(t)=\Phi(t-\lambda)+\sigma\frac{\mathbf{h}_{Idi}(t)-\mathbf{h}_{Iri}(t)}{\mathbf{B}_{I }}\end{array} \right. \tag{24}\]  
where \(\mathbf{M}\) is the desired inertia matrix, \(\mathbf{B}\) is the damping matrix, \(\lambda\) is the sampling period of the controller and \(\sigma\) is the update rate. The adaptive variable internal impedance law showns in Equation (7) is expressed as Equation (24). Where \(\mathbf{M}_{I}\) denotes the desired inertia matrix of the internal wrench, \(\mathbf{B}_{I}\) denotes the damping matrix of the internal wrench, and \(i\) denotes \(i\)-\(th\) robot. In our previous work [28], the adaptive variable impedance control has been proven stable and convergent and been used to track the dynamic force with unknown trajectory deviations.