# Multi-Robot Trajectory Planning and Position/Force Coordination Control in Complex Welding Tasks
## 4 Symmetrical Adaptive Variable Impedance Control for Multi-Robot Coordination
### Problem Formulation  
When the transfer robots are operating with a common object, the system becomes a closed-chain system. The closed-chain system which is constitute by the robots and the operated object is shown in Figure 4.  
In Figure 4, \(p_{i}(i=1,2)\), \(p_{m}\) denote the position vector of the robot end-effector with respect to the world coordinate system and the position vector of the operated object centroid with respect to the world coordinate system, respectively. \(r_{i}(i=1,2)\) represents the position vector of the center of mass which is relative to the coordinate system of the robot. The movement of the operated object can be described by the center of mass \(m\). And the Newton-Euler equation is shown as Equation (13). \[\left\{ \begin{array}{l}\mathbf{f}_{m}=\mathbf{M}\mathbf{c}-\mathbf{M}\mathbf{g}\\ \mathbf{n}_{m}=\mathbf{I}\mathbf{\omega}+\mathbf{\omega}\times\mathbf{I}\mathbf{\omega}\end{array} \right., \tag{13}\]  
where \(\mathbf{M}\) and \(\mathbf{I}\) are the mass and inertia matrices of the object, \(\mathbf{c}\) and \(\mathbf{\omega}\) are the position and the angular velocity vector of the object, \(\mathbf{g}\) is the acceleration of gravity, respectively. If the movement of the operated object is known (as shown in Equation (5)). According to Equation (13), the wrench which the transfer robots need to exert on the center of operated object can be obtained. The wrench diagram of the force exerting on the operated object is shown in Figure 5.  
The force formula of the transfer robots exerting on the operated object is shown in Equation (14). \[\left\{ \begin{array}{l}\mathbf{f}_{m}=\sum\limits_{i=1}^{2}\mathbf{f}_{i}\\ \mathbf{n}_{m}=\sum\limits_{i=1}^{2}\mathbf{n}_{i}+\sum\limits_{i=1}^{2}\mathbf{r}_{i} \times\mathbf{f}_{i}\end{array} \right. \tag{14}\]  
where \(\mathbf{f}_{i}\) and \(\mathbf{n}_{i}\) represent the force and moment of the \(i\)-th robot end-effector exerting on the operated object, respectively. According to the concept of \"virtual chain\", Equation (13) can be expressed in the form of Equation (15). Figure 4: The diagram of the closed-chain system. Figure 5: The wrench diagram of force exerting on the operated object. \[\mathbf{h}_{m}=\mathbf{W}\mathbf{h} \tag{15}\]  
where \(\mathbf{h}_{m}=\left[ \begin{array}{c}\mathbf{f}_{m}\\ \mathbf{n}_{m}\end{array} \right]\), \(\mathbf{h}=[\mathbf{f}_{1}{}^{T},\mathbf{n}_{1}{}^{T},\mathbf{f}_{2}{}^{T},\mathbf{n}_{2}{}^{T}]^{T}\), \(\mathbf{W}=\left[ \begin{array}{ccc}\mathbf{I}_{3}&\mathbf{0}_{3}&\mathbf{I}_{3}&\mathbf{0}_{3}\\ S(\mathbf{r}_{1})&\mathbf{I}_{3}&S(\mathbf{r}_{2})&\mathbf{I}_{3}\end{array} \right]\), \(\mathbf{S}(\mathbf{r}_{i})=\left[ \begin{array}{ccc}0&-\mathbf{r}_{iz}&\mathbf{r}_{iy}\\ \mathbf{r}_{iz}&0&-\mathbf{r}_{ix}\\ -\mathbf{r}_{iy}&\mathbf{r}_{ix}&0\end{array} \right]\), \(\mathbf{r}_{i}=[r_{ix}r_{iy}r_{iz}]^{T}\). In Equation (15), \(\mathbf{W}\) denotes the grip matrix and \(\mathbf{h}\) denotes the wrench matrix which is exerting on the contact point of the operated object. If the wrench is known, the wrench which is exerting on the centroid point of the operated object can be obtained according to the Equation (15). But the actual situation is the inverse problem of Equation (15), it can be attributed to the load distribution problem. From Equation (15), we can see that once the wrench \(\mathbf{h}_{m}\) at the center of the operated object is known, the wrench \(\mathbf{h}\) which the transfer robots need to exert on the contact point can be obtained by solving the pseudo-inverse matrix. Since \(\mathbf{W}\) is a row full rank matrix, theoretically there are infinitely many solutions for \(\mathbf{h}\). According to the conclusion in [20], the general form of the following equation is obtained.