# Cross-Coupled Contouring Control of Multi-DOF Robotic Manipulator
## 3 Stability Analysis  
The following notations are introduced to facilitate the stability analysis. \(\lambda_{m}\left(A\right)\) and \(\lambda_{M}\left(A\right)\) represent the smallest and the largest eigenvalues of a positive define matrix \(A\). If a square matrix \(A\) is positive definite, then it is denoted as \(A\succ 0\); If a square matrix \(A-B\) is positive definite, then it is denoted as \(A-B\succ 0\). For positive definite matrices, the following properties [21] will be used in this paper:  
1. If \(A\succ 0\), then \(A^{-1}\succ 0\); If \(A\geq B\succ 0\), then \(B^{-1}\geq A^{-1}\succ 0\). 2. If \(A\succ 0\) and \(B\succ 0\), then \(A+B\succ 0\), \(ABA\succ 0\), and \(BAB\succ 0\). According to P2, we have:  
\[ \begin{array}{ll}C\left(q,\dot{q}\right)\dot{q}_{d}&=C\left(q,\dot{q}_{d} \right)\dot{q}=C\left(q,\dot{q}_{d}\right)\dot{q}_{d}-C\left(q,\dot{q}_{d} \right)\dot{e}_{q}\\ &=C_{0}\dot{q}_{d}-C_{0}\dot{e}_{q}\end{array}  \tag{12}\]  
The dynamic model in Equation (1) can be rewritten as a function of the tracking errors in the following form. \[M\left(q\right)\ddot{e}_{q}+C\left(q,\dot{q}\right)\dot{e}_{q}+\left(K_{d}+ \overline{K}_{cd}\right)\dot{e}_{q}+\left(K_{p}+\overline{K}_{cp}\right)e_{q} =M\left(q\right)\ddot{q}_{d}+C\left(q,\dot{q}\right)\dot{q}_{d}+G\left(q \right)+F\left(\dot{q}\right) \tag{13}\]  
By applying Equation (12), Equation (13) is represented as the following simple form. \[M\ddot{e}_{q}+C\dot{e}_{q}+C_{0}\dot{e}_{q}+K_{dd}\dot{e}_{q}+K_{pp}e_{q}=\rho \tag{14}\]  
\[\text{with }\left\{ \begin{array}{l}K_{pp}=K_{p}+\overline{K}_{cp}\\ K_{dd}=K_{d}+\overline{K}_{cd}\\ \rho=M\ddot{q}_{d}+C_{0}\dot{q}_{d}+G\left(q\right)+F\left(\dot{q}\right)\end{array}  \right..\]  
It is assumed that the control gain matrices are positive definite. **Theorem 1**. : _For an \(n\)-DOF serial robotic manipulator described in Equation (1), if the proposed CC-PD control law in Equation (9) is applied to the system, the controlled robotic manipulator is globally stable for contouring control with bounded tracking errors, provided that the following conditions in Equation (15) are satisfied._  
\[\left\{ \begin{array}{l}\overline{K}_{d}\succ 0,\quad\overline{K}_{p}\succ M \succ 0\\ K_{pp}\succ 0,\quad K_{dd}\succ 0\\ K_{dd}+C_{0}-M\succ 0\\ \overline{K}_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0}\succ 0\\ \lambda_{m}\left(K_{dd}+C_{0}-M\right)>\frac{1}{2}\lambda_{M}\left(\overline{K }_{p}-K_{pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0}\right)>0\\ \lambda_{m}\left(K_{pp}\right)>\frac{1}{2}\lambda_{M}\left(\overline{K}_{p}-K_ {pp}+\overline{K}_{d}-K_{dd}+C^{T}-C_{0}\right)>0\end{array} \right. \tag{15}\]  
where \(\overline{K}_{p}\) and \(\overline{K}_{d}\) are user-defined symmetric positive definite matrices. Proof of Stability of CC-PD Controller  
To prove the stability of the proposed CC-PD contouring controller, the Lyapunov theorem [20; 22] is used by defining the following Lyapunov function candidate as:  
\[V=\frac{1}{2}\left[ \begin{array}{cc}e_{q}&\dot{e}_{q}\end{array} \right]^{T} L\left[ \begin{array}{c}e_{q}\\ \dot{e}_{q}\end{array} \right] \tag{16}\]  
where \(L=\left[ \begin{array}{cc}\overline{K}_{p}+\overline{K}_{d}&M\\ M&M\end{array} \right]\). As matrices \(\overline{K}_{p}\), \(\overline{K}_{d}\), and \(M\) are symmetric positive definite, then matrix \(L\) is also symmetric. From condition Equation (15) and property P5, it can be easily proven that:  
\[\left\{ \begin{array}{l}M^{-1}\succ\overline{K}_{p}^{-1}\succ 0\\ M^{-1}-\overline{K}_{p}^{-1}\succ 0\end{array} \right. \tag{17}\]  
Applying property P6 to Equation (17), we have:  
\[S=M-M\overline{K}_{p}^{-1}M\succ 0 \tag{18}\]  
From Equation (18), it can be seen that the Schur complement [21]\(S\) of matrix \(\overline{K}_{p}\) in matrix \(L\) is symmetric positive definite. According to the proposition introduced in [20; 22], it is demonstrated that matrix \(L\) is symmetric positive definite. In addition, the control gain matrix \(\overline{K}_{d}\) is symmetric positive definite.