## 3 Excitation Trajectory Generation  
The measurement method of the non-disassembly experiment needs to drive the robot to obtain the required data. Considering the calculation accuracy and the experimental feasibility, the collaborative robot trajectory, especially the excitation trajectory, must fulfill various conditions, such as maximum moving range, speed, and the acceleration of each joint. A six-axis robot has a highly complex dynamics calculation, and thus needs to be simplified. The identification of the dynamic parameters for each joint can be done individually, that is, when identifying the dynamic parameters of the last three joints, the other three joints can be locked. As a result, the difficulty of identification is greatly reduced and more targeted. The analysis, calculation, and experimental procedure of the first three joints is the same as the last three joints and can be done separately. Limited to this research process, this paper only focuses upon the dynamic parameters of the typical last three joints of the collaborative robot. In the experiment, the first five terms of the Fourier series, namely Formula (6), are used as the excitation trajectory. Because it is a periodic function, it is convenient to repeat this experiment as many times as needed. \[q_{i}(t)=q_{i,0}+\sum_{k}^{5}\Big{(}a_{i,k}\sin\Big{(}k\omega_{f}t\Big{)}+b_{ i,k}\cos\Big{(}k\omega_{f}t\Big{)}\Big{)} \tag{6}\]  
where \(q_{i}(t)\) is the \(i\)th joint trajectory function of time \(t\), \(\omega_{f}\) is the fundamental frequency, \(q_{i,0}\) is a constant offset. Each trajectory contains 11 parameters, namely \(q_{i,0}\), \(a_{i,k}\), \(b_{i,k}\), and \(\omega_{f}\) = 1 (\(i\) = 1, 2, 3, 4, 5). The quality of the excitation trajectory is closely related to the ability of the observation matrix to suppress noise and morbidity, which directly affects the accuracy of the identification parameters. Here, the objective function is defined as the condition number of the observation matrix \(\mathbf{W}\):  
\[J=\text{Cond}(\mathbf{W}) \tag{7}\]  
where \(\text{Cond}(\mathbf{W})=\sigma_{max}(\mathbf{W})/\sigma_{\text{min}}(\mathbf{W})\), \(\sigma_{max}(\mathbf{W})\) and \(\sigma_{min}(\mathbf{W})\) are the maximum and minimum singular value of \(\mathbf{W}\), respectively. The smaller the value of the objective function, the lower the sensitivity of the identification parameters to the measurement error is, and the higher the accuracy of parameters identification achieved by the experiment can be. Meanwhile, it is also necessary to consider the limitations of the actual robot performance. If the moving range, velocity, acceleration, and the status of start and stop points are taken as constraints, the optimization problem of the excitation trajectory can be described as:  
\[\left\{ \begin{array}{l}\min(J)\\ q_{min}\leqq_{i}(t)|\leq q_{max}&\forall i,t(a)\\ \dot{q}_{min}\leqq\dot{q}_{i}(t)|\leq\dot{q}_{max}&\forall i,t(b)\\ \ddot{q}_{min}\leqq\dot{q}_{i}(t)|\leq\ddot{q}_{max}&\forall i,t(c)\\ q_{i}(t_{0})=q_{i}(t_{M})=0&\forall i,t(d)\\ \dot{q}_{i}(t_{0})=\dot{q}_{i}(t_{M})=0&\forall i,t(e)\\ \ddot{q}_{i}(t_{0})=\ddot{q}_{i}(t_{M})=0&\forall i,t(f)\end{array} \right. \tag{8}\]  
where \(q_{min}\), \(\dot{q}_{min}\), \(\ddot{q}_{min}\), \(q_{max}\), \(\dot{q}_{max}\) are the boundary values of angle, speed and acceleration of each joint, respectively. \(t_{0}\), \(t_{M}\) are the start and end times of each period, respectively. Since the excitation trajectories are Fourier series, both periodicity and continuity are ensured, and the constraints at the end time \(t_{M}\) can also be omitted from the actual optimization calculation. Equation (8) is a nonlinear, multi-constrained optimization problem, which can be optimized using NSGA and other target optimization algorithms [19]. The optimized excitation trajectory is shown in Figure 3. As evident in Figure 3, the shift of the 4 th, the 5 th, and the 6 th joint smoothly change with time.