# Predictable Trajectory Planning of Industrial Robots with Constraints
## 4 Experiment  
The proposed predictable trajectory planning method and joint velocity/acceleration limits avoidance schemes were validated in the following experiment. The experiment used a 6-DOF industrial robot Efort robot (ER3A-3C (HD)) produced by Anhui Efort intelligent equipment Co., Ltd. The link parameters of the Efort robot are shown in Table 1. The joint velocity limits given by the manufacture are presented in Table 2. The joint acceleration limit is 2 \(\times\) 10\({}^{4}\) (rad/s\({}^{2}\)). An actual machining process is performed on the Efort robot. The coordinates and time stamps of the via points on the machined path are listed in Table 3. The paths are shown in Figure 1. The desired Cartesian velocity is 800 mm/s. To reduce the vibration, the Cartesian velocities at the corner points are set at 200 mm/s. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(i\) & \(\mathbf{\alpha}_{i-1}\) & \(\mathbf{a}_{i-1}\) (mm) & \(\mathbf{d}_{i}\)(mm) & \(\mathbf{\theta}_{i}\) \\ \hline
1 & 0 & 0 & 0 & \(\theta_{1}\) \\
2 & \(-\pi/2\) & 50 & 0 & \(\theta_{2}\) \\
3 & 0 & 270 & 0 & \(\theta_{3}\) \\
4 & \(-\pi/2\) & 70 & 299 & \(\theta_{4}\) \\
5 & \(\pi/2\) & 0 & 0 & \(\theta_{5}\) \\
6 & \(-\pi/2\) & 0 & 0 & \(\theta_{6}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Link parameters of the Efort robot. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Joint** & **1** & **2** & **3** & **4** & **5** & **6** \\ \hline limits(rad/s) & 4.01 & 4.01 & 4.36 & 5.59 & 5.59 & 7.33 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The joint velocity limits of the Efort robot. Before the experiment, we can predict the trajectories. The robot will move along the path shown in Figure 1 with the end-effector constraints. There will be an acceleration zone, a constant-velocity zone and a deceleration zone in a normal block. In a short block, there will be only an acceleration/deceleration zone or a constant-velocity zone. After avoiding the joint limits, the end-effector will move the desired path. The motion time and velocities will be scaled. In the constant-velocity zones, the trajectories are generated by Equation (19). The trajectories in the acceleration and deceleration zones are generated by Equation (17). The corresponding results are shown in Figure 2. In the first acceleration zone, the velocity creeps up from 0 to 200 mm/s in 0.2 s. The velocity runs to 800 mm/s within 0.04 s. The total motion time grows from 6.18 s to 6.49 s with the joint avoidance scheme. Corresponding velocities are also changed. The joint trajectories are shown in Figure 3. As shown in Table 4, the maximum velocity and the maximum acceleration of joint 3 are 5.06 rad/s and 2.32 \(\times\) 104 rad/s2, respectively. Both of the values exceed their limits. The method of joint velocity/acceleration limits avoidance introduced in Section 2.2 is employed to adjust the joint velocity and acceleration trajectories. Actually, there are three areas where the joint velocities exceed their limits. These trajectories can be modified to comply with the velocity and acceleration constraints by scaling the motion time. Scaling the neighboring zones using the scaling factor defined in Equation (12), the motion time is updated by the scaling factor as in Equation (13). The maximum velocity and the maximum acceleration of joint 3 reduced to 4.36 rad/s and 1.85 \(\times\) 104 rad/s2. The maximum joint velocities and accelerations are shown in Table 4. Table 4 shows that most of the values are reduced and all values are in the allowable range. The modified joint trajectories are shown in Figure 3. To see the details clearly, the first modified zone is taken as an example, as shown in Figure 4. In this zone, the motion time (0.48-0.76) (s) is scaled to (0.48-0.85) (s). The velocity of the constant-velocity zone is reduced from 800 mm/s to 592.57 mm/s, as shown in Figure 2. The Cartesian paths marked red in Figure 1 are the joint modified zones. The Cartesian path marked green in Figure 1 is above the plane.