introduction
As a crucial component of automated equipment, robotic arms are widely used in various fields such as industrial production, medical services, and military technology due to their operational flexibility. For different working environments, specific trajectory planning is required for each joint of the robotic arm to obtain the position and pose of its end effector. Therefore, accurate and rapid trajectory planning for robotic arms is particularly important. Trajectory tracking in the joint space of a robotic arm involves controlling variables such as the position, velocity, and acceleration of each joint to make the robotic arm move along the desired trajectory.
3. Experimental Research
3.1 Experimental Platform and Model Establishment
The Cyton II robotic arm, the subject of this study, is a six-degree-of-freedom robotic arm with six rotational joints. Its three-dimensional model is shown in Figure 1. The coordinate system diagram established according to the DH method is shown in Figure 2.
Figure 1. 3D model of Cyton II robotic arm
Figure 2. Schematic diagram of the coordinate system of the robotic arm
Table 1. Parameters of Robotic Arm Linkages
link | Angle variable θ n | link Spacing d n | Link length a n | Linkage torsion angle α |
Link 1 | θ 1 | d 1 | 0 | 0° |
Link 2 | θ 2 | 0 | a 2 | 90° |
Link 3 | θ 3 | d 3 | 0 | 0° |
Link 4 | θ 4 | 0 | a 4 | 90° |
Link 5 | θ 5 | d 5 | 0 | 0° |
Link 6 | θ 6 | 0 | 0 | 90° |
3.2 Steps for applying genetic algorithms
(1) Input a set of model points as initial values, including the starting point and ending point of the robotic arm, and require the coordinates of the path points.
(2) Define the fitness function
This paper uses a fourth-order Bezier curve fitting to randomly generate N control points, and simultaneously generates N-2 node vectors according to the given number of model points for error analysis. The number of control vertices is generally four unless otherwise specified. This paper uses the least squares method to calculate the error, i.e., minimizing the distance between the nodes of the approximating curve and the user-specified model points. The distance generated by the accumulated error between the coordinates of the m Bezier curve nodes and the actual tracked curve is...
(3) Determine the operation strategy and population size M, crossover probability Pc, and mutation probability Pm. In this paper, Pc=0.5 and Pm=0.1.
(4) Generate an initial population, calculate the fitness of individuals in the initial population, perform statistics on the fitness, and retain the optimal solution.
(5) Recombining individuals with high fitness using methods such as crossover and mutation to generate new populations.
(6) Return to step (4) to calculate the individual fitness of the new population.
(7) If the calculated population fitness function converges, then stop the herd; otherwise, continue to generate new populations. The maximum number of generations is expected to be Tmax=100.
(8) Use Matlab to plot the experimental results.
3.3 Simulation Experiment
When the end effector of the robotic arm moves from one point to another in Cartesian coordinate space, the shape points of each joint can be obtained through inverse kinematics of the robotic arm, using the known starting point, ending point, and intermediate points along the trajectory. The coordinates of each shape point in this paper are shown in Table 2 below.
Table 2 shows the numerical values of the nodes.
Node (rad) | Joint 1 | Joint 2 | Joint 3 | Joint 4 | Joint 5 | Joint 6 | |
1 | -0.4488 | -0.6283 | 1.0472 | -1.5708 | 0.5236 | 0.7854 | |
2 | -0.4360 | -0.6185 | 1.0421 | -1.5540 | 0.5124 | 0.7652 | |
3 | -0.3622 | -0.5616 | 1.0125 | -1.4571 | 0.4478 | 0.6489 | |
4 | -0.2048 | -0.4405 | 0.9496 | -1.2506 | 0.3101 | 0.4011 | |
5 | 0.0261 | -0.2627 | 0.8572 | -0.9475 | 0.1081 | 0.0374 | |
6 7 | 0.2992 0.5723 | -0.0524 0.1579 | 0.7480 0.6388 | -0.5890 -0.2306 | -0.1309 -0.3699 | -0.3927 -0.8228 | |
8 9 10 11 | 0.8032 0.9605 1.0344 1.0472 | 0.3357 0.4569 0.5137 0.5236 | 0.5464 0.4835 0.4539 0.4488 | 0.0725 0.2790 0.3759 0.3927 | -0.5719 -0.7096 -0.7742 -0.7854 | -1.1865 -1.4343 -1.5506 -1.5708 | |
Figures 3 through 8 show the simulation curves of displacement, angular velocity, and angular acceleration of the six joints of the robotic arm. The simulation results demonstrate that the method proposed in this paper enables the robotic arm to operate stably with a smooth and continuous trajectory.
Figure 3 shows the angular displacement, angular velocity, and angular acceleration curves of joint 1.
Figure 4 shows the angular displacement, angular velocity, and angular acceleration curves of joint 2.
Figure 5 shows the angular displacement, angular velocity, and angular acceleration curves of joint 3.
Figure 6 shows the angular displacement, angular velocity, and angular acceleration curves of joint 4.
Figure 7 shows the angular displacement, angular velocity, and angular acceleration curves of joint 5.
Figure 8 shows the angular displacement, angular velocity, and angular acceleration curves of joint 6.
4. Conclusion
The trajectory planning method studied in this paper mainly uses Bezier curves to approximate the motion trajectory of a robotic arm based on joint space. At the same time, the trajectory planning is optimized by taking advantage of the global and parallel characteristics of the genetic algorithm to meet the requirements of smooth operation and timely response of the robotic arm. Experimental tests show that the trajectory planning in joint space optimized by the genetic algorithm not only makes the trajectory smooth and continuous, but also improves the stability of the robotic arm operation.
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