In the era of global Industry 4.0 and the strategic framework of advanced manufacturing initiatives, the role of industrial robots has become increasingly critical in performing repetitive, high-precision, and efficient tasks. The evolution of robot technology has enabled widespread applications in material handling, palletizing, assembly, and other automated processes. However, challenges such as low operational efficiency, high energy consumption, and significant mechanical冲击persist, necessitating advanced optimization techniques. Traditional trajectory planning methods often fail to achieve multi-objective optimality, while conventional multi-objective particle swarm optimization (MOPSO) algorithms suffer from slow convergence and a tendency to陷入局部最优. To address these issues, this paper proposes an improved multi-objective particle swarm optimization (IMOPSO) algorithm, incorporating linear differential decreasing inertia weights and Gaussian mutation operators, to optimize time-energy-impact multi-objective trajectories for industrial robots. The IRB6700 industrial robot, known for its high payload capacity and precision, serves as the研究对象. This study encompasses kinematic analysis, trajectory planning via seventh-degree B-spline interpolation, and multi-objective optimization using IMOPSO, resulting in enhanced performance metrics validated through simulations.
The advancement of robot technology has revolutionized manufacturing by enabling complex operations with minimal human intervention. In this context, trajectory optimization plays a pivotal role in ensuring that robots operate smoothly, efficiently, and reliably. Multi-objective optimization, which balances conflicting criteria such as time minimization, energy reduction, and冲击mitigation, is essential for modern robotic systems. The integration of intelligent algorithms like PSO has shown promise, but standard MOPSO requires improvements to overcome limitations in global search capability and convergence speed. This research introduces IMOPSO to refine the trajectory planning process, leveraging mathematical modeling and simulation to demonstrate its efficacy. The following sections detail the kinematic analysis of the IRB6700 robot, the trajectory planning methodology, the IMOPSO algorithm, the multi-objective optimization model, and the simulation results, culminating in a comprehensive conclusion.
Kinematic Analysis of the IRB6700 Robot
Kinematic analysis forms the foundation for trajectory planning and optimization in robot technology. The IRB6700 is a six-degree-of-freedom (6-DOF) serial manipulator, characterized by its robust structure and dynamic responsiveness. Using the modified Denavit-Hartenberg (D-H) parameter method, the link coordinate systems are established to derive the forward and inverse kinematic equations. The D-H parameters for the IRB6700 robot are summarized in Table 1, which includes link lengths, joint angles, and offsets.
| Link i | di (mm) | αi-1 (°) | ai-1 (mm) | Joint Range (°) |
|---|---|---|---|---|
| 1 | 780 | 0 | 0 | -170 to 170 |
| 2 | 0 | -90 | 320 | -65 to 85 |
| 3 | 0 | 0 | 1125 | -180 to 70 |
| 4 | 1142.5 | -90 | 200 | -300 to 300 |
| 5 | 0 | 90 | 0 | -130 to 130 |
| 6 | 200 | -90 | 0 | -360 to 360 |
The transformation matrix between consecutive links, derived from the D-H parameters, is given by Equation (1). This matrix defines the position and orientation of each joint coordinate system relative to the previous one, facilitating the computation of the end-effector’s pose. For instance, the transformation from link i-1 to link i is expressed as:
$$^{i-1}T_i = \text{Rot}(x, \alpha_{i-1}) \cdot \text{Trans}(x, a_{i-1}) \cdot \text{Rot}(z, \theta_i) \cdot \text{Trans}(z, d_i)$$
where Rot and Trans represent rotation and translation matrices, respectively. The cumulative product of these matrices yields the end-effector transformation matrix \(^0T_6\), as shown in Equation (2):
$$^0T_6 = ^0T_1 \cdot ^1T_2 \cdot ^2T_3 \cdot ^3T_4 \cdot ^4T_5 \cdot ^5T_6 = \begin{bmatrix}
n_x & o_x & a_x & p_x \\
n_y & o_y & a_y & p_y \\
n_z & o_z & a_z & p_z \\
0 & 0 & 0 & 1
\end{bmatrix}$$
Here, \(\mathbf{n}\), \(\mathbf{o}\), and \(\mathbf{a}\) are the orientation vectors, and \(\mathbf{p}\) is the position vector of the end-effector. The elements of this matrix are computed using trigonometric functions of the joint angles, denoted as \(s_i = \sin(\theta_i)\) and \(c_i = \cos(\theta_i)\). For example, the components include:
$$n_x = c_1[c_2(c_4c_5c_6 – s_4s_6) – s_2s_5c_6] – s_1(s_4c_5c_6 + c_4s_6)$$
Similar expressions are derived for other elements, ensuring comprehensive kinematic modeling. Inverse kinematics involves determining the joint angles for a given end-effector pose. Using the Pieper criterion, which applies to robots with three consecutive axes intersecting, closed-form solutions are obtained. For instance, the angle for joint 1, \(\theta_1\), is computed as:
$$\theta_1 = \atan2(p_y – d_6 a_y, p_x – d_6 a_x)$$
Validation through MATLAB’s Robotics Toolbox confirms the accuracy of both forward and inverse kinematics, with computed values matching theoretical results for sample joint configurations. This kinematic analysis is crucial for subsequent trajectory planning and optimization in robot technology.
Trajectory Planning Using Seventh-Degree B-Spline Interpolation
Trajectory planning ensures that the robot follows a predefined path smoothly, avoiding abrupt movements that could lead to inefficiencies or damage. In this study, seventh-degree B-spline interpolation is employed to plan the robot’s trajectory through specified path points. B-spline curves offer local support, meaning that modifying a control point affects only a limited segment of the curve, enhancing flexibility and stability. Given a sequence of path points \((t_i, p_i)\) for \(i = 0, 1, \ldots, N\), where \(t_0\) and \(t_f\) are the start and end times, the B-spline curve is defined as:
$$P(u) = \sum_{j=0}^{m+6} Q_j N_{j,7}(u)$$
Here, \(Q_j\) are the control points, and \(N_{j,7}(u)\) are the seventh-degree B-spline basis functions, computed recursively using the de Boor algorithm:
$$N_{j,0}(u) = \begin{cases}
1 & \text{if } u \in [u_j, u_{j+1}) \\
0 & \text{otherwise}
\end{cases}$$
$$N_{j,k}(u) = \frac{u – u_j}{u_{j+k} – u_j} N_{j,k-1}(u) + \frac{u_{j+k+1} – u}{u_{j+k+1} – u_{j+1}} N_{j+1,k-1}(u)$$
The node vector \(U = [u_0, u_1, \ldots, u_{N+14}]\) is normalized using the cumulative chord length parameterization method, ensuring that the curve passes through all path points. This approach generates smooth trajectories for joint displacements, velocities, and accelerations, which are essential for minimizing energy consumption and mechanical冲击in robot technology applications. The resulting trajectories serve as the basis for multi-objective optimization using the IMOPSO algorithm.
Improved Multi-Objective Particle Swarm Optimization (IMOPSO) Algorithm
The standard MOPSO algorithm, inspired by bird flocking behavior, is widely used for multi-objective optimization due to its simplicity and fast convergence. However, it often suffers from premature convergence and local optima entrapment. To overcome these limitations, an improved version, IMOPSO, is proposed, incorporating linear differential decreasing inertia weights and Gaussian mutation operators. These enhancements balance global exploration and local exploitation, improving solution quality and diversity.
The inertia weight \(\omega\) in PSO controls the influence of previous velocities on the current update. In IMOPSO, a linear differential decreasing strategy is adopted, where \(\omega\) starts at a high value (e.g., 0.9) and decreases to a lower value (e.g., 0.4) over iterations. This is expressed as:
$$\omega(k) = \omega_{\text{start}} – (\omega_{\text{start}} – \omega_{\text{end}}) \times \left(\frac{k}{k_{\max}}\right)^2$$
Here, \(k\) is the current iteration, and \(k_{\max}\) is the maximum number of iterations. This quadratic decrease allows for gradual reduction in the early stages, promoting global search, and rapid decrease later, enhancing local refinement. Compared to constant or linear decreasing weights, this approach reduces the risk of local optima.
Additionally, a Gaussian mutation operator introduces random perturbations to particle positions, increasing population diversity. The mutation is applied as:
$$x_i^{(k+1)} = x_i^{(k)} + (x_{\max} – x_{\min}) \times \mathcal{N}(0, \sigma^2(k))$$
where \(\mathcal{N}(0, \sigma^2(k))\) is a Gaussian random variable with mean 0 and variance \(\sigma^2(k) = \sigma_0^2 \exp(-k / k_{\max})\). The distance between particles and the global best position influences the mutation magnitude, ensuring adaptive search behavior. The velocity update equation in IMOPSO is:
$$v_i^{(k+1)} = \omega(k) v_i^{(k)} + c_1 r_1 (P_{\text{best},i} – x_i^{(k)}) + c_2 r_2 (G_{\text{best}} – x_i^{(k)}) + G$$
where \(c_1\) and \(c_2\) are learning factors, \(r_1\) and \(r_2\) are random numbers, and \(G\) is the Gaussian perturbation term. This combined approach enables IMOPSO to efficiently navigate the solution space, producing a well-distributed Pareto front for multi-objective problems in robot technology.
Multi-Objective Optimization Model for Time-Energy-Impact
In robot technology, optimizing trajectories involves balancing multiple objectives under kinematic and dynamic constraints. The proposed multi-objective model minimizes time, energy consumption, and joint冲击simultaneously. The optimization objectives are formulated as follows:
1. Time Minimization: The total operation time \(f_1\) is minimized to enhance efficiency:
$$f_1 = \sum_{i=1}^{N} \Delta t_i = \sum_{i=1}^{N} (t_i – t_{i-1})$$
2. Energy Minimization: The energy consumption \(f_2\) is proportional to the square of joint accelerations, integrated over time:
$$f_2 = \frac{1}{T} \sum_{g=1}^{G} \int_{0}^{T} a_g^2(t) \, dt$$
where \(G\) is the number of joints, \(a_g(t)\) is the acceleration of joint \(g\), and \(T\) is the total time.
3. Impact Minimization: The mechanical冲击\(f_3\) is measured by the integral of squared jerk (rate of acceleration change):
$$f_3 = \frac{1}{T} \sum_{g=1}^{G} \int_{0}^{T} j_g^2(t) \, dt$$
where \(j_g(t)\) is the jerk of joint \(g\).
These objectives are subject to kinematic constraints, including joint limits, velocity limits, acceleration limits, and jerk limits:
$$\theta_{g,\min} \leq \theta_g(t) \leq \theta_{g,\max}$$
$$v_g(t) \leq v_{g,\max}$$
$$a_g(t) \leq a_{g,\max}$$
$$j_g(t) \leq j_{g,\max}$$
for each joint \(g = 1, 2, \ldots, 6\). The values for these constraints are derived from the IRB6700 specifications, as shown in Table 2.
| Joint | Max Velocity (°/s) | Max Acceleration (°/s²) | Max Jerk (°/s³) |
|---|---|---|---|
| 1 | 60 | 50 | 60 |
| 2 | 60 | 50 | 75 |
| 3 | 60 | 50 | 70 |
| 4 | 70 | 60 | 55 |
| 5 | 80 | 85 | 60 |
| 6 | 90 | 90 | 60 |
The optimization problem is to find the Pareto-optimal solutions that minimize \(f_1\), \(f_2\), and \(f_3\) simultaneously, using the IMOPSO algorithm. This approach ensures that the robot operates efficiently, with reduced energy usage and minimal wear and tear, advancing the capabilities of robot technology in industrial settings.
Simulation Experiments and Results Analysis
To validate the proposed IMOPSO algorithm, simulation experiments are conducted on the IRB6700 robot. The path points for the end-effector in Cartesian space are listed in Table 3, and the corresponding joint space positions, obtained through inverse kinematics, are provided in Table 4.
| Path Point | X-axis (mm) | Y-axis (mm) | Z-axis (mm) |
|---|---|---|---|
| 1 | 1645 | 0 | -562.5 |
| 2 | 1512.032 | 1897.503 | 1848.025 |
| 3 | -66.328 | 2539.982 | 1509.396 |
| 4 | -550.989 | 2597.748 | 535.198 |
| 5 | -971.16 | 2508.487 | 129.856 |
| 6 | -1735.74 | 1260.629 | -692.103 |
| Node | Joint 1 (°) | Joint 2 (°) | Joint 3 (°) | Joint 4 (°) | Joint 5 (°) | Joint 6 (°) |
|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |
| 2 | 50.90 | -13.22 | -98.81 | -187.70 | 60.41 | 157.44 |
| 3 | 89.54 | -9.87 | -88.10 | -206.06 | 80.74 | 127.22 |
| 4 | 99.34 | 22.20 | -107.12 | -232.44 | 50.36 | 113.84 |
| 5 | 110.57 | 24.20 | -97.17 | -245.67 | 8.95 | 147.97 |
| 6 | 144.01 | 60.00 | -120.01 | 0.00 | 0.00 | 0.00 |
The IMOPSO parameters are set as follows: population size of 100, maximum iterations of 200, inertia weight ranging from 0.9 to 0.4, learning factors \(c_1 = 1\) and \(c_2 = 2\), grid inflation factor of 0.1, and archive size controlled by a pruning factor of 2. The algorithm is tested on standard multi-objective functions (e.g., ZDT1, ZDT2, ZDT3) to evaluate its convergence and diversity. The results show that IMOPSO achieves lower inverted generational distance (IGD), spacing (SP), and generational distance (GD) metrics compared to other MOPSO variants, confirming its superior performance.
For the trajectory optimization, the Pareto front obtained from IMOPSO is depicted in Figure 1, illustrating the trade-offs between time, energy, and冲击. Three representative points (A, B, C) are selected from the Pareto front, with their objective values listed in Table 5.
| Point | Time (s) | Energy (°/s²) | Impact (°/s³) |
|---|---|---|---|
| A | 14.52 | 91.82 | 88.82 |
| B | 15.53 | 75.65 | 67.64 |
| C | 16.82 | 70.92 | 60.24 |
Point B is chosen as the desired solution, balancing all objectives. The optimized trajectory reduces the total time from 25 s to 17 s, while ensuring smooth velocity, acceleration, and jerk profiles. The joint displacements over time, plotted in Figure 2, demonstrate continuous and feasible movements. The end-effector trajectory, shown in Figure 3, confirms accurate path following. Comparative analysis of velocity, acceleration, and jerk before and after optimization (Figures 4-6) reveals that the optimized trajectories start and end with zero velocity and acceleration, minimizing冲击and energy consumption. These results underscore the effectiveness of IMOPSO in enhancing robot technology through multi-objective trajectory optimization.
Conclusion
This research addresses the challenges of low efficiency and suboptimal performance in industrial robots by proposing an IMOPSO-based approach for time-energy-impact multi-objective trajectory optimization. The IRB6700 robot undergoes comprehensive kinematic analysis, trajectory planning via seventh-degree B-spline interpolation, and optimization using IMOPSO with linear differential decreasing inertia weights and Gaussian mutation. The algorithm demonstrates improved convergence and diversity, producing a well-distributed Pareto front. Simulation results confirm a 28% reduction in operation time, along with smooth motion profiles that reduce energy consumption and mechanical冲击. This work advances robot technology by providing a robust framework for multi-objective optimization, applicable to various industrial scenarios such as welding and assembly. Future research will focus on incorporating precision constraints and extending the approach to other robotic systems, further solidifying the role of intelligent algorithms in evolving robot technology.