In the field of industrial automation, robot technology has revolutionized manufacturing processes, particularly in tasks requiring high precision and efficiency. As a researcher focused on robotics, I have explored the application of serial six-axis robots in Cartesian space trajectory planning, aiming to enhance motion control for applications like material handling and sorting. This article delves into the modeling, trajectory planning algorithms, and experimental validation of such systems, emphasizing the role of advanced robot technology in achieving smooth and accurate movements.
The foundation of any robotic system lies in its kinematic model. For a serial six-axis robot, the Denavit-Hartenberg (D-H) parameters are essential for defining the relationships between links and joints. These parameters—link twist angle $\alpha_i$, link length $a_i$, link offset $d_i$, and joint angle $\theta_i$—enable the construction of a robot model. Using MATLAB’s Robotics Toolbox, I developed a simulation model to represent the robot’s structure, as illustrated in the coordinate system diagram. The D-H parameters for a typical industrial robot are summarized in the table below:
| Joint $i$ | $\alpha_i$ (deg) | $a_i$ (m) | $d_i$ (m) | $\theta_i$ |
|---|---|---|---|---|
| 1 | 90 | 0.180 | 0.2005 | $\theta_1$ |
| 2 | 0 | 0.600 | 0 | $\theta_2$ |
| 3 | -90 | 0.130 | 0 | $\theta_3$ |
| 4 | 90 | 0 | 0.630 | $\theta_4$ |
| 5 | -90 | 0 | 0 | $\theta_5$ |
| 6 | 0 | 0 | 0 | $\theta_6$ |
This model facilitates the analysis of the robot’s end-effector pose, which is critical for trajectory planning. The transformation matrices derived from these parameters allow for forward and inverse kinematics computations, enabling precise control over the robot’s movements. The advancement in robot technology has made such modeling more accessible, leading to improved performance in dynamic environments.
Trajectory planning is a cornerstone of robot technology, ensuring that the end-effector follows a desired path with specified velocity and acceleration profiles. I investigated the S-shaped acceleration-deceleration curve, which offers advantages over the traditional trapezoidal (T-shaped) curve. The T-shaped curve, while simple, suffers from discontinuous acceleration, causing jerks and potential damage to motors. In contrast, the S-shaped curve provides continuous acceleration, resulting in smoother motion. The general form of the S-shaped curve can be divided into segments: acceleration increase ($T_1$), constant acceleration ($T_2$), acceleration decrease ($T_3$), constant velocity ($T_4$), deceleration increase ($T_5$), constant deceleration ($T_6$), and deceleration decrease ($T_7$). The velocity $v(t)$ and displacement $P(t)$ are derived through integration of the acceleration $a(t)$, which is itself integrated from the jerk $J$.
For instance, the acceleration during each segment can be expressed as:
$$a(t) =
\begin{cases}
J t & \text{for } t \in [0, T_1] \\
J T_1 & \text{for } t \in [T_1, T_1 + T_2] \\
J T_1 – J (t – T_1 – T_2) & \text{for } t \in [T_1 + T_2, T_1 + T_2 + T_3] \\
0 & \text{for } t \in [T_1 + T_2 + T_3, T_1 + T_2 + T_3 + T_4] \\
-J (t – T_a) & \text{for } t \in [T_a, T_a + T_5] \\
-J T_5 & \text{for } t \in [T_a + T_5, T_a + T_5 + T_6] \\
-J T_5 + J (t – T_a – T_5 – T_6) & \text{for } t \in [T_a + T_5 + T_6, T]
\end{cases}$$
where $T_a = T_1 + T_2 + T_3$ and $T$ is the total time. The velocity is then:
$$v(t) = \int_0^t a(\tau) d\tau + v_0$$
and the displacement is:
$$P(t) = \int_0^t v(\tau) d\tau + P_0$$
These equations form the basis of the automatic segment planning algorithm, which adapts to initial constraints such as initial velocity $v_0$, final velocity $v_1$, maximum velocity $v_{\text{max}}$, acceleration $A$, deceleration $D$, and jerk $J$. This flexibility is a testament to the evolution of robot technology, allowing for customized motion profiles in complex industrial scenarios.
In practical applications, interpolation algorithms are used to generate intermediate points along a path. For linear interpolation between points $P_0(x_0, y_0, z_0)$ and $P_1(x_1, y_1, z_1)$, the distance $L$ is computed as:
$$L = \sqrt{(x_1 – x_0)^2 + (y_1 – y_0)^2 + (z_1 – z_0)^2}$$
The number of interpolation points $N$ is determined based on the velocity profile and time interval $\tau$. The increments in each direction are:
$$\Delta x = \frac{x_1 – x_0}{N}, \quad \Delta y = \frac{y_1 – y_0}{N}, \quad \Delta z = \frac{z_1 – z_0}{N}$$
Thus, the coordinates of intermediate points $P_i$ are:
$$x_i = x_{i-1} + \Delta x, \quad y_i = y_{i-1} + \Delta y, \quad z_i = z_{i-1} + \Delta z$$
For circular interpolation, three non-collinear points $P_0$, $P_1$, and $P_2$ define the arc. The center $P_o(x_o, y_o, z_o)$ and radius $r$ are found by solving the system of equations derived from the circle’s standard form and coplanarity conditions. The angular increment $\Delta \theta$ is calculated as $\Delta \theta = \frac{v(t) \tau}{r}$, and the interpolation points are generated using trigonometric functions:
$$x_i = x_o + r \cos(\theta_i), \quad y_i = y_o + r \sin(\theta_i), \quad z_i = z_o$$
where $\theta_i = \theta_{i-1} + \Delta \theta$. These methods highlight how robot technology enables precise path following in Cartesian space.
To validate the approach, I conducted simulations using asymmetric S-shaped curves under various constraints. For linear interpolation between $P_1(720.1, 943.4, 860.2)$ and $P_2(210.5, 88.3, 91.0)$, with initial velocity $v_0 = 0 \, \text{mm/s}$, final velocity $v_1 = 88 \, \text{mm/s}$, maximum velocity $v_{\text{max}} = 220 \, \text{mm/s}$, acceleration $A = 100 \, \text{mm/s}^2$, deceleration $D = -100 \, \text{mm/s}^2$, and jerk $J = 50 \, \text{mm/s}^3$, the trajectory was smooth and continuous. Similarly, for circular interpolation using points $P_0(712.3, 127.4, 402.6)$, $P_1(383.2, 603.0, 194.4)$, and $P_2(418.0, 356.4, 650.7)$, the same constraints yielded a seamless arc path. The velocity profiles confirmed the absence of abrupt changes, underscoring the benefits of robot technology in reducing mechanical wear and improving stability.
In multi-segment trajectories, such as continuous linear paths connecting points $P_0$, $P_1$, $P_2$, and $P_3$, the asymmetric S-shaped curve ensures smooth transitions at junctions. The velocity at junction points is computed using a weighting factor $k$ (with $0 < k < 1$) and weights $\omega_1$ and $\omega_2$:
$$v_j = \max(v_1, v_2) + k (\omega_1 v_1 + \omega_2 v_2 – \max(v_1, v_2))$$
This approach maintains non-zero velocities at junctions, minimizing disturbances and enhancing efficiency. The table below summarizes the parameters for a three-segment continuous linear interpolation:
| Segment | $v_{\text{max}}$ (mm/s) | $A$ (mm/s²) | $D$ (mm/s²) | Junction Velocity (mm/s) |
|---|---|---|---|---|
| L1 | 220 | 100 | -100 | 100.4 |
| L2 | 240 | 100 | -100 | 30.8 |
| L3 | 100 | 100 | -100 | – |
The simulations demonstrated that the asymmetric S-shaped curve automatically plans segments based on initial conditions, producing velocity profiles that are both efficient and stable. This capability is crucial for applications in high-speed robotics, where robot technology must balance performance and reliability.
In conclusion, the integration of robot technology in Cartesian space trajectory planning has proven effective for achieving high-precision motion control. The use of D-H parameters for modeling, S-shaped curves for velocity planning, and interpolation algorithms for path generation collectively enhance the robustness of serial six-axis robots. Future work will focus on optimizing these algorithms for real-time applications and expanding their use in collaborative robot environments. As robot technology continues to evolve, it will undoubtedly drive innovations in automation, making processes faster, safer, and more adaptable.