In the realm of robotic systems and advanced automation, the demand for high-speed, precise motion control has led to significant interest in parallel mechanisms. Among these, cable-driven parallel mechanisms (CDPMs) have emerged as a promising alternative to traditional rigid-link counterparts due to their inherent advantages: reduced inertia, larger workspace, simpler structure, and higher dynamic performance. The core component in such systems is the end effector, whose movement dictates the overall efficiency and accuracy. However, planning smooth and continuous trajectories for the end effector remains a critical challenge, especially when dealing with discrete reference points that can introduce abrupt changes in motion parameters. Traditional interpolation methods often result in poor continuity and sudden accelerations, leading to vibrations, increased wear, and compromised precision. This article presents an in-depth exploration of an improved polynomial-based trajectory planning method for the end effector of a cable-driven parallel mechanism. By optimizing velocity profiles and employing least squares fitting, we enhance transition smoothness between discrete points, thereby improving overall motion performance. Through comprehensive kinematic modeling, co-simulation analysis, and experimental validation, we demonstrate the efficacy of this approach in reducing start-stop cycles, minimizing shocks, and shortening operation times. The insights gained here aim to contribute to the development of more efficient and reliable cable-driven systems for applications ranging from heavy machinery to robotic manipulation.
The cable-driven parallel mechanism under consideration utilizes multiple cables actuated by linear motors to control the position of the end effector in a planar environment. Unlike rigid links, cables offer flexibility and low mass, which reduce moving inertia and enable rapid movements. However, this flexibility introduces nonlinear dynamics that complicate motion control. The primary objective is to guide the end effector along a desired path—such as a parabolic curve—with minimal jerk and maximum smoothness. To achieve this, we first establish a kinematic model that relates cable lengths to end effector positions. Let us define the coordinate system: the workspace is represented in a 2D plane with origin O at the center. The cable attachment points on the frame are denoted as A_i with coordinates (x_i, y_i) for i = 1, 2, 3, 4, corresponding to four cables. The end effector is treated as a point mass P with coordinates (x, y). The inverse kinematics provide the cable lengths L_i as functions of P’s position:
$$L_i = \sqrt{(x – x_i)^2 + (y – y_i)^2}, \quad i = 1, 2, 3, 4.$$
Differentiating with respect to time yields the cable velocity and acceleration equations, essential for dynamic analysis. The velocity v_i of each cable is given by:
$$v_i = \dot{L}_i = \frac{1}{L_i} \left[ (x – x_i) \dot{x} + (y – y_i) \dot{y} \right],$$
and the acceleration a_i is:
$$a_i = \ddot{L}_i = \frac{1}{L_i} \left[ (x – x_i) \ddot{x} + (y – y_i) \ddot{y} + \dot{x}^2 + \dot{y}^2 \right] – \frac{\dot{L}_i^2}{L_i}.$$
These equations form the basis for simulating the system’s behavior. However, the nonlinearity of cables—due to elasticity and sag—makes purely analytical models insufficient. Therefore, we adopt a co-simulation approach combining MATLAB/Simulink for control logic and Adams for multi-body dynamics. This allows us to incorporate realistic cable properties, such as diameter and elastic modulus, into the simulation. The parameters used in our model are summarized in Table 1.
| Parameter | Value |
|---|---|
| Cable attachment points (A1, A2, A3, A4) coordinates | (-636 mm, 663 mm), (663 mm, 636 mm), (-663 mm, -636 mm), (636 mm, -663 mm) |
| Initial end effector position | (0 mm, 0 mm) |
| Mass of end effector | 300 g |
| Cable diameter | 4 mm |
| Cable elastic modulus | 3 GPa |
| Linear motor track length | 800 mm |
The trajectory planning for the end effector begins with selecting reference points along a desired path, such as a parabola with varying curvature. Traditional methods like linear interpolation force the end effector to pass exactly through each point, which can cause sudden direction changes and speed fluctuations. In contrast, fitting methods approximate the path without strict adherence to every point, offering smoother transitions. We propose an improved polynomial fitting technique that uses cubic polynomials to generate a continuous trajectory. Given n discrete reference points p_i(x_i, y_i), we seek a cubic function f(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 that minimizes the sum of squared errors between f(x_i) and y_i. This is achieved via the least squares method, solving for coefficients a_0, a_1, a_2, a_3. The minimization problem is formulated as:
$$\min_{a_0, a_1, a_2, a_3} \sum_{i=1}^{n} \left[ y_i – (a_0 + a_1 x_i + a_2 x_i^2 + a_3 x_i^3) \right]^2.$$
Setting partial derivatives to zero leads to a system of linear equations represented in matrix form:
$$\begin{bmatrix}
n & \sum x_i & \sum x_i^2 & \sum x_i^3 \\
\sum x_i & \sum x_i^2 & \sum x_i^3 & \sum x_i^4 \\
\sum x_i^2 & \sum x_i^3 & \sum x_i^4 & \sum x_i^5 \\
\sum x_i^3 & \sum x_i^4 & \sum x_i^5 & \sum x_i^6
\end{bmatrix}
\begin{bmatrix}
a_0 \\ a_1 \\ a_2 \\ a_3
\end{bmatrix}
=
\begin{bmatrix}
\sum y_i \\ \sum x_i y_i \\ \sum x_i^2 y_i \\ \sum x_i^3 y_i
\end{bmatrix}.$$
Solving this yields the optimal coefficients. For instance, with reference points from a parabolic path, we obtain a specific cubic equation. Once the path is defined, we apply a trapezoidal velocity profile to plan the motion of the end effector. The total path length s is computed using the arc length formula:
$$s = \int_{a}^{b} \sqrt{1 + \left( \frac{dy}{dx} \right)^2} \, dx,$$
where a and b are the x-coordinates of the start and end points. The motion is divided into three phases: constant acceleration, constant velocity, and constant deceleration. Let v_max be the maximum speed (e.g., 2 m/s) and k be the proportion of path allocated to acceleration and deceleration (e.g., k = 0.4). The acceleration a_acc is then:
$$a_{\text{acc}} = \frac{v_{\text{max}}^2}{2 s k},$$
with time intervals T1 for acceleration, T2 for constant velocity, and T3 for deceleration given by:
$$T_1 = \frac{v_{\text{max}}}{a_{\text{acc}}}, \quad T_2 = \frac{s(1 – 2k)}{v_{\text{max}}}, \quad T_3 = T_1.$$
The position along the path as a function of time, s(t), is derived for each phase, enabling calculation of the end effector’s coordinates at every simulation step. This approach ensures that the end effector maintains peak velocity for longer durations, enhancing efficiency while avoiding abrupt stops.
To evaluate the improved polynomial method, we conducted co-simulations using Adams and MATLAB/Simulink. The model includes the end effector as a cylindrical mass, linear motors as sliding blocks, and cables with realistic properties simulated via Adams’ cable-pulley module. The control inputs—motor displacements derived from inverse kinematics—are fed from Simulink, while Adams returns actual cable lengths and end effector positions. Simulation parameters: step size of 0.001 s, maximum speed of 2 m/s for the end effector. We compared our method against traditional interpolation using a set of discrete reference points from a parabolic trajectory. The coordinates of these points are listed in Table 2.
| Point ID | x-coordinate (mm) | y-coordinate (mm) |
|---|---|---|
| 1 | -281 | 268.52 |
| 2 | -187 | -48.22 |
| 3 | -94 | -236.38 |
| 4 | 0 | -300 |
| 5 | 94 | -236.38 |
| 6 | 187 | -48.22 |
| 7 | 281 | 268.52 |
Applying the improved polynomial fitting to these points yields the trajectory equation:
$$f(x) = -7.443 \times 10^{-22} x^3 + 0.006969 x^2 – 295.9.$$
Simulation results demonstrate significant improvements. The end effector’s path with our method is smooth and continuous, closely following the desired parabola without sharp turns. In contrast, interpolation produces a jagged trajectory with frequent direction changes. To quantify performance, we analyzed motor dynamics—focusing on Motor 1 as representative. Key metrics include displacement, velocity, and acceleration over time. The improved polynomial method reduces fluctuations in motor displacement, as shown in Table 3, which compares peak values and variances.
| Metric | Improved Polynomial Method | Traditional Interpolation | Improvement |
|---|---|---|---|
| Maximum displacement variation (mm) | ±15.2 | ±28.7 | 47% reduction |
| Number of start-stop cycles | 4 | 8 | 50% reduction |
| Peak acceleration (mm/s²) | 4.76 × 10⁴ | 6.68 × 10⁴ | 28.7% reduction |
| Total operation time (s) | 2.740 | 3.524 | 22.3% reduction |
The velocity profile for Motor 1 under our method shows four distinct start-stop events, compared to eight with interpolation. This halving of cycles minimizes mechanical shock and wear on the linear motors. Acceleration plots further confirm smoother transitions, with lower peak magnitudes reducing inertial forces on the end effector. Overall, the simulation indicates that the improved polynomial planning enhances motion continuity and efficiency, allowing the end effector to achieve higher average speeds while maintaining precision.
Experimental validation was performed on a physical prototype of the cable-driven parallel mechanism. The setup consists of four linear motors controlled via a PC-based system running TwinCAT 3 software. The end effector is a lightweight circular platform, and cables are routed through pulleys to the motors. We executed both trajectory planning methods—feeding position commands as inputs—and recorded motor responses using scope monitoring. High-speed camera measurements captured the actual path of the end effector for multiple runs. Results align closely with simulations: the improved polynomial method yields smoother motor displacements and velocities, with fewer abrupt changes. For instance, Motor 1’s velocity curve exhibits gradual ramps instead of sudden spikes, corroborating the reduction in start-stop cycles. Acceleration data from sensors also shows decreased variability, confirming minimized vibrations. The end effector’s trajectory, averaged over several trials, demonstrates consistent adherence to the planned parabolic path, with deviations within acceptable tolerances. These experimental outcomes validate the co-simulation model and underscore the practical benefits of our trajectory planning approach for real-world cable-driven systems.
In summary, this study presents a robust trajectory planning framework for the end effector of a cable-driven parallel mechanism. By leveraging improved polynomial fitting combined with trapezoidal velocity profiling, we address the limitations of traditional interpolation methods. The kinematic model establishes fundamental relationships between cable dynamics and end effector motion, while co-simulation incorporates nonlinear cable effects for accurate prediction. Key findings include: smoother end effector trajectories, reduced linear motor start-stop cycles by 50%, lower peak accelerations by approximately 28.7%, and shorter overall operation times by about 22.3%. Experimental tests confirm these improvements, highlighting the method’s efficacy in enhancing motion smoothness and system reliability. Future work could explore adaptive tuning of polynomial degrees or integration with real-time feedback control to further optimize end effector performance in dynamic environments. Ultimately, this research contributes to advancing cable-driven technology for applications demanding high-speed, precise manipulation of the end effector.
To delve deeper into the mathematical foundations, let us consider the optimization process for polynomial coefficients. The least squares solution not only minimizes errors but also ensures that the trajectory is differentiable, providing continuous velocity and acceleration for the end effector. This is crucial for dynamic stability. The general form of a cubic polynomial offers a balance between flexibility and computational efficiency; however, higher-order polynomials could be investigated for more complex paths. Additionally, the trapezoidal velocity profile can be refined using S-curve algorithms to further reduce jerk. In terms of system modeling, the co-simulation approach bridges gaps between ideal kinematics and real-world dynamics. The cables’ elasticity, modeled with Young’s modulus, introduces spring-like behavior that affects end effector positioning. This is accounted for in Adams through force elements, making simulations more realistic. For broader applicability, Table 4 summarizes key equations used throughout the analysis.
| Description | Equation |
|---|---|
| Inverse kinematics: cable length | $$L_i = \sqrt{(x – x_i)^2 + (y – y_i)^2}$$ |
| Cable velocity | $$v_i = \frac{1}{L_i} \left[ (x – x_i) \dot{x} + (y – y_i) \dot{y} \right]$$ |
| Cable acceleration | $$a_i = \frac{1}{L_i} \left[ (x – x_i) \ddot{x} + (y – y_i) \ddot{y} + \dot{x}^2 + \dot{y}^2 \right] – \frac{v_i^2}{L_i}$$ |
| Polynomial fitting coefficients | $$\mathbf{A} = (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{X}^T \mathbf{Y}, \text{ where } \mathbf{X} \text{ is the Vandermonde matrix}$$ |
| Path length integral | $$s = \int_{a}^{b} \sqrt{1 + (f'(x))^2} \, dx$$ |
| Trapezoidal profile acceleration | $$a_{\text{acc}} = \frac{v_{\text{max}}^2}{2 s k}$$ |
| Time in constant velocity phase | $$T_2 = \frac{s(1 – 2k)}{v_{\text{max}}}$$ |
The integration of these elements facilitates a holistic approach to end effector control. Moreover, the emphasis on the end effector’s motion quality cannot be overstated—it directly impacts tasks such as pick-and-place operations, machining, or inspection. By reducing abrupt accelerations, we extend the lifespan of mechanical components and improve energy efficiency. The improved polynomial method also offers scalability: for instance, adding more reference points or transitioning to 3D trajectories simply involves extending the polynomial dimensions and adjusting the kinematic model accordingly. In conclusion, the trajectory planning methodology outlined here provides a practical solution for enhancing the performance of cable-driven parallel mechanisms, ensuring that the end effector moves with the smoothness and precision required in modern automated systems.