This paper presents a smooth trajectory generation method for a four-degree-of-freedom parallel kinematic milling robot. The proposed approach integrates B-spline and Quaternion interpolation techniques to manage decoupled position and orientation data points. The synchronization of orientation and arc-length-parameterized position data is achieved through the fitting of smooth piece-wise Bezier curves, which describe the non-linear relationship between path length and tool orientation, solved via sequential quadratic programming. By leveraging the convex hull properties of Bezier curves, the method ensures spatial and temporal separation constraints for multi-agent trajectory generation. Unit quaternions are employed for orientation interpolation, providing a robust and efficient representation that avoids gimbal lock and facilitates smooth, continuous rotation. Modifier polynomials are used for position interpolation. Temporal trajectories are optimized using minimum jerk, time-optimal piece-wise Bezier curves in two stages: task space followed by joint space, implemented on a low-cost microcontroller. Experimental results demonstrate that the proposed method offers enhanced accuracy, reduced velocity fluctuations, and computational efficiency compared to conventional interpolation methods.
Over the last few decades, much attention has been given to the problems of tool path and trajectory generation of for generating freeform curves. More recently, parallel kinematic robots have emerged as trending topics in fields of additive [1] and subtractive manufacturing [2]. Parallel kinematic machine tools are significantly different from serial counterparts, since their multi degree of freedom are obtained from closed loop kinematic chains. This parallel formation offers superior rigidity, less mass, thus enabling the requirements for high-speed and high-precision machining [3].
Toolpath and trajectory generation are essential components of robotics and CNC machine automation. Toolpath generation must ensure high conformity and geometric smoothness of the defined path, while trajectory generation aims to produce optimal reference inputs for the control system and achieve smooth motion. The performance of planned trajectories is crucial for motion control since it exerts a significant and direct influence on the stability, reliability and productivity of the machinery [4].
In order to obtain a continuous tool path, many researches have been dedicated on implementation of parametric curves for multi axis tool path generation. Fleisig and Spence [5] used fifth-order polynomials for position splines and spherical Bezier splines for orientation, achieving coordinated motion through chord-length reparameterization.. Liu et al. [6] improved this by relating the orientation parameter to the interpolation arc length. In a different format, Langeron et al. [7] employed double Bsplines for tool position and orientation,. While, Yen et al. [8] decoupled position and orientation using fifth-order B-splines and solved a nonlinear optimization problem for parameter scheduling. The nonlinear relationship between B-spline parameters and arc length necessitates numerical solutions. Various methods have been developed to address this. The proportional interpolation method was first proposed by Bedi and augmented in [5,9], where the spline parameter is scheduled proportionally to the arc length. However, this straightforward method is only applicable to linear paths with constant speed. Taylor’s series approximation is the most commonly used among the above-mentioned methods [10][11][12][13]. However, Taylor Expansion (TE) cannot produce smooth geometry in case of sudden steep curvature changes. To avoid this drawback, others, [14,15] used predictor-corrector methods, though these are computationally intensive and may not converge. Erkorkmaz and Altintas [16] introduced polynomials to relate B-spline parameters to arc length. This method reduces feed-rate fluctuations and minimizes real-time computation by preprocessing coefficients, making it widely adopted [8,[17][18][19]. Trajectory generation for robotic machine tools typically involve mathematical formulations optimized for objectives like minimum time or jerk, subject to kinematic and geometric constraints. These problems are often solved using analytical methods (e.g., quadratic programming) or heuristic approaches (e.g., genetic algorithms, particle swarm optimization). The S-shape velocity with trapezoidal acceleration is widely used [20][21][22] for jerk-limited motion but fails to control sudden jerk changes, which can excite system resonances. Higher order polynomials [23,24] and trigonometric velocity scheduling [25,26] have been proposed to address this, though its computational cost of trigonomic equations limits practicality. Time-optimal velocity profiles, optimized under kinematic and geometric constraints, are another focus. Sencer et al. [27] used cubic B-splines and sequential quadratic programming to simplify constraints, while others, [28,29] approximated constraints with linear programming, though results were often overly conservative. Subsequently, Erkorkmaz et al. [30] combined linear programming with a windowing scheme to reduce computational load for long toolpaths. In an alternative approach, Gasparetto and Zanotto [31] proposed a hybrid time-jerk optimization method using sequential quadratic programming. Heuristic algorithms, such as genetic algorithms [32] and particle swarm optimization [33], have also been applied for near-optimal trajectory generation. However, their high computational cost and lower accuracy compared to classical methods make them less suitable for lowcost embedded systems, with a higher risk of constraint violations. While toolpath interpolation and feed-rate scheduling have been extensively studied, their application to parallel kinematic machines (PKMs) with more than three degrees of freedom remains underexplored. This work implements a proposed trajectory generation method on a 3T1R parallel mechanism previously developed by the authors [34,35]. A decoupled approach is used for toolpath parameterization, with quintic B-splines globally interpolating discrete tool pose data to ensure C³ continuity. Spatial tool
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