Smooth trajectory generation and hybrid B-splines-Quaternions based tool path interpolation for a 3T1R parallel kinematic milling robot
Authors: Sina Akhbari, Mehran Mahboubkhah
Organizations: Intelligent Automation Centre, Wolfson School of Mechanical, Electrical & Manufacturing Engineering, Loughborough University, Loughborough, UK · Department of Mechanical Engineering, University of Tabriz, 51666-16471, Tabriz, Iran
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.
We presented a program that automates the construction and solution of the cubic spline interpolation problem for a sequence of position-time waypoints, producing trajectories that are continuous in position, velocity, and acceleration and therefore avoid the infinite-acceleration problem inherent to piecewise-linear waypoint following. By assembling the interpolation and continuity conditions into a single linear system and solving directly for the spline coefficients, the program removes the need to manually derive per-segment equations for each new waypoint set. We illustrated the approach on free-form curve drawing and signature digitization, two tasks that use the same underlying machinery required for smooth robot and drone trajectory generation. Future work includes extending the implementation to multi-axis trajectories with explicit velocity and acceleration bounds, adopting the more computationally efficient tridiagonal solution method for larger waypoint sets, and evaluating the approach quantitatively against alternative trajectory generation methods.
This paper focuses on geometric methods for generating rational motions used in the design of single-loop rational linkages, 1-degree-of-freedom mechanisms that can execute prescribed spatial tasks. Building on established rational motion synthesis methods, we introduce a new interpolation scheme for seven 3D points based on cubic quaternionic Bezier curves. The resulting motion admits factorization, i.e. the synthesis of a spatial six-bar mechanism whose tool frame passes the specified seven points. To support engineering practice, we provide open-source CAD tools that implement also the other methods and provide fast visual evaluation of motion generation and mechanism synthesis.
Daniel Huczala, Severinas Zube, Martin Pfurner +2
Robotics Laboratory, Seoul National University, Seoul, Korea · Institute of Computer Science, Department of Mathematics and Informatics, Vilnius University, Vilnius, Lithuania · Unit of Geometry and Surveying, University of Innsbruck, Innsbruck, Austria +1
The paper presents a numerical approach for the end-effector trajectory smoothing of a parallel robot designed for minimally invasive pancreatic surgery. The approach is tailored for real-time master-slave control architecture and uses a 3D space mouse for command input for velocity control. The trajectory smoothing is achieved by generating S-curves in the end-effector velocity fields, thus controlling the accelerations, which in turn reduces tissue trauma in the minimally invasive procedures. Real-time control is enabled by segmenting the S-curves based on the command inputs from the 3D space mouse. A special case is considered where the acceleration time is constant for all command inputs. Numeric results demonstrate stable transitions (without abrupt changes) in both the end-effector parameter space and in the active joints parameters, thereby validating the proposed approach. Further work aims to test the approach on an experimental model and integrate it into AI-based training modules.
Iosif Birlescu, Alexandru Pusca, Bogdan Gherman +4