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This paper proposes a convex time-optimal trajectory planning method for industrial robotic manipulators with jerk constraints. To achieve smooth and efficient trajectories, the square of the pseudo velocity profile is constructed using a cubic uniform B-spline, and a linear relationship is defined with the control points of the B-spline to preserve convexity in the pseudo states. Bi-linear and non-convex jerk constraints are introduced in the optimization problem, and a convex restriction method is applied to achieve convexity. The proposed method is evaluated through three case studies: two contour following tasks and a pick-and-place task. Comparative optimization results demonstrate that the proposed method achieves time optimality and trajectory smoothness simultaneously in the reformulated and jerk-restricted optimization problem. The proposed method provides a practical approach to address the non-convexity of jerk constraints in trajectory optimization for industrial robotic manipulators. Note to Practitioners—This work was motivated by the challenge of generating smooth and time-optimal trajectories for industrial manipulators while considering jerk and dynamic constraints. Existing approaches typically use multi-objective methods that treat jerk constraints as soft constraints. In contrast, this work proposes a B-spline-based convex optimization method that treats jerk constraints as hard constraints. Three case studies are conducted, involving butterfly-type, door-type and ‘OPTEC’-type paths, with the UR5 cooperative robot. The numerical and experimental results demonstrate that the proposed method sacrifices some time optimality to achieve trajectory smoothness and computational efficiency due to the convex restriction reformulation of the jerk constraints. Moreover, the proposed method is applicable to general industrial manipulators. Supplementary video is available at https://youtu.be/gVz5IuPKD-o.

This paper proposes a compact and reconfigurable variable stiffness actuator (VSA) using disc spring which is named as SDS-VSA (symmetrical disc spring variable stiffness actuator). To enhance the actuator’s torque density, symmetric compression springs are integrated into the cam-roller-spring mechanism, and a disc spring combination design is employed instead of conventional springs. The disc spring configuration is tailored to achieve a broader stiffness range within a limited size, aligned with stiffness and dimensional requirements. Subsequently, the dynamics of the cam-roller-spring mechanism are derived. To tackle the challenge of strong coupling dynamics, a decoupled modeling method by introducing mismatched and matched disturbances is proposed. A back-stepping tracking controller and proportional-derivative (PD) controller with feedforward are proposed to track the link-side and stiffness motor-side trajectories, respectively. Tracking experiments under significant stiffness alteration are conduced to verify the performance of the prototype SDS-VSA.

To obtain excellent comprehensive performances of the planar parallel manipulator for the high-speed application, an integrated optimal design method, which integrated dimensional synthesis, motors/reducers selection, and control parameters tuning, is proposed, and the 3RRR parallel manipulator was taken as the example. The kinematic and dynamic performances of condition number, velocity index, acceleration capability, and low-order frequency are taken into accounts for the dimensional synthesis. Then, to match motors/reducers parameters and keep an economical cost, the constraint equations and the parameters library are built, and the cost is chosen as one of the optimization objectives. Also, to get high tracking accuracy, the dynamic forward plus proportional–derivative control scheme is introduced, and the tracking error is chosen as one of the optimization objectives. Hence, the optimization model including dimensional synthesis, motors/reducers selection and controller parameters tuning is established, which is solved by the genetic algorithm II (NSGA-II). The result shows that comprehensive performances can be effectively promoted through the proposed integrated optimal design, and the prototype was constructed according to the Pareto-optimal front.

An efficient dynamic modelling approach was presented for planar parallel manipulator with flexible links. To increase the accuracy of the model, an improved curvature-based finite element method (ICFE) was developed for discretisation of the flexible links. Then, a novel approach for analysis of the coupling between rigid-body motion and flexible-body motion was proposed, and compared to the regular geometrical method, the proposed method was accurate and easy to implement. With the aforementioned proposed methods, the Kane equation was integrated to formulate the dynamic model of a 3RRR planar parallel manipulator. Finally, comparison studies were performed to validate the proposed ICFE and the integrated dynamic modelling method. Compared to the regular curvature-based finite element method (CFE), the ICFE exhibits improved accuracy with equivalent degrees of freedom. Additionally, the proposed integrated dynamic model shows a good agreement with the Abaqus model. Therefore, it was concluded that the proposed dynamic modelling method herein was efficient and accurate for parallel manipulators with flexible links, demonstrating reasonable potentials for model based control.

To obtain the closed-form forward kinematics of parallel robots, researchers use algebra-based method to transform and simplify the constraint equations. However, this method requires a complicated derivation that leads to high-order univariate variable equations. In fact, some particular mechanisms, such as Delta, or H4 possess many invariant geometric properties during movement. This suggests that one might be able to transform and reduce the problem using geometric approaches. Therefore, a simpler and more efficient solution might be found. Based on this idea, we developed a new geometric approach called geometric forward kinematics (GFK) to obtain the closed-form solutions of H4 forward kinematics in this paper. The result shows that the forward kinematics of H4 yields an eighth degree univariate polynomial, compared with earlier reported 16th degree. Thanks to its clear physical meaning, an intensive discussion about the solutions is presented. Results indicate that a general H4 robot can have up to eight nonrepeated real solutions for its forward kinematics. For a specific configuration of H4, the nonrepeated number of real roots could be restricted to only two, four, or six. Two traveling plate configurations are discussed in this paper as two typical categories of H4. A numerical analysis was also performed for this new method.