Five-axis toolpaths are typically formatted as G01 commands by breaking curves into linear segments before being inputted into the computer numerical control systems. Despite the availability of various trajectory generation methods, real-time generation of smooth trajectories with limited smoothing error and constrained high-order kinematics remains a challenge. This paper proposes a velocity profile blending-based interpolation method for G01 paths of five-axis hybrid robots to obtain jerk-limited trajectories with constrained smoothing error. Each linear segment is represented by a 13-phase velocity profile instead of the widely adopted S-shape curve. The velocity profiles are locally blended to construct a smooth trajectory. For adjacent velocity profiles, the first three phases of the following profile strictly align with the last three phases of the preceding one. In the implementation, a bidirectional scanning algorithm is adopted to generate the velocity profile of each linear segment with the time constants of the overlapping phases pre-optimized. In the simulations and experiments, the effectiveness and benefits of the proposed method on G01 paths comprised of both long segments and short segments are validated through a comparative analysis with several representative trajectory generation methods. Note to Practitioners—This work aims to generate error-constrained and jerk-limited trajectories with high machining efficiency from G01 commands in real time for five-axis hybrid robots. Computer-aided manufacturing software typically exports five-axis toolpaths as G01 commands. Next, computer numerical control systems read the G01 commands and produce trajectories with limited error and constrained kinematics as reference commands to the drivers. To obtain smooth trajectories from G01 commands for five-axis mechanisms more efficiently, one-step trajectory generation methods based on finite impulse response (FIR) filtering are increasingly studied. Nevertheless, due to their simplistic velocity profiles, existing FIR filtering-based one-step trajectory generation approaches struggle to simultaneously enhance machining efficiency, limit blending errors, and constrain the mechanism’s high-order kinematics. Thus, we propose a more general velocity profile blending-based interpolation method that expresses the movements along linear segments as 13-phase velocity profiles rather than the S-shape velocity profiles generated by the FIR filters. Accordingly, a novel strategy is developed to efficiently determine the velocity profiles. The proposed method, implemented on a specific hybrid machining robot, was compared with a spline-based trajectory generation approach and three representative FIR filtering-based approaches. Simulations and experiments were performed, in which G01 paths constituted by both long and short segments were used to validate our method. The results suggest that our method offers several advantages at the same time: low computational cost, significant improvements in machining efficiency, strictly constrained blending error, and precisely limited robot kinematics. Furthermore, this method does not make any assumptions about the structure of the five-axis mechanism, allowing for its application to general five-axis mechanisms. In industrial applications, the proposed method can be embedded in the computer numerical control system for real-time implementation.

Increasing the viscosity of elastic joints can significantly improve the performance of elastic joint robots during physical human–robot interactions. However, current approaches for injecting viscous elements require an additional damper to be added in parallel with the elastic elements. In this paper, we propose a new concept called virtual viscous element injection (VVI), which enables a robot to exhibit viscoelasticity without altering its mechanical structure. VVI relies only on motor-side dynamics reshaping and state feedback. Interestingly, the VVI method allows high-resolution joint torque measurements in elastic joint robots, unlike in physical viscoelastic joint robots, which measure joint torque using higher-order derivatives of the positions. Furthermore, the VVI method is proved to preserve the passivity of robot dynamics, which provides numerous possibilities for the applications of combined passivity-based controllers. Specifically, we first emphasize the impedance control method using VVI. The results demonstrate that the VVI-DF method, which combines the direct feedback (DF) method with VVI, addresses the issue of excessive acceleration feedback in the controller. This provides looser constraints for achieving a high-gain torque loop in impedance control. Moreover, this paper also provides examples of the application of VVI combined with passivity-based position and torque controllers. Experiments and simulations demonstrate the effectiveness of the proposed methods. The proposed method can be extended to various robots, such as exoskeletons, and collaborative robots.

Multi-manipulator systems are becoming increasingly indispensable in modern manufacturing scenes. To achieve the highest accuracy in a dual-robot manufacturing system, it is essential to calibrate the transformation relationships between several coordinate frames, namely hand-to-eye, base-to-base, and tool-to-flange transformations. This process is classified as the problem of$\mathbf{AXB}=\mathbf{YCZ}$. In this article, we propose a simultaneous calibration method based on geometric algebra and geometric calculus as a solution to this problem. The proposed method consists of three steps. In the first step, an initial value for the rotation part is found in closed form. Starting from this reasonable initial value, a nonlinear constrained optimization problem is defined and solved using a stochastic gradient descent algorithm. The iterative part is formulated entirely in geometric algebra elements instead of matrices, and the gradient is calculated using geometric calculus instead of vector calculus. These substitutions enable the proposed method to find the optimal result with less than 1/3.5 of the computational footprint and to achieve approximately 4.5 times the runtime speed boost compared to previous methods. The translational component is derived last by solving a least square problem. Simulations and experiments are conducted to test the performance of the proposed method, and the results verify the proposed method’s capability to find optimal solutions and clear superiority in computational time.Note to Practitioners— A robotic manufacturing system requires calibration before deployment to achieve the best processing accuracy. In a dual-robot scenario, determining the hand-eye, base-base, and tool-flange transformation relationships is necessary. In this article, we propose a novel method that can perform this calibration efficiently and effectively. The proposed method employs a new mathematical tool called geometric algebra to encode rotations, which enables this method to find the optimal result with significantly less time and memory. Further experiments show that the proposed method achieves identical accuracy performance but runs approximately 4.5 times faster than previous approaches. Such outstanding efficiency makes this proposed method a great approach for systems that require frequent calibration. The final measurement experiments also showed that the system calibrated by the proposed method can achieve an accuracy of 0.3mm.

In this article, the stiffness mapping of compliant robotic systems is generalized to the special Euclidean group SE(3). A geometric framework is proposed to unify the existing stiffness models. We analyze the symmetry and exactness relationship between joint and Cartesian stiffness matrices in this framework. To verify the theoretical results, motions of different types of manipulators, including serial and parallel ones, are tested in simulations. Based on the conservative property of the stiffness matrix, an impedance control strategy to achieve variable stiffness is proposed. In addition, a feasible stiffness identification method is developed using the skew-symmetric structure of the stiffness matrix.

Off-line programming of the polishing tool trajectory for complex workpieces is challenging due to the nontrivial material removal model and the polishing accuracy requirement. Current tool trajectory planning methods are mainly developed for some simple surfaces but cannot handle the increasingly complicated industrial parts, such as the wheel hubs. This article first develops a numerical contact mechanics model for the point-sampled complex workpieces. The contact pressure distribution and the material removal depths on the workpiece point cloud can be predicted efficiently. A novel high-priority subregion searching algorithm is developed to track the most-worth-polishing workpiece points. By selecting the path pattern as direction-parallel, the path direction, tool dwell times, and the path spacings inside each extracted subregion are optimized to minimize the deviation from the desired material removal depths. The effectiveness of the proposed method is verified by performing disk polishing simulations on workpieces with different shapes. A robotic polishing experiment is also conducted on a wheel hub. Both simulation and experimental results show that reasonable tool trajectories can be generated on the workpiece, and the desired material removal depths can be achieved. Note to Practitioners—In robotic polishing industries, it is crucial to plan the tool trajectory (tool path and feed velocity) to achieve desired material removal depths on the workpiece surface, which means high surface quality. In this article, a model-based tool trajectory planning method for robotic polishing of complex surfaces that are represented by the point cloud form is presented. The advantage of using the point cloud is that workpiece surfaces with varying curvatures and complex features, e.g., grooves and holes, need not be expressed explicitly. The proposed method generates high-priority subregions according to the updated material removal distribution dynamically. In this work, the polishing path pattern is chosen as direction-parallel. Based on an efficient numerical contact mechanics and material removal model, the path locations and the tool dwell times inside each subregion are optimized to minimize the deviation between the actual and the desired material removal depths. When the desired material removal depths are attained in an extracted subregion, the algorithm finds the next high-priority subregion until the whole workpiece is well polished. The trajectory planning method can be integrated into an industrial robot with the force-control module. Future work is to integrate the roughness model into the tool trajectory planning method.

In this article, an artificially weighted spanning tree coverage (AWSTC) algorithm is proposed for the distributed path planning of multiple flying robots. To balance well the efficiency, redundancy, and robustness in the cooperative coverage problem, each robot simultaneously constructs its spanning tree, which grows toward the center of inertia of the uncovered area and keeps away from the trees of its partners. Based on this, each robot covers a concerned area with almost equal trajectory length and computational burden. To guarantee dynamical consistency, a trajectory-smoothing method is developed utilizing Bézier curve transition. As transition in the spanning tree path planning is always conducted around the corner with right angle, geometric and velocity profiles of this kind of transition can be preevaluated as a basic component and then connected to establish more complex real-time trajectories according to the constructed spanning trees. Numerical evaluations and real-time flight experiments are carried out at last. Results demonstrate that the proposed strategy can generate a smooth trajectory for an area coverage problem while ensuring efficiency and robustness. In particular, the efficiency of the proposed algorithm is quite close to the typical centralized spanning tree coverage (STC) algorithm, while ensuring the fulfillment of the coverage task regardless of any breakdown in the individual robot.Note to Practitioners—In applications such as infrastructure inspection and plant protection, the concerned area or terrain is required to be fully covered with the smallest possible time consumption. In view of these real-world requirements, multiple unmanned systems with proper path planning provide a promising solution that can significantly enhance the efficiency as well as the robustness, compared with a single-robot system. Especially, centralized STC algorithms, concerning the computation efficiency and trajectory redundancy of multiple unmanned systems, demonstrate satisfactory effectiveness, and thereby receive a lot of attention. Unfortunately, most of the STC algorithms are proposed in a centralized control framework. Therefore, it is worth to study its decentralized version that could exploit the advantages of the distributed multiple robots. With this consideration, a decentralized AWSTC algorithm is proposed in this article. This spanning tree-based strategy artificially assigns priority weight for each cell, with respect to each individual robot, in the concerned area. With specially designed weight assignment, each robot tends to construct a spanning tree with cells that are uncovered but keep away from the trees of its partners. Based on this, the task burden as well as computational cost of each robot are almost equal. To improve the real-time performance, a Bézier transition trajectory-generation method is also used. With proper parameter selection, robots can generate smooth trajectory with constant velocity in real-time missions. Numerical evaluations and experiments with real-time flight demonstrate the effectiveness of this methodology.

This paper presents a trajectory tracking control strategy based on the subspace stabilization approach to accurately manipulate an under-actuated flying robot from a known initial state to the desired terminal state. To facilitate the development of this tracking strategy, the dynamical model of the quadrotor is firstly proposed. Subsequently, an optimal trajectory generation algorithm is adopted to generate dynamically consistent trajectories regarding the initial and terminal state constraints in specific missions. Then, a trajectory tracking control strategy based on the subspace stabilization approach is developed considering the lumped disturbances and time delays. The developed control strategy is applied for ball juggling of a highly under-actuated quadrotor, which is a popular flying robot in recent years. Real-time experimental results show that the quadrotor can be accurately manipulated from a known initial state to the desired terminal state within a given time horizon. In the consecutive juggling tasks, the quadrotor with a racket of radius 0.065 m can consecutively juggle the ball for averagely 4 hits in each rally, and a longest rally achieved by the developed control strategy is 14 hits. The feasibility of the developed control strategy is also preliminarily verified through the cooperative juggling between two quadrotors. All of these results demonstrate the effectiveness of the developed control strategy.

The aim of this paper is to develop an integral equation based spectral method for prediction of chatter stability in low radial immersion milling. First, the delay-differential equation with time-periodic coefficients governing the dynamic milling process is transformed into the integral equation. Then, the duration of one tooth period is divided into the free vibration and the forced vibration processes. While the former one has an analytical solution, the discretization technique is explored to approximate the solution of the latter one. After the forced vibration duration being equally discretized, the Gauss-Legendre formula is used to discretize the definite integral, in the meantime the Lagrange interpolation is adopted for approximating the state item and the time-delay item by using the corresponding discretized state points and time-delay state points. The approximate Floquet transition matrix is thereafter constructed to predict the milling stability based on the Floquet theory. The benchmark examples are utilized to verify the proposed method. Compared with previous time domain methods, the proposed method enables higher rate of convergence. The results also demonstrate that the proposed method is high-effective.