Trajectory Planning Of Robot Manipulators Research Articles

An Optimal Control Approach to the Minimum-Time Trajectory Planning of Robotic Manipulators

Trajectory planning is a classic problem in robotics, with different approaches and optimisation objectives documented in the literature. Most of the time, the path is assumed, i.e., pre-defined, and optimisation consists of finding the timing of motion under a number of constraints. The focus of this work is on the minimum-time manoeuvring of robotic manipulators. A nonlinear optimal control approach is proposed that does not require the provision of either a pre-defined path or a pre-defined control structure and allows the inclusion of dynamic constraints. The solution (path and timing of motion) is obtained by transforming the optimal control problem into a nonlinear programming problem. The proposed approach is applied to a two-link manipulator for illustration purposes. The optimisation is carried out both without and with obstacles. The minimum-distance and minimum-time solutions are compared, and some classic results are obtained, including the trapezoidal pattern of the joint velocity and the bang–bang structure of the control torques. The effects of limitations on the jerks of actuators and the rate of change in torque inputs are discussed. The application to a four-link manipulator is also included to show the ‘scalability’ of the approach, together with a comparison with a classic path-and-motion-planning method, to highlight the characteristics and performance of the proposed approach. Finally, the possibility of enforcing a number of via-points along the path is demonstrated. The proposed method allows the computation of the path and motion simultaneously with the computation time, which is 1–30 times the manoeuvre time, on a standard PC with the current implementation.

Smooth and proximate time-optimal trajectory planning of robotic manipulators

The simultaneous achievement of high accuracy and productivity in robotics has been extensively investigated using the trajectory planning approach. We propose a smooth and proximate time-optimal trajectory planning method. First, we generated the velocity limit curve by expressing joint torque and joint velocity constraints as a function of the path coordinate. Then, we calculated the time-optimal trajectory using a dynamic programming technique and fitted the smooth cubic spline. The smoothing factor of the cubic spline curve was optimized by applying a binary recursive. The proposed method can guarantee the continuity and smoothness of joint torque. The simulation results showed that the proposed method can reduce joint-jerk and improve the tracking accuracy of the controller.

Trajectory Planning of Robot Manipulator Based on RBF Neural Network.

Robot manipulator trajectory planning is one of the core robot technologies, and the design of controllers can improve the trajectory accuracy of manipulators. However, most of the controllers designed at this stage have not been able to effectively solve the nonlinearity and uncertainty problems of the high degree of freedom manipulators. In order to overcome these problems and improve the trajectory performance of the high degree of freedom manipulators, a manipulator trajectory planning method based on a radial basis function (RBF) neural network is proposed in this work. Firstly, a 6-DOF robot experimental platform was designed and built. Secondly, the overall manipulator trajectory planning framework was designed, which included manipulator kinematics and dynamics and a quintic polynomial interpolation algorithm. Then, an adaptive robust controller based on an RBF neural network was designed to deal with the nonlinearity and uncertainty problems, and Lyapunov theory was used to ensure the stability of the manipulator control system and the convergence of the tracking error. Finally, to test the method, a simulation and experiment were carried out. The simulation results showed that the proposed method improved the response and tracking performance to a certain extent, reduced the adjustment time and chattering, and ensured the smooth operation of the manipulator in the course of trajectory planning. The experimental results verified the effectiveness and feasibility of the method proposed in this paper.

A General Framework for Optimal Tuning of PID-like Controllers for Minimum Jerk Robotic Trajectories

The minimum jerk principle is commonly used for trajectory planning of robotic manipulators. However, since this principle is stated in terms of the robot’s kinematics, there is no guarantee that the joint controllers will actually track the planned acceleration and jerk profiles because the tuning of the controllers’ gains is decoupled from the trajectory planning. Bearing this in mind, in this paper we introduce a comprehensive framework for optimal estimation of the gains of PID-like controllers for tracking minimum-jerk (MJ) robot trajectories. The proposed methodology relies mainly on a novel variant of error-based performance indices (ISE, ITSE, IAE and ITAE) which are adapted to the tracking of MJ trajectories. Furthermore, the particle swarm optimization (PSO) algorithm is used to search for optimal values for the gains of the controllers of all joints simultaneously. The resulting approach is much simpler than recent developments based on more complex performance indices, in which joint controllers were individually optimized. The proposed approach is general enough to easily encompass the tuning of fractional PID controllers and a comprehensive set of experiments are reported comparing the performances of standard and fractional PID controllers for the task of interest.

A New Trajectory-Planning Beetle Swarm Optimization Algorithm for Trajectory Planning of Robot Manipulators

Based on a heuristic optimization algorithm, this paper proposes a new algorithm named trajectory-planning beetle swarm optimization (TPBSO) algorithm for solving trajectory planning of robots, especially robot manipulators. Firstly, two specific manipulator trajectory planning problems are presented as the practical application of the algorithm, which are point-to-point planning and fixed-geometric-path planning. Then, in order to verify the effectiveness of the algorithm, this paper develops a control model and conducts numerical experiments on two planning tasks. Moreover, it compares with existing algorithms to show the superiority of our proposed algorithm. Finally, the results of numerical comparisons show that algorithm has a relatively faster computational speed and better control performance without increasing computational complexity.

Optimal time-jerk trajectory planning for industrial robots

A methodology for time-jerk synthetic optimal trajectory planning of robotic manipulators is described in this paper. The trajectory is interpolated in the joint space by means of 5th-order B-spline and then optimized by the elitist non-dominated sorting genetic algorithm (NSGA-II) for two objectives, namely, traveling time and mean jerk along the whole trajectory. 5th-order B-spline interpolation technique enables the trajectory to be constrained in the kinematic limits of velocity, acceleration, and jerk while satisfying the continuity of jerk. NSGA-II as a multi-objective optimization technique is used to address the time-jerk optimal trajectory planning problem. The obtained Pareto optimal front provides decision-makers flexible selections on non-dominated solutions for industrial applications. Two performance measures are presented to evaluate the strength of the Pareto optimal front and to select the best optimal solution respectively. Simulations and experiments validate the effectiveness and practicability of the proposed methodology in comparison with those provided by another important trajectory planning methodology.

An efficient stochastic approach for robust time-optimal trajectory planning of robotic manipulators under limited actuation

SUMMARYThis paper focuses on the problem of robust time-optimal trajectory planning of robotic manipulators to track a given path under a probabilistic limited actuation, that is, the probability for the actuation to be limited is no less than a given bound κ. We give a general and practical method to reduce the probabilistic constraints to a set of deterministic constraints and show that the deterministic constraints are equivalent to a set of linear constraints under certain conditions. As a result, the original problem is reduced to a linear optimal control problem which can be solved with linear programming in polynomial time. In the case of κ = 1, the original problem is proved to be equivalent to the linear optimal control problem. Overall, a very general, practical, and efficient algorithm is given to solve the above problem and numerical simulation results are used to show the effectiveness of the method.

A convex relaxation for the time-optimal trajectory planning of robotic manipulators along predetermined geometric paths

A convex relaxation for the time-optimal trajectory planning of robotic manipulators along predetermined geometric paths

Tractable Algorithm for Robust Time-Optimal Trajectory Planning of Robotic Manipulators under Confined Torque

In this paper, the problem of time optimal trajectory planning under confined torque and uncertain dynamics and torque parameters along a predefined geometric path is considered. It is shown that the robust optimal solution to such a problem can be obtained by solving a linear program. Thus a tractable algorithm is given for robust time-optimal path-tracking control under confined torque.

A Mathematical Modeling and Optimization Approach for Trajectory Planning of Robot Manipulators

This paper presents a method for the problem of optimal trajectory planning of redundant robot manipulators in the presence of fixed obstacles. Quadrinomial and quintic polynomials are used to describe the segment of the trajectory. Cultural based PSO algorithm (CBPSO) is proposed to design a collision-free trajectory for planar redundant manipulators. CBPSO optimizes the trajectory and ensures that obstacle avoidance can be achieved. Simulations are carried out for different obstacles to prove the validity of the proposed algorithm. Different test data generated by GA, QPSO and CBPSO are provided with a tabular comparison. Simulation studies show CBPSO has potential online usage in engineering and distinct fast computation speed compared with the other two algorithms. Results demonstrate the effectiveness and capability of the proposed method in generating optimized collision-free trajectories.

Time-optimal trajectory planning of robot manipulators in point-to-point motion using an indirect method

In this study, an iterative method for computing the time-optimal point-to-point control of robot manipulators is studied, subject to limits on the actuator torques and actuator jerks. This method uses the indirect solution of an open-loop optimal control problem so that the optimal control problem is translated to a non-linear two-point boundary value problem (TPBVP). Since there are many difficulties in finding the switching points and solving the TPBVP, a simple iterative method is proposed. This method is based on solving the minimum-energy problem without the necessity of finding the switching point. So, the resultant TPBVP can be solved using the usual algorithms. In addition, since the solution of the TPBVPs is sensitive to the initial guess, a simple procedure for making proper initial guesses is introduced. Furthermore, since the inverse kinematic of redundant manipulators is an ill-posed problem, the general form for boundary conditions corresponding to the final time is considered. In addition, it is shown that the proposed method has no sensitivity for specifying the time-optimal trajectory of manipulators that may contain singular arcs.

Global Time-energy Optimal Planning of Industrial Robot Trajectories

A technique for optimal trajectory planning of robot manipulators is presented. It consists of linking two points in the operational space while minimizing a cost function, taking into account dynamic equations of motion as well as bounds on joint velocities, accelerations, jerks and force/torques. The cost function is used as a weighted balance of traveling time and mechanical energy of the actuators. Also, a novel ranking technique for the penalty function is designed to deal with the constraints. Furthermore, the environment-gene evolutionary immune clonal algorithm (EGICA) is proposed to solve the optimization problem. The use of the above-mentioned strategy makes the algorithm have a certain learning ability and enhances its global searching ability, thus improving solution quality and algorithm efficiency. The algorithm is tested with Stanford robot in simulation, and the result shows that the presented method has higher search efficiency and can obtain better solution in comparison with the existing methods.

Optimal trajectory planning for industrial robots

An analysis of the results of an algorithm for optimal trajectory planning of robot manipulators is described in this paper. The objective function to be minimized is a weighted sum of the integral squared jerk and the execution time. Two possible primitives for building the trajectory are considered: cubic splines or fifth-order B-splines. The proposed technique allows to set constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. The described method is then applied to a 6-d.o.f. robot (a Cartesian gantry manipulator with a spherical wrist); the results obtained using the two different primitives are presented and discussed.

Multi-objective dynamic optimal trajectory planning of robot manipulators in the presence of obstacles

In this paper, a real-world test problem is presented and made available for the use of evolutionary multi-objective community. The generation of manipulator trajectories by considering multiple objectives and obstacle avoidance is a non-trivial optimisation problem. In this paper two multi-objective evolutionary algorithms viz., elitist non-dominated sorting genetic algorithm (NSGA-II) and multi-objective differential evolution (MODE) algorithm are proposed to address this problem. Multiple criteria are optimised to two simultaneous objectives. Simulations results are presented for industrial robots with two degrees of freedom (Cartesian robot (PP) with two prismatic joints) and six degrees of freedom (PUMA 560 robot), by considering two objectives optimisation. Two methods (normalized weighting objective functions and average fuzzy membership function) are used to select the best optimal solution from Pareto optimal fronts. Two multi-objective performance measures namely solution spread measure and ratio of non-dominated individuals are used to evaluate the strength of Pareto optimal fronts. Two more multi-objective performance measures namely optimiser overhead and algorithm effort are used to find computational effort of NSGA-II and MODE algorithms. The Pareto optimal fronts and results obtained from various techniques are compared and analysed.

A technique for time-jerk optimal planning of robot trajectories

A technique for optimal trajectory planning of robot manipulators is presented in this paper. In order to get the optimal trajectory, an objective function composed of two terms is minimized: a first term proportional to the total execution time and another one proportional to the integral of the squared jerk (defined as the derivative of the acceleration) along the trajectory. This latter term ensures that the resulting trajectory is smooth enough. The proposed technique enables one to take into account kinematic constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. Moreover, it does not require the total execution time of the trajectory to be set a priori. The algorithm has been tested in simulation yielding good results, also in comparison with those provided by another important trajectory planning technique.

A new method for smooth trajectory planning of robot manipulators

A new method for smooth trajectory planning of robot manipulators is described in this paper. In order to ensure that the resulting trajectory is smooth enough, an objective function containing a term proportional to the integral of the squared jerk (defined as the derivative of the acceleration) along the trajectory is considered. Moreover, a second term, proportional to the total execution time, is added to the expression of the objective function. In this way it is not necessary to define the total execution time before running the algorithm. Fifth-order B-splines are then used to compose the overall trajectory. With respect to other trajectory optimization techniques, the proposed method enables one to set kinematic constraints on the robot motion, expressed as upper bounds on the absolute values of velocity, acceleration and jerk. The algorithm has been tested in simulation yielding good results, which have also been compared with those provided by another important trajectory planning technique.

Novel integrated optimization algorithm for trajectory planning of robot manipulators based on integrated evolutionary programming

Optimal trajectory planning for robot manipulators plays an important role in implementing the high productivity for robots. The performance indexes used in optimal trajectory planning are classified into two main categories: optimum traveling time and optimum mechanical energy of the actuators. The current trajectory planning algorithms are designed based on one of the above two performance indexes. So far, there have been few planning algorithms designed to satisfy two performance indexes simultaneously. On the other hand, some deficiencies arise in the existing integrated optimi2ation algorithms of trajectory planning. In order to overcome those deficiencies, the integrated optimization algorithms of trajectory planning are presented based on the complete analysis for trajectory planning of robot manipulators. In the algorithm, two object functions are designed based on the specific weight coefficient method and ' ideal point strategy. Moreover, based on the features of optimization problem, the intensified evolutionary programming is proposed to solve the corresponding optimization model. Especially, for the Stanford Robot,the high-quality solutions are found at a lower cost.

Trajectory Modeling of Robot Manipulators in the Presence of Obstacles

This paper presents two different strategies for the problem of the optimal trajectory planning of robot manipulators in the presence of fixed obstacles. The first strategy is related to the situation where the trajectory must pass through a given number of points. The second strategy corresponds to the case where only the initial and final points are given. The optimal traveling time and the minimum mechanical energy of the actuators are considered together to build a multiobjective function. The trajectories are defined using spline functions and are obtained through offline computation for online operation. Sequential unconstrained minimization techniques (SUMT) have been used for the optimization. The obstacles are considered as three-dimensional objects sharing the same workspace performed by the robot. The obstacle avoidance is expressed in terms of the distances between potentially colliding parts. Simulation results are presented and show the efficiency of the general methodology used in this paper.

Trajectory planning of robot manipulators by using algebraic and trigonometric splines

In this paper, the use of algebraic and trigonometric splines for the trajectory planning of robot manipulators is discussed. First, the two methods are analyzed and compared in detail; then, a strategy, which involves a combined use of the two schemes to perform sudden changes in a predefined trajectory (e.g. in case of obstacle avoidance) is proposed. Results show that the main interest in using trigonometric splines lies especially in the task of connecting two separate pieces of cubic splines, as overshoots are significantly reduced, although the continuity of velocity, acceleration and (in case) jerk is guaranteed.

Optimal trajectory planning of robot manipulators in the presence of moving obstacles

A method for computing the optimal motions of robot manipulators in the presence of moving obstacles is presented. The algorithm considers the nonlinear manipulator dynamics, actuator constraints, joint limits and obstacle avoidance. The optimal traveling time and the minimum mechanical energy of the actuators are considered together to build a multicriterion function. Sequential unconstrained minimization techniques have been used for the optimization. Given the initial and final points the trajectories are defined using spline functions and are obtained through off-line computation for on-line operation. The obstacles are considered as objects sharing the same workspace performed by the robot. The obstacle avoidance is expressed in terms of the distances between potentially colliding parts and the motion is represented using translation and rotational matrices. Numerical applications involving a Stanford manipulator are presented.