Time-optimal Feedrate Research Articles

An Efficient Linear Programming-Based Time-Optimal Feedrate Planning Considering Kinematic and Dynamics Constraints of Robots

An Efficient Linear Programming-Based Time-Optimal Feedrate Planning Considering Kinematic and Dynamics Constraints of Robots

A jerk-limited heuristic feedrate scheduling method based on particle swarm optimization for a 5-DOF hybrid robot

Compared with machine tools, five degrees-of-freedom (DOFs) hybrid robots have been widely concerned in the manufacturing of large complex surface parts due to their characteristics of high flexibility and large workspace. It is of great significance to schedule the time-optimal feedrate that satisfies the high-order constraints (e.g., jerk or jounce) in the toolpath and joint systems to achieve the high-precision and high-efficiency machining of the robot. To overcome the complexity of five-axis feedrate scheduling and improve the optimality of machining time, this paper proposes a jerk-limited heuristic feedrate scheduling (HFS) method with near-optimal time. Firstly, the analytical equations between all constraints and the parametric feedrate are derived, and the mathematical model for control points optimization of the parametric feedrate profile expressed by a B-spline curve is established. Then, combined with the global search particle swarm optimization (GSPSO) algorithm, a proposed moving window planning method optimizes a small number of control points so that the feedrate curve has enough rising space. Subsequently, to obtain the near time-optimal feedrate, the local search PSO (LSPSO) algorithm within the moving window is developed to locally adjust the non-uniformly inserted control points. Compared with the existing methods, the HFS method is beneficial to further optimize the machining time with a faster computation speed, and ensure the stability of the robot machining. Finally, simulations and experiments on the developed TriMule-800 hybrid robot verify the effectiveness of this method.

Generalised NURBS interpolator with nonlinear feedrate scheduling and interpolation error compensation

Current nonuniform rational B-spline interpolating methods encounter problems pertaining to the interpolating error, feedrate profile, and machining efficiency; the optimal feedrate profile may not be obtained owing to the complex kinematics of multi-axis machine tools with hybrid structures. To address these problems and obtain an optimal feedrate profile with minimum machining time, this paper establishes a generalised representation for the nonlinear feedrate scheduling model, with static and dynamic feedrate constraints, through a numerical convex transformation. Based on the generalised model representation, a time-optimal feedrate profile can then be solved globally for both series- and parallel-structured mechanisms without problems caused by profile linear simplification or imprecise feedrate evolution. Considering the feedrate fluctuation problem caused by the difference between the theoretical arc length of the curve interpolation and the actual tool path, this paper proposes a novel co-lateral triangle deviation (CTD)-based parameter compensation interpolation algorithm. With the CTD calculation, which is related to the actual feed motion in real time, the proposed interpolation algorithm can determine the interpolating compensation distance for each interpolating segment and obtain the interpolation points using a second-order Taylor series expansion with a minimal interpolating error. Carried on a hybrid mechanism with complex kinematic constraints, the experimental and comparison results validated the effectiveness of the proposed method in feedrate scheduling and feedrate fluctuation reduction.

A Third-Order Constrained Approximate Time-Optimal Feedrate Planning Algorithm

Robot end-effector tracking of a specific spatial path to fulfill certain requirements is a topic of great interest. First, considering both dynamic constraints and kinematic constraints, the planned velocity has characteristics of bounded high-order derivatives to ensure smooth motion. Second, to meet high-efficiency requirements, the feedrate of the end-effector should be planned within the best possible time. Although some studies have shown that they obtained the time-optimal solution by considering the second-order constraints, this is still an open problem for third-order constraints. Therefore, this article proposes an approximate time-optimal feedrate (ATOF) planning algorithm that considers third-order constraints. This algorithm is primarily based on the bidirectional scanning method, which includes two steps. First, we calculate the second-order time-optimal velocity (TOV2) curve under the third-order maximum velocity curve. Then, considering TOV2, we propose an ATOF algorithm that includes single-direction speed planning and speed curve smoothing. It is more practical for not only considering the Coulomb viscous friction and torque rate but also avoiding difficult calculations at the singular point and solves discontinuous joint-acceleration under second-order constraints. Finally, we design two sets of experiments to verify the higher calculation efficiency and effectiveness of the proposed ATOF algorithm.

GA-based time-optimal feedrate scheduling on NURBS tool path with geometric and drive constraints

GA-based time-optimal feedrate scheduling on NURBS tool path with geometric and drive constraints

Global time-optimal B-spline feedrate scheduling for a two-turret multi-axis NC machine tool based on optimization with genetic algorithm

Feedrate scheduling is of great significance in numerical control machining for the improvement of the surface finish and machining efficiency. For multi-axis machine tools with more than two rotary axes, the feedrate scheduling issue considering the geometry and drive constraints is highly complicated due to the nonlinear kinematic relationship between the cutter location and drive axes. In this paper, a general optimization-based time-optimal feedrate scheduling method for two-turret machine tools with three-rotary axes and one short linear axis is proposed. The closed-form solution for the inverse kinematics of the machine tool is developed to analytically determine the preferable motion of each drive axis for a desired motion of the cutter. By using the B-spline method to represent the feedrate profile, the optimization-based feedrate scheduling model is built, where the control points and knot vector are designated as the decision variables. Two alternative operations, namely the progressive knot insertion and genetic-algorithm-based optimization, are conducted to determine the knot vector and control points for a time-optimal feedrate, respectively. To improve the robustness and computational efficiency of the algorithm, a feedrate profile segmentation criteria is established for concurrent computing. Without any simplification of the constraints, the proposed method can be employed to produce a global time-optimal feedrate efficiently. Typical simulation cases and experiments are carried out to verify the effectiveness and time-optimality of the proposed method.

Time-optimal feedrate scheduling with actuator constraints for 5-axis machining

Cycle time minimization is one of the major goals to achieve in manufacturing. Maximizing feedrate is the direct solution; however, physical motions need to be under the specified motion limits to avoid high-frequency vibration, causing machining error. In this paper, a time-optimal feedrate scheduling approach with corner smoothing for 5-axis G1 toolpaths is presented for 5-axis machining. One key feature of the proposed method is to describe the orientation toolpath in the spherical coordinate system (SCS), instead of the commonly used unit sphere. With this, the position and orientation of the 5-axis toolpath are integrated to become a hyper-curve in a 5-dimensional (5-D) space. The feedrate scheduling and corner smoothing are designed for this 5-D hyper-curve. Thus, no synchronization of position and orientation feedrate scheduling is needed, making the feedrate scheduling easier. A quintic B-spline corner smoothing method is utilized to smoothen sharp corners in the toolpath. Then, the S-shape feedrate profile of each block is optimized under the actuator motion constraints, with the objective of minimizing the cycle time. Particle swarm optimization (PSO) is used to provide the optimized solution. Experiments are conducted to validate the proposed approach, and the results are compared with two other existing approaches. It is found that the proposed method can achieve shorter cycle time and less contour errors, showing the effectiveness of the proposed approach.

Modeling and simulation of trajectory smoothing and feedrate scheduling for vibration-damping CNC machining

The mathematical models of tool path smoothing and feedrate scheduling in Computer Numerical Control (CNC) system are critical for high-precision manufacture. Inthe existing studies, G2/G3 (curvature-continuous/-smooth) trajectory smoothing, and jerk-limited/-continuous feedrate scheduling schemes have been developed. Nonetheless, there are exist some requirements that cannot be satisfied simultaneously, including confined chord error, G01 points interpolated, analytical curvature extremum, real-time performance, kinematic time-optimality, and the smoothness of tool-path and feedrate profile. Recently the scholars found the potentials of the tool path with high-order geometric continuity and the feedrate with high-order kinematic constraints, respectively in increasing the smoothness of tool path and reducing the impact caused by axis actuators during the process. Aiming to reduce the vibration and guarantee high machining efficiency, this work proposes G4 (curvature-variation-smooth) interpolative trajectory model with confined chord error and analytical curvature extrema for trajectory smoothing, and employs jerk-smooth (jerk-differentiable) feedrate mode to perform time-optimal feedrate scheduling. Finally, a real-time tool path processing strategy under various geometric and kinematic constraints is developed. The simulation shows approximation error, curvature extrema, and feedrate fluctuation are reduced compared with G2 transition and G3 interpolation schemes. The experimental results demonstrate advantages of the proposed method in vibration damping, surface quality, compared with the previous works.

Contour error–bounded parametric interpolator with minimum feedrate fluctuation for five-axis CNC machine tools

In this paper, a new generalized parametric interpolation method is proposed for optimized five-axis machining with the consideration of both the machine contour errors and the feedrate fluctuation elimination. An analytic processing is presented for linearization of contour errors with respect to feedrate limit. For machine configuration, explicit analytical modeling of the contour error and the tracking error with respect to feedrate is presented; thus, the error constraint problem is nicely converted to a kinematic constraint problem. On this basis, an accurate feedrate upper limit with confined contour errors is further determined by using a shifted Frenet frame with linear computational complexity. With the consideration of both the motion smoothness and machining efficiency, a time-based optimization algorithm is proposed for time-optimal feedrate scheduling. For eliminating the feedrate fluctuation, a new real-time interpolation algorithm is developed for free derivation between the path parameter and the arc length for smoothed five-axis tool path generation. Laboratory testing experiments were conducted for validation and were presented in the paper. And the experimental results indicate that the proposed feedrate interpolation method is capable of confining both the tool tip contour error and tool orientation contour error simultaneously, as well as maintaining a satisfactory interpolation performance in both computation efficiency and accuracy. The presented methods can be used for five-axis machining and the feedrate optimization of complex part machining.

An iterative feed rate scheduling method with confined high-order constraints in parametric interpolation

Feed rate scheduling of complex tool path is critical for improvement of processing efficiency by fully exploiting the dynamic performance of machine tool in CNC machining. To overcome the complexity and low efficiency of conventional feedrate scheduling algorithm in computation, an iterative feed rate optimization method is proposed. In this work, the constraints that can reach the third-order derivative of motion in trajectory and drive system are established according to the differential operation of tool path and the feed rate profile. Besides, the feed rate expressed in the form of B-spline curve is adjusted iteratively by raising the control points from a zero-speed velocity curve. Specifically, in each iteration process, the control points are raised one by one through binary search technique to approach the permissive maximum with the constraints bounded. The iteration process terminates when the maximum increment of control points is less than the predefined threshold and then a near time-optimal feed rate is achieved. The scheduled feed rate derived by the iterative optimization method possesses both superior time-optimality and enough motion smoothness to the constraints without leading to excessive computation load. At last, the validity of the proposed method is verified by the simulation cases and the experiments.

An offline predictive feedrate scheduling method for parametric interpolation considering the constraints in trajectory and drive systems

The ability of drive unit limits the velocity, acceleration, and jerk of the axis, and the process and tool path geometry constrain the feedrate, acceleration, and jerk of the trajectory. It is of great difficulty and importance to schedule a suitable feedrate profile to achieve the higher machining efficiency and satisfy all the constraints. Due to the complexity of the feedrate scheduling algorithm, this paper presents an offline predictive feedrate scheduling method considering the constraints of trajectory system and drive system. The constraints in the drive system are transformed to the trajectory system appropriately, which reduces the difficulty of feedrate scheduling. The one-dimensional high-order time-optimal problem is solved by a numerical calculation method according to the bang-bang control method. The predictive method is applied to find the switching points of the jerk. Then a time-optimal feedrate profile under all the constraints is obtained. Both simulations and experiments are carried out to prove the effectiveness of the proposed offline predictive feedrate scheduling method.

Feedrate optimization for smooth minimum-time trajectory generation with higher order constraints

The generation of a time-optimal feedrate trajectory under various machine and process-related constraints has received significant attention in CNC machining and robotics applications. While most of the existing feedrate planning algorithms take velocity and acceleration into the consideration as capability constraints, the introduction of higher order dynamic states, such as jerk and/or jounce, makes the feedrate planning and optimization extremely challenging, as the dimension of the planning problem is increased accordingly. This paper proposes a heuristic trajectory planning algorithm that can provide a near-optimal minimum time trajectory for problems with higher order dynamic states. The algorithm starts with a non-time-optimal but feasible velocity trajectory, which is interpolated from a number of knot points by piecewise spline interpolation with high-order continuity. Then, the trajectory is improved by scanning and increasing the velocity at each knot points while maintaining the feasibility of the resulting trajectory. A near-optimal trajectory is achieved when the improvement in travel time from the last scan iteration is smaller than a given value. The algorithm supports the incorporation of higher order dynamic states (up to the fifth derivative of displacement) in constraints for optimization without sacrificing the computational efficiency. Examples including linear and curved toolpath are presented to illustrate the effectiveness of this algorithm for high-speed contouring.

Convex optimisation method for time-optimal feedrate planning with complex constraints

Time-optimal feedrate planning theory has been widely applied in high-speed machining to improve machining efficiencies with a specified trajectory accuracy by fully exploiting the capabilities of high-speed computer numerical control machine tools. Reasonable and effective constraints conforming to practical situations are necessary conditions for obtaining optimal planning results. In this article, complex constraints, such as the motor torque constraint, the contour error constraint, the friction error constraint, the instruction velocity constraint and the smooth velocity constraint, are investigated in depth. Moreover, an efficient convex optimisation method is developed to address the time-optimal feedrate planning problem with these complex constraints. The experimental results from two feedrate planning tests and the computational efficiency for a test trajectory demonstrate the validity, stability and high efficiency of the proposed approach.

A Smooth Trajectory Generation Algorithm for Addressing Higher-Order Dynamic Constraints in Nanopositioning Systems

The generation of a time-optimal feedrate trajectory has received significant attention in CNC machining and robotics applications. Most of the existing feedrate planning algorithms take velocity and acceleration into the consideration for capability constraints. The introduction of higher order dynamic states, such as jerk and/or jounce into the feedrate scheduling problem makes generating computationally efficient solutions while simultaneously guaranteeing optimality a challenging problem, as the dimension of the planning problem is increased accordingly. This paper proposes a heuristic trajectory planning algorithm that can provide a near optimal trajectory for problems with higher order dynamic states. The algorithm starts with a non-optimal but feasible velocity trajectory, which is interpolated from a number of knot points by piece-wise spline interpolation with high order continuity. Then the trajectory is improved by scanning the interpolating knot points and increasing the velocity at each knot points while maintaining the feasibility of the resulting trajectory. A near optimal trajectory is achieved when the improvement in travel time is neglectable from the last scan iteration. The algorithm supports the incorporation of high order dynamic states (up to fifth order derivative of position) in constraints for optimization without sacrificing the computational efficiency. Examples including linear and curved toolpaths are presented to illustrate the effectiveness of this algorithm for high-speed contouring.

Time optimal feedrate generation with confined tracking error based on linear programming

In this paper, the problem of time optimal feedrate generation under confined feedrate, axis accelerations, and axis tracking errors is considered. The main contribution is to reduce the tracking error constraint to constraints about the axis velocities and accelerations, when the tracking error satisfies a second order linear ordinary differential equation. Based on this simplification on the tracking error, the original feedrate generation problem is reduced to a new form which can be efficiently solved with linear programming algorithms. Simulation results are used to validate the methods.

Efficient time-optimal feedrate planning under dynamic constraints for a high-order CNC servo system

In this paper, the time-optimal feedrate planning problem under confined feedrate, axis velocity, axis acceleration, axis jerk, and axis tracking error for a high-order CNC servo system is studied. The problem is useful in that the full ability of the CNC machine is used to enhance the machining productivity while keeping the machining precision under a given level. However, the problem is computationally challenging. The main contribution of this paper is to approximate the problem nicely by a finite-state convex optimization problem which can be solved efficiently. The method consists of two key ingredients. First, a relationship between the tracking error and the input signal in a high-order CNC servo system is established. As a consequence, the tracking error constraint is reduced to a constraint on the kinematic quantities. Second, a novel method is introduced to relax the nonlinear constraints on kinematic quantities to linear ones. Experimental results are used to validate the proposed method.

Efficient algorithm for time-optimal feedrate planning and smoothing with confined chord error and acceleration

In this paper, an efficient algorithm is proposed to generate a smooth near-time-optimal feedrate function along a parametric tool path for three-axis CNC machining under a feedrate bound, an acceleration bound for each axis, and a chord error bound. The algorithm first gives a discrete and computationally efficient algorithm to find a sequence of globally optimal velocity points under the feedrate, acceleration, and chord error bounds. A linear programming strategy is proposed to smooth the velocity points sequence. Finally, the velocity points sequence is fitted into a cubic spline to obtain a near-time-optimal and smooth feedrate function. Simulation and experiment results for Non-uniform rational basis spline curves are presented to illustrate the feasibility of the algorithm.

Time-optimal traversal of curved paths by Cartesian CNC machines under both constant and speed-dependent axis acceleration bounds

Algorithms are developed to compute the feedrate variation along a curved path, that ensures minimum traversal time for a 3-axis CNC machine subject to both fixed and speed-dependent axis acceleration bounds arising from the output-torque characteristics of the axis drive motors. For a path specified by a polynomial parametric curve, the time-optimal feedrate is determined as a piecewise-analytic function of the curve parameter, with segments that correspond to saturation of the acceleration along one axis under constant or speed-dependent limits. Break points between the feedrate segments may be computed by numerical root-solving methods. For segments that correspond to fixed acceleration bounds, the (squared) optimal feedrate is rational in the curve parameter. For speed-dependent acceleration bounds, the optimal feedrate admits a closed-form expression in terms of a novel transcendental function whose values may be efficiently computed, for use in real-time control, by a special algorithm. The optimal feedrate admits a real-time interpolator algorithm, that can drive the machine directly from the analytic path description. Experimental results from an implementation of the time-optimal feedrate on a 3-axis CNC mill driven by an open-architecture software controller are presented. The algorithm is a significant improvement over that proposed in [Timar SD, Farouki RT, Smith TS, Boyadjieff CL. Algorithms for time-optimal control of CNC machines along curved tool paths. Robotics Comput Integrated Manufacturing 2005;21:37–53], since the addition of motor voltage constraints precludes the possibility of arbitrarily high speeds along linear or near-linear path segments.

Algorithms for time–optimal control of CNC machines along curved tool paths

The problem of specifying the feedrate variation along a curved path, that yields minimum traversal time for a 3-axis CNC machine subject to constraints on the feasible acceleration along each axis, is addressed. In general, this time–optimal feedrate incurs “bang–bang control,” i.e., maximum acceleration/deceleration is demanded of at least one axis throughout the motion. For a path defined by a polynomial parametric curve r( ξ), we show that the (square of the) time–optimal feedrate can be determined as a piecewise-rational function of the curve parameter ξ, with break-points corresponding to the roots of certain polynomial equations. Furthermore, this type of feedrate function is amenable to a real-time interpolator algorithm that drives the machine directly from the analytic curve description, eliminating the need for linear/circular G code approximations. The theoretical and computational aspects of such time–optimal feedrate functions are presented, together with experimental results from their implementation on a 3-axis mill driven by an open-architecture software controller.