Minimum-time Trajectory Planning Research Articles

Continuous advantage learning for minimum-time trajectory planning of autonomous vehicles

Continuous advantage learning for minimum-time trajectory planning of autonomous vehicles

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.

Optimal Time Trajectory Planning and Control of Space Cable Robot based on " Brachistochrone " Theory

This paper presents optimal-time trajectory planning of Samen cable robot using optimal control theory. The optimal-time trajectory planning solution method is based on the calculus of variations method. This issue can be solved by the Pontryagin’s Minimum Principle using the Bang-Bang control. To do so, a time-cost function is considered. By turning optimal-time trajectory planning into a problem-solving procedure with a two-point boundary value problem, the solving stages begin. In addition, the beginning and the end of the path are certain, which can be solved with numerical algorithms. One of the most important and effective methods of minimum time trajectory planning is the brachistochrone method. In this study, the brachistochrone curve is introduced and solved by both brachistochrone method and minimum time trajectory planning using optimal control. First, the path generation for the robot is described using optimization methods; subsequently, with respect to appropriate cost functions, a proportional-integral-derivative controller (PID) and a sliding mode for tracking generated paths are presented. By testing the algorithms in the simulation environment, good and acceptable results are obtained. Simulation results demonstrate proper performance of the proposed method.

Minimum-time trajectory planning under dynamic constraints for a wheeled mobile robot with a trailer

This paper focuses on the minimum-time trajectory planning in a structured environment for a wheeled mobile robot with a trailer. The constraints which are accounted for include: obstacle avoidance, boundary conditions on the position/orientation of both the tractor and the trailer, maximum articulation angle between the two platforms, kinodynamic capacities of the actuators, and dynamic stability of the system. The proposed technique is based on the random-profile approach, which uses a versatile master–slave optimization scheme. It is robust enough to handle different scenarios (free or imposed path) and different structures of the system (on-axle or off-axle articulation). Simulation results show that it converges, in reasonable runtimes, towards good-quality solutions which are saturating the constraints. A preliminary experimental test suggests that the planned trajectories are realistically achievable.

Minimum-Time Trajectory Planning for a Differential Drive Mobile Robot Considering Non-slipping Constraints

We propose a real-time minimum-time trajectory planning strategy with obstacle avoidance for a differential-drive mobile robot in the context of robot soccer. The method considers constraints important to maximize the system’s performance, such as the actuator limits and non-slipping conditions. We also present a novel friction model that regards the imbalance of normal forces on the wheels due to the acceleration of the robot. Theoretical guarantees on how to obtain a minimum-time velocity profile on a predetermined parametrized curve considering the modeled constraints are also presented. Then, we introduce a nonlinear, non-convex, local optimization using a version of the Resilient Propagation algorithm that minimizes the time of the curve while avoiding obstacles and respecting system constraints. Finally, employing a new proposed benchmark, we verified that the presented strategy allows the robot to traverse a cluttered field (with dimensions of 1.5 m $$\times $$ 1.3 m) in 2.8 s in 95% of the cases, while the optimization success rate was 85%. We also demonstrated the possibility of running the optimization in real-time, since it takes less than 13.8 ms in 95% of the cases.

Smooth Trajectory Generation for Predefined Path With Pseudo Spectral Method

In view of smooth trajectory generation for a 3-axis machine tool, many methods have been presented. Among them, the optimal control based method is increasingly concerned, because it is considered to be able to make full use of kinematic abilities of machine tools. Under the unified framework of optimal control, the feedrate can be adjusted flexibly by adding or removing axial constraints and tangential constraints. But the problem of smooth trajectory generation (PSTG) based on optimal control for a machine tool is not easily to be solved. In this article, to efficiently solve the PSTG, it is divided into two sub-problems: the problem of minimum time trajectory planning (PMTTP) and the pseudo problem of smooth trajectory generation (PPSTG). Since both sub-problems are convex, the existence of unique solutions can be guaranteed. Then, the PMTTP and the PPSTG are transformed into nonlinear programming (NLP) problems with radau-pseudo-spectral (RPM) method successively. Due to convexity, the two NLP problems can be efficiently solved with mature optimization methods. In addition, the RPM method allows two sub-problems to have different Legendre-Gauss-Radau (LGR) points, thereby further saving computational costs. Finally, three different predefined paths are employed to test the proposed method, and simulation results show the effectiveness of proposed method.

Smooth minimum time trajectory planning with minimal feed fluctuation

Smooth and effective feedrate profile is very important for high-speed and high-precision machining. In this paper, a two-stage feedrate scheduling scheme (TFSS) is proposed to efficiently generate the smooth feedrate profile. The minimum time trajectory planning (MTTP) is firstly employed to generate the initial feedrate profile considering constraints of both the kinematics and the chord error. To improve the computational efficiency, the MTTP is transformed into a problem of linear programming by applying direct transcription method. Then, all the sharp corners along the initial feedrate profile are identified, especially those that are interacted by each other. The jerk-limited method of eliminating interaction (JMEI) at sharp corners is applied to smooth the final feedrate profile. Finally, the improved feed correction polynomials are applied to reduce the fluctuations owing to non-arc-length parameterization. To illustrate the validity and rationality of the proposed scheme, two curves were employed to verify the proposed method. The simulation and experimental results demonstrate the efficiency of the proposed scheme for the nonuniform rational basis spline (NURBS) tool paths.

Near-optimal probabilistic search via submodularity and sparse regression

The goal of search is to maximize the probability of target detection while covering most of the environment in minimum time. Existing approaches only consider one of these objectives at a time and most optimal search problems are NP-hard. In this research, a novel approach for search problems is proposed that considers three objectives: (1) coverage using the fewest sensors; (2) probabilistic search with the maximal probability of detection rate (PDR); and (3) minimum-time trajectory planning. Since two of three objective functions are submodular, the search problem is reformulated to take advantage of this property. The proposed sparse cognitive-based adaptive optimization and PDR algorithms are within ($$1-1/e$$1-1/e) of the optimum with high probability. Experiments show that the proposed approach is able to search for targets faster than the existing approaches.

Minimum-Time Trajectory Planning and Control of a Pick-and-Place Five-Bar Parallel Robot

This paper presents trajectory planning optimization and real-time control of a special five-bar parallel robot. Planning is based on a cubic spline stochastic approach that minimizes trajectory time and selects the best combination of working mode regions to circumvent all parallel singularities, allowing the size of the workspace to be increased. Identification of the dynamic model of the robot and its actuators allows a precise implementation of the trajectory planning and real-time control approach. The optimization algorithm achieves a fast trajectory and a controller that operates with an error of less than 0.7° for both actuated joints of the robot.

Minimum-Time Trajectory Planning for Underactuated Overhead Crane Systems With State and Control Constraints

In this paper, we propose a novel offline minimum-time trajectory planning (MTTP) approach for underactuated overhead cranes. To the best of our knowledge, it is the first optimal solution to the MTTP problem for overhead crane systems, which simultaneously takes into account various constraints, including the bounded swing angle for the payload, bounded velocity, acceleration, and even jerk for the trolley. Different from existing approaches, by means of system discretization and augmentation, the quasi-convex optimization technique is successfully adopted to find the minimum-time solution while satisfying all the aforementioned constraints. Extensive simulation and experiments with comparisons to previously published methods are conducted to show the superior performance of the proposed method. Note that the results derived in this paper also serve as promising guidance in engineering applications, since it provides a performance limit, namely, the possible highest efficiency for automatic or manual operation of overhead cranes.

Trajectory Planning of Unicycle Mobile Robots With a Trapezoidal-Velocity Constraint

We propose an efficient stochastic scheme for minimum-time trajectory planning of a nonholonomic unicycle mobile robot under constraints on path curvature, velocities, and torques. This problem, which is known to be complex, often requires important runtimes, particularly if obstacles are present and if full dynamics is considered. The proposed technique is a fast variant of the random-profile approach recently applied to wheeled-mobile robots. It incorporates a trapezoidal-velocity-profile constraint that helps reduce the number of unknown parameters and that speeds up the calculation steps. Results are presented for two- and three-wheel mobile robots in free/constrained workspaces. A comparison with reference solutions, which were obtained independently, shows that the proposed variant is able to achieve almost the same quality of calculated trajectories while reducing the runtime considerably.

Minimum-time trajectory planning of spatial cable-suspended robots along a specified path considering both tension and velocity constraints

In this article, minimum-time trajectory planning problems of spatial non-redundant and redundant cable suspended robots are addressed. Both cable tensions and cable velocities are considered to be limited. The equations of motion of such systems are reduced to a system of second order differential equations in terms of path parameter s using the specified desired path. The bounds on the cable tensions and cable velocities are transformed into bounds on the acceleration and velocity along the path. By assuming bang-bang control, the switching points in the plane are obtained. Then the cable tensions as the inputs are found versus path parameter and subsequently versus time.

A hierarchical neuro-fuzzy system to near optimal-time trajectory planning of redundant manipulators

In this paper, the problem of minimum-time trajectory planning is studied for a three degrees-of-freedom planar manipulator using a hierarchical hybrid neuro-fuzzy system. A first neuro-fuzzy network named NeFIK is considered to solve the inverse kinematics problem. After a few pre-processing steps characterizing the minimum-time trajectory and the corresponding torques, a second neuro-fuzzy controller is built. Its purpose is to fit the robot dynamic behavior corresponding to the determined minimum-time trajectory with respect to actuators models, torque nominal values, as well as position, velocity, acceleration and jerk boundary conditions. A Tsukamoto Neuro-Fuzzy Inference network is designed to achieve the online control of the robot. The premise parameters (antecedent membership functions parameters) as well as rule-consequence parameters are learned and optimized, generating the optimal-time trajectory torques, representing the robot dynamic behavior. Simulation results are presented and discussed.

Novel minimum time trajectory planning in terrain following flights

A new methodology has been proposed to enhance inverse dynamics applications in the process of trajectory planning and optimization in terrain following flights (TFFs). The new approach uses a least square scheme to solve a general two-dimensional (2-D) TFF in a vertical plane. In the mathematical process, Chebyshev polynomials are used to model the geographical data of the terrain in a given route in a manner suitable for the aircraft at hand. The aircraft then follows the modeled terrain with sufficient clearance. In this approach the terrain following (TF) problem is effectively converted to an optimal tracking problem. Results show that this method provides a flexible approach to solve the TFF problem especially in conditions where the existing terrain to be flown over is not mathematically well behaved

モレイ型多関節ロボットアームの最短時間軌道計画(冗長自由度ロボット・メカニズムとその制御)

モレイ型多関節ロボットアームの最短時間軌道計画(冗長自由度ロボット・メカニズムとその制御)

Optimized feed scheduling in three axes machining. Part I: Fundamentals of the optimized feed scheduling strategy

An optimized feed scheduling strategy is proposed in this paper to maximize the metal removal rate in 3-axis machining while guaranteeing the machining accuracy. The tool path is assumed defined by a cubic parametric form. In part I of this paper, the fundamentals of this strategy are presented. This strategy integrates the feed drive dynamics, described by the acceleration/deceleration (Acc/Dec) profile, with the minimum-time trajectory planning in order to achieve the desired feed rate at the appropriate position. An optimum use of the feed drive capabilities is considered to track the changes in the cutting geometry along the tool path and to ensure an acceptable contour error. Therefore, this strategy combines different constraints and various criteria in modifying the feed rate to maintain a near-constant cutting force resulting in a highly non-linear problem. The constraints include the cutting force magnitude, the feed rate boundaries, the contour error and the characteristics of the (Acc/Dec) profile of the different feed drive systems. The criteria are the maximum production rate, the machining accuracy and safety. In part II of this paper, the effectiveness of this strategy is demonstrated using ball end mill operation on a workpiece that provides variable cutting geometry along a non-linear tool path. The performance of this strategy in terms of productivity, machining safety, and machining accuracy, is compared to a feed scheduling strategy based on control points as well as to milling with constant feed rate.

Minimum-time trajectory planning of mechanical manipulators under dynamic constraints

The paper presents a global optimization approach to the trajectory planning problem of mechanical manipulators. The purpose is to obtain a minimum-time cubic spline trajectory subject to constraints given by limited joint torques and torque derivatives taking into account the non-linear manipulator dynamics. It is shown how, without conservativeness, a semi-infinite optimization problem emerges. Conditions ensuring that the formalized problem admits a solution are given. The estimated global solution can be actually obtained by means of an hybrid genetic/interval algorithm that guarantees the feasibility of the found solution. The methodology is illustrated with numerical details for a two-link planar arm and a PUMA six-link manipulator; for the former, comparisons with an alternative optimization solver are exposed.

ENHANCEMENT OF PRODUCTIVITY BY INTELLIGENT PROGRAMMING OF FEED RATE IN 3-AXIS MILLING

An optimized feed scheduling strategy (OFSS) is proposed in this paper to maximize the metal removal rate in 3-axis milling while guaranteeing the machining accuracy. This strategy integrates the feed drive dynamics, described by the acceleration/deceleration (Acc/Dec) profile, with the minimum-time trajectory planning in order to achieve the desired feed rate at the appropriate position. An optimum use of the feed drive capabilities is considered to track the changes in the cutting geometry along the tool path. Therefore, this strategy combines different constraints and various criteria in modifying the feed rate to maintain near-constant cutting force resulting in highly non-linear problem. The constraints include the cutting force, the feed rate boundaries, the contour error and the characteristics of the (Acc/Dec) profile. The criteria are the maximum production rate, the machining accuracy and safety. The performance of the OFSS in terms of these criteria, is compared to two end milling operations where the trajectory planning disregards the feed drive dynamics. The first is based on a feed scheduling strategy using control points (FSCP). The second is a milling operation with nominal feed rate. By increasing the feed rates, the OFSS improves the machining accuracy, reduces the machining time, and allows a better regulation of the cutting force.

Optimal trajectory planning and sliding mode control for robots using evolution strategy

Although robots have some kinematic and dynamic constraints such as the limits of the position, velocity, acceleration, jerk, and torque, they should move as fast as possible to increase the productivity. Researches on the minimum-time trajectory planning and control based on the dynamic constraints assume the availability of full dynamics of robots. However, the dynamic equation of robot may not often be exactly known. In this case, the kinematic approach for the minimum-time trajectory planning is more meaningful. We also have to construct a controller to track precisely the minimum-time trajectory. But, finding a proper controller is also difficult if we do not know the explicit dynamic equations of a robot.This paper describes an optimization of trajectory planning based on a kinematic approach using the evolution strategy (ES), as well as an optimization of a sliding mode tracking controller using ES for a robot without dynamic equations.

Global minimum-time trajectory planning of mechanical manipulators using interval analysis

The paper addresses the global minimum-time trajectory planning of an m -joint mechanical manipulator. Using a joint space scheme with given intermediate points to be interpolated by piecewise cubic polynomials, a novel bisecting-plane algorithm is proposed to schedule the times between adjacent knots under velocity, acceleration, and jerk constraints. This algorithm, which is proved to be globally convergent with certainty within an arbitrary precision, uses an interval procedure (a subroutine adopting tools and ideas of interval analysis) in proving that a local minimum is actually a global one. A worked example for the six-joint case is exposed, and computational results of a C+ + implementation are included.