Smooth time-optimal tool trajectory generation for CNC manufacturing systems

An optimization approach is proposed in this paper for generating smooth and time-optimal path constrained tool trajectory for Cartesian computer numerical control (CNC) manufacturing systems. The desired smooth time-optimal trajectory generation (STOTG) problem is formulated as a general optimal control problem. And axis jerk (derivative of acceleration with respect to time) constraints are introduced into this problem to remove discontinuities of the acceleration profiles. The desired smoothness of the trajectory can be accomplished by adjusting the values of jerk constraints. A control vector parameterization (CVP) method is applied to convert the optimal control problem into a nonlinear programming (NLP) problem which can be solved conveniently and effectively. The third derivative of the path parameter with respect to time (pseudo-jerk) and jerk act as optimization variables. The pseudo-jerk is approximated as piecewise constant, thus for at least second-order continuous parametric path, the resulted optimized trajectory with respect to time is also at least second-order continuous. Sequential quadratic programming (SQP) method is used to solve the NLP problem, through which numerical solution is obtained. Non-smooth (i.e. without considering jerk constraints) time-optimal trajectory generation (non-STOTG) problem is also considered in this paper for the purpose of comparison. Solutions of time-optimal trajectory generation (TOTG) problems for two test paths are performed to verify the effectiveness of the proposed approach.     

Algebraic solution to minimum-time velocity planning

The paper poses the problem of minimum-time velocity planning subject to a jerk amplitude constraint and to arbitrary velocity/acceleration boundary conditions. This problem which is relevant in the field of autonomous robotic navigation and also for inertial one-dimensional mechatronics systems is dealt with an algebraic approach based on Pontryagin’s Maximum Principle. The exposed complete solution shows how this time-optimal planning can be reduced to the problem of determining the positive real roots of a quartic equation. An algorithm that is suitable for real-time applications is then presented. The paper includes detailed examples also highlighting the special cases of this planning problem.     

敏捷卫星时间最优姿态机动研究综述

敏捷卫星是新一代的对地观测卫星,凭借其出色的机动性能带来了巨大的军事利益与商业利益.它最大的优势是具有快速姿态机动的能力,其研究的重点之一也正是快速姿态机动问题,需要通过优化来获得最短时间的姿态机动策略.本文围绕敏捷卫星的时间最优姿态机动问题,分别从时间最优姿态机动的优化求解和时间最优解的特性两个方面对该问题的研究现状进行了综述.     

FIR filter-based online jerk-constrained trajectory generation

In the context of human–robot manipulation interaction for service or industrial robotics, the robot controller must be able to quickly react to unpredictable events in dynamic environments. In this paper, a FIR filter-based trajectory generation methodology is presented, combining the simplicity of the analytic second-order trajectory generation, i.e. acceleration-limited trajectory, with the flexibility and computational efficiency of FIR filtering, to generate on the fly smooth jerk-constrained trajectories. The proposed methodology can generate synchronized (fixed-time) and time-optimal jerk-limited trajectories from arbitrary initial velocity and acceleration conditions within $20$ microsecond. Other jerk-constrained trajectories such as jerk-time fixed trajectories, which are particularly suitable for vibration reduction, can be easily generated. Experimental validations carried out on a seven axis Kuka LBR iiwa are presented.     

Optimal 3D trajectory generation in delivering missions under urban constraints for a flying robot

Interest in applying flying robots especially quadcopters for civil applications, in particular for delivering purposes, has dramatically grown in the recent years. In fact, since quadcopters are capable of vertical takeoff and landing, they can be widely employed for nearly any aerial task where a human presence is hazardous or response time is critical. In this regard, quadcopters come to be very beneficial in delivering packages; accordingly, generating an optimal flight trajectory plays a preponderant role for meeting this vision. This paper is concerned with generation of a time-optimal 3D path for a quadcopter under municipal restrictions in delivering tasks. To this end, the flying robot’s dynamics is first modeled through Newton–Euler method. Subsequently, the problem is formulated as a time-optimal control problem such that the urban constraints, which are safe-margins of high-rise buildings located throughout the course, are first modeled and then imposed to the trajectory optimization problem as inequality constraints. After discretizing the trajectory by means of Hermit–Simpson method, the optimal control problem is transformed into a nonlinear programming problem and finally is solved by the direct collocation technique. Extensive simulations demonstrate the efficacy of the proposed method and correspondingly verify the effectiveness of the suggested method in generation of optimum 3D routes while all constraints and mission requirements are satisfied.     

Time-optimal path generation for continuous and quasi-continuous path control of industrial robots

For a given continuous path, the problem of designing a time-optimal time-parametrization is considered. First, algorithms are presented which, under rather mild assumptions, yield the exact solution within two computational steps consisting of a forward and a backward computation. Then, the problem of quasi-continuous robot motion is investigated in detail. An algorithm of the same type results, but the computational burden is considerably reduced by making appropriate use of the special structure of the problem. By this, on-line use becomes feasible.Work supported by Oesterreichischer Fonds zur Foerderung der wissenschaftlichen Forschung.     

车型机器人的时间最优轨迹

基于几何最优控制理论, 本文提出了一种新的几何推理方法, 这种方法能够有效地推导出车型机器人给定两点间的时间最优轨迹类型的充分集合. 同时也为此类非线性问题提供了一种新的思路. 首先根据旁氏极大值原理和李代数推导出切换函数的结构特性, 根据这种结构特性建立一个切换坐标系并引入一个新的向量, 该向量在此坐标系中的旋转轨迹与时间最优轨迹具有一一对应关系. 进而得到一个结论: 如果该向量的始末旋转位置和方向一致, 那么将唯一的确定一条最优轨迹. 这是第一次得到一个能够直接应用于计算一条精确的最优轨迹的结论.     

Time-Optimal Control of T--S Fuzzy Models via Lie Algebra

This paper investigates a geometric property of time-optimal problem in the Takagi–Sugeno (T--S) fuzzy model via Lie algebra. We will focus on the existence of a time-optimal solution, singularity of switching function, and number of switching. These inherent problems are considered because of their rich geometric properties. The sufficient condition for the existence of a time-optimal solution reveals the controllability of T--S fuzzy model that can be found by the generalized rank condition. The time-optimal controller can be found as the bang–bang type with a finite number of switching by applying the maximum principle. In the study of the singularity problem, we will focus on the switching function whenever it vanishes over a finite time interval. Finally, we show that the bounded number of switching can be found if the T--S model (also a nonlinear system) is solvable.      

Kinematic Control of Omni-directional Robots for Time-optimal Movement between Two Configurations

The main goal of this study is to investigate the time-optimal control problem of an omni-directional mobile robot between two configurations. In the proposed method, this problem is formulated and solved as a constrained nonlinear programming (NLP) one. During the optimization process, the count of control steps is fixed initially and the sampling period is treated as a variable to be determined. The goal is to minimize the sampling period such that it is below a specific minimum value, which is set in advance considering the accuracy of discretization. To generate initial feasible solutions of the NLP problem, a systematic approach is also proposed. Since different initial feasible solutions can be generated, the optimization process of the NLP problem can be started from many different points to find the optimal solution. To show the feasibility of the proposed method, simulation and experimental results are included for illustration.     

三维环境中多机器人动态目标主动协作观测方法

动态目标的多移动机器人主动协作观测方法是指以获取较优的观测结果为目的, 对携带同构/异构观测传感器的多个机器人系统的观测数据进行有效融合并同时对其行为进行协调优化的方法. 本文主要研究了三维环境中的多机器人动态目标主动协作观测的问题. 首先, 以扩展集员估计方法(Extended set-membership filter, ESMF)为基础, 将信息融合过程与算法本身存在的集合运算环节相结合, 提出了一种高精度的多机器人观测信息融合方法. 该方法在保证较高观测精度的同时, 并没有显著增加单机器人扩展集员估计算法的计算量, 因此具有较高的实时性. 此外, 利用最优观测角度的概念, 通过引入相对速度空间(Relative velocity coordinates, RVCs), 设计了多移动机器人协调行为优化方法, 该方法可以将多机器人协调行为优化问题转化为线性规划问题, 以实现具有较高实时性的多机器人三维动态目标主动协作观测. 最后, 为了验证所研究方法的可行性与有效性, 进行了三维空间动态目标协作观测仿真实验.     

Multi-UAV Velocity and Trajectory Scheduling Strategies for Target Classification by a Single Human Operator

This work addresses the problem of enabling a single human operator to individually inspect targets for a fixed amount of time in a reconnaissance mission. The task of the operator is to classify the targets as friends or foes in real time, as they appear in video feeds from multiple UAVs. In order to account for cognitive limitations, the human is modeled as a single processing unit that can only execute one task at a time. A task is defined as a target inside the field of view of a given UAV, that needs to be inspected. Under the assumptions of this model, a linear program (LP) formulation is used to optimally find each task’s arrival time and latency in the system such that the human operator can inspect each target individually for some time Δt. Previous work by the authors investigated the idea of using UAV velocity modifications to meet the timing schedule specified by the LP solution. In this paper, the idea of UAV trajectory changes is introduced by modeling the UAVs as Dubins vehicles. Modifications to the bounds on the LP constraints are derived based on Dubins trajectories. The new bounds ensure that the LP solution returns a timing schedule achievable via maneuvers that combine velocity and trajectory changes to the UAVs’ flight plans. An on-line algorithm is developed that constructs and commands these velocity and trajectory changes in real time when conflicts arise. Correctness properties of this algorithm are analyzed and discussed for mission scenarios where the location of the targets is unknown and targets are discovered by the UAVs in real time.     

A phase-plane approach to time-optimal control of single-DOF mechanical systems with friction

In this paper, we attempt to solve the time-optimal control problem for single-degree-of-freedom (DOF) mechanical systems with friction, while taking into account not only the velocity-dependent control input constraint but also the state constraint. Direct application of the Pontryagin's maximum principle (PMP) leads to a sixth-order nonlinear two-point boundary-value problem (TPBVP) which is very difficult to solve numerically. In this context, we take a phase-plane analysis without resorting to the PMP. Thereby, the exact time-optimal solution can be obtained simply by solving a set of first-order differential equations with continuous right-hand sides. We also present some simulation results to demonstrate the practical use of the time-optimal solution.     

A Discrete Method for Time-Optimal Motion Planning of a Class of Mobile Robots

This paper addresses the time-optimal motion planning (TOMP) problem between two configurations for a mobile robot with two independently driven wheels. Different from previous methods, in which one needs to solve a set of differential equations, a discrete method is proposed to solve this problem. The first step is to transform the TOMP problem into a nonlinear programming (NLP) problem by an iterative procedure, in which the sampling period and the control inputs are chosen as variables, and the traveling time is to be minimized. Since it is usually hard to find initial feasible solutions of an NLP problem, a method that combines the concepts of genetic algorithms (GAs) and penalty functions is also proposed. In this manner, the NLP problem can be solved since initial feasible solutions can be generated easily. Simulation results are included to show the validity of the proposed method.     

基于混合遗传算法的工业机器人最优轨迹规划

为兼顾工业机器人工作效率与轨迹的平稳性,提出一种基于混合遗传算法的二次轨迹规划方案.通过最优时间轨迹规划得到最小执行时间,在最小执行时间内进行最优冲击轨迹规划,进而规划出一条既高效又平滑的运动轨迹.采用五次均匀B样条在关节空间进行快速插值,不仅保证了各关节速度和加速度连续性还保证了各关节冲击的连续性.连续平滑的冲击可以减少机械振动,延长机器人的工作寿命.选用PUMA560为对象进行仿真与实验,结果表明,该方案可以获得比较理想的机器人运动轨迹,所提出的混合遗传算法能有效提高全局寻优的性能和算法运行的稳定性.     

Energy-optimal trajectory planning for car-like robots

When a battery-powered robot needs to operate for a long period of time, optimizing its energy consumption becomes critical. Driving motors are a major source of power consumption for mobile robots. In this paper, we study the problem of finding optimal paths and velocity profiles for car-like robots so as to minimize the energy consumed during motion. We start with an established model for energy consumption of DC motors. We first study the problem of finding the energy optimal velocity profiles, given a path for the robot. We present closed form solutions for the unconstrained case and for the case where there is a bound on maximum velocity. We then study a general problem of finding an energy optimal path along with a velocity profile, given a starting and goal position and orientation for the robot. Along the path, the instantaneous velocity of the robot may be bounded as a function of its turning radius, which in turn affects the energy consumption. Unlike minimum length paths, minimum energy paths may contain circular segments of varying radii. We show how to efficiently construct a graph which generalizes Dubins’ paths by including segments with arbitrary radii. Our algorithm uses the closed-form solution for the optimal velocity profiles as a subroutine to find the minimum energy trajectories, up to a fine discretization. We investigate the structure of energy-optimal paths and highlight instances where these paths deviate from the minimum length Dubins’ curves. In addition, we present a calibration method to find energy model parameters. Finally, we present results from experiments conducted on a custom-built robot for following optimal velocity profiles.     

On optimal spacecraft damping

The problem of spacecraft damping (damping of initial angular velocity to zero) for a minimal time is studied. Two variants of formulation of the optimization problem are considered; these variants differ in the form of constraints on the control torque. Analytical solution to the formulated problem is obtained in the closed form and numerical expressions for synthesis of optimal angular velocity control program are given. Similar problem of time-optimal angular acceleration of the spacecraft to the given value is also solved. Procedure for determination of the control torque at the initial time instant for the problem of acceleration of the spacecraft to the required angular velocity is presented. Numerical example of solution of the problems of buildup and damping of spacecraft rotation velocity for a minimal time is given.     

Time-optimal control of robotic manipulators with limit heat characteristics of the actuator

In this paper, we present a time-optimal control scheme for a robot manipulator to track a predefined geometric path, subject to constraints due to the limit heat characteristics of the actuator (the DC motor was assumed to be the actuator used). Constraints due to the rated torque bounds and the rated velocity bounds of the motor would not be valid for continuous use of the manipulator, since the required mechanical output of the actuator (DC motor) exceeds its maximum power capacity and greatly exceeds its heat-converted power limit. The heat-converted power of the DC motor is thus considered as the actuation bound and the time-optimal trajectories are generated subject to this bound. Computer simulation was also executed to demonstrate the effectiveness of the proposed scheme in comparison to former schemes that used the rated torque and the rated velocity.     

Time-optimal tool motion planning with tool-tip kinematic constraints for robotic machining of sculptured surfaces

A time-optimal motion planning method for robotic machining of sculptured surfaces is reported in this paper. Compared with the general time-optimal robot motion planning, a surface machining process provides extra constraints such as tool-tip kinematic limits and complexity of the curved tool path that also need to be taken into account. In the proposed method, joint space and tool-tip kinematic constraints are considered. As there are high requirements for tool path following accuracy, an efficient numerical integration method based on the Pontryagin maximum principle is adopted as the solver for the time-optimal tool motion planning problem in robotic machining. Nonetheless, coupled and multi-dimensional constraints make it difficult to solve the problem by numerical integration directly. Therefore, a new method is provided to simplify the constraints in this work. The algorithm is implemented on the ROS (robot operating system) platform. The geometry tool path is generated by the CAM software firstly. And then the whole machine moving process, i.e. the feedrate of machining process, is scheduled by the proposed method. As a case study, a sculptured surface is machined by the developed method with a 6-DOF robot driven by the ROS controller. The experimental results validate the developed algorithm and reveal its advantages over other conventional motion planning algorithms for robotic machining.     

Safe proactive plans and their execution

We present in this paper a methodology for computing the maximum velocity profile over a trajectory planned for a mobile robot. Environment and robot dynamics as well as the constraints of the robot sensors determine the profile. The planned profile is indicative of maximum speeds that can be possessed by the robot along its path without colliding with any of the mobile objects that could intercept its future trajectory. The mobile objects could be arbitrary in number and the only information available regarding them is their maximum possible velocity. The velocity profile also enables one to deform planned trajectories for better trajectory time. The methodology has been adopted for holonomic and non-holonomic motion planners. An extension of the approach to an online real-time scheme that modifies and adapts the path as well as velocities to changes in the environment such that both safety and execution time are not compromised is also presented for the holonomic case. Simulation and experimental results demonstrate the efficacy of this methodology.     

Minimum-Energy Trajectory Generation for Cornering with a Fixed Heading for Three-Wheeled Omni-Directional Mobile Robots

The minimum-energy trajectory generation problem of cornering with a fixed heading is solved for three-wheeled omni-directional mobile robots (TOMRs). To maximize the total operation time of a mobile robot with carried batteries having finite energy, we have chosen a practical cost function to be the total energy drawn from the batteries. Then, we formulate the minimum-energy trajectory generation problem of executing a cornering motion with a fixed heading for TOMRs with given dynamics including actuator motors. The optimal control theory using a Hamiltonian function and a numerical method are used to obtain the minimum-energy trajectory, which gives the velocity profile in analytic form. Performance analyses are conducted with various simulations and the consumed energy using obtained minimum-energy trajectory is compared with a typical conventional trajectory with a trapezoidal velocity profile, which reveals that an energy savings of up to 18.7 % is achieved. To validate the actual performance of our trajectory, we implemented and tested an accurate trajectory following system which utilizes a resolved acceleration controller.