计算机数控系统光滑时间最优轨迹规划

基于控制向量参数化(CVP)方法, 研究了计算机数控(CNC)系统光滑时间最优轨迹规划方法. 通过在规划问题中引入加加速度约束, 实现轨迹的光滑给进. 引入时间归一化因子, 将加加速度约束的时间最优轨迹规划问题转化为固定时间的一般性最优控制问题. 以路径参数对时间的三阶导数(伪加加速度)和终端时刻为优化变量, 并采用分段常数近似伪加加速度, 将最优控制问题转化为一般的非线性规划(NLP)问题进行求解. 针对加加速度、加速度等过程不等式约束, 引入约束凝聚函数, 将过程约束转化为终端时刻约束, 从而显著减少约束计算. 构造目标和约束函数的Hamiltonian函数, 利用伴随方法获得求解NLP问题所需的梯度.     

Feed-rate and trajectory optimization for CNC machine tools

The key task performed by CNCs is the generation of the time-sequence of set-points for driving each physical axis of the machine tool during program execution. This interpolation of axes movement must satisfy a number of constraints on axes dynamics (velocity, acceleration, and jerk), and on process outcome (smooth tool movement and precise tracking of the nominal tool-path at the desired feed-rate). This paper presents an algorithm for CNC kernels that aims at solving the axes interpolation problem by exploiting an Optimal Control Problem formulation. With respect to other solutions proposed in the literature, the approach presented here takes an original approach by assuming a predefined path tracking tolerance—to be added to the constraints listed above—and calculating the whole trajectory (path and feed-rate profile) that satisfies the given constraints. The effectiveness of the proposed solution is benchmarked against the trajectory generated by an industrial, state-of-the-art CNC, proving a significant advantage in efficiency and smoothness of axes velocity profiles.     

Smooth and time‐optimal trajectory planning for industrial manipulators along specified paths

This article presents a method for determining smooth and time‐optimal path constrained trajectories for robotic manipulators and investigates the performance of these trajectories both through simulations and experiments. The desired smoothness of the trajectory is imposed through limits on the torque rates. The third derivative of the path parameter with respect to time, the pseudo‐jerk, is the controlled input. The limits on the actuator torques translate into state‐dependent limits on the pseudo‐acceleration. The time‐optimal control objective is cast as an optimization problem by using cubic splines to parametrize the state space trajectory. The optimization problem is solved using the flexible tolerance method. The experimental results presented show that the planned smooth trajectories provide superior feasible time‐optimal motion. © 2000 John Wiley & Sons, Inc.     

Time-optimal and Smooth Trajectory Planning for Robot Manipulators

This paper presents a practical time-optimal and smooth trajectory planning algorithm and then applies it to robot manipulators. The proposed algorithm uses the time-optimal theory based on the dynamics model to plan the robot’s motion trajectory, constructs the trajectory optimization model under the constraints of the geometric path and joint torque, and dynamically selects the optimal trajectory parameters during the solving process to prominently improve the robot’s motion speed. Moreover, the proposed algorithm utilizes the input shaping algorithm instead of the jerk constraint in the trajectory optimization model to achieve a smooth trajectory. The input shaping of trajectory parameters during postprocessing not only suppresses the residual vibration of the robot but also takes the signal delay caused by traditional input shaping into account. The combination of these algorithms makes the proposed time-optimal and smooth trajectory planning algorithm ensure absolute time optimality and achieve a smooth trajectory. The results of an experiment on a six-degree-of-freedom industrial robot indicate the validity of the proposed algorithm.

     

基于Radau伪谱法的重复使用运载器再入轨迹优化

为了提高数值解法的收敛速度,本文利用Radau伪谱法求解重复使用运载器的再入轨迹优化问题.该方法在一组Legendre-Gauss-Radau点上构造全局Lagrange插值多项式对状态变量和控制变量进行逼近,在动力学方程中状态变量对时间的导数可由插值多项式的导数来近似,故可将动力学方程约束转化为在Legendre-Gauss-Radau点上的代数微分方程约束.因此,可将连续时间的最优控制问题转化为有限维的非线性规划(NLP)问题,之后通过稀疏NLP求解器SNOPT即可对其进行求解.最后的仿真结果显示,通过该方法优化后的再入轨迹成功满足过程约束与边界约束.由于该方法的高效率和高精度特性,可将其应用于轨迹快速优化工程实际问题中.     

Suboptimal Robot Joint Interpolation Within User-Specified Knot Tolerances

Approximation of a desired robot path can be accomplished by interpolating a curve through a sequence of joint-space knots. A smooth interpolated trajectory can be realized by using trigonometric splines. But, sometimes the joint trajectory is not required to exactly pass through the given knots. The knots may rather be centers of tolerances near which the trajectory is required to pass. In this article, we optimize trigonometric splines through a given set of knots subject to user-specified knot tolerances. The contribution of this article is the straightforward way in which intermediate constraints (i.e., knot angles) are incorporated into the parameter optimization problem. Another contribution is the exploitation of the decoupled nature of trigonometric splines to reduce the computational expense of the problem. The additional freedom of varying the knot angles results in a lower objective function and a higher computational expense compared to the case in which the knot angles are constrained to exact values. The specific objective functions considered are minimum jerk and minimum torque. In the minimum jerk case, the optimization problem reduces to a quadratic programming problem. Simulation results for a two-link manipulator are presented to support the results of this article.     

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 trajectories for cooperative multi-manipulator systems

We present two schemes for planning the time-optimal trajectory for cooperative multi-manipulator system (CMMS) carrying a common object. We assume that the desired path is given and parameterizable by an arclength variable. Both approaches take into account the dynamics of the manipulators and object. The first approach employs linear programming techniques, and it allows us to obtain the time-optimal execution of the given task utilizing the maximum torque capacities of the joint motors. The second approach is a sub-time-optimal method that is computationally very efficient. In the second approach the given load is divided into a share for each robot in the CMMS in a manner in which the trajectory acceleration/deceleration is maximized, hence the trajectory execution time is minimized. This load distribution approach uses optimization schemes that degenerate to a linear search algorithm for the case of two robots manipulating a common load, and this results in significant reduction of computation time. The load distribution scheme not only enables us to reduce the computation time, but also gives us the possibility of applying this method in real-time planning and control of CMMS. Further, we show that for certain object trajectories the load distribution scheme yields truly time-optimal trajectories.     

Near Time-Optimal Control of Redundant Manipulators along a Specified Path with Jerks Constraint

A time-optimal control scheme for a general type of closed-chain manipulator is proposed. The considered manipulator is composed of multiple serial manipulators that are connected to each other and single manipulators may be kinematically redundant. Also, the limit on the actuator torques and actuator jerks are considered. The jerk constraints create a smooth trajectory for reducing strain on robot actuators and satisfy torque limitations of industrial actuators. Inclusion of the jerk constraints increases the traversal time, hence, a method is introduced to optimize this time. To this end, a simple method to find switching points is investigated.     

Newton-conjugate gradient （CG） augmented Lagrangian method for path constrained dynamic process optimization

In this paper, a Newton-conjugate gradient (CG) augmented Lagrangian method is proposed for solving the path constrained dynamic process optimization problems. The path constraints are simplified as a single final time constraint by using a novel constraint aggregation function. Then, a control vector parameterization (CVP) approach is applied to convert the constraints simplified dynamic optimization problem into a nonlinear programming (NLP) problem with inequality constraints. By constructing an augmented Lagrangian function, the inequality constraints are introduced into the augmented objective function, and a box constrained NLP problem is generated. Then, a linear search Newton-CG approach, also known as truncated Newton (TN) approach, is applied to solve the problem. By constructing the Hamiltonian functions of objective and constraint functions, two adjoint systems are generated to calculate the gradients which are needed in the process of NLP solution. Simulation examples demonstrate the effectiveness of the algorithm.     

On inverse kinematics with inequality constraints: new insights into minimum jerk trajectory generation

The problem of inverse kinematics is revisited in the present paper. The paper is focusing on the problem of solving the inverse kinematics problem while minimizing the jerk of the joint trajectories. Even though the conventional inverse kinematics algorithms have been proven to be efficient in many applications, it has been proven that constraints on the accelerations or the jerk cannot be guaranteed, and even yields to divergence or makes the problem unsolvable. The proposed algorithm yields smooth velocity and acceleration trajectories, which are highly desired features for industrial robots. The algorithm uses the joint jerk as the control parameter instead of the classical use of the joint velocity as result constraints on the jerk function can be easily incorporated. To validate the proposed approach, we have conducted several simulations scenarios. The simulation results have revealed that the proposed method can efficiently solve the inverse kinematics problem while considering constraints on the joint acceleration and jerk.     

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.     

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.     

飞行器上升段轨迹的优化设计

针对飞行器上升段轨迹优化求解困难的问题,提出一种基于正交配点的优化求解方法。该方法以第二类切比雪夫正交多项式的零点作为系统控制变量和状态变量的离散点,利用拉格朗日插值多项式对状态和控制变量进行拟合。通过对多项式的求导将动力学微分方程约束转化为代数约束,从而把无限维的最优控制问题转化为一个有限维的非线性规划(Nonlinear Programming,NLP)问题。随后,利用序列二次规划(Sequential Quadratic Program-ming,SQP)方法求解转化后的NLP问题,获得最优的飞行轨迹。最后,飞行器上的仿真结果验证了所提方法的有效性。研究成果可为飞行器的制导控制提供可行的飞行轨迹,有一定的工程应用价值。     

Control variable parameterisation with penalty approach for hypersonic vehicle reentry optimisation

An efficient trajectory optimisation approach combining the classical control variable parameterisation (CVP) with a novel smooth technology and two penalty strategies is developed to solve the trajectory optimal control problems. Since it is difficult to deal with path constraints in CVP method, the novel smooth technology is firstly employed to transform the complex constraints into one smooth constraint. Then, two penalty strategies are proposed to tackle the converted path and terminal constraints to decrease the computational complexity and improve the constraints satisfaction. Finally, a nonlinear programming problem, which approximates the original trajectory optimisation problem, is obtained. Error analysis shows that the proposed method has good convergence property. A general hypersonic cruise vehicle trajectory optimisation example is employed to test the performance of the proposed method. Numerical results show that the path and terminal conditions are well satisfied and better trajectory profiles are obtained, showing the effectiveness of the proposed method.     

Smooth trajectory planning for a parallel manipulator with joint friction and jerk constraints

In order to achieve better tracking accuracy effectively, a new smooth and near time-optimal trajectory planning approach is proposed for a parallel manipulator subject to kinematic and dynamic constraints. The complete dynamic model is constructed with consideration of all joint frictions. The presented planning problem can be solved efficiently by formulating a new limitation curve for dynamic constraints and a reduced form for jerk constraints. The motion trajectory is planned with quartic and quintic polynomial splines in Cartesian space and septuple polynomial splines in joint space. Experimental results show that smaller tracking error can be obtained. The developed method can be applied to any robots with analytical inverse kinematic and dynamic solutions.     

Time-Optimal Control of a Hovering Quad-Rotor Helicopter

The time-optimal control problem of a hovering quad-rotor helicopter is addressed in this paper. Instead of utilizing the Pontryagin's Minimum Principle (PMP), in which one needs to solve a set of highly nonlinear differential equations, a nonlinear programming (NLP) method is proposed. In this novel method, the count of control steps is fixed initially and the sampling period is treated as a variable in the optimization process. The optimization object is to minimize the sampling period such that it will be below a specific minimum value, which is set in advance considering the accuracy of discretization. To generate initial feasible solutions of the formulated NLP problem, genetic algorithms (GAs) are adopted. With the proposed method, one can find a time-optimal movement of the helicopter between two configurations. To show the feasibility of the proposed method, simulation results are included for illustration.     

A comparison principle for state-constrained differential inequalities and its application to time-optimal control

In this paper, we present a comparison principle that characterizes the maximal solutions of state-constrained differential inequalities in terms of solutions of certain differential equations with discontinuous right-hand sides. For the sake of completeness, we show through some set-valued analysis that the differential equations determining the maximal solutions have the unique solutions in the Carathe/spl acute/odory sense, in spite of discontinuity of their right-hand sides. We apply our comparison principle to the explicit characterization of the solution to a time-optimal control problem for a class of state-constrained second-order systems which includes the dynamic equations of robotic manipulators with geometric path constraints as well as single-degree-of-freedom mechanical systems with friction. Specifically, we show that the time-optimal trajectory is uniquely determined by two curves that can be constructed by solving two scalar ordinary differential equations with continuous right-hand sides. Hence, the time-optimal trajectory can be found in a computationally efficient way through the direct use of the well-known Euler or Runge-Kutta methods. Another interesting feature is that our method to solve the time-optimal control problem works even when there exist an infinite number of switching points. Finally, some simulation results using a two-degrees-of-freedom (DOF) robotic manipulator are presented to demonstrate the practical use of our complete characterization of the time-optimal solution.     

Three-dimensional trajectory optimization of soft lunar landings from the parking orbit with considerations of the landing site

Minimum fuel, three-dimensional trajectory optimization from a parking orbit considering the desired landing site is addressed for soft lunar landings. The landing site is determined by the final longitude and latitude; therefore, a two-dimensional approach is limited and a three-dimensional approach is required. In addition, the landing site is not usually considered when performing lunar landing trajectory optimizations, but should be considered in order to design more accurate and realistic lunar landing trajectories. A Legendre pseudospectral (PS) method is used to discretize the trajectory optimization problem as a nonlinear programming (NLP) problem. Because the lunar landing consists of three phases including a de-orbit burn, a transfer orbit phase, and a powered descent phase, the lunar landing problem is regarded as a multiphase problem. Thus, a PS knotting method is also used to manage the multiphase problem, and C code for Feasible Sequential Quadratic Programming (CFSQP) using a sequential quadratic programming (SQP) algorithm is employed as a numerical solver after formulating the problem as an NLP problem. The optimal solutions obtained satisfy all constraints as well as the desired landing site, and the solutions are verified through a feasibility check.     

LP-based velocity profile generation for robotic manipulators

In this paper we examine the minimum-time velocity profile generation problem which belongs to the second stage of the decoupled robot motion planning. The time-optimal profile generation problem can be translated to a convex optimal control task through a nonlinear change of variables. When the constraints of the problem have special structure, the time-optimal solution can be obtained by linear programming (LP). In this special case, the velocity of the robot along the path is maximised, instead of time minimising. The benefit of the LP solution is the lower computational time. Validation of the LP algorithm is also presented based on simulation results.