A dynamic programming approach to trajectory planning of robotic manipulators

This paper presents a solution to the problem of minimizing the cost of moving a robotic manipulator along a specified geometric path subject to input torque/force constraints, taking the coupled, nonlinear dynamics of the manipulator into account. The proposed method uses dynamic programming (DP) to find the positions, velocities, accelerations, and torques that minimize cost. Since the use of parametric functions reduces the dimension of the state space from2nfor ann- jointed manipulator, to two, the DP method does not suffer from the "curse of dimensionality." While maintaining the elegance of our previous trajectory planning method, we have developed the DP method for the general case where 1) the actuator torque limits are dependent on one another, 2) the cost functions can have an arbitrary form, and 3) there are constraints on the jerk, or derivative of the acceleration. Also, we have shown that the DP solution converges as the grid size decreases. As numerical examples, the trajectory planning method is simulated for the first three joints of the PACS arm, which is a cylindrical arm manufactured by the Bendix Corporation.     

Path-constrained control of a redundant manipulator in a task space

This study addresses the problem of controlling a redundant manipulator with both state and control dependent constraints. The task of the robot is to follow by the end-effector a prescribed geometric path given in the task space. The control constraints resulting from the physical abilities of robot actuators are also taken into account during the robot movement. Provided that a solution to the aforementioned robot task exists, the Lyapunov stability theory is used to derive the control scheme. The numerical simulation results, carried out for a planar manipulator whose end-effector follows a prescribed geometric path given in a task space, illustrate the trajectory performance of the proposed control scheme.     

Path following by the end-effector of a redundant manipulator operating in a dynamic environment

This paper addresses the problem of generating at the control-loop level a collision-free trajectory for a redundant manipulator operating in dynamic environments which include moving obstacles. The task of the robot is to follow, by the end-effector, a prescribed geometric path given in the work space. The control constraints resulting from the physical abilities of robot actuators are also taken into account during the robot movement. Provided that a solution to the aforementioned robot task exists, the Lyapunov stability theory is used to derive the control scheme. The numerical simulation results for a planar manipulator whose end-effector follows a prescribed geometric path, given in both an obstacle-free work space and a work space including the moving obstacles, illustrate the trajectory performance of the proposed control scheme.     

Optimum motion planning in joint space for robots using genetic algorithms

The optimum motion planning in joint space (OMPJS) for robots, which generally consists of two subproblems, optimum path planning and optimum trajectory planning, was considered as a whole in the paper. A new method for optimum motion planning problem based on an improved genetic algorithm is proposed, which is more general, flexible and effective. This approach incorporates kinematics constraints, dynamics constraints, and control constraints of robotic manipulator. The simulation results for a two and a three degrees of freedom robots are presented and discussed. The simulations are based on genetic algorithm class library WGAClass 1.0 developed by us with Borland C++ 3.1.     

基于遗传算法的机械臂实时避障路径规划研究

针对模块化机械臂在运行时可能与工作空间中的障碍物发生碰撞的问题, 提出一种基于遗传算法的避障路径规划算法。首先采用D-H(Denavit-Hartenberg)表示法对机械臂进行建模, 并进行运动学和动力学分析, 建立机械臂运动学和动力学方程。在此基础上, 利用遗传算法分别在单/多个障碍物工作环境中, 以运动的时间、移动的空间距离和轨迹长度作为优化指标, 实现机械臂避障路径规划的优化。通过仿真验证了基于遗传算法的机械臂避障路径规划算法的有效性与可行性, 该算法提高了运行中的机械臂有效避开工作空间中障碍物的效率。     

A hybrid strategy for the time- and energy-efficient trajectory planning of parallel platform manipulators

In planning the trajectories of motor-driven parallel platform manipulators, the objective is to identify the trajectory which accomplishes the assigned motion with the minimal travel time and energy expenditure subject to the constraints imposed by the kinematics and dynamics of the manipulator structure. In this study, the possible trajectories of the manipulator are modeled using a parametric path representation, and the optimal trajectory is then obtained using a hybrid scheme comprising the particle swarm optimization method and the local conjugate gradient method. The numerical results confirm the feasibility of the optimized trajectories and show that the hybrid scheme is not only more computationally efficient than the standalone particle swarm optimization method, but also yields solutions of a higher quality.     

The control of robot manipulators with bounded input

The problem of tracking a desired trajectory in the state space of ann-link robotic manipulator with bounds on the allowable input torque is considered. Using a so-called optimal decision strategy (ODS), a pointwise optimal control law is derived which, at each timet, minimizes the deviation between the vector of joint accelerations and a desired joint acceleration vector, subject to the input constraints. The design of the optimal control law is reduced to the solution of a quadratic programming problem which is solved using the primal-dual method. The solution gives an on-line feedback control scheme for trajectory following in the presence of input constraints. In addition, we extend the above optimal decision strategy to the case where the controller design is based on a simplified model or where the plant itself is imprecisely known. The resulting torque optimization scheme may be incorporated into any existing control scheme to account for input bounds. This has important implications for the problem of deriving robust control schemes that take into account parameter uncertainty and model simplification. Simulations are presented for the case of a three-link manipulator with bounded torque, and our results are compared to the computed torque method. Our simulations show that by optimally adjusting the input torque to each joint when one or more of them saturates, significant improvement in tracking performance can result.     

Selection of near-minimum time geometric paths for robotic manipulators

A number of trajectory planning algorithms exist for calculating the joint positions, velocities, and torques which will drive a robotic manipulator along a given geometric path in minimum time. However, the time depends upon the geometric path, so the traversal time of the path should be considered again for geometric planning. There are algorithms available for finding minimum distance paths, but even when obstacle avoidance is not an issue, minimum (Cartesian) distance is not necessarily equivalent to minimum time. In this paper, we have derived a lower bound on the time required to move a manipulator from one point to another, and determined the form of the path which minimizes this lower bound. As numerical examples, we have applied the path solution to the first three joints of the Bendix PACS arm and the Stanford arm. These examples do indeed demonstrate that the derived approximate solutions usually require less time than Cartesian straight-line (minimum-distance) paths and joint-interpolated paths.     

Robust trajectory planning for robotic manipulators under payload uncertainties

A number of trajectory planning algorithms are available for determining the joint torques, positions, and velocities required to move a manipulator along a given geometric path in minimum time. These schemes require knowledge of the robot's dynamics, which in turn depend upon the characteristics of the payload which the robot is carrying. In practice, the dynamic properties of the payload will not be known exactly, so that the dynamics of the robot, and hence the required joint torques, must be calculated for a nominal set of payload characteristics. But since these trajectory planners generate nominal joint torques which are at the limits of the robot's capabilities, moving the robot along the desired geometric path at speeds calculated for the nominal payload may require torques which exceed the robot's capabilities. In this paper, bounds on joint torque uncertainties are derived in terms of payload uncertainties. Using these bounds, a new trajectory planner is developed to incorporate payload uncertainties such that all the trajectories generated can be realized with given joint torques. Finally, the trajectory planner is applied to the first three joints of the Bendix PACS arm, a cylindrical robot to demonstrate its use and power.     

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.     

基于hp自适应伪谱法的空间机器人路径规划

针对空间机器人运动过程中基座姿态产生较大扰动的问题,基于hp自适应高斯伪谱法提出了一种以基座所受反作用力矩最小为目标函数的空间机器人路径规划方法.首先,综合考虑空间机器人运动过程中存在的关节角度约束、关节角速度约束、控制力矩约束及初始状态和终端状态约束等约束条件,将空间机器人路径规划问题看成满足一系列约束条件和边界条件并实现特定性能指标最优的最优控制问题.其次,结合hp自适应高斯伪谱法（hp-AGPM）与非线性规划技术,求解带有边界约束和路径约束的优化控制问题,得到满足约束且性能指标最优的空间机器人运动轨迹.最后,以平面2自由度空间机械臂为例对所设计方法进行仿真验证,并与其他伪谱法进行对比分析.仿真结果表明：本文算法能在10.6 s的时间内规划出满足各约束条件且容许偏差低于10^-6的最优运动轨迹,并且在计算速度和配点数量上都优于其他伪谱法.     

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.     

基于深度学习的多机械手轨迹规划系统设计

目前提出的多机械手轨迹规划系统路径规划精准度低,避障能力差。基于深度学习对多机械手的规划系统进行设计,通过研究传统系统中存在精确度、智能性不足的缺点,在设计的系统分别引入了相应的解决条件,在硬件结构的设计中本文应用ISL-320型号的伺服电机提升多机械手的动力功能,应用SKT64系列的芯片提升多机械手的路径精准度;在应用程序设计上应用拟合算法与叠加算法对规划路径中的节点精准运算,在提升系统整体精准度的同时提升了系统的智能程度。实验结果表明,基于深度学习的多机械手轨迹规划系统路径与标准路径十分接近,说明该方法的规划精准度较高,避障能力得到有效增强。     

Dynamically-Stable Motion Planning for Humanoid Robots

We present an approach to path planning for humanoid robots that computes dynamically-stable, collision-free trajectories from full-body posture goals. Given a geometric model of the environment and a statically-stable desired posture, we search the configuration space of the robot for a collision-free path that simultaneously satisfies dynamic balance constraints. We adapt existing randomized path planning techniques by imposing balance constraints on incremental search motions in order to maintain the overall dynamic stability of the final path. A dynamics filtering function that constrains the ZMP (zero moment point) trajectory is used as a post-processing step to transform statically-stable, collision-free paths into dynamically-stable, collision-free trajectories for the entire body. Although we have focused our experiments on biped robots with a humanoid shape, the method generally applies to any robot subject to balance constraints (legged or not). The algorithm is presented along with computed examples using both simulated and real humanoid robots.     

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.

     

Optimal path generation for a simulated autonomous mobile robot

The paper deals with a set of algorithms including path planning, trajectory planning, and path tracking for a tricycle type wheeled mobile robot. Path planning is carried out with parametric polynomial interpolation using an optimization algorithm based on robot geometric constraints. Trajectory characteristics are then derived from the planned geometric path with time varying parameters. A sliding mode control algorithm combined with an adaptive control law are used to track the planned trajectory. The technique deals with an environment free of obstacles. However, it can be easily integrated in a piecewise non colliding path generation. Simulation results are presented to show the validity of the different algorithms.Bissé Emmanuel is a Ph.D. student at the École Polytechnique de Montréal, Department of Mechanical Engineering.Bentounes Mohamed is a Ph.D. student at the École Polytechnique de Montréal, Department of Mechanical Engineering.Boukas El-Kébir is a Professor at the École Polytechnique de Montréal, Department of Mechanical Engineering.     

State Space Constrained Iterative Learning Control for Robotic Manipulators

Real‐life work operations of industrial robotic manipulators are performed within a constrained state space. Such operations most often require accurate planning and tracking a desired trajectory, where all the characteristics of the dynamic model are taken into consideration. This paper presents a general method and an efficient computational procedure for path planning with respect to state space constraints. Given a dynamic model of a robotic manipulator, the proposed solution takes into consideration the influence of all imprecisely measured model parameters, making use of iterative learning control (ILC). A major advantage of this solution is that it resolves the well‐known problem of interrupting the learning procedure due to a high transient tracking error or when the desired trajectory is planned closely to the state space boundaries. The numerical procedure elaborated here computes the robot arm motion to accurately track a desired trajectory in a constrained state space taking into consideration all the dynamic characteristics that influence the motion. Simulation results with a typical industrial robot arm demonstrate the robustness of the numerical procedure. In particular, the results extend the applicability of ILC in robot motion control and provide a means for improving the overall trajectory tracking performance of most robotic systems.     

Suboptimal control of industrial manipulators with a weighted minimum time-fuel criterion

Even if a manipulator does not have to follow a prespecified path (i.e., a time history of position and velocity) due to the complexity and nonlinearity of the manipulator dynamics, control of manipulators has been conventionally divided into two subproblems, namely path planning and path tracking, which are then separately and independently solved. This may result in mathematically tractable solutions but cannot offer a solution that utilizes manipulators' maximum capabilities (e.g., operating them at their maximum speed). To combat this problem, we have developed a suboptimal method for controlling manipulators that provides improved performance in both their operating speed and use of energy. The nonlinearity and the joint couplings in the manipulator dynamics-a major hurdle in the design of robot control-are handled by a new concept of averaging the dynamics at each sampling interval. With the averaged dynamics, we have derived a feedback controller which has a simple structure allowing for on-line implementation with inexpensive mini- or microcomputers, and offers a near minimum time-fuel (NMTF) solution, thus enabling manipulators to perform nearly up to their maximum capability and efficiency. As a demonstrative example, we have simulated the proposed control method with a dynamic model of the Unimation PUMA 600 series manipulator on a DEC VAX-11/780. The simulation results agree with the expected high performance nature of the control method.     

Nonlinear Performance Index (npi): A Tool for Manipulator Dynamics Improvement

The precise control of manipulators depends nonlinearly on the velocity of the motion as well as on manipulator configuration and commanded acceleration requiring complex control strategies. This paper presents a useful tool for identifying and quantifying nonlinear effects appearing during the motion of any manipulator, the Nonlinear Performance Index (npi). The npi takes into account not only the geometrical parameters defining the manipulator but also its structural dynamics through the use of inertial parameters like mass, inertia, centre of mass... The npi can be used in the design stage for analysing and reducing these undesirable nonlinear effects in any general motion or in the trajectory planning looking for paths along which more precise control is expected. The last part of the paper shows how this design optimisation and path planning has been applied to the Agribot, a fruit picking robot designed at the IAI.     

Motion control and obstacle avoidance of a mobile robot with an onboard manipulator

Navigation and control of autonomous mobile vehicles with onboard manipulator systems are currently being investigated for intelligent manufacturing applications. A systematic approach for modeling and base motion control of a mobile vehicle with an onboard robot arm is presented. Feedback linearization is used to take into account the complete dynamics with non-holonomic constraints, yet methods from potential field theory are incorporated to provide resolution among possibly conflicting performance goals (e.g. path following and obstacle avoidance). The feedback linearization provides an inner loop that accounts for possible motion of the onboard arm. The two cases of maintaining a desired course and speed, and following a desired Cartesian trajectory are considered. The outer control loop is designed using potential field theory, with the two objectives of homing and avoiding an obstacle. This simple result obtained using potential functions provides very naturally the necessary intelligence for online resolution of conflicting performance objectives. It gives capabilities to these autonomous vehicles for maintaining a desired course and speed or tracking a Cartesian trajectory, avoiding obstacles during the course of travel, and initiating new online path planning when the size of the object is large so that unnecessary wandering in the work space is avoided.