Reliable computation of minimum-time motions for manipulators moving in obstacle fields using a successive search for minimum-overload trajectories

In this paper we present a practical method for finding obstacle-free minimum-time motions for manipulators subject to the limits of velocity-dependent actuator forces. An optimal motion-planning problem is converted into a finite-dimensional nonlinear programming problem by means of parameter optimization with quintic B-splines. We introduce the concept of the minimum-overload trajectory in which the motion time is specified to be faster than the actuators can handle, and the actuator overloads are minimized with the motion time fixed. By successive searches for minimum-overload trajectories, minimum-time motions are determined for three example manipulators without simplifying any of the kinematic, dynamic or geometric properties of the manipulators or the obstacles. In the resultant minimum-time motions, almost all of the joint actuators are close to saturation during the motions. © 2005 Wiley Periodicals, Inc.     

Geometrically non-linear formulation of flexible multibody systems in terms of beam elements: Geometric stiffness

The occurrence of strong deflections and major axial forces in many applications involving flexible multibodies entails including non-linear terms coupling deformation-induced axial and transverse displacements in the motion equation. The formulations, including such terms, are known as geometrically non-linear formulations. The authors have developed one such formulation that preserves higher-order terms in the strain energy function. By expressing such terms as a function of selected elastic coordinates, three stiffness matrices and two non-linear vectors of elastic forces are defined. The first matrix is the conventional constant-stiffness matrix, the second is the classical geometric stiffness matrix and the third is a second-order geometric stiffness matrix. The aim of this work is to define the third matrix and the two non-linear vectors of elastic forces by using the finite-element method.     

Observer-Based Optimal Control of Flexible Stewart Parallel Robots

The design and simulation of two optimal control schemes for a parallel flexible link manipulator of the Stewart type is presented. The first control scheme combines a nonlinear rigid model of the flexible manipulator with a linear rigid observer, whereas the second scheme uses a nonlinear flexible manipulator model with a linear flexible observer. The majority of the results available in the literature do not address the optimal control problem through the use of observers, as it is done in this paper, for the control of parallel robots. The simulation results have shown in both cases that indeed optimal state-obscrver-based control is a good candidate for controlling parallel-link manipulators in practice. As expected, the second scheme (flexible model plus flexible observer) gives better results than the first one, achieving faster trajectory tracking. The effects of white noises, applied forces, and zero gravity environment were taken under consideration.     

Joint space trajectory planning for flexible manipulators

A trajectory planning approach for controlling flexible manipulators is proposed. It is demonstrated that choosing actual joint angles as the generalized rigid coordinates is the key to applying the proposed approach. From the observation of the special structure of the input matrix, the concepts of motion-induced vibration and inverse dynamics under a specified motion history of the joints are formed naturally. Based on the above concepts, trajectory planning in joint space is proposed by using the optimization technique to determine the motion of joints along a specified path in joint space or work space and for general point-to-point motion. The motion for each joint is assumed to be in a class consisting of a fifth-order polynomial and a finite terms of Fourier series. This parameterization of motion allows the optimal trajectory planning to be formulated as a standard nonlinear programming problem, which avoids the necessity of solving a two-point-boundary-value problem and using dynamic programming. Setting the accelerations to zero at the initial and the final times is used to obtain smoother motion to reduce the spillover energy into unmodeled high-frequency dynamics. A penalty term on vibration energy contained in the performance index is used to minimize the vibration of the system modeled by lower frequency only. The final simulation results show the effectiveness of the proposed approach and the advantage for proper trajectory planning. © 2995 John Wiley & Sons, Inc.     

Time‐optimal constant speed motion program for multiple cooperating manipulators

This article presents a systematic approach for synthesizing the time‐optimal constant speed motion program for multiple manipulators moving a commonly held object along a specified Cartesian trajectory. In this approach, the motion program is constructed by using piecewise polynomials to blend the acceleration, constant speed, and deceleration periods. The polynomials are interpolated according to the boundary and continuity conditions to obtain a smooth and continuous profile. With this formulation, it is shown that the final form of the motion program can be established in terms of the initial acceleration, the constant operation speed, and the finial deceleration. The optimum values of these terms to allow the given trajectory to be executed in minimum time are determined based on the parametric dynamic equations of the system and the torque constraints of the actuator. This approach is conceptually straightforward and can be applied to various multirobot systems with nonlinear actuator constraints. ©1999 John Wiley & Sons, Inc.     

Chain Vibration and Dynamic Stress in Three-Dimensional Multibody Tracked Vehicles

A three-dimensional computational finite element procedure for the vibration and dynamic stress analysis of the track link chains of off-road vehicles is presented in this paper. The numerical procedure developed in this investigation integrates classical constrained multibody dynamics methods with finite element capabilities. The nonlinear equations of motion of the three-dimensional tracked vehicle model in which the track link s are considered flexible bodies, are obtained using the floating frame of reference formulation. Three-dimensional contact force models are used to describe the interaction of the track chain links with the vehicle components and the ground. The dynamic equations of motion are first presented in terms of a coupled set of reference and elastic coordinates of the track links. Assuming that the structural flexibility of the track links does not have a significant effect on their overall rigid body motion as well as the vehicle dynamics, a partially linearized set of differential equations of motion of the track links is obtained. The equations associated with the rigid body motion are used to predict the generalized contact, inertia, and constraint forces associated with the deformation degrees of freedom of the track links. These forces are introduced to the track link flexibility equations which are used to calculate the deformations of the links resulting from the vehicle motion. A detailed three-dimensional finite element model of the track link is developed and utilized to predict the natural frequencies and mode shapes. The terms that represent the rigid body inertia, centrifugal and Coriolis forces in the equations of motion associated with the elastic coordinates of the track link are described in detail. A computational procedure for determining the generalized constraint forces associated with the elastic coordinates of the deformable chain links is presented. The finite element model is then used to determine the deformations of the track links resulting from the contact, inertia, and constraint forces. The results of the dynamic stress analysis of the track links are presented and the differences between these results and the results obtained by using the static stress analysis are demonstrated.     

柔性臂漂浮基空间机器人建模与轨迹跟踪控制

利用拉格朗日法和假设模态方法建立了末端柔性的两臂漂浮基空间机器人的非线性动力学方程．通过坐标变换,推导出一种新的以可测关节角为变量的全局动态模型，并在此基础上运用基于模型的非线性解耦反馈控制方法得到关节相对转角与柔性臂的弹性变形部分解耦形式控制方程．最后，讨论了柔性臂漂浮基空间机器人的轨迹跟踪问题，并通过仿真实例计算,表明该模型转换及控制方法对于柔性臂漂浮基空间机器人末端轨迹跟踪控制的有效性．     

A Perturbation Method for Dynamic Analysis and Simulation of Flexible Manipulators

This paper presents a perturbation method for the dynamicsimulation of flexible manipulators. In this method thevibrational motion of the manipulator is modeled as a first-orderperturbation of the nominal rigid link motion. For that purposethe flexible dynamic model is split into two parts. A rigidifiedsystem describes the nominal rigid link motion of themanipulator. A linear system linearized about the nominaltrajectory describes the vibrational motion. These equations arecomputationally more efficient than the non-linear dynamicequations and offer more insight in the dynamic phenomena of thesystem. The method is based on a full non-linear finite elementformulation which treats the general case of coupled largedisplacements motion and small elastic motion. A planar one linkmanipulator and a spatial two link manipulator with flexiblelinks are used for case studies. A comparison is made between thenon-linear and the perturbation analyzes. The perturbation methodappears to be a very efficient approach for dynamic analyzes offlexible manipulators and yields accurate results even forsystems with low frequency elastic modes.     

A systematic formulation of dynamic equations for robot manipulators with elastic links

Although a variety of formulation schemes for the dynamic equations of robot manipulators with rigid links can be found in the literature, an efficient method of the formulation for robot manipulators with elastic links is not well known. Accordingly, this work presents the derivation of the equations of motion for application to mechanical manipulators with elastic links. The formulation is conducted analytically using Hamilton's principle. The resultant equations consist of the terms of inertial, Coriolis, centrifugal, gravitational, and exerted forces. They are expressed in terms of a set of independent generalized coordinates. In contrast to conventional variational approaches, the present method provides an efficient and systematic way for obtaining the compact symbolic equations of flexible manipulator systems. Two examples are presented to illustrate the proposed methodology. Firstly, a three-link flexible manipulator with three revolute joints is studied. A flexible manipulator consisting of a prismatic joint and a discrete mass is the second model.     

Optimal Trajectory Planning for Minimum Vibration of Flexible Redundant Cooperative Manipulators

The optimal path planning for two flexible cooperating manipulators carrying a solid object on a prescribed tip trajectory has been studied using kinematic resolution. The formulation has been derived using the Pontryagin minimum principle that results in a two-point boundary value problem. Also, a numerical technique based on converting the abstract optimization to parametric optimization problem has been proposed. The problem has been solved and compared for globally minimized elastic deformation of flexible links and joints as well as minimization of joints velocities while the end-effector moves along the specified path.     

Robust adaptive controller design for flexible joint manipulators

In this paper the control problem for robot manipulators with flexible joints is considered. A reduced-order flexible joint model is constructed based on a singular perturbation formulation of the manipulator equations of motion. The concept of an integral manifold is utilized to construct the dynamics of a slow subsystem. A fast subsystem is constructed to represent the fast dynamics of the elastic forces at the joints. A composite control scheme is developed based on on-line identification of the manipulator parameters which takes into account the effect of certain unmodeled dynamics and parameter variations. Stability analysis of the resulting closed-loop full-order system is presented. Simulation results for a single link flexible joint manipulator are given to illustrate the applicability of the proposed algorithm.     

High-speed positioning of an industrial robot based on Preview-Learning control

In this study, a numerical procedure for designing kinematic parameters of SCARA-type manipulators is proposed to yield such a design that the resulting manipulator has the fastest cycle time for a given task. To achieve this goal, an optimization problem is formulated to minimize the cycle time by determining geometric parameters such as the link lengths and the locations of manipulators as well as the trajectory. The representative task to get the cycle time is defined as CP (continuous path) motion along the path crisscrossing the standard working area. A gradient projection algorithm is used to obtain the optimal design with the assumption that each actuator should exert a torque and angular velocity within the capacity of specific commercially available direct-drive motors. SCARA-type manipulators of both absolute coordinate and relative coordinate types are designed to reduce the cycle times. The results show that the absolute coordinate manipulator produces a shorter cycle time than the relative coordinate manipulator in optimal designs.     

Adaptive compliant force–motion control of coordinated non-holonomic mobile manipulators interacting with unknown non-rigid environments

Most studies on the coordination control of multiple mobile manipulators system assume exact knowledge of system dynamics and deal only with motion control. However, actual applications may involve the tasks in which multiple coordinated mobile manipulators system interacts with rigid or non-rigid working surfaces. In this paper, we consider multiple mobile manipulators grasping a rigid object in contact with a deformable working surface, whose geometric and real physical parameters are unknown but boundedness of physical parameters is known. The contact forces are nonlinear and difficult to model. A neuro-adaptive control for coordinated mobile manipulators is proposed for robust force/motion tracking. The control law is based on the philosophy of the parallel approach, in which the control problem is divided into three subspaces and the adaptive techniques are employed to deal with the uncertain environmental constraints, disturbances, and unknown robotic and object dynamics. The proposed adaptive force–motion controller guarantees the tracking errors of motion and force trajectories converge to zero. Simulation examples are presented to verify the effectiveness of the proposed control.     

Existence and uniqueness of solutions of the inverse dynamics of multilink flexible arms: Convergence of a numerical scheme

The inverse dynamics of flexible manipulators consists in finding the joint torques that drive the end effector of a manipulator along a given trajectory. A proof of the existence and uniqueness of solutions of the inverse dynamics of planar open-chain multilink flexible manipulators and the convergence of an algorithm previously proposed by the second author for the numerical solution of the problem are presented in this article. Higher-order strain terms leading to the geometric stiffness matrix in finite element models are included in the formulation and their influence on the inverse dynamics solution is studied.     

Dynamic Modeling Based on Real-Time Deflection Measurement and Compensation Control for Flexible Multi-Link Manipulators

The problem of tip trajectory tracking control is considered in this paper for flexible multi-link manipulators. An integrated optical laser sensor system is utilized to measure the tip deformations of the flexible links. The Lagrangian assumed-mode method incorporating the measured linear displacements and angular deflections of flexible links is used to derive the dynamic model of the flexible manipulator. To reduce as far as possible the tip tracking/positioning errors caused by the link flexibility, an error compensation approach is proposed. The additional compensation amounts of joint variables are calculated kinematically in terms of the measured deformations, and are added to the nominal commands generated by the computed torque controller. The simulation results demonstrate the effectiveness of the proposed approach.     

New optimization method to solve motion planning of dynamic systems: application on mechanical manipulators

In this paper, a new method is proposed to solve a nonlinear optimal control problem and determine the Dynamic Load-Carrying Capacity (DLCC) of fixed and mobile manipulators in point-to-point motion. Solution methods for designing nonlinear optimal controller in closed loop form are usually based on indirect methods, but the proposed method is a combination of direct and indirect methods. The optimal control law with state feedback form, for nonlinear dynamic systems, is given by the solution to the nonlinear Hamilton–Jacobi–Bellman (HJB) equation. The Galerkin procedure and a nonlinear optimization algorithm are used to solve this equation numerically. Another innovation of this paper is optimal trajectory planning, which is done simultaneously with the controller design procedure. Finally, a new algorithm is developed to find DLCC of manipulators and the related optimal trajectory using proposed method. The validity of the method is demonstrated via simulation and experimental tests for a fixed manipulator and two-link wheeled mobile manipulator named Scout.     

Experimental Control of a Two-Dof Flexible Robot Manipulator by Optimal and Sliding Methods

This paper presents an experimental study of a robust control scheme for flexible-link robotic manipulators. The design is based on a simple strategy for trajectory tracking which exploits the two-time scale nature of the flexible part and the rigid part of the dynamic equations of this kind of robotic arms: A slow subsystem associated with the rigid motion dynamics and a fast subsystem associated with the flexible link dynamics. Two experimental approaches are considered. In a first test an LQR optimal design strategy is used, while a second design is based on a sliding-mode scheme. Experimental results on a laboratory two-dof flexible manipulator show that this composite approach achieves good closed-loop tracking properties for both design philosophies, which compare favorably with conventional rigid robot control schemes.     

柔性机械臂运动轨迹的鲁棒自适应控制

本文针对多连杆柔性机械臂的运动轨迹问题，讨论了动力学建模，控制系统结构设计以及鲁棒自适应控制法，运用假设模记方法得到了柔性机械臂动力学所似方程，通过对柔性机械臂动力学特性分析，建立了等价动力学模型，依此提出了一种鲁棒自适应控制算法，并给出仿真研究结果。