Analytical development of dynamic equations of motion for a three-dimensional flexible link manipulator with revolute and prismatic joints

In this paper, a mathematical model capable of handling a three-dimensional (3D) flexible n-degree of freedom manipulator having both revolute and prismatic joints is considered. This model is used to study the longitudinal, transversal, and torsional vibration characteristics of the robot manipulator and obtain kinematic and dynamic equations of motion. The presence of prismatic joints makes the mathematical derivation complex. In this paper, for the first time, prismatic joints as well as revolute joints have been considered in the structure of a 3D flexible n-degree of freedom manipulator. The kinematic and dynamic equations of motion representing longitudinal, transversal, and torsional vibration characteristics have been solved in parametric form with no discretization. In this investigation, in order to obtain an analytical solution of the vibrational equations, a novel approach is presented using the perturbation method. By solving the equations of motion, it is shown that mode shapes of the link with prismatic joints can be modeled as the equivalent clamped beam at each time instant. As an example, this method is applied to a three degrees of freedom robot with revolute and prismatic joints. The obtained equations are solved using the perturbation method and the results are used to simulate vibrational behavior of the manipulator.     

Measurement and analysis of structural dynamics properties of robotic joint transmission system

The flexibility of a robotic manipulator is considered an important factor, especially the joint compliance for improving the accuracy and operating speed of a robot application. In this article, two methods are employed to identify the structural dynamic characteristics of each joint transmission system of an ITRI-U type robot. The driving system of each joint is modeled as a mass-spring-damper mechanism that has a second-order dynamic mathematical equation. Then, the response experiments of hammer impact excitation and motor driving actuation are done and the difference between them compared. Those are used to simulate two different operation situations, the robot colliding with the environment and the discontinuing dynamic operation impacts. The system's parameters of each joint axis are obtained by the system identification technique. From the experimental results, the angular error of the motor shaft can be compensated by the control gain of the motor controller. However, the small damping ratio of the robotic mechanism limits the magnitude of servo gain. If the servo gain increases, the robot arm has oscillation phenomenon during dynamic operation. In addition, the error due to joint elasticity cannot be overcome by the current industrial robotic controller. Therefore, a suitable controller should be designed to compensate the dynamic effects of joint compliance with the identified parameters.     

Modelling the dynamic characteristics of a robot arm joint

The dynamic behavior of a robot arm has been modelled, taking into account the elasticity and damping of the joints and the backlash introduced by the gear pairs of the transmission mechanism. The links of the robot arm are considered to be rigid and its joints rotational. A technique for automatic formulation of the differential equations of the lumped mass dynamic model of the gear train, incorporating the backlash nonlinearity, has been developed. Each link and the accompanying joint comprise a module. The dynamic equations of the manipulator are formulated through an iterative technique. The algorithm will provide automatic formulation of the dynamic equations of the model of the arm including the referenced nonlinearities of the joint drives. Finally, an application of the algorithm to a planar manipulator arm with two links and two rotational joints is presented.     

Dynamic modeling and vibration characteristics analysis of flexible-link and flexible-joint space manipulator

With the development of space technology, lighter and larger space manipulators will be born, of which flexible characteristics are more obvious. The manipulator vibration caused by the flexibility not only reduces the efficiency of the manipulator but also affects the accuracy of the operation. The flexibility of space manipulator mainly comes from structural flexibility of links and transmission flexibility of harmonic gear reducer in joints. The vibrations generated by these two kinds of flexibility are coupled and transformed mutually, making the dynamics characteristics of space manipulator system complicated. Therefore it is difficult to assess respective effects of these flexibilities on vibrations of the manipulator tip. And the characteristics of integrated vibration of manipulator tip with different link and joint stiffnesses are not very clear. In this paper, the dynamic equations of multi-link multi-DOF flexible manipulator are established. Then, vibration responses of the tip under different elastic modulus, damping and joint stiffness were studied, and vibration characteristics of the tip with both link and joint were also analyzed. Moreover, the effects of motion planning on the vibration of the tip were analyzed. Finally, the vibration characteristics of the manipulator with flexible joints and links are verified by a two-degree-of-freedom manipulator experimental system. Dynamics analysis results presented some useful rules for the path planning and control to suppress the vibration of the flexible space manipulator.     

Design and Control of a Hydraulic Simulator for a Flexible-Joint Robot

For the first time, a novel experimental hydraulic system that simulates joint flexibility of a single-rigid-link flexible-joint robot manipulator, with the ability of changing the joint flexibility's parameters, was designed and implemented in this study. Such a system could facilitate future control studies of robot manipulators by reducing investigation time and implementation cost of research. It could also be used to test the performance of different strategies to control the movement of flexible-joint manipulators. A hydraulic rotary servo motor was used to simulate the action of a flexible-joint robot manipulator, which was a challenging task, since the control of angular acceleration was required. In this study, a single-rigid-link elastic-joint robot manipulator was mathematically modeled and implemented in which joint flexibility parameters such as stiffness and damping could be easily changed. This simulation is referred to as a 'function generator' to drive a hydraulic robot manipulator. In this study the desired angular acceleration of the manipulator was used as the input to the hydraulic rotary motor and the objective was to make the hydraulic system follow the desired acceleration in the frequency range specified. A hydraulic actuator robot was built and tested. The results indicated that if the input signal had a frequency in the range of 5–15 Hz and damping ratio of 0.1 (typical values for flexible joints), the experimental setup was able to reproduce the input signal with acceptable accuracy. Owing to the inherent noise associated with the measurement of acceleration and some severe nonlinearities in the rotary motor, control of the experimental test system using classical methods was a challenging task that had not been anticipated.     

Stability and control aspects of flexible link robot manipulators during constrained motion tasks

The effect of robotic manipulator structural compliance on system stability and trajectory tracking performance and the compensation of this structural compliance has been the subject of a number of publications for the case of robotic manipulator noncontact task execution. The subject of this article is the examination of dynamics and stability issues of a robotic manipulator modeled with link structural flexibility during execution of a task that requires the robot tip to contact fixed rigid objects in the work environment. The dynamic behavior of a general n degree of freedom flexible link manipulator is investigated with a previously proposed nonlinear computed torque constrained motion control applied, computed based on the rigid link equations of motion. Through the use of techniques from the theory of singular perturbations, the analysis of the system stability is investigated by examining the stability of the “slow” and “fast” subsystem dynamics. The conditions under which the fast subsystem dynamics exhibit a stable response are examined. It is shown that if certain conditions are satisfied a control based on only the rigid link equations of motion will lead to asymptotic trajectory tracking of the desired generalized position and force trajectories during constrained motion. Experiments reported here have been carried out to investigate the performance of the nonlinear computed torque control law during constrained motion of the manipulator. While based only on the rigid link equations of motion, experimental results confirm that high-frequency structural link modes, exhibited in the response of the robot, are asymptotically stable and do not destabilize the slow subsystem dynamics, leading to asymptotic trajectory tracking of the overall system. © 1992 John Wiley & Sons, Inc.     

Fault detection on robot manipulators using artificial neural networks

Nowadays, gas welding applications on vehicle’s parts with robot manipulators have increased in automobile industry. Therefore, the speed of end-effectors of robot manipulator is affected on each joint during the welding process with complex trajectory. For that reason, it is necessary to analyze the noise and vibration of robot’s joints for predicting faults. This paper presents an experimental investigation on a robot manipulator, using neural network for analyzing the vibration condition on joints. Firstly, robot manipulator’s joints are tested with prescribed of trajectory end-effectors for the different joints speeds. Furthermore, noise and vibration of each joint are measured. And then, the related parameters are tested with neural network predictor to predict servicing period. In order to find robust and adaptive neural network structure, two types of neural predictors are employed in this investigation. The results of two approaches improved that an RBNN type can be employed to predict the vibrations on industrial robots.     

欠驱动冗余度空间机器人优化控制

欠驱动控制是空间技术中容错技术的重要方面.本文研究了被动关节中有制动器的欠驱动冗余度空间机器人系统的运动优化控制问题.从系统动力学方程出发,分析了欠驱动冗余度空间机器人的优化能力和控制方法;给出了主、被动关节间的耦合度指标;提出了欠驱动冗余度空间机器人系统的“虚拟模型引导控制”方法,在这种方法中采用与欠驱动机器人机构等价的全驱动机器人作为模型来规划机器人的运动,使欠驱动系统在关节空间中逼近给出的规划轨迹,实现了机器人末端运动的连续轨迹运动优化控制;通过末关节为被动关节的平面三连杆机器人进行了仿真,仿真的结果证明了提出算法的有效性.     

Robust adaptive control of an underactuated free-flying space robot under a non-holonomic structure in joint space

This paper introduces a robust adaptive control scheme for an underactuated free-flying space robot under non-holonomic constraints. An underactuated robot manipulator is defined as a robot that has fewer joint actuators than the number of total joints. Because, if one of the joints is out of order, it is so hard to repair the joint, especially in space, the control of such a robot manipulator is important. However, it is difficult to control an underactuated robot manipulator because of the reduced dimension of the input space, i.e. the non-holonomic structure of the underactuated system. The proposed scheme does not need to assume that the exact dynamic parameters must be known. It is analysed in joint space to control the underactuated robot mounted on the space station under parametric uncertainties and external disturbances. The simulation results have shown that the proposed method is very feasible and robust for a two-link planar free-flying space robot with one passive joint.     

Microcomputer-based real-time control of a pantograph mechanism robot

This paper presents a technique for microcomputerbased real-time control of a three-axis pantograph robot. The dynamic model of a pantograph type manipulator, which includes both the kinematic and kinetic equations of motion. is first established by applying Lagrange's method. In order to improve the computational efficiency, the nonlinear and coupled coefficients of the equations of motion are then simplified by using a piecewise planar surface fitting scheme. The pantograph manipulator possesses three independently powered joints. Each joint provides position and rate feedback to a proportional and derivative (PD) type servo controller. Simulation results show that the proposed simplification and control algorithms are very suitable for microcomputer-based real-time control of a simple pantograph type manipulator.     

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.     

Computed torque control of fully-actuated nondeterministic multibody systems

In this paper we are interested in the dynamic behavior of a slider-crank mechanism with single and two revolute clearance joints. Due to the clearance existence in the revolute joints, it is important to choose an appropriate contact force model in analyzing the dynamic response of a slider-crank mechanism with clearances. The dynamic equations are established by combining the Newton–Euler equations with modified contact force model and improved Coulomb friction force model, and the Baumgarte stabilization approach is used to improve the numerical stability. According to numerical and experimental results, the method of continuous contact can be verified to be reasonable. Comparing dynamic the response between one clearance joint and two clearance joints in a crank-slider mechanism, it is easy to find a significant mutual coupling region due to the presence of two clearance joints by simply contact figures. The dynamic response in a mechanism with two clearance joints is not a simple superposition of that in mechanism with one clearance joint. Therefore, all the joints in a multibody system should be modeled as clearance joints.     

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．     

Modelling of Complete Robot Dynamics Based on a Multi-Dimensional, RBF-like Neural Architecture

A neural network based identification approach of manipulator dynamics is presented. For a structured modelling, RBF-like static neural networks are used in order to represent and adapt all model parameters with their non-linear dependences on the joint positions. The neural architecture is hierarchically organised to reach optimal adjustment to structural apriori-knowledge about the identification problem. The model structure is substantially simplified by general system analysis independent of robot type. But also a lot of specific features of the utilised experimental robot are taken into account.A fixed, grid based neuron placement together with application of B-spline polynomial basis functions is utilised favourably for a very effective recursive implementation of the neural architecture. Thus, an online identification of a dynamic model is submitted for a complete 6 joint industrial robot.     

Noise analysis of robot manipulator using neural networks

Due to a lot of robot manipulators application in industry, low noise degree is very important criteria for robot manipulator's joints. In this paper, joint noise problem of a robot manipulator with five joints is investigated both theoretically and experimentally. The investigation is consisted of two steps. First step is to analyze the noise of joints using a hardware and software. The hardware is a part of noise sensors. The second step; according to experimental results, some neural networks are employed for finding robust neural noise analyzer. Five types of neural networks are used to compare each other. From the results, it is noted that the proposed RBFNN gives the best results for analyzing joint noise of the robot manipulator.     

Investigation of joint clearance effects on the dynamic performance of a planar 2-DOF pick-and-place parallel manipulator

In this study, the effects of joint clearance on the dynamic performance of a planar 2-DOF pick-and-place parallel manipulator are investigated. The parallel manipulator is modeled by multi-body system dynamics. The contact effect in revolute joints with clearance is established by using a continuous analysis approach that is combined with a contact force model considering hysteretic damping. The evaluation of the contact force is based on Hertzian contact theory that accounts for the geometrical and material properties of the contacting bodies. Furthermore, the incorporation of the friction effect in clearance joints is performed using a modified Coulomb friction model. By numerical simulation, variations of the clearance joint's eccentric trajectory, the joint reaction force, the input torque, the acceleration, and trajectory of the end-effector are used to illustrate the dynamic behavior of the mechanism when multiple clearance revolute joints are considered. The results indicate that the clearance joints present two obvious separation leaps in a complete pick-and-place working cycle of the parallel manipulator, following a collision. The impact induces system vibration and thus reduces the dynamic stability of the system. The joint clearances affect the amplitudes of the joint reaction force, the input torque, and the end-effector's acceleration, additionally the joint clearances degrade the kinematic and dynamic accuracy of the manipulator's end-effector. Finally, this study proposes related approaches to decrease the effect of joint clearances on the system's dynamic properties for such parallel manipulator and prevent “separation-leap-impact” events in clearance joints.     

Robot manipulator kinematic compensation using a generalized jacobian formulation

The kinematic error compensation of robot manipulators is at present being attempted by improving the precision of the nominal robot kinematic parameters. This paper addresses the problem of kinematic compensation using a new mathematical joint model proposed to account for shortcomings in existing methods. The corrected manipulator transformation is formulated in terms of “generalized Jacobians”: relating differential errors at the joints to the differential change in the manipulator transformation. The details of application are discussed for a particular industrial manipulator.     

Application of a Perturbation Method for Realistic Dynamic Simulation of Industrial Robots

This paper presents the application of a perturbation method for the closed-loop dynamic simulation of a rigid-link manipulator with joint friction. In this method the perturbed motion of the manipulator is modelled as a first-order perturbation of the nominal manipulator motion. A non-linear finite element method is used to formulate the dynamic equations of the manipulator mechanism. In a closed-loop simulation the driving torques are generated by the control system. Friction torques at the actuator joints are introduced at the stage of perturbed dynamics. For a mathematical model of the friction torques we implemented the LuGre friction model that accounts both for the sliding and pre-sliding regime. To illustrate the method, the motion of a six-axes industrial Stäubli robot is simulated. The manipulation task implies transferring a laser spot along a straight line with a trapezoidal velocity profile. The computed trajectory tracking errors are compared with measured values, where in both cases the tip position is computed from the joint angles using a nominal kinematic robot model. It is found that a closed-loop simulation using a non-linear finite element model of this robot is very time-consuming due to the small time step of the discrete controller. Using the perturbation method with the linearised model a substantial reduction of the computer time is achieved without loss of accuracy.     

Real-time implementation of a robust adaptive controller for arobotic manipulator based on digital signal processors

Presents an approach to the design and real-time implementation of an adaptive controller for a robotic manipulator based on digital signal processors. The Texas Instruments DSP (TMS320C31) chips are used in implementing real-time adaptive control algorithms to provide enhanced motion control performance for robotic manipulators. In the proposed scheme, adaptation laws are derived from the direct model reference adaptive control principle based on the improved Lyapunov second method. The proposed adaptive controller consists of an adaptive feedforward and feedback controller and PI-type time-varying auxiliary control elements. The proposed control scheme is simple in structure, fast in computation, and suitable for real-time control. Moreover, this scheme does not require any accurate dynamic modeling nor values of manipulator parameters and payload. Performance of the proposed adaptive controller is illustrated by simulation and experimental results for an industrial robot with four joints in the joint space and Cartesian space     

Fuzzy self-adaptive radial basis function neural network-based control of a seven-link redundant industrial manipulator

This paper proposes a method for the identification of dynamics and control of a multi-link industrial robot manipulator using Runge-Kutta-Gill neural networks (RKGNNs). RKGNNs are used to identify an ordinary differential equation of the dynamics of the robot manipulator. A structured function neural network (NN) with sub-networks to represent the components of the dynamics is used in the RKGNNs. The sub-networks consist of shape adaptive radial basis function (RBF) NNs. An evolutionary algorithm is used to optimize the shape parameters and the weights of the RBFNNs. Due to the fact that the RKGNNs can accurately grasp the changing rates of the states, this method can effectively be used for long-term prediction of the states of the robot manipulator dynamics. Unlike in conventional methods, the proposed method can even be used without input torque information because a torque network is part of the functional network. This method can be proposed as an effective option for the dynamics identification of manipulators with high degrees-offreedom, as opposed to the derivation of dynamic equations and making additional hardware changes as in the case of statistical parameter identification such as linear least-squares method. Experiments were carried out using a seven-link industrial manipulator. The manipulator was controlled for a given trajectory, using adaptive fuzzy selection of nonlinear dynamic models identified previously. Promising experimental results are obtained to prove the ability of the proposed method in capturing nonlinear dynamics of a multi-link manipulator in an effective manner.