Inverse Jacobian Regulator With Gravity Compensation: Stability and Experiment

Task-space regulation of robot manipulators can be classified into two fundamental approaches, namely, transpose Jacobian regulation and inverse Jacobian regulation. In this paper, two inverse Jacobian regulators with gravity compensations are presented, and the stability problems are formulated and solved. It is shown that the inverse Jacobian systems can be stabilized, and there exists a region of attraction such that the system remains stable. Our results show that the two fundamental approaches are two dual controllers, in the sense that the transpose Jacobian matrix can be replaced by the inverse Jacobian matrix and vice versa. The theoretical results are verified experimentally by implementing the inverse Jacobian regulators on an industrial robot, PUMA560.     

An Adaptive Regulator of Robotic Manipulators in the Task Space

This note addresses the problem of position control of robotic manipulators both nonredundant and redundant in the task space. A computationally simple class of task space regulators consisting of a transpose adaptive Jacobian controller plus an adaptive term estimating generalized gravity forces is proposed. The Lyapunov stability theory is used to derive the control scheme. The conditions on controller gains ensuring asymptotic stability are obtained herein in a form of simple inequalities including some information extracted from both robot kinematic and dynamic equations. The performance of the proposed control strategy is illustrated through computer simulations for a direct-drive arm of a SCARA type redundant manipulator with the three revolute kinematic pairs operating in a two-dimensional task space.     

Asymptotically stable visual servoing of manipulators via neural networks

In this article we present a class of position control schemes for robot manipulators based on feedback of visual information processed through artificial neural networks. We exploit the approximation capabilities of neural networks to avoid the computation of the robot inverse kinematics as well as the inverse task space–camera mapping which involves tedious calibration procedures. Our main stability result establishes rigorously that in spite of the neural network giving an approximation of these mappings, the closed‐loop system including the robot nonlinear dynamics is locally asymptotically stable provided that the Jacobian of the neural network is nonsingular. The feasibility of the proposed neural controller is illustrated through experiments on a planar robot. © 2000 John Wiley & Sons, Inc.     

A simple PID control for asymptotic visual regulation of robot manipulators

This paper addresses the asymptotic regulation problem of robot manipulators with a vision‐based feedback. A simple image‐based transpose Jacobian proportional‐integral‐derivative (PID) control is proposed. The closed‐loop system formed by the proposed PID control and robot system is shown to be asymptotically stable by using Lyapunov's direct method and LaSalle's invariance theorem. Advantages of the proposed control include the absence of dynamical model parameters in the control law formulation and the control gains are easily chosen according to simple inequalities including some well‐known bounds extracted from robot dynamics and kinematics. Simulations performed on a two degree‐of‐freedom manipulator are provided to illustrate the effectiveness of the proposed approach. Copyright © 2010 John Wiley & Sons, Ltd.     

Collision‐free control of robotic manipulators in the task space

This paper addresses the problem of position control of robotic manipulators in the task space with obstacles. A computationally simple class of task space regulators consisting of a transpose Jacobian controller plus an integral term including the task error and the gradient of a penalty function generated by obstacles is proposed. The Lyapunov stability theory is used to derive the control scheme. Through the use of the exterior penalty function approach, collision avoidance of the robot with obstacles is ensured. The performance of the proposed control strategy is illustrated through computer simulations for a direct‐drive arm of a SCARA type manipulator operating in both an obstacle‐free task space and a task space including obstacles. © 2005 Wiley Periodicals, Inc.     

Experimental evaluation of Cartesian stiffness control on a seven degree-of-freedom robot arm

The programmability of Cartesian stiffness in Cartesian servo control algorithms that do not use explicit force feedback is examined. A number of Cartesian algorithms are implemented and evaluated on a commercial seven degree-of-freedom robot arm, using the NASREM robot control system testbed. It is found that Cartesian servo algorithms which use the transpose of the Jacobian and model-based gravity compensation, provide easy programmability and accurate reproduction of stiffnesses over a wide range. When dynamic behavior is a consideration, dynamic damping control, augmented to include a parameterization of the manipulator self-motion, provides superior performance and programmability.     

Closed‐loop inverse kinematics algorithm for constrained flexible manipulators under gravity

This work deals with the inverse kinematics problem for a flexible robot manipulator under gravity in contact with a stiff surface. This problem consists of finding the joint and deflection variables for a given tip position and constraint force. The solution algorithm is based on the well‐known closed‐loop inverse kinematics (CLIK) scheme, using the Jacobian transpose, developed for rigid manipulators. The Jacobian employed in the algorithm is obtained by correcting the equivalent rigid manipulator Jacobian with two terms that account for the static deflections due to gravity and contact force, respectively. The algorithm is tested in a number of case studies on a planar two‐link arm. ©1999 John Wiley & Sons, Inc.     

Feedback control for robotic manipulator with an uncertain Jacobian matrix

In most applications of robots, a desired path for the end‐effector is usually specified in task space such as Cartesian space. One way to move the robot along this path is to solve the inverse kinematics problem to generate the desired angles in joint space. However, it is a very time consuming task to solve the inverse kinematics problem. Furthermore, in the presence of uncertainty in kinematics, it is impossible to derive the desired joint angle from the desired end‐effector path and the Jacobian matrix of the mapping from joint space to task space. In this article, a feedback control law using an uncertain Jacobian matrix is proposed for setpoint control of robots. Sufficient conditions for the bound of the estimated Jacobian matrix and stability conditions for the feedback gains are presented to guarantee the stability and passivity of the robots. A gravity regressor with an uncertain Jacobian matrix is also proposed for gravitational force compensation when the gravitational force is uncertain. Simulation results are presented to illustrate the performance of the proposed controllers. ©1999 John Wiley & Sons, Inc.     

Task-space neuro-sliding mode control of robot manipulators under Jacobian uncertainties

Cartesian robot control is an appealing scheme because it avoids the computation of inverse kinematics, in contrast to joint robot control approach. For tracking, high computational load is typically required to obtain Cartesian robot dynamics. In this paper, an alternative approach for Cartesian tracking is proposed under assumption that robot dynamics is unknown and the Jacobian are uncertain. A neuro-sliding second order mode controller delivers a low dimensional neural network, which roughly estimates inverse robot dynamics, and an inner smooth control loop guarantees exponential tracking. Experimental results are presented to confirm the performance in a real time environment.     

Applications of neural networks for coordinate transformations in robotics

The use of artificial neural networks is investigated for application to trajectory control problems in robotics. The relative merits of position versus velocity control is considered and a control scheme is proposed in which neural networks are used as static maps (trained off-line) to compute the inverse of the manipulator Jacobian matrix. A proof of the stability of this approach is offered, assuming bounded errors in the static map. A representative two-link robot is investigated using an artificial neural network which has been trained to compute the components of the inverse of the Jacobian matrix. The controller is implemented in the laboratory and its performance compared to a similar controller with the analytical inverse Jacobian matrix.     

Adaptive force–environment estimator for manipulators based on adaptive wavelet neural network

This study focuses on the accurate tracking control and sensorless estimation of external force disturbances on robot manipulators. The proposed approach is based on an adaptive Wavelet Neural Network (WNN), named Adaptive Force-Environment Estimator (WNN-AFEE). Unlike disturbance observers, WNN_AFEE does not require the inverse of the Jacobian transpose for computing the force, thus, it has no computational problem near singular points. In this scheme, WNN estimates the external force disturbance to attenuate its effects on the control system performance by estimating the environment model. A Lyapunov based design is presented to determine adaptive laws for tuning WNN parameters. Another advantage of the proposed approach is that it can estimate the force even when there are some parametric uncertainties in the robot model, because an additional adaptive law is designed to estimate the robot parameters. In a theorem, the stability of the closed loop system is proved and a general condition is presented for identifying the force and robot parameters. Some suggestions are provided for improving the estimation and control performance. Then, a WNN-AFEE is designed for a planar manipulator as an example, and some simulations are performed for different conditions. WNN_AFEE results are compared attentively with the results of an adaptive force estimator and a disturbance estimator. These comparisons show the efficiency of the proposed controller in dealing with different conditions.     

Global positioning of robot manipulators with mixed revolute and prismatic joints

The existing controllers for robot manipulators with uncertain gravitational force can globally stabilize only robot manipulators with revolute joints. The main obstacles to the global stabilization of robot manipulators with mixed revolute and prismatic joints are unboundedness of the inertia matrix and the Jacobian of the gravity vector. In this note, a class of globally stable controllers for robot manipulators with mixed revolute and prismatic joints is proposed. The global asymptotic stabilization is achieved by adding a nonlinear proportional and derivative term to the linear proportional-integral-derivative (PID) controller. By using Lyapunov's direct method, the explicit conditions on the controller parameters to ensure global asymptotic stability are obtained.     

Transputer arrays for the on-line computation of robot jacobians

On-line computation of forward and inverse Jacobian matrices is essential in robot manipulator controllers, where high-speed robot motion is required. The complexity of Jacobian calculation is such that the computational burden is large, and parallel processing is necessary if on-line computation is to be achieved. Various algorithms and parallel-processing networks suitable for this are considered. All algorithms have been implemented on transputer networks and computation times measured. The paper emphasises the importance of including communication overheads in comparisons of the computational efficiency of alternative algorithms and processor networks. Theoretical processing times based on computer cycle times and arithmetic operation counts are shown to be a false basis for comparison. Whilst considering the specific case of computation of Jacobian matrices for a robot manipulator, the paper provides a useful example of the considerations and constraints involved in distributing any algorithm across a multi-processor network.     

Constraint finite‐time control of redundant manipulators

This work addresses the problem of the accurate task‐space control subject to finite‐time convergence. Dynamic equations of a redundant manipulator are assumed to be uncertain. Moreover, globally unbounded disturbances are allowed to act on the manipulator when tracking the trajectory by the end effector. Furthermore, the movement is to be accomplished in such a way as to optimize some performance index. Based on suitably defined task‐space non‐singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of inverse‐free robust controllers consisting of a Jacobian transpose component plus a compensating term, which seem to be effective in counteracting uncertain dynamics, unbounded disturbances and (possible) kinematic singularities met on the robot trajectory. The numerical simulations carried out for a redundant manipulator of a Selective Compliant Articulated Robot for Assembly (SCARA) type consisting of three revolute kinematic pairs and operating in a two‐dimensional task space illustrate performance of the proposed controllers. Copyright © 2016 John Wiley & Sons, Ltd.     

Model reference adaptive impedance control in Cartesian coordinates for physical human–robot interaction

In this paper, a nonlinear model reference adaptive impedance controller is proposed and tested. The controller provides asymptotic tracking of a reference impedance model for the robot end-effector in Cartesian coordinates applicable to rehabilitation robotics or any other human–robot interactions such as haptic systems. The controller uses the parameters of a desired stable reference model which is the target impedance for the robot’s end-effector. It also considers uncertainties in the model parameters of the robot. The asymptotic tracking is proven using Lyapunov stability theorem. Moreover, the adaptation law is proposed in joint space for reducing the complexity of its calculations; however, the controller and the stability proof are all presented in Cartesian coordinates. Using simulations and experiments on a two DOFs robot, the effectiveness of the proposed controller is investigated.     

A novel contour error compensator for 3‐PRPS platform

Using a mathematical model to represent the nonlinear characteristics in dynamics of robot manipulators is rather difficult. To reduce the Cartesian space contour error, this study presents a novel contour error compensator influenced by the parameter and unstructured uncertainties in robot manipulators. The proposed compensator is based on the strategy of a Cartesian space cross‐coupled control and the transform relations between Cartesian space and joint space. In addition, the joint space compensated control effort derives reducing the Cartesian space contour error. Consequently, the contour error can be reduced via the theoretical analysis. Moreover, a PC based, 3‐PRPS platform control system is constructed to closely examine the effects of the controller. Experiment results indicate that the controller can reduce the contour error as expected. Furthermore, the forward and the inverse kinematics are derived, along with the forward kinematics solved using the numerical method. The work space of the platform is also described in a three‐dimensional Cartesian space. © 2000 John Wiley & Sons, Inc.     

A design of digital adaptive control systems for space robot manipulators using a transpose of the generalized Jacobian matrix

We propose a digital control method for space robot manipulators using the transpose of the generalized Jacobian matrix. The method, however, is developed on the supposition that all the physical parameters of the robot manipulator are known. Therefore, if the end-effector of the manipulator captures an object whose mass is unknown, the stability of the control system cannot be maintained because the physical parameters are changed. This article presents the adaptive control version.This work was presented, in part, at the 9th International Symposium on Artificial Life and Robotics, Oita, Japan, January 28–30, 2004     

Fast sensor corrections of robot motion paths processed in real-time by control algorithms running in parallel

Robot motion controls (especially for non-Cartesian kinematics of the robot) are realized with strong real-time demands. In industrial use they are primarily designed to work without sensor feedback. Within the control different levels of coordinates are necessary that rank from the Cartesian world to the realization of motions by the joints. A connection of the sensor to the Cartesian level results in a long reaction-time which often is insufficient either for the data rate of the sensor data or for the demands of the application. This paper describes a method which offers advantages because of parallel integration of the sensor in the robot motion control. These advantages are: great modularity by decentralization, fast reactions in real-time by parallel computing of the sensor data, implementability of various, application-specific control algorithms for sensor data, a relatively slow communication with the robot motion control on the Cartesian level and fast, immediate sensor influence on the robot joints. Since sensor corrections are performed by the use of a differential method (inverse Jacobian matrix) fast corrections have to be limited to certain amounts. Nevertheless corrections of robot motion paths can extend to any value by overlaying the fast corrections by additive slow correction values.     

具有不确定雅可比矩阵机器人的鲁棒非线性PID控制器的抗饱和失效设计

对于雅克比阵不确定的操作机器人笛卡尔空间操作任务, 提出一种鲁棒非线性PID控制器的抗饱和设计方案, 解决了PID控制中的积分饱和问题. 该控制器通过引入有界递增分段连续函数于PID控制器中的积分环节, 限制了积分器的积分作用, 从而克服了积分环节对闭环系统的不利影响: 一方面使得闭环系统是渐进稳定的, 另一方面又保证了其鲁棒性; 特别是, 相比于其它的抗饱和设计方法, 显得更加简单有效.