A Class of Transpose Jacobian‐based NPID Regulators for Robot Manipulators with an Uncertain Kinematics

Transpose Jacobian‐based controllers present an attractive approach to robot set‐point control in Cartesian space that derive the end‐effector posture to a specified desired position and orientation with neither solving the inverse kinematics nor computing the inverse Jacobian. By a Lyapunov function with virtual artificial potential energy, a class of complete transpose Jacobian‐based Nonlinear proportional‐integral‐derivative regulators is proposed in this paper for robot manipulators with uncertain kinematics on the basis of the set of all continuous differentiable increasing functions. It shows globally asymptotic stability for the result closed‐loop system on the condition of suitable feedback gains and suitable parameter selection for the corresponding function set as well as artificial potential function, and only upper bound on Jacobian matrix error and Cartesian dynamics parameters are needed. The existing linear PID (LPID) regulators are the special cases of it. Nevertheless, in the case of LPID regulators, only locally asymptotic stability is guaranteed if the corresponding conditions are satisfied. Simulations demonstrate the result and robustness of transpose Jacobian‐based NPID regulators. © 2002 Wiley Periodicals, Inc.     

Cooperative manipulation of a floating object by some space robots with joint velocity controllers: application of a tracking control method using the transpose of the generalized Jacobian matrix

We have studied the cooperative manipulation of floating object by several space robots. For these cooperative motions, we have reported that a tracking control method using the transpose of the generalized Jacobian matrix (GJM) can be utilized for robots with joint torque controllers. For cooperative motions by some robots with joint velocity controllers, we proposed a tracking control method using the transpose of the GJM. Simulation results show the effectiveness of the proposed control method.     

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.     

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.     

Cooperative manipulation of a floating object by some space robots: application of a tracking control method using the transpose of the generalized Jacobian matrix

In future space missions, it is considered that many tasks will be achieved by cooperative motions of space robots. For free-floating space robots with manipulators, we have proposed a digital tracking control method using the transpose of the generalized Jacobian matrix (GJM). In this paper, the tracking control method using the transpose of the GJM is applied to cooperative manipulations of a floating object by space robots. Simulation results show the effectiveness of the control method. This work was presented in part at the 12th International Symposium on Artificial Life and Robotics, Oita, Japan, January 25–27, 2007     

Digital tracking control of space robots using a transpose of the generalized Jacobian matrix

We have proposed discrete time-control methods using the transpose of the generalized Jacobian matrix (GJM) for free-floating space robots having manipulators. The control methods are robust for singular configurations of robots. Since the methods belong to a class of constant-value control, in this article we propose a digital trajectory tracking control method using the transpose of the GJM. Computer simulations show the effectiveness of the proposed method. This work was presented in part at the 11th International Symposium on Artificial Life and Robotics, Oita, Japan, January 23–25, 2006     

Collaborative Multi-MSA Multi-Target Tracking and Surveillance: a Divide &; Conquer Method Using Region Allocation Trees

This work deals with the problem of the accurate task space control subject to finite-time convergence. Kinematic and dynamic equations of a rigid robotic manipulator are assumed to be uncertain. Moreover, unbounded disturbances, i.e., such structures of the modelling functions that are generally not bounded by construction, are allowed to act on the manipulator when tracking the trajectory by the end-effector. Based on suitably defined task space non-singular terminal sliding vector variable and the Lyapunov stability theory, we derive a class of absolutely continuous (chattering-free) robust controllers based on the estimation of a Jacobian transpose matrix, which seem to be effective in counteracting uncertain both kinematics and dynamics, unbounded disturbances and (possible) kinematic and/or algorithmic singularities met on the robot trajectory. The numerical simulations carried out for a 2DOF robotic manipulator with two revolute kinematic pairs and operating in a two-dimensional task space, illustrate performance of the proposed controllers.     

Inverse mapping of continuous functions using local and globalinformation

This paper presents a method for solving inverse mapping of a continuous function learned by a multilayer feedforward mapping network. The method is based on the iterative update of input vector toward a solution, while escaping from local minima. The input vector update is determined by the pseudo-inverse of the gradient of Lyapunov function, and, should an optimal solution be searched for, the projection of the gradient of a performance index on the null space of the gradient of Lyapunov function. The update rule is allowed to detect an input vector approaching local minima through a phenomenon called "update explosion". At or near local minima, the input vector is guided by an escape trajectory generated based on "global information", where global information is referred to here as predefined or known information on forward mapping; or the input vector is relocated to a new position based on the probability density function (PDF) constructed over the input vector space by Parzen estimate. The constructed PDF reflects the history of local minima detected during the search process, and represents the probability that a particular input vector can lead to a solution based on the update rule. The proposed method has a substantial advantage in computational complexity as well as convergence property over the conventional methods based on Jacobian pseudo-inverse or Jacobian transpose.     

基于雅可比转秩-RRT法的冗余机械手路径规划

针对核反应堆检修用机械手在末端任务给定的无碰撞路径规划问题进行研究.提出一种将雅可比转秩控制算法与快速搜索随机树法相结合的混合算法,通过选择距目标点工作空间距离最近的位姿点作为树的扩充节点,避免对逆运动学进行求解.利用雅可比转秩控制算法计算出最优扩充方向,采用二分梯度下降扩充方法对末端工具速度加以限制,避免关节速度发生突变.仿真结果表明该混合算法的快速性和有效性.     

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.     

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.     

Inverse statics analysis of planar parallel manipulators via Grassmann-Cayley algebra

The wrench Jacobian matrix plays an important role in the statics and singularity analysis of planar parallel manipulators (PPMs). The Jacobian matrix can be calculated based on the conventional Plücker coordinate method. However, this method cannot be applied when two links are in parallel. A new approach is proposed for the analysis of the forward and inverse wrench Jacobian matrix using Grassmann-Cayley algebra (GCA). A symbolic formula for the inverse statics analysis is obtained based on the Jacobian. The proposed method can be applied when two links are in parallel. The approach is explained in detail based on a planar 3-RPR PPM example, and the analysis procedure for nine other PPMs is also presented. This novel approach to deriving the statics can be applied to spatial parallel manipulators and redundant cases of PPMs.

     

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.     

A closed-loop jacobian transpose scheme for solving the inverse kinematics of nonredundant and redundant wrists

The solution of the inverse kinematic problem is of the utmost importance in robotic manipulator control. This article proposes a closed-loop scheme for solving the inverse kinematic problem for nonredundant and redundant wrists based on the computation of the Jacobian transpose. The manipulability measure is suitably introduced as a constraint for redundant wrists, by taking advantage of the null space of the Jacobian matrix. The resulting algorithm provides a computational tool to solve a specified orientation trajectory into a joint trajectory. Numerical results with two spherical wrists show the excellent performance of the scheme.     

实时SAR成像系统矩阵转置方法研究与实现

合成孔径雷达（SAR）是一种高分辨率成像雷达,而矩阵转置是实时SAR成像信号处理中一个很重要的操作,矩阵转置的效率高低将直接决定整个SAR成像信号处理系统的性能。对于矩阵转置,可采用行进列出或列进行出、两页式或三页式转置等方法进行处理,但这些方法处理时间较长,转置效率较低。在现有矩阵转置方法的基础上,提出了一种新的矩阵转置方法。在实际硬件平台上利用提出的矩阵转置方法进行了实时SAR成像处理,所得结果的矩阵转置效率为78%,整个SAR成像处理时间为10秒。测试结果表明,该方法对解决矩阵转置问题是行之有效的。     

Digital control of space robot manipulators with velocity type joint controller using transpose of generalized Jacobian matrix

For free floating space robots having manipulators, we have proposed a discrete-time tracking control method using the transpose of Generalized Jacobian Matrix (GJM). Control inputs of the control method are joint torques of the manipulator. In this paper, the control method is augmented for angular velocity inputs of the joints. Computer simulations have shown the effectiveness of the augmented method. This work was presented in part and awarded as Best Paper Award at the 13th International Symposium on Artificial Life and Robotics, Oita, Japan, January 31–February 2, 2008     

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.     

Fast and robust numerical method for inverse kinematics with prioritized multiple targets for redundant robots

In this paper, a new numerical method for inverse kinematics with prioritized multiple targets is proposed. The proposed method is constructed based on the virtual spring model and joint-based damping control. The targets are prioritized by adjusting the effect of the virtual springs. The proposed method has the following three features. First, it does not require complex calculations such as a Jacobian matrix projection into the null space. Second, it can solve prioritized inverse kinematics problems in the position level without integrating the joint velocity. Third, it is robust to parameter variations and singular configurations. The second feature is motivated by the background that most industrial robots in factories are used as position-controlled robots. Simulation experiments using a 9-DOF redundant robot show that the proposed method is faster and more robust than the conventional method. The proposed method is expected to be useful for helping to avoid collisions between links and obstacles using the redundancy.     

Inverse velocity analysis for line guidance five-axis robots

In this paper, inverse velocity problem for five-axis robots is investigated. The conventional method for a five-axis robot is to pseudo-inverse the 6×5 Jacobian matrix. The solution, primarily based on six freedoms inverse velocity analysis, is just an approximation with a least-square error. A five-axis robot can exactly guide an axis-symmetrical tool in 3-D space. Two exact solutions are provided for five-axis robots. One is based on the screw motion of the tool. The other is based on spherical angles of the tool to derive a 5×5 Jacobian matrix. A new type of singular configuration is discovered and is called the task singularity. The moving path of the line shaped tool is constructed as a ruled surface. Analysis of the angular acceleration shows the surface constructed based on the spherical angle representation has better characteristic. It is concluded that for five-axis robots, the tool position is better represented by five parameters rather than six parameters in order to get better solutions for inverse velocity as well as the motion planning.     

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.