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.     

A general force/torque relationship and kinematic representation for flexible link manipulators

A technique to determine the relationship between the generalized joint torque and the generalized force exerted by the end-effector of a flexible link manipulator on the environment is presented. First, a systematic technique which accounts for the links' bending and torsion is used to determine the kinematic equations of the manipulator. A procedure to determine a “pseudo” Jacobian matrix which accounts for the links' flexibility is then developed. In the determination of the pseudo Jacobian matrix, partial derivatives are not used, thus reducing inaccuracies resulting from approximating the bending and torsion of the links. Using the principle of virtual work, a force and torque relationship between the joints and end-effector is obtained. To demonstrate the validity of the technique presented, computer simulations of the resulting force/torque relationship are compared to results obtained experimentally.     

Error-model-based robot calibration using a modified CPC model

Selection of a proper robot kinematic model is a critical step in error-model-based robot calibration. The Denavit-Hartenberg (DH) model exhibits singularities in calibration of robots having consecutive parallel joint axes. The complete and parametrically continuous (CPC) modeling technique is one of the more versatile alternative modeling conventions designated to fit the needs of manipulator calibration. No modeling convention is, however, perfect. One “user-unfriendly” aspect of the CPC model is a condition handling technique needed, when constructing the error model, to avoid model singularities due to the adoption of the direction vectors of the joint axes as link parameters.This paper presents a modification to the CPC model which brings the model closer to the DH model. Rather than using the direction vectors of joint axes, the modified CPC (MCPC) model employs angular parameters to acommodate the required rotations for each link transformation. This modification results in a much simplified error model. The model, like the CPC model, is capable of completely describing the geometry and motion of the manipulator in a reference coordinate frame. Its error model possesses a minimum number of parameters to span the entire geometric error space and it can be made singularity-free by proper selection of the tool axis. This paper presents a calibration study of the PUMA robot using the MCPC model. A moving stereo camera system was employed for end-effector pose measurements. The MCPC error model was then used for kinematic identification. Results on the PUMA arm show that the MCPC performs well for robot calibration. The well-defined structure and user friendliness of the MCPC model may facilitate the implementation of robot calibration techniques on the factory floor.     

Dynamics analysis of a 3-RRP spherical parallel manipulator using the natural orthogonal complement

In the present research, application of the Natural Orthogonal Complement (NOC) for the dynamic analysis of a spherical parallel manipulator, referred to as SST, is presented. Both inverse and direct dynamics are considered. The NOC and the SST fully parallel robot are explained. To drive the NOC for the SST manipulator, constraints between joint variables are written using the transformation matrices obtained from three different branches of the robot. The Newton–Euler formulation is used to model the dynamics of each individual body, including moving platform and legs of the manipulator. D’Alembert’s principle is applied and Newton–Euler dynamical equations free from non-working generalized constraint forces are obtained. Finally two examples, one for direct and one for inverse dynamics are presented. The correctness and accuracy of the obtained solution are verified by comparing with the solution of the virtual work method as well as commercial multi-body dynamics software.     

Improved manipulator perfomance through local d-h calibation

This article deals with the subject of increasing the positioning accuracy of a manipulator beyond what is typically obtained via Denavit-Hartenberg (D-H) calibration. It is commonly accepted that one method for accomplishing this requires including “extra” terms (such as static deflection) in the D-H model. Another technique, which is experimentally verified and quantified in this article, is that of local calibration. Rather than continually introducing more and more modeling terms, we restrict the workvolume of the manipulator. One contribution of this article is an experimental investigation of local calibration. Most data indicates that local calibration offers advantages and it quantifies that the approximate improvement on one particular PUMA 560 robot is as much as four times better than global calibration. Some data indicates that it is not always true that “localizing” the calibration is an improvement. The article also discusses the problem of recognizing bad calibration data. We demonstrate a technique for identifying and removing “bad” data. Finally, the article shows that the locality of workvolume must include a consideration of the joint motions as well. For example, it presents results when data is taken with the arm in a single configuration, and in multiple configurations. © 1995 John Wiley & Sons, Inc.     

Arm Dynamics Simulation

The ability to mathematically model the movement of a robot manipulator is a prerequisite to the understanding of the key factors that influence a manipulator's performance. This paper presents a manipulator model which has been used to simulate and control a real robot arm. A method of describing the arm by its rotational characteristics. a set of equations called the “Inverse Arm” and an algorithm called the “Forward Arm” are presented. The Forward Arm simulates the movement of an arm and the Inverse Arm provides a means of computing the correct voltages to apply to an arm to achieve a desired movement.     

Adaptive tracking control of rigid manipulators using only position measurements

This article presents a new approach to trajectory tracking control of uncertain rigid manipulators using only position measurements. The proposed control strategy is an adaptive scheme that is very general and computationally efficient, requires virtually no information regarding the manipulator dynamic model, and is implementable without calculation of the robot inverse dynamics or inverse kinematic transformations. It is shown that the controller ensures semiglobal uniform boundedness of all signals in the presence of bounded disturbances, and that the ultimate size of the tracking errors can be made arbitrarily small. Additionally, it is demonstrated that the proposed strategy can be used as the basis for developing controllers for “cascaded” robotic systems, such as manipulators with significant actuator dynamics or joint flexibility. The efficacy of this approach to manipulator control is illustrated through both computer simulations and hardware experiments. © 1997 John Wiley & Sons, Inc.     

Torque Optimizing Control with Singularity-Robustness for Kinematically Redundant Robots

A new control method for kinematically redundant manipulators having the properties of torque-optimality and singularity-robustness is developed. A dynamic control equation, an equation of joint torques that should be satisfied to get the desired dynamic behavior of the end-effector, is formulated using the feedback linearization theory. The optimal control law is determined by locally optimizing an appropriate norm of joint torques using the weighted generalized inverses of the manipulator Jacobian-inertia product. In addition, the optimal control law is augmented with fictitious joint damping forces to stabilize the uncontrolled dynamics acting in the null-space of the Jacobian-inertia product. This paper also presents a new method for the robust handling of robot kinematic singularities in the context of joint torque optimization. Control of the end-effector motions in the neighborhood of a singular configuration is based on the use of the damped least-squares inverse of the Jacobian-inertia product. A damping factor as a function of the generalized dynamic manipulability measure is introduced to reduce the end-effector acceleration error caused by the damping. The proposed control method is applied to the numerical model of SNU-ERC 3-DOF planar direct-drive manipulator.     

基于干扰观测器的机械臂广义模型预测轨迹跟踪控制

为实现对多自由度机械臂关节运动精确轨迹跟踪,提出一种基于非线性干扰观测器的广义模型预测轨迹跟踪控制方法。针对机械臂轨迹跟踪运动学子系统,采用广义预测控制（Generalized Predictive Control,GPC）方法设计期望的虚拟关节角速度。对于机械臂轨迹跟踪动力学子系统,考虑机械臂的参数不确定性和未知外界扰动,利用GPC方法设计关节力矩控制输入,基于非线性干扰观测器方法实时估计和补偿系统模型中的不确定性。在李雅普诺夫稳定性理论框架下证明了机械臂关节角位置和角速度的跟踪误差最终收敛于零的小邻域。数值仿真验证了所提出控制方法的有效性和优越性。     

一种基于指数积的移动机械臂联合标定方法

提出一种基于指数积的移动机械臂联合标定方法,以实现移动平台和机械臂两者间位姿标定与机械臂运动学参数标定模型的统一.机械臂运动学参数标定使用最多的是基于D-H参数法,但D-H参数法无法克服相邻关节平行或接近平行时的奇异性问题,以及建模过程复杂、建模后的模型通用性差等问题.基于指数积的移动机械臂联合标定方法建模时不会因为关节轴平行而出现奇异性问题,建模过程简单.通过对整个系统的运动学方程进行微分运算,获得末端位姿误差和移动机械臂零位状态旋量误差及关节旋量误差的线性化模型.利用伴随矩阵方式建立关节旋量理论值与关节旋量实际值的关系,并通过改变伴随矩阵实现基于最小二乘法的参数辨识计算过程中参数更新.使用高精度激光跟踪仪作为测量工具,通过实验验证所提出方法的有效性.     

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.     

基于接近觉的机械臂避障路径规划

提出了一种基于接近觉的机械臂避障路径规划方法。首先,针对排列稀疏的接近觉传感器阵列的障碍物感知问题,定义连杆“运动方向”的概念,优先读取连杆运动方向上的传感器以缩短读取周期的长度,并将距离传感器最近的障碍物表面建模为“假想圆锥”,过其顶点作“安全平面”,连杆在运动过程中不能与其发生碰撞。其次,针对避障路径规划问题,采用基于势函数与关节空间的人工势场法,并根据所提出的障碍物感知方法提出了基于“绕行”的改进方法以解决人工势场法的局部最优问题。最后,在UR10机器人的小臂连杆上验证本文方法,实验结果证明了本文方法的有效性、鲁棒性与实时性。     

A study on industrial robotic manipulator model using model based predictive controls

In this study, a single input single output (SISO) neural generalized predictive control (NGPC) was applied to a six joint robotic manipulator. The SISO generalized predictive control (GPC) was also used for comparison. Modeling of the dynamics of the robotic manipulator was made by using the Lagrange–Euler equations. The cubic trajectory principle is used for position reference and velocity reference trajectories. A simulation program was prepared by using Delphi 6.0. All computations for manipulator dynamics model, GPC-SISO, and NGPC-SISO were done on PC with 1.6 GHz Centrino CPUs by using this program. The parameter estimation algorithm used in the GPC-SISO is Recursive Least Squares. The minimization algorithm used in the NGPC-SISO is Newton–Raphson. According to the simulation results, the results of the NGPC-SISO algorithm were better than those of the GPC-SISO algorithm.     

New inverse kinematic algorithms for redundant robots

New inverse kinematic algorithms for generating redundant robot joint trajectories are proposed. The algorithms utilize the kinematic redundancy to improve robot motion performance (in joint space or Cartesian space) as specified by certain objective functions. The algorithms are based on the extension of the existing “joint-space command generator” technique in which a null space vector is introduced which optimizes a specific objective function along the joint trajectories. In this article, the algorithms for generating the joint position and velocity (PV) trajectories are extensively developed. The case for joint position, velocity, and acceleration (PVA) generation is also addressed. Application of the algorithms to a four-link revolute planar robot manipulator is demonstrated through simulation. Several motion performance criteria are considered and their results analyzed.     

Impact dynamics of flexible-joint robots

In this paper the dynamic analysis of a flexible-joint robot colliding with its environments is presented. The system considered here is an n-rigidlink manipulator driven by n DC-motors through n revolute flexible-joints. The flexibility of each flexible joint is modelled as a linearly elastic torsional spring. 2n generalized coordinates are introduced to identify the configuration of the robot. The concept of impulse potential energy is introduced, and the generalized impulse-momentum equations along with the equation involving coefficient of restitution are used to establish the complete mathematic model for dealing with the case of a flexible-joint manipulator colliding with its environments (such as the ground, another manipulator, and so on). Solving for this mathematic model, one can obtain the jump discontinuities in system generalized velocities and the impulses at the impact points explicitly. To validate the algorithm presented in this paper, the dynamic simulation of a robot involving impact with its environments is given as an example.     

Generalized algorithm for computing the 4-3-4 N-axis robotic trajectory

A generalized inversion technique and algorithm is presented for the purpose of computing the 9n coefficients pertaining to the three segments—i.e., liftoff, transitory, and setdown—of a motion trajectory of an n-axis robot manipulator. The technique and the computer program presented is general enough to be immediately applicable to an n-axis robot manipulator if only the desired initial and final position, velocity, acceleration, as well as the intermediate “via” positions are given. A number of examples are presented for a wide spectrum of these constraint conditions on the desired motion trajectories.     

Control-orientated dynamic modeling of forging manipulators with multi-closed kinematic chains

The dynamic model, particularly with reference to controller design, is an important issue in mechanical control and design. However, this model is often difficult to achieve in complex multi-closed-loop mechanisms, such as parallel mechanisms or forging manipulators. A new approach on the dynamic modeling of a multi closed-chain mechanism in a forging manipulator, which applies screw theory and the reduced system model, is proposed in this paper. The proposed method not only allows a straightforward calculation of actuator forces but also obtains the dynamic equation of the multi-closed-loop mechanism easily. The structure of dynamic model obtained is similar to that of standard Lagrangian formulations, which can extend vast control strategies developed for serial robots to complex multi-closed-loop mechanisms. A complex multi-closed-loop mechanism on a forging manipulator is decomposed into several serial mechanisms or simpler subsystems. The Lagrangian equations associated with each subsystem are directly derived from the local generalized coordinates of the sub-mechanisms. Jacobian matrices are used to interpret the differential equations of the sub-mechanisms into the generalized coordinate or the actuated pairs according to the D’Alembert principle. Hessian matrices are also applied to form a standard Lagrangian formulation. The screw theory is introduced to overcome the difficulties of solving transformed Jacobian matrices, thereby simplifying the calculation of the matrices. Computation difficulties of transformation matrices may decrease considerably by choosing suitable generalized coordinates instead of direct actuator variables. The full dynamics of the complex multi-closed-loop mechanism in a forging manipulator is presented. Simulations and experiments illustrate the reliability of the proposed method and the correctness of the dynamic model of the forging manipulator.     

机器人最佳轨迹规划和图形仿真的研究

本文介绍了基于 IBM-PC 微机的机器人轨迹规划的图形仿真系统.该系统可用于机器人运动学分析和最佳轨迹规划,并能产生机械手和广义物体组成的环境物的三维几何模型.系统以交互方式工作.在微机的 CRT 上,机械手连杆的运动具存动态效果.基于 MINIMAX 准则,我们提出了一种动力学性能指标用于轨迹规划.根据该指标所进行的轨迹规划可以获得最小的关节驱动力矩.最后,文中给出了 PUMA 560 机械手的仿真结果.     

Minimization of joint reaction forces of kinematic chains by a multi-objective approach

The paper describes a procedure of simultaneous minimization of joint reaction forces of planar kinematic chains. The best design is sought using the methods of mathematical programming. The optimization model of the kinematic chain is based on a multi-objective approach where the objective functions may be related to the generalized joint forces. The design variables are the geometric parameters of the links. The implementation of the optimization model is illustrated with two examples. The first example considers the minimization of joint reaction forces of an open loop system of a two-arm manipulator. The second example considers optimization of a closed loop system representing a four-bar mechanism being an actual part of a hydraulic support employed in the mining industry.     

Characterising trader manipulation in a limit-order driven market

Use of trading strategies to mislead other market participants, commonly termed trade-based market manipulation, has been identified as a major problem faced by present day stock markets. Although some mathematical models of trade-based market manipulation have been previously developed, this work presents a framework for manipulation in the context of a realistic computational model of a limit-order market. The Maslov limit order market model is extended to introduce manipulators and technical traders. We show that “pump and dump” manipulation is not possible with traditional Maslov (liquidity) traders. The presence of technical traders, however, makes profitable manipulation possible. When exploiting the behaviour of technical traders, manipulators can wait some time after their buying phase before selling, in order to profit. Moreover, if technical traders believe that there is an information asymmetry between buy and sell actions, the manipulator effort required to perform a “pump and dump” is comparatively low, and a manipulator can generate profits even by selling immediately after raising the price.