Robot calibration using the CPC error model

The complete and parametrically continuous (CPC) robot kinematic modeling convention has no model singularities and allows the modeling of the robot base and tool in the same manner by which the internal links are modeled. These two properties can be utilized to construct robot kinematic error models employing the minimum number of kinematic error parameters. These error parameters are independent and span the entire geometric error space. The BASE and TOOL error models are derived as special cases of the regular CPC error model. The CPC error model is useful for both kinematic identification and kinematic compensation. This paper focuses on the derivation of the CPC error models and their use in the experimental implementation of robot calibration.     

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.     

Analysis of unified error model and simulated parameters calibration for robotic machining based on Lie theory

In robotic machining process, the kinematic errors of serial structure and compliance errors caused by external cutter-workpiece interactions can result in considerable deviation of the desired trajectory. Therefore, this paper proposes an efficient calibration methodology by establishing a unified error model about kinematic errors and compliance errors based on Lie theory, which simultaneously calibrates the kinematic parameters and joint compliances of a serial machining robot. In this methodology, the propagation law of kinematic errors is investigated by analysis of the kinematic error model, and the corresponding equivalent kinematic error model is thus obtained, in which the joint offset errors are regarded as one source of twist (joint twist and reference configuration twist) errors. On this basis, with the segmentation and modelling of the joint compliance errors caused by the link self-weight and cutting payloads, the unified error model is developed by linear superposition of configuration errors of the robotic end-cutter, calculated from the kinematic errors and compliance errors respectively. Meanwhile, to improve the accuracy of parameters calibration, the observability index is adopted to optimize the calibration configurations so as to eliminate the twist error constraints. The calibrated kinematic parameters and joint compliances are obtained eventually, and used to compensate the kinematic and compliance errors of the serial machining robot. Finally, to validate the effectiveness of the proposed unified error model, simulation analysis is performed using a 6-DOF serial machining robot, namely KUKA KR500. The comparisons among calibrated parameters show that the unified error model is more computationally efficient with optimal calibration configurations, rendering it suitable for the calibration of kinematic parameters and joint compliances in actual machining applications.     

Kinematic calibration of modular reconfigurable robots using product-of-exponentials formula

A modular reconfigurable robot system is a collection of individual link and joint components that can be assembled into different robot geometries for specific task requirements. However, the machining tolerance and assembly errors at the module interconnections affect the positioning accuracy of the end-effector. This article describes a novel kinematic calibration algorithm for modular robots based on recursive forward dyad kinematics. The forward kinematic model derived from the Product-of-Exponentials formula is configuration independent. The error correction parameters are assumed to be in the relative initial positions of the dyads. Two calibration models, namely the six- and seven-parameter methods, are derived on the grounds of the linear superposition principle and differential transformation. An iterative least square algorithm is employed for the calibration solution. Two simulation examples of calibrating a three-module manipulator and a 4-DOF SCARA type manipulator are demonstrated. The result has shown that the average positioning accuracy of the end-effector increases two orders of magnitude after the calibration. © 1997 John Wiley & Sons, Inc.     

机器人几何连杆参数辨识

本文给出了一种通用的机器人几何连杆参数估计方法.基于机器人运动学模型和运动学 误差模型,可以推导出包含未知参数的线性输入输出方程;然后采用最小二乘法得到参数估计 值.该方法已应用于PUMA机器人参数辨识.     

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.     

机器人理论关节参数的标定方法及软件设计

本文研究了机器人机构理论控制模型中采用的D-H关节参数的标定方法,并编制了相应的计算机应用软件。应用这一方法和软件标定机器人的理论关节参数,对机器人实施误差综合补偿,提高机器人的精度,是特别重要的。对PUMA 562机器人进行了实测与标定。     

Geometric calibration of industrial robots using enhanced partial pose measurements and design of experiments

The paper deals with geometric calibration of industrial robots and focuses on reduction of the measurement noise impact by means of proper selection of the manipulator configurations in calibration experiments. Particular attention is paid to the enhancement of measurement and optimization techniques employed in geometric parameter identification. The developed method implements a complete and irreducible geometric model for serial manipulator, which takes into account different sources of errors (link lengths, joint offsets, etc). In contrast to other works, a new industry-oriented performance measure is proposed for optimal measurement configuration selection that improves the existing techniques via using the direct measurement data only. This new approach is aimed at finding the calibration configurations that ensure the best robot positioning accuracy after geometric error compensation. Experimental study of heavy industrial robot KUKA KR-270 illustrates the benefits of the developed pose strategy technique and the corresponding accuracy improvement.     

基于视觉标志间相对位姿的可变形臂标定方法

针对家庭服务机器人工作的非结构化环境, 本文设计了一种可以根据任务需求相应地调整连杆形状的可变形操作臂.该操作臂工作空间易于拓展、灵活度较高且成本低廉.但连杆形状的改变给操作臂的建模和控制带来了困难.首先, 可变形臂的运动学参数发生了巨大且无规律的变化, 使得固结在操作臂连杆上的关节坐标系可能脱离操作臂本体, 变得不可测量.其次, 为适应不同任务需求, 可变形臂的连杆形状需要经常改变, 而传统标定方法往往追求更高的标定精度而非标定效率.最后, 可变形臂的标定方法必须低成本且易于在家庭环境中实施, 而基于激光等传感器的标定方法设备价格昂贵, 对实验环境要求严格, 不便于在家庭中实施.因此, 一种廉价、快速、易于实施的标定方法是可变形臂应用的基础.本文分别基于Denavit-Hartenberg(DH)模型和旋量模型提出了基于视觉标志块间相对位姿测量的标定算法, 该算法在标志块处建立虚拟关节, 通过测量不同标志块间的相对位姿可快速、高效地获取可变形臂的运动学参数.实验说明了两种标定方法的有效性, 同时还表明旋量模型更适合可变形臂的建模.最后, 本文给出了利用可变形臂进行点触任务操作的实例, 展示出可变形操作臂在家庭使用中的优势.     

A new approach to improve absolute positioning accuracy of robot manipulators

This article describes a new calibration system for robot manipulators which improves their absolute positioning accuracy by using parameter-estimation algorithms based on the Newton method. When 3D position data of the specified points on a manipulator and the joint encoder values are input to the calibration system, the system estimates the offset values of joint encoders, link lengths, and position and orientation of the manipulator base coordinate system with respect to the world coordinate system which is difficult to obtain by conventional calibration methods. This calibration system can be applied to various manipulator types by just changing the basic kinematic equations. The system employs an algebraic programming system called REDUCE to automatically reduce the manipulator kinematic equation and partial differential calculus in the Newton method. For efficiency, first only the arm part with three degrees of freedom and then the hand part are calibrated. The experimental results demonstrate the effectiveness of this system by reducing the robot's absolute positioning errors to the order of repeatability errors.     

Improving the absolute positioning accuracy of robot manipulators

The absolute positioning accuracy of robot manipulator can be increased substantially by updating the nominal link parameters in the control software. This paper presents a general method to estimate the link parameter errors for any serial link manipulator (i.e., n links, and any combination of revolute and prismatic joints). The parameters are estimated through a linear kinematic model which relates the link bias errors to the end-effector positioning error. Only end-effector measurements are required instead of individual link measurements to implement this method.     

基于末端圆周运动的可变形臂旋量参数快速标定算法

针对家庭服务机器人工作的非结构化环境,设计了一种可以根据任务需求相应地调整连杆形状的可变形操作臂．该操作臂工作空间易于拓展、灵活度较高且成本低廉．但臂形的改变也给操作臂的建模和控制带来了困难．首先,可变形臂的运动学参数发生了巨大且无规律的变化,使得固结在连杆上的坐标系脱离连杆本体,变得不可测量．其次,为适应不同任务需求,可变形臂的连杆形状需要经常改变,而传统标定方法往往追求更高的标定精度而非标定效率,因而需要开发一种耗时较短的标定方法．最后,可变形臂的标定方法必须简便而易于在家庭环境中实施．针对上述问题,本文提出了一种基于末端圆周运动的可变形臂旋量参数快速标定算法．对于任意一个旋转关节,单独转动该关节时,操作臂末端的轨迹形成了一个圆弧,且该关节轴垂直于圆弧所在平面（旋转平面）并经过圆弧的圆心．基于该性质,利用随机抽样一致性（RANSAC）算法和最小二乘算法来拟合操作臂末端轨迹,从而获取操作臂的旋量参数初值．在获取到初值的基础上,提出了一种扩展卡尔曼滤波（EKF）算法进一步优化旋量参数,提高标定精度．仿真和实验结果验证了本文方法的有效性．     

Kinematic calibration for collaborative robots on a mobile platform using motion capture system

For modern robotic applications that go beyond the typical industrial environment, absolute accuracy is one of the key properties that make this possible. There are several approaches in the literature to improve robot accuracy for a typical industrial robot mounted on a fixed frame. In contrast, there is no method to improve robot accuracy when the robot is mounted on a mobile base, which is typical for collaborative robots. Therefore, in this work, we proposed and analyzed two approaches to improve the absolute accuracy of the robot mounted on a mobile platform using an optical measurement system. The first approach is based on geometric operations used to calculate the rotation axes of each joint. This approach identifies all rotational axes, which allows the calculation of the Denavit–Hartenberg (DH) parameters and thus the complete kinematic model, including the position and orientation errors of the robot end-effector and the robot base. The second approach to parameter estimation is based on optimization using a set of joint positions and end-effector poses to find the optimal DH parameters. Since the robot is mounted on a mobile base that is not fixed, an optical measurement system was used to dynamically and simultaneously measure the position of the robot base and the end-effector. The performance of the two proposed methods was analyzed and validated on a 7-DoF Franka Emika Panda robot mounted on a mobile platform PAL Tiago-base. The results show a significant improvement in absolute accuracy for both proposed approaches. By using the proposed approach with the optical measurement system, we can easily automate the estimation of robot kinematic parameters with the aim of improving absolute accuracy, especially in applications that require high positioning accuracy.     

Visual motor control of a 7 DOF robot manipulator using a fuzzy SOM network

A fuzzy self-organizing map (SOM) network is proposed in this paper for visual motor control of a 7 degrees of freedom (DOF) robot manipulator. The inverse kinematic map from the image plane to joint angle space of a redundant manipulator is highly nonlinear and ill-posed in the sense that a typical end-effector position is associated with several joint angle vectors. In the proposed approach, the robot workspace in image plane is discretized into a number of fuzzy regions whose center locations and fuzzy membership values are determined using a Fuzzy C-Mean (FCM) clustering algorithm. SOM network then learns the inverse kinematics by on-line by associating a local linear map for each cluster. A novel learning algorithm has been proposed to make the robot manipulator to reach a target position. Any arbitrary level of accuracy can be achieved with a number of fine movements of the manipulator tip. These fine movements depend on the error between the target position and the current manipulator position. In particular, the fuzzy model is found to be better as compared to Kohonen self-organizing map (KSOM) based learning scheme proposed for visual motor control. Like existing KSOM learning schemes, the proposed scheme leads to a unique inverse kinematic solution even for a redundant manipulator. The proposed algorithms have been successfully implemented in real-time on a 7 DOF PowerCube robot manipulator, and results are found to concur with the theoretical findings.     

Accurate error compensation for a MR-compatible surgical robot based on a novel kinematic calibration method

Kinematic calibration is an effective and economical way to improve the accuracy of surgical robot, and in most cases, it is a necessary procedure before the robot is put into operation. This study investigates a novel kinematic calibration method where the effect of controller error is taken into account when formulating the model based on screw theory, which is applied to the kinematic control of magnetic resonance compatible surgical robot. Based on screw theory, the kinematic error model is established for the relationship between error of controller and the deviation of the measured pose of the end-effector. Therefore, the error of controller can be figured out and parameters of controller can be adjusted accordingly. Control strategy based on the kinematic calibration framework is proposed. According to artificial neural network, the deviation of end-effector in arbitrary configuration can be effectively obtained. Comparative experiments are carried out to show the validity and effectiveness of the proposed framework with the help of commercial visual system and joint encoders.     

神经外科机器人定位精度研究

针对机器人辅助神经外科手术对高精度的要求，对“黎元”神经外科机器人的定位精度进行了研究．考虑到关节轴的微小偏差，基于修正的DH运动学模型对机器人的实际几何参数进行了辨识．采用基于误差反传（BP）算法的多层前馈神经网络对各关节的传动误差进行了补偿．利用高精度的三坐标测量臂对“黎元”神经外科机器人的定位精度进行了测量．结果表明，该机器人绝对定位精度最大值为1.63 mm，平均值为1.04 mm，较以前有了很大提高，能够满足多数神经外科手术的需要．     

Uncalibrated Eye-to-Hand Visual Servoing Using Inverse Fuzzy Models

A new uncalibrated eye-to-hand visual servoing based on inverse fuzzy modeling is proposed in this paper. In classical visual servoing, the Jacobian plays a decisive role in the convergence of the controller, as its analytical model depends on the selected image features. This Jacobian must also be inverted online. Fuzzy modeling is applied to obtain an inverse model of the mapping between image feature variations and joint velocities. This approach is independent from the robot's kinematic model or camera calibration and also avoids the necessity of inverting the Jacobian online. An inverse model is identified for the robot workspace, using measurement data of a robotic manipulator. This inverse model is directly used as a controller. The inverse fuzzy control scheme is applied to a robotic manipulator performing visual servoing for random positioning in the robot workspace. The obtained experimental results show the effectiveness of the proposed control scheme. The fuzzy controller can position the robotic manipulator at any point in the workspace with better accuracy than the classic visual servoing approach.     

Generalized joint model for robot manipulator kinematic calibration and compensation

A generalized model that goes beyond the usual assumption of “ideal” joint behavior is proposed. The “real” joint has five ancillary degrees of freedom besides the dominant motion. The resulting manipulator transformation with its greater degree of sophistication is expected to help in calibration and compensation of the various kinematic contributions to robot inaccuracy. The procedure to compute this generalized manipulator transformation is presented. The generalized model also results in manipulator differential relationships and these are discussed.     

Estimation model for kinematic calibration of manipulators with a parallel structure

This article provides an estimation model for calibrating the kinematics of manipulators with a parallel geometrical structure. Parameter estimation for serial link manipulators is well developed, but fail for most structures with parallel actuators, because the forward kinematics is usually not analytically available for these. We extend parameter estimation to such parallel structures by developing an estimation method where errors in kinematical parameters are linearly related to errors in the tool pose, expressed through the inverse kinematics, which is usually well known. The method is based on the work done to calibrate the MultiCraft robot. This robot has five linear actuators built in parallel around a passive serial arm, thus making up a two-layered parallel-serial manipulator, and the unique MultiCraft construction is reviewed. Due to the passive serial arm, for this robot conventional serial calibration must be combined with estimation of the parameters in the parallel actuator structure. The developed kinematic calibration method is verified through simulations with realistic data and real robot kinematics, taking the MultiCraft manipulator as the case. © 1994 John Wiley & Sons, Inc.     

Simultaneous kinematic calibration,localization, and mapping (SKCLAM) for industrial robot manipulators†

Recently, the demand for more accurate, productive, and economical robot manipulators is increasing in the robotics industry. However, a manipulator will produce kinematic errors during production. Thus low-cost kinematic calibration is demanded. Moreover, environmental mapping is also demanded to plan the motions of the manipulator. In this paper, we proposed a simultaneous kinematic calibration, localization, and mapping (SKCLAM) method, which can simultaneously calibrate the kinematic parameters of an industrial robot manipulator using a commercial RGB-D camera attached to its end effector to reconstruct its surroundings. In our method, the kinematic calibration is achieved with feature detection and epipolor geometry. Synthetic and real data experiments were conducted to verify the SKCLAM method. We succeeded in reducing the kinematic errors of the manipulator and reconstructing dense 3D maps of the workspace in the experiments.