机械外文翻译--应用坐标测量机的机器人运动学姿态的标定

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1、应用坐标测量机的机器人运动学姿态的标定Morris R. Driels, Lt W. Swayze USN and Lt S. Potter USNDepartment of Mechanical Engineering, Naval Postgraduate School, Monterey, California, US这篇文章报到的是用于机器人运动学标定中能获得全部姿态的操作装置坐标测量机（CMM）。运动学模型由于操作器得到发展, 它们关系到基坐标和工件。 工件姿态从实验测量中的引出是讨论, 同样地是识别方法学。允许定义观察策略的完全模拟实验已经实现。 实验工作是描写参数辨认和精确确认。

2、推论原则是那方法能得到在重复时近连续地校准机器人。关键字：机器人标定；坐标测量； 参数辨认；模拟学习； 精确增进1. 前言机器手有合理的重复精度 (0.3毫米)而知名, 仍有不好的精确(10.0 毫米)。为了实现机器手精确性，机器人可能要校准也是好理解 1. 在标定过程中， 几个连续的步骤能够精确地识别机器人运动学参数，提高精确性。 这些步骤为如下描述：1 操作器的运动学模型和标定过程本身是发展，和通常有标准运动学模型的工具实现的2。 作为结果的模型是定义基于厂商的运动学参数设置错误量, 和识别未知的,实际的参数设置。2 机器人姿态的实验测量法(部分的或完成) 是拿走为了获得联系到实际机器人的

3、参数设置数据。3 实际的运动学参数识别是系统地改变参数设置和减少在模型阶段错误量的定义。 一个接近完成辨认由分析不同中间姿态变量P和运动学参数K的微分关系决定： 于是等价转化得：两者择一, 问题可以看成为多维的优化问题，这是为了减少一些定义的错误功能到零点，运动学参数设置被改变。这是标准优化问题和可能解决用的众所周知的3 方法。4 最后一步是机械手控制中的机器人运动学识别和在学习之下的硬件系统的详细资料。包含实验数据的这张纸用于标度过程。 可获得的几个方法是可用于完成这任务, 虽然他们相当复杂，获得数据需要大量的成本和时间。这样的技术包括使用可视化的和自动化机械 4, 5, 6，伺服控制激光干

4、涉计 7，有关声音的传感器8 和视觉传感器 9。理想测量系统将获得操作器的全部姿态(位置和方向)，因为这将合并机械臂各个位置的全部信息。上面提到的所有方法仅仅用于唯一部分的姿态, 需要更多的数据是为了标度过程到进行。2理论文章中的理论描述，为了操作器空间放置的各自的位置，全部姿态是可测量的，虽然进行几个中间测量，是为了获得姿态。测量姿态使用装置是坐标测量机(CMM),它是三轴的，棱镜测量系统达到0.01毫米的精确。机器人操作器是能校准的，PUMA 560，放置接近于CMM，特殊的操作装置能到达边缘。图1显示了系统不同部分安排。在这部分运动学模型将是发展, 解释姿态估算法，和参数辨认方法。2.1

5、 运动学的参数在这部分，操作器的基本运动学结构将被规定，它关系到完全坐标系统的讨论, 和终点模型。从这些模型，用于可能的技术的运动学参数的识别将被规定，和描述决定这些参数的方法。那些基础的模型工具用于描写不同的物体和工件操作器位置空间的关系的方法是Denavit-Hartenberg方法2，在Hayati 10有调整计划，停泊处11 和Wu 12 当二连续的接缝轴是名义上地平行的用于说明不相称模型 13。如图2这中方法存在于物体或相互联系的操作杆结构中，和运动学中需要从一个坐标到另一个坐标这种同类变化是定义的。这种变化是相同形式的上面的关系可以解释通过四个基本变化操作实现坐标系n-1到结构坐标

6、系n的变化。只有需要找到与前一个的关系的四个变化是必需的，在那个时候连续的轴是不平行的，定义为零点。当应用于一个结构到下一个结构的等价变化坐标系与更改Denavit-Hartenberg系相一致时，它们将被书写成矩阵元素实现运动学参数功能的矩阵形状。这些参数是变化的简单变量：关节角，连杆偏置, 连杆长度，扭角，矩阵通常表示如下：对于多连接的, 例如机械操作臂,各自连续的链环和两者瞬间的位置描写在前一个矩阵变化中。这种变化从底部链环开始到第n链环因此关系如下：图3表示出PUMA机器人在Denavit-Hartenberg系中每一连杆，完全坐标系和工具结构。变化从世界坐标系到机器人底部结构需要仔细

7、考虑过，因为潜在的参数取决于被选择的改变类型。 考虑到图4,世界坐标，在D-H系中定义的从世界坐标到机器人基坐标，坐标是PUMA机器人定义的基坐标和机器人第二个D-H结构中坐标。我们感兴趣的是从世界坐标到必需的最小的参数数量。实现这种变化有两种路径：路径1，从到D-H变化包括四个参数，接着从到的变化将牵连二个参数和的变化图3图4最后，另外从到的D-H变化中有四个参数其中和两个参数是关于轴Z0因此不能独立地识别， 和是沿着轴Z0因此也不能是独立地识别。因此，用这路径它需要从世界坐标到PUMA机器人的第一个坐标有八个独立的运动学参数。路径2，同样地二中择一，从世界坐标到底部结构坐标的变化可以是直接

8、定义。因此坐标变换需要六个参数，如Euler形式：下面是从到DH变化中的四个参数，但与相关联，与相关联，减少成两个参数。很显然这种路径和路径1一样需要八个参数，但是设置不同。上面的方法可能使用于从世界坐标系到PUMA机器人的第二结构的移动中。在这工作中，选择路径2。工具改变引起需要六个特殊参数的改变的Euler形式：用于运动学模型的参数总数变成30，他们定义于表12.2 辨认方法学运动学的参数辨认将是进行多维的消去过程, 因此避免了雅可比系统的标定，过程如下：1. 首先假设运动学的参数, 例如标准设置。2. 为选择任意关节角的设置。3. 计算PUMA机器人末端操作器。4. 测量PUMA机器人末

9、端操作器的位姿如关节角，通常标准的和预言的位姿将是不同的。5. 为了最好使预言位姿达到标准的位姿，在整齐的方式更改运动学的参数。这个过程应用于不是单一的关节角设置而是一定数量的关节角，与物理测量数量等同的全部关节角设置是需要，必须满足在这儿Kp是识别的运动学参数的数量N是测量位姿的数Dr是测量过程中自由度的数量文章中，给定了自由度的数量，赠值为因此全部位姿是测量的。在实践中，更多的测量应该是在实验测量法去掉补偿结果。优化程序使用命名为ZXSSO，和标准库功能的IMSL14。2.3 位姿测量法显然它是从上面的方法确定PUMA机器人全部位姿是必需的为了实现标定。这种方法现在将详细地描写。如图5所示

10、，末端操作器由五个确定的工具组成。 考虑到借助于工具坐标和世界坐标中间各个坐标的形式，如图6这些坐标的关系如下：是关于世界坐标结构的第i个球的4x1列向量坐标, Pi是关于工具坐标结构第i个球的4x1坐标的列向量, T是从世界坐标结构到工具坐标结构变化的4x4矩阵。设定Pi，测量出，然后算出T，使用于在标定过程的位姿的测量。它是不会很简单，但是不可能由等式(11)反求出T。上面的过程由四个球A, B, C和D来实现，如下：或为由于P, T和P全部相符合，反解求的位姿矩阵在实践中当PUMA机器人放置在确定的位置上，对于CMM由四个球决定Pi是困难的。准确的测量三个球第四球根据十字相乘可以获得考虑

11、到决定的球中心坐标的是基于球表面点的测量,没有分析可获到的程序。 另外，数字优化的使用是为了求惩罚函数的最小解这里是确定球中心，是第个球表面点的坐标且是球的半径。在测试过程中，发现只测量四个表面上的点来确定中心点是非常有效的。Full-Pose Calibration of a Robot Manipulator Using a Coordinate-Measuring MachineMorris R. Driels, Lt W. Swayze USN and Lt S. Potter USNDepartment of Mechanical Engineering, Naval Postgra

12、duate School, Monterey, California, USThe work reported in this article addresses the kinematiccalibration of a robot manipulator using a coordinate measuringmachine (CMM) which is able to obtain the full pose ofthe end-effector. A kinematic model is developed for themanipulator, its relationship to

13、 the world coordinate frame andthe tool. The derivation of the tool pose from experimentalmeasurements is discussed, as is the identification methodology.A complete simulation of the experiment is performed, allowingthe observation strategy to be defined. The experimental workis described together w

14、ith the parameter identification andaccuracy verification. The principal conclusion is that themethod is able to calibrate the robot successfully, with aresulting accuracy approaching that of its repeatability.Keywords: Robot calibration; Coordinate measurement; Parameteridentification; Simulation s

15、tudy; Accuracy enhancement1. IntroductionIt is well known that robot manipulators typically havereasonable repeatability (0.3 ram), yet exhibit poor accuracy(10.0 mm). The process by which robots may be calibratedin order to achieve accuracies approaching that of themanipulator is also well understo

16、od 1. In the calibrationprocess, several sequential steps enable the precise kinematicparameters of the manipulator to be identified, leading toimproved accuracy. These steps may be described as follows:1. A kinematic model of the manipulator and the calibrationprocess itself is developed and is usu

17、ally accomplished withstandard kinematic modelling tools 2. The resulting modelis used to define an error quantity based on a nominal(manufacturers) kinematic parameter set, and an unknown,actual parameter set which is to be identified.2. Experimental measurements of the robot pose (partial orcomple

18、te) are taken in order to obtain data relating to theactual parameter set for the robot.3.The actual kinematic parameters are identified by systematicallychanging the nominal parameter set so as to reducethe error quantity defined in the modelling phase. Oneapproach to achieving this identification

19、is determiningthe analytical differential relationship between the posevariables P and the kinematic parameters K in the formof a Jacobian, and then inverting the equation to calculate the deviation ofthe kinematic parameters from their nominal valuesAlternatively, the problem can be viewed as a mul

20、tidimensionaloptimisation task, in which the kinematic parameterset is changed in order to reduce some defined error functionto zero. This is a standard optimisation problem and maybe solved using well-known 3 methods.4. The final step involves the incorporation of the identifiedkinematic parameters

21、 in the controller of the robot arm,the details of which are rather specific to the hardware ofthe system under study.This paper addresses the issue of gathering the experimentaldata used in the calibration process. Several methods areavailable to perform this task, although they vary in complexity,

22、cost and the time taken to acquire the data. Examples ofsuch techniques include the use of visual and automatictheodolites 4, 5, 6, servocontrolled laser interferometers 7,acoustic sensors 8 and vidual sensors 9. An ideal measuringsystem would acquire the full pose of the manipulator (positionand or

23、ientation), because this would incorporate the maximuminformation for each position of the arm. All of the methodsmentioned above use only the partial pose, requiring moredata to be taken for the calibration process to proceed.2. TheoryIn the method described in this paper, for each position inwhich

24、 the manipulator is placed, the full pose is measured,although several intermediate measurements have to be takenin order to arrive at the pose. The device used for the posemeasurement is a coordinate-measuring machine (CMM),which is a three-axis, prismatic measuring system with aquoted accuracy of

25、0.01 ram. The robot manipulator to becalibrated, a PUMA 560, is placed close to the CMM, and aspecial end-effector is attached to the flange. Fig. 1 showsthe arrangement of the various parts of the system. In thissection the kinematic model will be developed, the poseestimation algorithms explained,

26、 and the parameter identificationmethodology outlined.2.1 Kinematic ParametersIn this section, the basic kinematic structure of the manipulatorwill be specified, its relation to a user-defined world coordinatesystem discussed, and the end-point toil modelled. From thesemodels, the kinematic paramete

27、rs which may be identifiedusing the proposed technique will be specified, and a methodfor determining those parameters described.The fundamental modelling tool used to describe the spatialrelationship between the various objects and locations in themanipulator workspace is the Denavit-Hartenberg met

28、hod2, with modifications proposed by Hayati 10, Mooring11 and Wu 12 to account for disproportional models 13when two consecutive joint axes are nominally parallel. Asshown in Fig. 2, this method places a coordinate frame oneach object or manipulator link of interest, and the kinematicsare defined by

29、 the homogeneous transformation required tochange one coordinate frame into the next. This transformationtakes the familiar form The above equation may be interpreted as a means totransform frame n-1 into frame n by means of four out ofthe five operations indicated. It is known that only fourtransfo

30、rmations are needed to locate a coordinate frame withrespect to the previous one. When consecutive axes are notparallel, the value of/3. is defined to be zero, while for thecase when consecutive axes are parallel, d. is the variablechosen to be zero.When coordinate frames are placed in conformance w

31、iththe modified Denavit-Hartenberg method, the transformationsgiven in the above equation will apply to all transforms ofone frame into the next, and these may be written in ageneric matrix form, where the elements of the matrix arefunctions of the kinematic parameters. These parameters aresimply th

32、e variables of the transformations: the joint angle0., the common normal offset d., the link length a., the angleof twist a., and the angle /3. The matrix form is usuallyexpressed as follows:For a serial linkage, such as a robot manipulator, a coordinateframe is attached to each consecutive link so

33、that both theinstantaneous position together with the invariant geometryare described by the previous matrix transformation. Thetransformation from the base link to the nth link will thereforebe given byFig. 3 shows the PUMA manipulator with theDenavit-Hartenberg frames attached to each link, togeth

34、erwith world coordinate frame and a tool frame. The transformationfrom the world frame to the base frame of themanipulator needs to be considered carefully, since there arepotential parameter dependencies if certain types of transformsare chosen. Consider Fig. 4, which shows the world framexw, y, z,

35、 the frame Xo, Yo, z0 which is defined by a DHtransform from the world frame to the first joint axis ofthe manipulator, frame Xb, Yb, Zb, which is the PUMAmanufacturers defined base frame, and frame xl, Yl, zl whichis the second DH frame of the manipulator. We are interestedin determining the minimu

36、m number of parameters requiredto move from the world frame to the frame x, Yl, z. Thereare two transformation paths that will accomplish this goal:Path 1: A DH transform from x, y, z, to x0, Yo, zoinvolving four parameters, followed by another transformfrom xo, Yo, z0 to Xb, Yb, Zb which will invol

37、ve only twoparameters b and d in the transformFinally, another DH transform from xb, Yb, Zb to Xt, y, Zwhich involves four parameters except that A01 and 4 areboth about the axis zo and cannot therefore be identifiedindependently, and Adl and d are both along the axis zo andalso cannot be identified

38、 independently. It requires, therefore,only eight independent kinematic parameters to go from theworld frame to the first frame of the PUMA using this path.Path 2: As an alternative, a transform may be defined directlyfrom the world frame to the base frame Xb, Yb, Zb. Since thisis a frame-to-frame t

39、ransform it requires six parameters, suchas the Euler form:The following DH transform from xb, Yb, zb tO Xl, Yl, zlwould involve four parameters, but A0 may be resolved into4, 0b, , and Ad resolved into Pxb, Pyb, Pzb, reducing theparameter count to two. It is seen that this path also requireseight p

40、arameters as in path i, but a different set.Either of the above methods may be used to move fromthe world frame to the second frame of the PUMA. In thiswork, the second path is chosen. The tool transform is anEuler transform which requires the specification of sixparameters:The total number of param

41、eters used in the kinematic modelbecomes 30, and their nominal values are defined in Table12.2 Identification MethodologyThe kinematic parameter identification will be performed asa multidimensional minimisation process, since this avoids thecalculation of the system Jacobian. The process is as foll

42、ows:1. Begin with a guess set of kinematic parameters, such asthe nominal set.2. Select an arbitrary set of joint angles for the PUMA.3. Calculate the pose of the PUMA end-effector.4. Measure the actual pose of the PUMA end-effector forthe same set of joint angles. In general, the measured andpredic

43、ted pose will be different.5. Modify the kinematic parameters in an orderly manner inorder to best fit (in a least-squares sense) the measuredpose to the predicted pose.The process is applied not to a single set of joint angles butto a number of joint angles. The total number of joint anglesets requ

44、ired, which also equals the number of physicalmeasurement made, must satisfywhereKp is the number of kinematic parameters to be identifiedN is the number of measurements (poses) takenDr represents the number of degrees of freedom present ineach measurementIn the system described in this paper, the n

45、umber of degreesof freedom is given bysince full pose is measured. In practice, many more measurementsshould be taken to offset the effect of noise in theexperimental measurements. The optimisation procedure usedis known as ZXSSO, and is a standard library function in theIMSL package 14.2.3 Pose Mea

46、surementIt is apparent from the above that a means to determine thefull pose of the PUMA is required in order to perform thecalibration. This method will now be described in detail. Theend-effector consists of an arrangement of five precisiontoolingballs as shown in Fig. 5. Consider the coordinates

47、ofthe centre of each ball expressed in terms of the tool frame(Fig. 5) and the world coordinate frame, as shown in Fig. 6.The relationship between these coordinates may be writtenaswhere Pi is the 4 x 1 column vector of the coordinates ofthe ith ball expressed with respect to the world frame, P isth

48、e 4 x 1 column vector of the coordinates of the ith ballexpressed with respect to the tool frame, and T is the 4 4homogenious transform from the world frame to the toolframe.then T may be found, and used as the measured pose in thecalibration process. It is not quite that simple, however, sinceit is

49、 not possible to invert equation (11) to obtain T. Theabove process is performed for the four balls, A, B, C andD, and the positions ordered asor in the formSince P, T and P are all now square, the pose matrix maybe obtained by inversionIn practice it may be difficult for the CMM to access fourbails

50、 to determine P when the PUMA is placed in certainconfigurations. Three balls are actually measured and a fourthball is fictitiously located according to the vector cross productRegarding the determination of the coordinates of thecentre of a ball based on measured points on its surface,no analytica

51、l procedures are available. Another numericaloptimisation scheme was used for this purpose such that thepenalty functionwas minimised, where (u, v, w) are the coordinates of thecentre of the ball to he determined, (x/, y, z) are thecoordinates of the ith point on the surface of the ball and ris the ball diameter. In the tests performed, it was foundsufficient to measure only four points (i = 4) on the surfaceto determine the ball centre.19