Numerical Plotting of Surfaces of Positioning Accuracy of Manipulators
Alok Kumar~
and
Kenneth J. Waldron~
Received for publication 25 November 1980
Abstract A mathematical model of the random positioning errors of mechanical manipulators is developed. This model is applied to a computer program to plot the generating curves of equal error surfaces for given manipulator geometries.
Introduction MECHANICAL manipulators are designed for versatility. This versatility raises problems of quantitative characterization and evaluation. For design optimization purposes, it is very important to be able to measure how good one manipulator geometry is with respect to another, under different performance criteria. Since, almost all performance characteristics are functions of the position of the manipulator, one approach is to summarize performance data throughout the reachable workspace by generating surfaces on which a given performance parameter has a specific value. These surfaces are surfaces of revolution if the first joint is revolute and are non-circular cylinders with generators parallel to the first joint axis if the first joint is prismatic. Hence, in either case the surface can be represented as a contour on a generating plane on which a given performance parameter has specific values. A method is presented in this work, to generate these contours. As an example, the performance criterion of positioning accuracy will be used.
Current practice in the industrial use of automatic manipulators is overall open loop control. There is, normally, closed loop positioning of the individual joints. However direct feedback of hand position is not, at present, usual. An important limiting factor on tasks a manipulator can perform in this mode is positioning accuracy. The error in positioning arises because of, one: dimensional errors in the individual members, two: dynamic errors due to elastic deflections, mechanical clearance, etc and three: errors in positioning the joints accurately. Typically, automatic manipulators are "taught" by moving them through the steps of the desired motion, while the outputs of the position transducers are being recorded. By teaching the manipulator in this manner it is being caliberated and error in positioning the hand due to the first class of errors is partially eliminated. But since the position error due to errors in positioning the joints is random in nature, it cannot be eliminated. Also, because of differences in the load in the "teaching" and "operating" modes, dynamic errors may be important. In this work the method is used to estimate only the errors in positioning of the hand due to errors in positioning the joints. This is often referred to as "incremental accuracy".
Two cases of positioning errors of the hand are considered in this work. First a simple case in which the error in positioning the hand in only one direction is important. This type of error is important when the manipulator is being used as gauging device with the hand replaced by a probe. The contours of specified probabilities of error within a certain magnitude with the probe being in a given orientation are plotted. These contours give the regions in which the probability of error being within a certain magnitude is more than an equal to a specified probability. The other case, is that in which the error in positioning a reference point on the hand is important.
'fAssistant Professor, University of Wisconsin, Platteville, Wisconsin, U.S.A. :~Department of Mechanical Engineering, Ohio State University, Columbus, OH 43210, U.S.A.
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This is of interest when manipulators are being used as production line tools. The contours of specified probabilities with the hand being in a given orientation; and of magnitude of error less than specified value irrespective of direction are plotted to characterize the error in positioning the reference point on hand.
A mathematical model has been developed by authors to calculate the expected errors of position and orientation of the hand at any point in workspace, given the expected error of positioning the individual joints of manipulator[4, 5]. In this work, the model is used to derive equations, which can be used to plot the contours of expected errors in positioning the hand either in a specified direction or regardless of direction. Because of the highly non-linear characteristics of the equations, they are solved numerically to plot the contours. The parameter perturbation method [2] is used in conjunction with the Newton-Raphson method to solve the simultaneous non-linear equations.
Theory In this work it is assumed that the dimensions of the links are precisely known and that all
positioning errors are the result of errors in joint position. In principal dimensional errors can be reduced by calibration and dynamic errors can be eliminated by computation whereas errors due to inaccuracy in joint positioning are always present. A joint position error is treated as a smal] displacement of a joint from its desired position and the method of instantaneous screw axes is used to study them. Although this method is usually associated with velocity analysis, by multiplying velocities by small time interval one obtains small displacements and so the method is equally applicable to small displacements. The matrix transformation method is used to model mathematically the position of the manipulator in terms of its joint positions.
The errors in joint position have been treated as random variables. The statistical model is essentially the same as that given in Ref. [4]. It is summarized below and the governing equations which give contours of probabilities of a reference point on the hand being within a certain magnitude of error from the desired position are derived. A modified form of the notation for mechanism displacements developed by Denavit and Hartenberg[l] is used. The essential dimensions of each link and the joints mounted on it are shown in Fig. I. Reference frame N is fixed to link N and is aligned with its z axis along the axis of joint N + 1, which is one of the two joints mounted on link N. Its X axis is aligned along the common normal
AXIS N+I
N/
Figure 1.
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between joint axis N and N + 1. The transformation0 ~, ~ e position of a point p. in frame N to P.-i in reference frame N - 1 can be written in the form
P.- I = u.(v.p. +s. ) (1)
where
cO. 0 v . = c 2 . - a 2 s . = 0
0 1 s2. c2..J r.
The joint angle 0. is variable for a revolute joint and offset r. is variable for a prismatic joint. These are the only joint types used in manipulators.
The relationship between the position of a point relative to hand member K and the fixed frame, member 0, as shown in Fig. 2 is
Po = u l ( v l u 2 ( v 2 u 3 - - ( v t P k + s t ) + " " + s2) + s l ) (2)
If the orientation of the hand is known then eqn (2) can be written in the form
where :co, Yo, Zo are the components of position vector po and
q = UlVlU2V2 • • • UtVk. (4)
Matrix q is orthogonal (q r= q-I) and gives the orientation of hand. Only three of the component equations of the matrix eqn (4) are independent. The components of position vector Po are three variables and the relationship between them gives the surface on which a certain performance parameter has a specific value. Hence, if the orientation of the hand is known, eqns (3) and (4) give six equations in (K+ 3) unknowns: K joint variables plus the three component of position vector Po. For a six degree of freedom manipulator we get six equations
LINK K ~'~---- REFERENCE POINT
/ ~ AXIS K
LINK K-I . . . 4 AXIS K- I
l -
AXIS 3
LINK O ~ / - ~ (BASE) I LINK I ~ LINK 2
AXIS 2 Figure 2.
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in nine unknowns. If we have one more equation, the seven equations in nine unknowns can be reduced to one equation in three unknowns, the components of position vector Po.
This one equation in three unknowns gives the relationship which determines the surface. The seventh equation is derived by finding the relationship between the values of the performance parameters and the variables.
Since the performance parameter is positioning accuracy, let us consider the statistical model[4] derived for a mechanical manipulator. The displacement of a reference point in the hand Po due to a small displacement about joint N can be described by means of a motor (80,, 8p,)[3]. If joint N is revolute
8o, = 80,~o, (5)
and
8p, = 8Ono~,x(po- po)
where ton is a unit vector in the direction of axis N, Pn is the position of any point on that axis and 80n is the angular displacement about the revolute joint N.
If joint N is prismatic
88, = 0 (6)
and
~p. = 8r, con
where 8r, is the linear displacement along the prismatic joint. con and Pn can be expressed in frame 0. pn is taken as the position vector of the origin in
frame N - 1 f ixed on Nth axis. Then
~n = uzvlu~ • . • unvnk (7)
and
Pn = U I V I U 2 • • • UnS. "}" " " " " i " U I V l U 2 S 2 " ] " I ta lS I
where [k] r = [001] because the z axis of the frame lies along the joint axis N. The motor (80, 8pal describing the overall displacement of the hand is the resultant of the
motors due to the displacements about the respective joints. Thus
K
80 = ~ l aOn
K 8po = ~ i 8p,. (8)
The errors in positioning the joints are modelled as normally distributed random variables with zero mean, since the zero mean corresponds to the joint being in desired position. The mean of each component of 80 and &p0 is the weighted sum of the means of joint error and hence is zero. Similarly the variance of each component of 80 and 8po is a weighted sum of the variances of joint variables.
Resolved into the axis direction of any convenient fixed frame, the components of the error in position of the hand at a point Po assumes the form
K
xj = ~ X0~i (9) i = 1
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where j - 1, 2, 3 gives the components of error in three mutually perpendicular direction of the frame, 8~ = 80s for a revolute and 8~ = 8r~ for a prismatic joint and X~j functions of all the joint variables.
The three variances and three covariances for the errors in positioning the hand are given by
k
v[x~xt] = ~ Xi~av(gi) (10)
where j = 1,2,3 and ! = 1,2,3. When j = ! we get three variances and when j# l we get three covariances as v[xjxt] = v[xlx~].
The principal variances and the directions in which they occur can be determined by solving the eigenvalue problem
v u = ~ u (11)
where elements of matrix v are given by eqn (10). The square roots of the principal variances are the principal standard deviations Tx, Ty and
T~. If the axes are rotated into the directions of the eigenvectors, the three resultant random variables x, y, z are mutually independent with standard deviation Tx, T, and T~ respectively and zero mean. The trivariate normal probability density function takes the form
1 [ l /x2+y2+22 '~ "] f(x, y, z) = (2~.)312TxTyTz exp L - 2 ~ '~ ~y "~ J J" (12)
The probability of positioning the hand at po within an error of magnitude R in any direction is
= f~ f(x, y, z) dv (13) P(Po)
where v is a sphere with center at the point at which the manipulator hand is to be positioned and radius R.
The first case of interest is that in which the error in positioning the hand in a certain direction is important. Let x~ be the direction in which error is to be estimated. If Tx: is the standard deviation in the xt direction, the one dimensional normal distribution function is
1 f(xt) = X/(2crTxt) exp (- x212T21) (14)
and the probability that error in positioning a reference point on the hand in the given direction is between +- R from the desired position is
p = f(xl) dxt. (15) R
Therefore, if this probability P is to have a specified value, the value of the standard deviation Txt is determined uniquely from the numerical tables of areas under a normal distribution curve[6] for a given R. The standard deviation can be expressed, from (10), as follows
6
T21 = v[xlxl] = ~ X2 v(Si) (16) i=l
where xa are functions of the six joint variables only. Hence, if the probability of positioning a point in the hand within an error of + R in a given direction, with a certain orientation of hand is specified, one gets seven equations, three from vector eqn (3), three independent equations
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from matrix eqn (4) and one equation given the relationship between the joint variables and specified probability (16). There are nine variable in these equations, six joint variables for a six degree of freedom manipulator and three components of position vector Po. This can be reduced, in principle, to one equation in three variables Xo, Yo, Zo to give the equation of the surface of interest.
Similarly, for the second case, where there is to be a specified probability that the error in positioning a point on the hand is to be within + R of the desired position irrespective of direction one gets eqn (13). When the probability is specified, it gives a function of Tx, Ty and Tz which in turn are functions of the three variances and three covariances given by eqn (10). These variances and covariances are functions of the joint variables. Hence eqn (13) can be reduced to a relationship between the six joint variables and the specified probability. Thus, when the orientation of the hand is given, once again one gets seven equations given by (3), (4) and (13) in nine unknowns, which can in principal be reduced to one equation in three unknowns x0, yo, Zo to give the surface of specified probability.
In general these seven equations in nine unknowns cannot be reduced to the closed form equation of a surface in three variables, hence numerical methods are used to plot the surfaces. If the first joint is a revolute these surfaces are surfaces of revolution generated by curves in the XoZo plane. In this work, they are plotted as planar curves by taking Yo equal to zero, varying Zo over the workspace of the manipulator and solving the seven equations given by (3), (4) and either (13) or (16) in seven unknowns, six joint variables and xo. The relationship between Xo and Zo gives the desired curves of specified probability.
Similarly, if the first joint is prismatic, the surfaces are cylinder surfaces with the cylinder generators being parallel to first joint. These cylindric surfaces can be generated by the planar curves in the XoYo plane. Thus the desired surfaces can be represented by contours in the xoYo
plane and can be obtained by taking Zo equal to zero, varying Yo over the workspace of the manipulator and solving the seven equations in seven unknowns, six joint variables and Xo. The relationship between Xo and Yo gives the desired contours of specified probability.
Thus the problem is reduced to solving seven non-linear equations in seven unknowns for a six degree of freedom manipulator. Because of the highly non-linear characteristics of these equations the parameter perturbation method [2] in conjunction with Newton-Raphson iteration is used for their solution. For Newton's iteration method a 7 x 7 matrix of derivatives of each equations with respect to seven variables is derived for a 6-R manipulator. Software is developed in FORTRAN to solve the equations. All the calculations was done on the Honeywell 66--60 system of University of Houston.
X6
AXIS 6 Z5 ' Z6
X' 4
X ~ 7 x~ --z3
2 AXISx X2/Zo ~ ' AX'IS 4 AXIS 5
,XI
AXIS 3
AXIS I Figure 3.
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An Example To illustrate, a 6-R manipulator of the type shown in Fig. 3 is taken. The manipulator has a
reach of one meter and the parameters of the manipulator are
a~=0 for i = 1 , 2 . . . 6
r~ =0 for i = 1,2,4,6 and r3 = 60cm, rs=4Ocm
ui = 90 ° for i = 1 ,2 . . . 5 and a6 = 0 °.
The variances of errors in positioning of the six revolute joints of the manipulator were taken to be 0.0005 rads. The magnitude of error in positioning a point, R, was taken to be 0.625 cm for both cases. The contours are plotted in Figs. 4 and 5, for different probabilities and for each of the two cases.
zc PLOT OF CONTO4JRS OF POSiTIONiNG PROBABILITY
6° 0
50
,)lll,,o, I0 20 30 4 0 50 60 70 80 go I00 "-- x o
Figure 4,
Z o
I PLOT OF CONTOURS OF POSITIONING PROBABILITY so I ., O~EN'rAT~ON OF .ANO 0 . [I 0 0 ]
- - 0 I O[ I J
50
I 1 I I I J I I I I I I0 20 30 40 ,50 60 70 80 90 I00 ------ zO
Figure 5.
368
Concluding Remarks Although the method presented in this work was developed to plot positioning accuracy, it is
a very versatile and useful tool to quantitatively characterize the performance of manipulators in terms of their geometry. Once a performance criterion is expressed as a function of the joint variables, one can plot, using the same software, the surfaces on which the performance parameters have specific values. These surfaces help the designer to identify the regions in which the performance parameter has more/less than and equal to the specified values. These regions can be used to compare, evaluate and operate the manipulators.
In this work, the method is used to plot the contour surfaces of the purely geometric criterion positioning accuracy, but it can be used for such diverse criteria as the forces and torques on the hand of a manipulator in a certain direction for static force analysis or natural frequencies of the manipulator in dynamic analysis. The method can be used to assist in the design, or performance optimization problems by choosing the appropriate performance cri- teria. Another advantage of the method is, that one does not need to do a separate position analysis, hence it can be, used for manipulators of general geometry.
Acknowledgement--The authors wish to acknowledge the support of National Science Foundation, Grant No ENG77- 22747 during the course of this work.
References 1. J. Denavit and R. S. Hartenbcrg, A kinematic notation of lower pair mechanisms based on matrices. J. Appl. Mech. 22,
Trans ASME 77, 215-221 (1955). 2. F. Freudenstein and B. Roth, Numerical solution of systems of non linear equations. J. Assoc. Comput. Mach. 10(4)
(1963). 3. K. J. Waldron, The use of motors in spatial kinematics. Proc. IFToMMInt. Syrup. Link. and Comput. Design Method B,
535-545, Bucharest (1973). 4. K. J. Waldron, Positioning accuracy of manipulators. Proc. NSF Workshop on the impact on the Academic Community
of required Research Activity for Generalized Robotic Manipulators. University of Florida (Feb. 1978). 5. K. J. Waldron and A, Kumar, Development of a theory of errors for manipulators. Proc. of the 5th World Cong. Theory
Mach. Mech. A, 821.-825 0979). 6. G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers. McGraw Hill, New York (1968).
RESTITUTION NUMERIQUE DE SURFACES DE PRECISION DE POSITIONNEMENT DE MANIPULATEURS
A. Kumar et K.J. Waldron
n~mum~ -- Afin de porter & l'optimum le oalcul de manipulateurs, il est tr~s important de
pouvoir meaurer l'applicabilit§ d'une g~om~trie de manipulateur par rapport ~ une autre.
Puisque pratiquement routes les caract~ristiques de performances sont des fonctions de la
position du manipulateur, une faqon d'aborder ce probl~me est de r~sumer les donn~es de
performances dens tout l'eapace de travail ~ pottle par la g~n6ration de surfaces de r~vo-
lution si le premier joint est un de rotation, et elles sont des cylindres non-circulaires
ayant leur g6n~ratrice parall~le ~ l'axe du premier joint s'il est prismatique. Ce papier
pr6sente une m~thode de g~n~ration de ces contours. Un exemple utilisant le crit~re de per-
formances de la precision de positionnement est pr~sent~.
