Method for kinematic calibration of stewart platforms

Method for Kinematic Calibration of Stewart Platforms Hanqi Zhuang and Zvi S. Roth Robotics Center and Department of Electrical Engineering Florida Atlantic University Boca Raton, FL 33431

Received May 27, 1992; accepted October 26, 1992

A Stewart platform is a six degrees of freedom parallel manipulator composed of six variable-length legs connecting a fixed base to a movable plate. Like all parallel manipulators, Stewart platforms offer high forcehorque capability and high structural rigidity in exchange for small workspace and reduced dexterity. Because the solution for parallel manipulators' forward kinematics is in general much harder than their inverse kinematics, a typical control strategy for such manipulators is to specify the plate's pose in world coordinates and then solve the individual leg lengths. The accuracy of the robot critically depends on accurate knowledge of the device's kinematic parameters. This article focuses on the accuracy improvement of Stewart platforms by means of calibration.

Calibration of Stewart platforms consists of construction of a kinematic model, measurement of the position and orientation of the platform in a reference coordinate frame, identification of the kinematic parameters, and accuracy compensation. A measurement procedure proposed in this article allows a great simplification of the kinematic identification. The idea is to keep the length of the particular leg, whose parameters are to be identified, fixed while the other legs change their lengths during the measurement phase. By that, redundant parameters can be eliminated systematically in the identification phase. The method also shows the estimation of each leg's parameters separately because the measurement equations are fully decoupled, which results in a drastical reduction of the computational effort in the parameter identification. Simulation results assess the performance of the proposed approach. @ 1993 John Wiley & Sons, Inc.

Journal of Robotic Systems 10(3), 391-405 (1993) 0 1993 by John Wiley & Sons, Inc. ccc 0741 -2223/93/030391-015

392 Journal of Robotic Systems-1 993

1. INTRODUCTION

The majority of today's industrial robots are serial manipulators. Such robots usually have large workspace and high maneuverability on the one hand and relatively low accuracy and load/weight ratio due to their cantilever structure on the other hand. Attention has recently been given to parallel manipulators in applications that demand high precision, high speed, and large payloads with relatively small workspace. A Stewart platform consists of six variable-length legs connecting a plate to a base. Such a parallel manipulator was first used for the design of flight simulators.' To date, much of the effort on parallel manipulators in general and Stewart platforms in particular has focused on kinematic, dynamic, control, and design issues.*-1° Accuracy enhancement through kinematic calibration is important in applications that require off-line programming or transfer of programmed applications from one robot to another.

In the robot calibration literature (ref. 1 1 and references cited therein), little attention thus far has been given to nonserial manipulators. Everett and LinI2 investigated kinematic calibration of manipulators with closed kinematic chains. Their approach, which is relatively complex for Stewart platform calibration, is based on constrained optimization involving large number of redundant parameters. In this article, a simple calibration strategy is proposed exclu- sively for the calibration of Stewart p1atf0rrns.l~

As in serial manipulator calibration, the calibration of Stewart platforms consists of four steps: construction of a robot kinematic model, measurement of the position and orientation of the platform in a reference coordinate frame, identification of unknown kinematic parameters in the robot model, and compensation of pose errors based on the calibrated model.

A unique feature of the Stewart platform calibration method proposed in this article is in its measurement procedure: To identify each leg's kinematic parameters, the length of the respective leg is kept fixed while other legs stretch or shrink. This approach allows the systematic elimination of redundant parameters in the kinematic identification phase. Another advantage of the method is that the parameters of each leg can be estimated individually, thus drastically reducing the complexity of the parameter estimation problem.

It is assumed in this article that each leg of the Stewart platform contains a single degree of freedom ideal prismatic joint and that the ball joints connecting each leg to the base and plate are perfectly machined so that rotations of the leg are about the respective ball joint center. These assumptions are realistic as a recent error analysis14 has shown that the effect of the joint inaccuracy is secondary to base and plate kinematic errors, The kinematic identification goal is to estimate the coordinates of these ball joint centers in a reference frame.

Zhuang and Roth: Kinematic Calibration of Stewart Platforms 393

2. PROBLEM STATEMENT

The robot, illustrated in Figure 1, is composed of six variable-length legs, a base, and a plate. The base and plate are arbitrary convex hexagons. Each vertex of the base supports one leg and each vertex of the platform connects to one leg.

A base coordinate frame {B) is placed at an arbitrary location on the base. The z-axis of {B} is chosen to be perpendicular to the surface of the nominal base plane. Similarly, a plate coordinate frame { P } is placed somewhere on the plate with its z-axis being perpendicular to the surface of the nominal plate plane (Fig. 1). Two more coordinate frames, the world frame and the tool frame, can be placed at convenient locations. Because the transformations from the tool to the platform frames and from the base to the world frames are fixed and application dependent, these are omitted for better clarity of the discussion and with no severe loss of generality.

The corner points of the base are denoted as B I , Bz, . . . , B6 and those of the plate as P I , P2, . . . , Ps. The coordinates of B;, i = 1, 2, . . . , 6, in {B} are described by the vectors bl, 62, . . . , 6 6 and those of P I , Pz, . . . , P6 in { P } are described by P I , p2, . . . , p6. Each of the leg lengths is denoted by i;, i = 1, 2, . . . , 6. 6; and pi are the kinematic parameter vectors of the ith leg.

Figure 1. Stewart platform coordinate frames and parameters.

394 Journal of Robotic Systems-1993

Let R and t denote the orientation and position of the plate coordinate frame with respect to the base coordinate frame. R is a 3 X 3 matrix and t is a 3 X 1 vector; both are functions of the leg variables l i , i = 1, 2, . . . , 6. Under the assumptions of perfect connecting ball joints and perfect legs’ prismatic joints, the following constraint equations represent the kinematics of the Stewart platform:

1; = llbi - Rpi - tIl2 i = 1 , 2, . . . , 6 (1)

where 1).11 denotes a Euclidean norm. The actual leg length is the sum of the nominal joint command li,0 and a fixed offset AIi. Thus

(2) (li.0 + = IJbi - Rpi - tll’ i = 1 , 2, . . , , 6

li,o is provided by the leg position transducer. Four typical problems can now be stated as follows:

1. Thefonvard kinematicsproblem. In eq. (11, given the leg lengths l i and the leg parameters bi and pi for i = 1,2, . . . ,6 , find the platform pose {R, t}.

For this problem to be tractable, it is essential to assume that both the base and platform are semiregular planar hexagons. Even in this case, numerical solutions are generally needed. Only in the particular case of a six-legged triangular platform was a partial closed-form solution obtained .9

2. The inverse kinematicsproblem. In eq. (l) , given the platform pose { R , t} and the leg parameters bi and pi, find the leg lengths 1; for i = 1,2, . . . ,6. A unique solution is trivially provided by eq. (1) itself.

i = 1 , 2, . . . , 6, be recorded by the leg-position transducers and the platform pose {Rj, t j } be measured using end-point sensors, at each of the calibration measurement configurationsj = 1 ,2 , . . . , m, where m is the number of pose measurements. Estimate the leg parameters bi and pi as well as the leg length offsets Ali for i = 1 , 2, . . . , 6. Remarks 1. Neither the plate corner points { P I , Pz, . . . , P6} nor the base corner

points {B, , B2, . , . , B6) are necessarily coplanar, taking into account fabrication and assembly tolerances or any other fixed deformations.

2. Whenever {RJ, ti} are measured with respect to { W}, the world coordinate system, rather than {B}, the estimated base parameters bi will also be represented in {W}.

4. The accuracy compensation problem. In eq. (2), given the platform pose {R, t}, the identified leg parameters bi and p i , and the identified leg length offsets Ali, find the leg lengths li.0 for i = 1 , 2, . , . , 6. This problem is identical to the inverse kinematic problem.

The article therefore focuses primarily on problem 3, the kinematic identification problem. Some discussion of a numerical solution to problem 1, the forward kinematics, is given as part of the simulation studies.

3 . The kinematic identijicationprobfem. In eq. (2), let the leg lengths


3. CALIBRATION STRATEGY

The kinematic identification is formulated as a nonlinear optimization problem under certain assumptions about the measurement procedure. The problem can be solved by a standard iterative algorithm that requires the elimination of redundant variables and the use of an identification Jacobian matrix.

3.1. Measurement Strategy for the Kinematic Identification

By eq. (2), there exist seven parameters for each leg. Ali, however, is redundant because it can be computed given the leg parameters, platform pose, and leg length measurements. Six leg parameters, three for each plate corner and three for each base corner, sufficiently represent geometric errors of each leg of the Stewart platform. The elimination of Ali from eq. (2) is in general not easy because of the nonlinearity of the equation. A measurement procedure is proposed next in which Ali is naturally eliminated.

The key idea is to keep one leg length of the platform fixed while the other five legs move freely. Figure 2 illustrates the measurement strategy. Let the ith leg length be fixed while the platform changes its configuration. As the robot changes its pose, if a sufficient number of poses are measured enough equations for the estimation of the fixed leg parameters bi and pi are generated. Thus, if li

b

Figure 2. Changes of configurations while the length of leg i is kept fixed.

396 Journal of Robotic Systems-1993

is kept fixed then by eq. (2)

llbi - Rjpi - t’l( = I(bi - Rkpi - tkJ(

i = 1, 2, . . . , 6; j , k E (1, 2, . . . , m}, and j < k (3)

where m is the number of configurations that are taken while the ith leg length is fixed and Rj and tj are the measured orientation and position of the platform at its jth configuration. This is similar for Rk and tk. Let s be the resulting number of algebraic equations, s = (m - l )m/2. For instance, if m = 7 then s = 21.

Let 6 = f;.(bi, pi) = ((bi - R’pi - tjl12. Equation ( 3 ) can be rewritten as

6 = f k j, k E (1, 2 , . . . , m}, a n d j < k (4)

The minimum number of configurations needed for the identification of the ith leg vector is revealed by the following simple observation.

Lemma 1. The number of independent equations in eq. (4) is minim - 1, 6).

Proofi The proof is by mathematical induction. For m = 2, fi = f2 is the only independent equation. For m = p - 1

& = f k j , k E { 1 , 2 , . . . , p - l } , and j<k ( 5 )

Let p - 2 be the number of independent equations in eq. (5). By increasing rn to m = p, p - 1 equations are added:

fj=& j = l , 2 , . . . , p - 1 (6)

However, only f i = f , is independent while the other p - 2 equations can be obtained by combining& = f , with eq. (5). Thus, the number of independent equations in (5) and (6) is p - 2 + 1 = p - 1.

Because there are six unknown scalar parameters in eq. (4), the maximum number of independent equations in eq. (4) is 6.

Corollary. The minimum number of configurations for the estimation of bi, and pi is seven.

Any selection of 6 independent equations of 21 equations provided by 7 configurations will do, such a s h = J;.+l forj = 1,2, . . . ,6. However, with no increase of the program complexity and for better noise reduction all of the measurement equation combinations may be used for leg parameter identification when the number of measurement configurations are relatively small (as shown by simulations).

The kinematic identification problem of Stewart platforms can now be formulated in terms of six parallel optimization problems: Find bi and pi such that each one of the cost functions Ji is minimized, where


m m-l

By eq. (2), let

= - live i = 1, 2, . . . , 6; j = 1, 2, . , . , m (8)

After the leg parameters 6; and pi are identified, the leg length offsets Al; can easily be computed. Because li,0 is constant for all m measurements, the least- squares estimate of Al; is

i = l , 2 , . . . , 6 (9)

We now focus on the estimation of b; and pi.

3.2. Identification Jacobian for Stewart Platforms

The nonlinear optimization problems described by eq. (7) may be solved by standard gradient-based algorithms because nominal leg parameters of Stewart platforms are usually available. Let x; = [bf, pTlT and h;(x;) = [hi.I(Xi), hi.2(xi), . . . , h; . s (~ ; ) lT , where

hi,&) f,(xi) - f k ( d i = 1, 2, . . . , 6; j , k E (1, 2, . . . , m}; and j < k (10)

The index a in eq. (lo), which depends o n j and k , runs from 1 to s, where s is the number of algebraic equations in eq. (3). hi is the s x I measurement residual vector for the ith leg and xi (6 X I ) is the ith leg parameter vector. Let xy be the ith nominal leg parameter vector. Expanding R; about x! and neglecting second- and higher-order terms yield

where Vhi = ah;/&; is the s x 6 idenf$cafion Jacobian matrix and 6x; is the perturbationx; - xy. Iterative procedures such as the Gauss-Newton or Leven- berg-Marquardt algorithms” can then be applied to update the leg parameter vector.

The analytic form of the Jacobian matrix is obtained by differentiating hi with respect to b; and pi. the ath element of Ri(xi), is

hi,, = (bi - Rjp; - r j ) . (bi - Rjp; - d ) - (bi - Rkpi - r k ) . (bi - Rkpi - r k ) = tr[tJtjT] - 2 tr[b;pfRjT] - 2 tr[bitjT] + 2 tr[RjpitjT]

- tr[tktkTl + 2 tr[bipTRkT] + 2 tr[b;tkTl - 2 tr[Rkp;tkT] (12)


where tr[.] denotes the trace of a matrix. Then

where

dhi9,/dbi = -2(Rj - Rk)pi - 2(d - r k )

ahi,,lapi = -2RJT(bi - t j ) + 2RkT(bi - tk)

By Lemma 1 and its corollary, a necessary condition for the identification Jacobian to be full rank is that the number of measurement configurations be equal to or greater than seven.

In addition, as can be seen from eq. (13), for a better condition number of the identification Jacobian the Stewart platform has to significantly change its orientation from one measurement configuration to another. The extreme case of RJ = Rk = I (i.e., as the platform moves horizontally) yields

resulting in a singular Jacobian matrix. In practical platforms, due to link length and ball joint angle constraints the roll-pitch angles of the plate are rather limited (typically no greater than f t5"14). As a result, condition numbers of the identification Jacobian are expected to be relatively large. The selection of optimal configurations is an open research problem, as is the study of identification Jacobian singularities.

4. SIMULATION STUDY

4.1. Iterative Algorithm for Forward Kinematic Solution For the identification simulation of each leg parameter vector, a sequence of

poses with a fixed leg length needs to be generated. Because a closed-form solution for the forward kinematics of a general Stewart platform has not yet been found, an iterative solution had to be used in the simulation study.

This problem too requires nonlinear optimization. Let the rotation matrix R be a function of three parameters r$ (for instance, a set of Euler angles). Let p = [+T , PIT; p is a 6 x 1 vector. The cost function can be written as

where


The problem is then to minimize J by a proper choice of p. Similar to the kinematic identification problem, an iterative nonlinear optimization algorithm may be utilized to find the pose vector p.’O

4.2. Simulation Results

For the simulation study, the base and plate were both taken to be nominally semiregular planar hexagons, as shown in Figure 3. The radii of the base and platform are 1 .O and 0.6 m, respectively.

To simulate an actual Stewart platform, 5% uniformly distributed random numbers were added to the leg parameters bi and p i . Table I lists the first leg’s nominal and “actual” parameters. By fixing the first leg length to 1 m, seven measurement configurations over the range 1.2 2 0.125 m of the other leg lengths were randomly selected for identification of the first leg parameter vector after verifying that these have valid forward kinematic solutions. The Stewart platform poses corresponding to the selected configurations were com-

(b)

Figure 3. Geometry of the simulated Stewart platform. (a) Base geometry. (b) Platform geometry.


Table I. First leg’s nominal and actual parameters (unit: meter).

Parameter type bl, bl, bl,Z PI J P1.Y Pl.2 Nominal 0.3420 0.9397 O.oo00 0.3000 0.51% O . o o 0 0 Actual 0.3171 0.9451 -0.0154 0.2925 0.5392 0.0163

puted iteratively by minimizing the cost function (14) using the Gauss-Newton algorithm. In each iteration, the Jacobian matrix was approximated using a finite difference and the matrix inversion was performed by applying the singular value decomposition (SVD) technique. The Gauss-Newton algorithm was also used to identify the leg parameters. In this case, the identification Jacobian was computed from eq. (13), although the finite difference method worked as well. By assuming zero measurement noise, the algorithm converged to the true leg parameter vector after few (typically three) iterations (Fig. 4).

Simulations with artificial noisy data were performed to study the sensitivity of the algorithm to pose measurement noise and the efficiency of the algorithm in terms of number of measurements. Different types of uniformly distributed noise processes were added to plate position and/or orientation measurements (listed in Table 11).

Orientation measurement typically involves the processing of position measurements of multiple rigidly connected target points. Fitting a rotation matrix to a set of point measurements may introduce additional errors.

L e

Y a

0 1 2 3 4 5

Number of iterations

Figure 4. First leg’s rms parameter errors in measurement noise-free case, where the number of configurations = 7.


Table If. Measurement noise types. Noise type Orientation noise (rad) Position noise (mm)

I U[-O.ooOo2, O.ooOo21 No Noise 2 U[-0.0002, 0.00021 No Noise 3 No Noise U[-0.002, 0.0021 4 No Noise U[-0.02, 0.021 5 U[-0.0002, 0.00021 U[ -0.02, 0.021

Sets of 18 calibration configurations were randomly taken for those moving legs in a range of 1.2 2 0.125 m. Estimated parameters are all in the neighbor- hood of the “true” parameters when the number of measurements is larger than or equal to seven.

Thirty uniformly sampled poses, different from those used for identification, were used to verify the performance of the algorithm. The forward kinematic solution algorithm described in Section 4.1 was applied to produce Stewart platform poses using the true parameters and identified parameters. An estimated position error is defined in terms of the Euclidian norm of the difference between the true position and an estimated position using identified parameters. A similar definition is used for estimated orientation error. Figures 5 and 6 illustrate the verification results under different noise and measurement condi- tions.

By examining the simulation results, the following observations are made:

By calibration, platform position and orientation errors can both be significantly reduced if measurement noise is small. To reduce the effect of measurement noise, a sufficiently large number of measurement configurations must be taken. As a rule of thumb, about twice as many as the necessary number of configurations should be used for robust leg parameter estimation. Simulation results indicate that both orientation and position measurement uncertainties contribute significantly to estimation errors. Figure 5 shows that as the intensity level of measurement noise increases by a factor of 10 estimated orientation and position errors also roughly increase by the same factor.

5. CONCLUDING REMARKS

This article proposed an approach for calibration of Stewart platforms. With the proposed measurement and identification strategy, redundant leg length offsets of Stewart platform can be effectively eliminated, which paves the way for a kinematic calibration. The technique also allows the estimation of each leg parameter separately, by which the complexity of the corresponding identification algorithms is greatly reduced.

Simulation results have shown that to successfully calibrate a Stewart platform the accuracy of the pose measurement device must be at least 50 times

402

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Journal of Robotic Systems-1993

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better than the calibration accuracy. A major reason behind this phenomenon is that because each leg’s parameters are identified independently these parameters may not be optimal globally. Statistically, not all available information is used for the parameter identification. However, the parameters identified using the proposed method can serve as an initial condition for a more comprehen- sive optimization procedure such as the one given in ref. 14. Another limitation with the proposed method is that because the Stewart platform has to be moved in the joint level measurement configurations have to be carefully planned. This is due to the fact that, unlike serial manipulators, not all combinations of leg lengths have forward kinematic solutions. More study is needed to improve Stewart platform calibration efficiency, including optimal selection of platform calibration configurations and development of more suitable identification strat- egies. l4

There is an interesting duality between the method proposed in this article and the joint axis identification methods used for serial robot A common measurement strategy in these methods is to move or release the serial manipulator one joint at a time successively, fixing the other joints. In Stewart platform calibration, one fixes one leg at a time while moving all other joints.

This study focused only on geometric parameter identification of Stewart platform with ideal ball joints. More effort is needed in exploring singularities of the identification Jacobian and the effect of nongeometric errors on pose errors.

The authors express their sincere gratitude to Dr. Ming Z. Huang for fruitful discussions on Stewart platform kinematics.

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Method for kinematic calibration of stewart platforms

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