CMM Kinematic errors model solution

2.3.1 Introduction

Considering a CMM with perfect geometry, the scale reading on each of the three machine axis will be considered as accurate for the coordinate measurement. However, in practice, the most improbable is that CMMs with perfect geometry in manufacturing or production line. CMMs, in practice have imperfect geometry in the following regards

 Straightness of each axis  Squareness of pairs of axes

 Rotation describing the pitch, roll and yaw

It is vital these that these errors, also called systematic or parametric errors are taken in to account for in order to fully realise the accuracy potential of CMMs. Most manufacturers of CMMs ensure their CMM machines undergo a detailed and comprehensive calibration procedure as a final check before delivering them to their customers. On the customer site, an acceptance test is conducted to the customer by the manufacturer to show that the CMMs performance is within the international or national specification. Moreover, these standards include clauses that ensure other tests are conducted at certain intervals to maintain the CMM conformance to the specification. The 'other' tests are of two types:

Periodic re-verification test – This requires repeating the acceptance test yearly with the goal of identifying and compensating for the drift in CMM performance from the first acceptance testing. The re-verification measurement task is usually different from the routine measurement made with CMM.

Interim Checking - This is done between formal periodic re-verification test with the goal of ensuring that the CMM measurements are accurate, thereby giving the user confidence in the measurement results. The degree of measurement complexity varies between user and application of CMM machine. For some machine few test is carried to maintain measurement confidence while some require a complete re-calibration to maintain measurement confidence. NPL (1998) and Cox, Forbes and Peggs, (1997) have

30

classified the different level of checking on the degree of confidence which CMMs needed to be checked. Table 2.2 shows the listing.

Table 2.3 Showing classified level of the CMM degree of confidence (NPL 1998).

Class Degree of confidence

Class 1 Verification of all degrees of freedom of the CMM accounting for any influence

of temperature variations

Class 2 Verification of the entire working volume of the CMM using a standard artefact

such as a ball- or hole-plate (a two-dimensional plate with regularly spaced features, usually holes or spheres);

Class 3 Verification of the space envelope of interest using a master artefact representing

the work piece features measured in the user's usual application;

Class 4 Verification of just the working volume of interest using standard artefacts, for

example, length bars

2.3.2 CMM kinematic error model and capabilities

This is based on the superposition of behaviours from the three coordinate axes with each describe as a function of the corresponding reading scale. For example, six functions of x could be used in describing the behaviour of a stylus assembly with the reading scale of y and z kept fixed. The six functions comprise of three stating the angle of rotation that describe the pitch roll and yaw of x axis and three giving the location of the fixed point on the housing. Similarly, keeping x and z fixed six function are generated for y axis and keeping, keeping x and y, six functions for z-axis. Kinematic model is used to describe the machine in terms of the six functions of each axis. Considering the x axis, which could be describe by

[ 𝑥𝑥 𝑦_𝑥 𝑧_𝑥] = [ 𝑥(1,0,0)𝑇_{+ 𝛿} 𝑥𝑥(𝑥, 𝑎) 𝛿_𝑥𝑦(𝑥, 𝑎) 𝛿𝑥𝑧(𝑥, 𝑎) ] (2.1)

31

and the orientation is described by a rotation matrix Rx(x, r) defined as a product of plane rotations. The (𝛿𝑥𝑥) term is sometimes described as the x-axis scale error while (𝛿𝑥𝑦) and (𝛿𝑥𝑧) give the x-axis straightness errors. Similarly, the location and orientation along the y-axis is described by

[ 𝑥_𝑦 𝑦_𝑦 𝑧_𝑦] = [ 𝛿𝑦𝑥(𝑦, 𝑏) 𝑦(0,1,0)𝑇_{+ 𝛿} 𝑦𝑦(𝑦, 𝑏) 𝛿_𝑦𝑧(𝑦, 𝑏) ] (2.2)

and R(y, s). This information can be combined to provide an estimate of location as a function of x and y

𝑥𝑥𝑦 = 𝑥𝑥(𝑥 , 𝑎 ) + 𝑅𝑥 (𝑥, 𝑟 ) 𝑥𝑦 (𝑦 , 𝑏 ) (2.3)

This formulation supposes that the y-motion depends on (or is carried by) the x- motion and moreover that at x = y = 0 the axes are orthogonal and the two rotation matrices Rx and Ry are aligned. Adding the z-axis (assumed to be carried by the y-axis) and the probe offset under similar assumptions, the model takes the form

𝑥 =

𝑥_𝑥(𝑥 , 𝑎 ) + 𝑅𝑥 (𝑥, 𝑟 )𝑥𝑦 (𝑦, 𝑏 ) + 𝑅𝑥 (𝑥, 𝑟 )𝑅𝑦 (𝑦, 𝑠 )𝑥𝑧(𝑧, 𝑐) + 𝑅𝑥 (𝑥, 𝑟 )𝑅𝑦 (𝑧, 𝑡 )𝑃 (2.4)

Thus the kinematic model is specified in terms of 18 individual error functions, three positional and three rotational for each of the three axes. The basic components are therefore functions of a single variable and these can be specified using polynomials or splines, for example. The correct formulation depends on the architecture of the CMM, but the common CMM designs are covered by the model; see, for example, Zhang et al. (1988). If all three rotation matrices are set to the identity matrix, then x* is given by

𝑥∗ _{= 𝑥 + 𝑃}

𝑥 + [𝛿𝑥𝑥(𝑥) + 𝛿𝑦𝑥(𝑦) + 𝛿𝑧𝑥(𝑧)] (2.5)

Where the actual position of the probe in x-axis is 𝑃_𝑥 and similarly for y* and z*, showing that the positional correction for x includes a sum of functions of x, y and z. If each of these functions includes a constant term, there will be degrees of freedom in the model which cannot be determined from the data. The simplest way to resolve this is to

32

constrain 𝛿_𝑦𝑥(0) = 𝛿_𝑥𝑥(0) = 0. In fact, similar constraints have to be placed on all six types of component as will become clear when we consider a linearized version of the kinematic model. If we evaluate the kinematic model at points along each axis we obtain 𝑥∗_{([𝑥, 0,0]}𝑇_{) = 𝑥} 𝑥(𝑥, 𝑎) + 𝑅𝑥(𝑥, 𝑟)𝑃 (2.6) 𝑥∗_{([0, 𝑦, 0]}𝑇_{) = 𝑥} 𝑦(𝑦, 𝑏) + 𝑅𝑥(𝑦, 𝑠)𝑃 (2.7) 𝑥∗_{([0,0, 𝑧]}𝑇_{) = 𝑥} 𝑧(𝑧, 𝑐) + 𝑅𝑧(𝑧, 𝑡)𝑃 (2.8)

showing that each of the three sets of parameters can be determined by measurements along the three axes. Thus, a measurement strategy which includes measurements along or parallel to each of the three axes using three or more probe offsets will be sufficient to determine the model parameters, assuming that there are enough measurements along each axis. Often the kinematic model is used not as given above but in a simpler, linearized form. Employing the linearization, then R is given by:

𝑅̃_𝑥𝑦𝑧 = R_xR_yR_z = [− 1 (wx + wy+ wz) −(vx + vy+ vz) (wx + wy+ wz) 1 (ux + uy+ uz) (v_x + v_y+ v_z) −(u_x + u_y+ u_z) 1 ] (2.9) R_xx_y = [ δ_yx(y) + w_x (x)y δyy(y) + y δ_yz(y) − ux(x)y ] (2.10) R_xR_yx_z = [ δ_zx(z) − (v_x (x) + v_y (y)z) δzy(z) + (ux (x) + uy (y)z) δ_zz(y) + z ] (2.11)

33 𝑥∗ _{= 𝑥 + 𝑃} 𝑥 + [𝛿𝑥𝑥(𝑥) + 𝛿𝑦𝑥(𝑦) + 𝛿𝑧𝑥(𝑧)] + 𝑤𝑥 (𝑥)𝑦 − (𝑣𝑥 (𝑥) + 𝑣𝑦 (𝑦)) 𝑧 + (𝑤𝑥(𝑥) + 𝑤𝑦(𝑦) + 𝑤𝑧 ( 𝑧)) 𝑃𝑦 − (𝑣𝑥(𝑥) + 𝑣𝑦(𝑦) + 𝑣𝑧 (𝑧)) 𝑃𝑧 (2.12) 𝑦∗ _{= 𝑦 + 𝑃} 𝑦 + [𝛿𝑥𝑦(𝑥) + 𝛿𝑦𝑦(𝑦) + 𝛿𝑧𝑦(𝑧)] + (𝑢𝑥 (𝑥) + 𝑢𝑦 (𝑦)) 𝑧 − (𝑤_𝑥(𝑥) + 𝑤𝑦(𝑦) + 𝑤𝑧 ( 𝑧)) 𝑃𝑥− (𝑢𝑥(𝑥) + 𝑢𝑦(𝑦) + 𝑢𝑧 (𝑧)) 𝑃𝑧 (2.13) 𝑧∗ _{= 𝑧 + 𝑃} 𝑧 + [𝛿𝑥𝑧(𝑥) + 𝛿𝑦𝑧(𝑦) + 𝛿𝑧𝑧(𝑧)] − 𝑢𝑥 (𝑥)𝑦 + (𝑣𝑥 (𝑥) + 𝑣𝑦 (𝑦) + + 𝑣𝑧 (𝑧)) 𝑃𝑥− (𝑢𝑥(𝑥) + 𝑢𝑦(𝑦) + 𝑢𝑧 (𝑧)) 𝑃𝑦 (2.14)

where x* is the positional correction for x, y* is the positional correction for y, and z* is the positional correction for z.