Performance analysis of cross coupled controllers for CNC machines based upon precise real time contour error measurement

Transcription

1 Performance analysis of cross coupled controllers for CNC machines based upon precise real time contour error measurement Jeremy R. Conway, Charlie A. Ernesto, Rida T. Farouki Department of Mechanical and Aerospace Engineering, University of California, Davis, CA 95616, USA. Mei Zhang Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing , CHINA. Abstract Experimental results from the implementation of a cross coupled control scheme on a 3 axis CNC mill governed by an open architecture software controller are presented. Unlike prior cross coupling schemes, which depend on osculating circle approximations to the commanded toolpath for real time contour error estimation, the proposed scheme is based on essentially exact contour error computations for free form curved paths. The implementation illustrates the feasibility of precise real time contour error computation with a modest (300 MHz) CPU and a 1024 Hz sampling frequency. For a P type controller and paths with strong curvature variation, the new scheme ensures much better diminution of contour error than earlier methods. The contour error is observed to decrease monotonically as the relative gain is increased, up to a critical gain value that incurs instability. For PI type controllers, the comparative contour error reduction is more modest, due to their effective suppression of steady state position error along the path.

2 Keywords: CNC machines, contour error, feedrate, high speed machining, open architecture controller, real time interpolator, control gain, instability. e mail addresses: jerconway@ucdavis.edu, caernesto@ucdavis.edu, farouki@ucdavis.edu, zhangm@amss.ac.cn

3 1 Introduction The axes of CNC machines are typically driven by independent servosystems, that compare the actual position of each axis (measured by encoders) with the desired position (computed by the real time interpolator from the prescribed path geometry and feedrate) in each servo sampling interval [1]. Positioning errors of the axes generally incur both geometrical deviations from the desired path (contour error) and temporal discrepancies along it (feed error). In most applications, suppression of contour error is more critical than feed error, to minimize dimensional inaccuracies of the machined part. It is now widely recognized [4, 5, 21, 22, 23, 29, 31, 32, 37, 38, 39, 40] that a cross coupled controller, in which the actuating signal for each machine axis is influenced by the behavior of the other axes, can offer significantly better reduction of contour error than autonomous controllers for each machine axis. Efficient and accurate algorithms for real time computation of contour error with respect to general curved paths are a key requirement for cross coupled controllers. Past studies have considered only contour error computation for simple (linear or circular) paths or, second order estimates (based on the osculating circle approximation) for general curved paths [22, 38]. The current availability of much faster processors and open architecture software for real time control of CNC machines permits more sophisticated (and essentially exact) contour error computation for general free form curved paths. This is a non trivial task, since precise computation of contour error for polynomial or rational curves involves a polynomial root finding problem that reflects the global geometry/topology of the toolpath. The goal of this paper is to demonstrate the feasibility of implementing algorithms to address this problem on CNC machines, and to illustrate the improvements in path tracking accuracy they offer, compared to independent axis controllers and cross coupled controllers based on the osculating circle approximation. Accurate tracking is most challenging when executing paths with severe curvature variations at extreme feedrates, a situation frequently encountered in high speed machining [8, 20, 28, 33]. The experiments reported here show that the improvements in tracking performance obtained through exact real time measurement of contour error, as compared to approximations based on the osculating circle, depend on the controller type. The sampling frequency f Hz (corresponding to a sampling interval t = 1/f sec.) is typically high enough for the osculating circle at reference point r k for time k t to closely approximate the commanded path in its vicinity. With a P type controller, 1

4 however, significant steady state axis position errors may develop, that cause the actual machine position p k at time k t to deviate appreciably from the commanded position r k. If the curvature varies rapidly along the toolpath, the osculating circle at r k gives inaccurate estimates of the true contour error (i.e., the distance of p k from the exact commanded path), due to the position error e k = r k p k. A PI type controller, on the other hand, can effectively subdue the steady state axis position errors, and the improvement obtained by using exact contour measurement is therefore less pronounced. The plan for this paper is as follows. In 2 the problem of contour error measurement along curved paths is reviewed, and an algorithm for precise computation of contour error with respect to free form polynomial or rational curves is briefly described. An implementation of this algorithm on a 3 axis CNC milling machine governed by an open architecture software controller is then discussed in 3. Experimental results from this implementation are presented in 4, in terms of diminution of the contour error and fidelity to the prescribed feedrate, using both P and PI type controllers. The dependence of the machine performance on the controller gains is also studied, up to the onset of dynamic instability. Finally, 5 summarizes key results of this paper, and identifies problems that deserve further attention. 2 Contour error measurement Let the desired path for a CNC machine be specified as a differentiable plane parametric curve 1 r(ξ) = (x(ξ), y(ξ)). Introducing a unit vector z orthogonal to the plane of r(ξ), we may define the unit tangent and normal vectors and the curvature at each point of r(ξ) by t(ξ) = r (ξ) r (ξ), n(ξ) = z t(ξ), κ(ξ) = [r (ξ) r (ξ) ] z r (ξ) 3. (1) The radius of curvature is defined by ρ(ξ) = 1/κ(ξ), and the osculating circle (circle of curvature) is the circle with radius ρ(ξ) and center c(ξ) = r(ξ) + ρ(ξ)n(ξ). Note that κ(ξ) and ρ(ξ) are signed quantities. According to (1), the normal n(ξ) points locally to the left of the curve r(ξ) as we traverse it in the direction 1 We consider only planar paths here: the extension to spatial paths is straightforward. 2

5 of increasing ξ. Then κ(ξ), ρ(ξ) are positive or negative according to whether n(ξ) points toward or away from the center of curvature c(ξ). Now suppose the CNC machine is to traverse the curve r(ξ) at a specified (constant or variable) feedrate V. The function of the real time interpolator algorithm is to compute the reference point parameter values ξ 1, ξ 2,... that define the commanded machine positions at times t, 2 t,... in accordance with the feedrate V. The parametric speed of the curve r(ξ) is defined by σ(ξ) = r (ξ) = x 2 (ξ) + y 2 (ξ) = ds dξ, (2) i.e., it is the derivative of the arc length s with respect to the parameter ξ. Since the feedrate is the derivative V = ds dt of arc length with respect to time, we have dξ dt = ds dξ dt ds = V σ. The role of the real time CNC interpolator is to determine a reference point (i.e., commanded machine position) within each sampling interval of the servo system. This amounts to computing the parameter values ξ k such that ξk 0 σ V dξ = k t, k = 1, 2,... Many algorithms have been developed to address this problem for example, see [6, 14, 15, 19, 25, 27, 36]. 2.1 Approximate contour error measurement Let r(ξ k ) = (x r, y r ) be the reference point computed at time t k = k t by the real time interpolator. At this point, the tangent and normal vectors may be expressed as t = (cos θ, sin θ) and n = ( sin θ, cosθ) where θ is the tangent angle, and the osculating circle C has radius ρ(ξ k ) and center c = (x c, y c ) = r(ξ k ) + ρ(ξ k )n(ξ k ) = (x r ρ sin θ, y r + ρ cosθ). (3) 3

6 Suppose p = (x p, y p ) is the actual machine position, as measured by the axis encoders, at time t k. The variable gain cross coupling controller introduced in [22] estimates the contour error i.e., the distance of p from r(ξ) by approximating r(ξ) with its osculating circle C at the reference point r(ξ k ). Namely, if R = ρ(ξ k ) is the magnitude of the radius of curvature at that point, the estimated contour error ε is defined by ε = p c R = (x p x c ) 2 + (y p y c ) 2 R. (4) Note that this is a signed quantity ε is the distance of p from the nearest point of C, and ε is negative or positive according to whether p lies inside or outside the osculating circle (see Figure 1). Now let e x = x r x p and e y = y r y p denote the axis position errors, and e = (e x, e y ) the position error vector. By writing (x p, y p ) = (x r e x, y r e y ) and using (3), the estimated contour error (4) can be formulated in terms of e x, e y and tangent angle θ (see Figure 1) as ε = (ρ sin θ e x ) 2 + (ρ cosθ + e y ) 2 R. It is customary to assume that e x, e y ρ. Invoking the approximation 1 + δ 1 + 1δ for δ 1, one can then write 2 [ (ρ sin θ ex ) 2 + (ρ cosθ + e y ) 2 R 1 + e y cosθ e x sin θ + e2 x + ] e2 y. ρ 2ρ 2 Noting that R/ρ = sign(ρ), the osculating circle estimate (4) of the contour error can then be cast as the quasi linear expression ε C x e x + C y e y in the axis position errors e x, e y. Here, the quantities [ C x = sign(ρ) sin θ e ] x, C y = sign(ρ) 2ρ [ cosθ + e ] y 2ρ are regarded [22] as variable gains dependent on the local path geometry and current axis position errors. This basic scheme for estimating contour error has subsequently been adopted by many authors [23, 37, 38, 40]. The estimated contour error vector ε at time t k is the displacement from the actual machine position p to the point of the osculating circle C closest to it namely, ε = (ε x, ε y ) = εu, where u = c p c p. (5) 4

7 c osculating circle r(ξ) R r(ξ k ) θ ε p e x e y Figure 1: Estimation of the distance of the point p from the curve r(ξ) using an osculating circle approximation at the current reference point r(ξ k ). The contour error ε is estimated as the distance of p from the osculating circle. Here u is a unit vector in the direction from the actual machine position p to the center of curvature c corresponding to the current reference point r(ξ k ). From (4) (5), the components of (5) are ε x = [ 1 R c p ] (x c x p ), ε y = [ 1 ] R (y c y p ). c p The above scheme for estimating contour error, based on approximating the curve by its osculating circle at the current reference point, was motivated by the need to obtain real time estimates using the relatively slow processors available [22] in the early 90s. For paths with moderate (and moderately varying) curvature, and controllers that ensure axis position errors of modest magnitude, it is capable of good results. When these conditions are not met, however, the contour error estimates may become inaccurate. Let t, n, κ, and κ be the tangent, normal, curvature and the arc length derivative of curvature at the reference point r(ξ k ). Then the displacement r corresponding to an arc length increment s may be expressed [30] as r = s t ( s)2 κn ( s)3 ( κn κ 2 t) + The osculating circle approximation at r(ξ k ) corresponds to retaining just the s and ( s) 2 terms in this expansion, while the ( s) 3 term defines the 5

8 lowest order deviation of the curve from the osculating circle. The relative magnitude of these terms is 1 6 ( s)3 ( κn κ 2 t) s t ( s)2 κn = κ 2 /κ 4 6 (κ s)2, (6) (κ s)2 4 and if κ 1/R and κ 1/RL, with R being a typical radius of curvature and L a typical length scale for its variation, this ratio can be written as 1 6 ( ) 2 s R 1 + (R/L) ( s/r)2. Here s can be regarded as a measure of the magnitude of the position error vector e = r(ξ k ) p. Thus, if s/r and R/L are not 1, the osculating circle at the reference point r(ξ k ) is a poor approximation to the curve r(ξ) in its vicinity. We now proceed to describe essentially exact algorithms for real time contour error computation along curved paths. Although the method is computationally more expensive, the implementation and results described in Sections 3 4 demonstrate that it is within the scope of modern processors and the sampling frequencies of typical CNC systems. 2.2 Precise contour error measurement As previously noted, use of the osculating circle approximation to estimate contour error can be inaccurate when the path r(ξ) has large and/or rapidly varying curvature, and when the actual machine position p = (x p, y p ) differs significantly from the current reference point r(ξ k ), at which the osculating circle is defined. These shortcomings may be circumvented by implementing point/curve distance function algorithms to exactly compute contour error an approach made viable through dramatic increases in processor speed for CNC systems, and increasing adoption of open architecture control software. In keeping with modern CAD systems, we assume that the path r(ξ) = (x(ξ), y(ξ)) to be executed by the CNC machine is a polynomial or rational curve defined on ξ [ 0, 1 ]. If p = (x p, y p ) is the actual machine position at any given instant, the exact contour error with respect to r(ξ) is defined by ε = distance(p, r(ξ)) = min p r(ξ) = min p r(ξ i), (7) ξ [0,1] 0 i N+1 6

9 where ξ 1,..., ξ N are the distinct (odd multiplicity) roots on ξ (0, 1) of F(x p, y p, ξ) = [ x p x(ξ) ] x (ξ) + [ y p y(ξ) ] y (ξ), (8) and we set ξ 0 = 0 and ξ N+1 = 1. When r(ξ) is a degree n polynomial curve, F(x p, y p, ξ) is of odd degree 2n 1 in ξ, and thus has at least one real root. Real roots of (8) identify points of r(ξ) where p lies on the normal line of the curve. As we traverse the curve, the value of p r(ξ) attains a local stationary value at ξ 1,...,ξ N since d dξ p r(ξ) = F(x p, y p, ξ) p r(ξ). We consider only odd multiplicity roots of (8), since even multiplicity roots identify values of p r(ξ) that are stationary but not extremal. The value of distance(p, r(ξ)) is the smallest of the interior extremal distances and the distances to the endpoints r(0) and r(1). To distinguish minima from maxima of p r(ξ), note that its second derivative may be written as where d 2 dξ p r(ξ) = F (x p, y p, ξ) 2 p r(ξ) F (x p, y p, ξ) = [p r(ξ) ] r (ξ) r (ξ) 2 F 2 (x p, y p, ξ) p r(ξ) 3, = [ x p X(ξ) ] X (ξ) + [ y p Y (ξ) ] Y (ξ) X 2 (ξ) Y 2 (ξ). Since F(x p, y p, ξ i ) = 0 for i = 1,...,N we note that ξ i identifies a minimum of p r(ξ), with a positive second derivative, if F (x p, y p, ξ i ) < 0 i.e., [ x p X(ξ i ) ] X (ξ i ) + [ y p Y (ξ i ) ] Y (ξ i ) < X 2 (ξ i ) + Y 2 (ξ i ). When the minimum in (7) is realized for i = m, we call r(ξ m ) a footpoint of p on the curve r(ξ). If 1 m N it is an interior footpoint, while if m = 0 or N + 1 it is a terminal footpoint. Usually the footpoint is unique, but for certain special locations of p there may be more than one footpoint. For a generic position of p, the identity of the root ξ m that realizes the minimum in (7) is preserved as p moves relative to the curve r(ξ), within some neighborhood of that position i.e., ξ m has an analytic dependence on 7

10 the location of p within some neighborhood. 2 Hence, as p moves, the value of ξ m can be updated by means of a numerical analytic continuation (e.g., predictor corrector) method. As p passes through certain special locations, however, the identity of the root yielding the minimum in (7) may change, and it cannot be updated by analytic continuation. These critical locations of p may be categorized as follows: (a) When p crosses the normal line to r(ξ) at ξ = 0 or 1, a real root of (8) appears or disappears on ξ [ 0, 1 ] from ξ < 0 or ξ > 1. (b) Locations of p that coincide with the center of curvature for the current footpoint incur a change in the number of real roots of (8) through the appearance/disappearance of complex conjugate root pairs. (c) Locations of p may have non unique footpoints these are points for which the minimum in (7) is realized by at least two distinct real roots ξ i, ξ j [ 0, 1 ] of (8) with p r(ξ i ) = p r(ξ j ), and they correspond to points p on the self bisector (or medial axis) of r(ξ). Figure 2: Self bisector (left) and evolute (right) for the cubic r(ξ) = (ξ, ξ 3 ). Figure 2 illustrates the self bisector and evolute i.e., locus of centers of curvature for the simple cubic r(ξ) = (ξ, ξ 3 ). For higher order curves, these loci of critical locations for the point p relative to the curve r(ξ) are much more complicated, and in general they admit no simple (i.e., rational) 2 This follows from the implicit function theorem [2] of calculus. 8

11 parameterization. In principle, using an analytic continuation procedure to update the footpoint parameter ξ m as p moves is the most efficient approach to computing the point/curve distance function, but the need to constantly check for coincidence of p with the self bisector of r(ξ) or with the center of curvature for the current footpoint greatly complicates this task. A simpler though less efficient alternative is to continuously track all (real and complex) roots of (8), and select the real root yielding the smallest value of p r(ξ). This approach is much easier to implement and, although less efficient, is still amenable to use in real time motion control when appropriate iterative methods are adopted. To track all the roots ξ 1,...,ξ 2n 1 of (8) as p moves in discrete steps p from some initial point p 0 = (x 0, y 0 ), we require initial values for these roots, when p = p 0 i.e., we need to determine all (real and complex) roots of the polynomial Q(ξ) = F(x 0, y 0, ξ) (9) of degree 2n 1. In the present context, methods that isolate each root of (9) within a complex plane rectangle [3, 7, 35] are most suitable these are based on the ability to count the number of roots in complex rectangles using the principle of the argument and Sturm sequences. Once the roots of (9) have been sufficiently isolated, iterative methods can be invoked to determine each root to machine precision, using the rectangle center as a starting value. We choose Laguerre s method [16, 17, 18, 24, 26] since it exhibits faster (cubic) and more robust convergence than Newton s method. For an initial approximation ξ (0) to a root of a degree m polynomial f(ξ), the Laguerre iteration is defined by ξ (r+1) = ξ (r) mf(ξ (r) ) f (ξ (r) ) ±, r = 0, 1, 2,... (10) g(ξ (r) ) where we set g(ξ) = (m 1)[ (m 1)f 2 (ξ) mf(ξ)f (ξ) ], and the sign in (10) is chosen to minimize ξ (r+1) ξ (r). Although Laguerre s method also uses the second derivative of f(ξ) and a square root extraction, its faster and more robust convergence easily compensates for the additional computational cost per iteration, as compared to Newton s method. From the initial values, we need an efficient means to update the roots of (8) as p moves in small steps p = ( x p, y p ). This process is facilitated 9

12 by the fact that, for small p, the roots for location p are excellent starting approximations to the roots for location p + p. We again use Laguerre s method for updating the roots (all roots are regarded as complex values in this method, the real roots corresponding to those with zero imaginary part). To compute the updated roots we identify f(ξ) with F(x p + x p, y p + y p, ξ), and as the starting values for the Laguerre iterations, we use the converged roots of F(x p, y p, ξ) from the previous location of p. Typically, Laguerre s method gives convergence to machine precision in one or two iterations for steps p of modest size. Once the roots corresponding to p+ p are known, p + p r(ξ) is evaluated for each real root on ξ (0, 1) and at ξ = 0, 1 the smallest of these values then determines distance(p + p,r(ξ)). Knowing the footpoint parameter value ξ m, the contour error vector may be defined as ε = r(ξ m ) p. Note that ε is nominally a non negative quantity, but we can convert it into a signed quantity by associating with it the sign of n ε, where n is the curve normal at the footpoint r(ξ m ). For a cubic curve r(ξ), the polynomial (8) has five roots that must be simultaneously tracked, while for a quintic it has nine. Note that computing the roots of (9) for the initial position p 0 = (x 0, y 0 ) can be done off line (if a single curve is to be executed). The key advantage of this approach is its ease of implementation since all roots of (8) are tracked, no consideration of critical locations of p relative to r(ξ) is necessary. To minimize the effect of floating point rounding errors, all polynomial computations are performed in the numerically stable Bernstein basis [10, 12, 13, 34] on ξ [ 0, 1 ] Figure 3: Left: a cubic Bézier curve r(ξ), together with sample positions for the moving point p. Right: the variation of the function distance(p,r(ξ)). Figure 3 shows a cubic r(ξ) with sample positions for p. Also shown is the variation of the distance of p from r(ξ), as determined using Laguerre 10

13 iterations to compute all five roots of (8). A tolerance of on the value of (8) required at most two iterations for convergence at each step Figure 4: Left: a quintic Bézier curve r(ξ) with a sampling of positions for the moving point p. Right: the point/curve distance function, distance(p,r(ξ)). Figure 4 shows an example using a quintic curve r(ξ) together with the graph of the distance of p from r(ξ), determined using Laguerre iterations to compute all nine roots of (8). Again, a tolerance of on the value of (8) was used, and at most two iterations yielded convergence at each step. Im Im Re Re Figure 5: Variation of the roots of (8) in the complex plane with the motion of the point p for the curves shown in Figures 3 and 4. In the former case, there is always a unique real root that identifies the footpoint of p on r(ξ). In the latter case, the number of real roots alternates between three and one as complex conjugate roots emerge from or enter the real axis five times. Figure 5 shows the variation of the roots of (8) in the complex plane as 11

14 p moves for the curves shown in Figures 3 and 4. In the former case, there is no change in the identity of the root of (8) specifying the footpoint of p on r(ξ), since there is always a unique real root. In the latter case, there are initially three real roots, and three complex conjugate pairs. As p moves, the number of real roots alternates between three and one a total of five times, as a pair of real roots meet and become complex conjugates, and vice versa. 3 Implementation on CNC mill The cross coupled controller algorithms were implemented on a 3 axis CNC milling machine manufactured by MHO Corporation of Oakland, California. The three machine axes are powered by independent Yaskawa servomotors driving precision ground ball screws. The mill is controlled by the OpenCNC software, developed by MDSI, Inc. of Ann Arbor, Michigan, running on an off the shelf PC with a modest (300 MHz) CPU. OpenCNC provides variable dictionaries and interface specifications that facilitate modification of existing functions in the control software, or the insertion of user developed functions. The sampling frequency of the controller is f = 1024 Hz. During each run, real time position encoder data is stored in memory for subsequent analysis of the machine performance. From the position encoder data and the known exact path geometry, it is possible to compute the actual machine contour error in each sampling interval t = 1/f s. Real time data on axis velocities and accelerations may also be computed by first and second order differencing of the position data, and the feedrate along the path can be determined as the magnitude of the velocity vector v whose components are specified by the individual axis velocities (v x, v y ). Figure 6 illustrates the cross coupled control scheme for a single machine axis (the x axis). The actuating signal u x determines the actual position x p through the axis dynamics transfer function G(s). The axis position error is the difference e x = x r x p between the reference position x r and the actual position x p measured by the axis encoder, while ε x is the x component of the contour error vector ε (i.e., the vector from the machine location p to its footpoint on the commanded path r(ξ)). The actuating signal u x is the sum of e x and ε x, as modulated by individual transfer functions H e (s) and H ε (s). The transfer functions are as follows: H e (s) = K p [ 1 + K i s ], H ε (s) = k cc H e (s), G(s) = 12 K (Js + B)s. (11)

15 x r + Σ e x ε x H e (s) H ε (s) + + Σ u x G(s) x p Figure 6: Schematic block diagram for cross coupled control of a single CNC machine axis (the x axis) see the text for an explanation of the symbols. H e (s) is a basic PI controller acting on the axis position error e x. The transfer function H ε (s) acting on the x component of the contour error may differ in form, but we choose to make it identical except for a scale factor k cc, which we call the relative gain. The axis dynamics transfer function G(s) depends on the inertia J and damping B, while K is an overall gain determined by the current amplifier and motor torque gains, and the axis transmission ratio (i.e., the axis translation per unit rotation of the motor shaft). The nominal machine controller employs only proportional control with the rather high gain K p = 40, 000 chosen to reduce the axis position errors as much as possible, without verging on instability. With this high gain, the contour errors are typically quite small and difficult to accurately measure. In the experiments reported below, we adopt the smaller gain K p = 5, 000 to permit higher, and thus more easily measured, contour errors. 4 Experimental results Figure 7 shows the curve r(ξ), ξ [ 0, 1 ] used in the controller performance tests. This curve is a Pythagorean hodograph (PH) quintic interpolant [11] to the first order Hermite data r(0) = (2.0, 2.0), r(1) = (13.2, 3.6), r (0) = (48, 40), r (1) = (40, 48), where coordinates are in inches. In terms of the complex model [9] for planar PH curves, the hodograph r (ξ) = [w 0 (1 ξ) 2 + w 1 2(1 ξ)ξ + w 2 ξ 2 ] 2 13

16 of this curve is the square of the complex quadratic polynomial with Bernstein coefficients w 0 = i, w 1 = i, w 2 = i. Hence, the parametric speed (2) is the quartic with Bernstein coefficients σ 0 = w 0 2, σ 1 = Re(w 0 w 1 ), σ 2 = 1 3 [ 2 w Re(w 2 w 0 ) ], σ 3 = Re(w 1 w 2 ), σ 4 = w 2 2, and the arc length s(ξ) is the quintic with Bernstein coefficients s 0 = 0 and s k = 1 k 1 σ j for k = 1,..., 5. 5 The total curve arc length is then S = s(1) = s 5 = in. The variation of the parametric speed σ and curvature κ with the arc length s along the curve is shown in Figure 8. Because it exhibits a region of strong curvature (and extreme curvature variation), the test curve presents a severe challenge to accurate contour error estimation. The constant feedrate V = 500 in/min. was employed in all the experiments. Figure 9 illustrates the measured contour error when the curve in Figure 7 is traversed at 500 in/min using P type (K p = 5000, K i = 0) cross coupled controllers that employ the osculating circle approximation of contour error (Section 2.1) and the exact contour error measure (Section 2.2), with values k cc = 0, 1, 2, 4, 8 for the relative gain in (11). For k cc = 0 (no cross coupling) the two methods give identical results, with a peak contour error in. As k cc is increased, however, they exhibit markedly different behavior. For the exact contour error method, the measured peak contour error decreases monotonically to in, in, in, in when k cc = 1, 2, 4, 8 (with k cc = 8 the controller is on the verge of instability). For the osculating circle approximation, increasing k cc to 1 incurs an initial reduction of the peak contour error from in to in, but no further systematic diminution is observed for higher values: the observed peak contour errors for k cc = 2, 4, 8 are in, in, in. This phenomenon is explained below. 14 j=0

17 y (in) x (in) Figure 7: PH quintic test curve used in cross coupled controller experiments parametric speed σ curvature κ arc length s (in) arc length s (in) Figure 8: Variation of the parametric speed σ (left) and curvature κ (right) with the arc length s, along the PH quintic test curve illustrated in Figure 7. 15

18 k cc = 8 contour error (in) k cc = k cc = contour error (in) k cc = time (sec) Figure 9: Measured contour error in executing the test curve in Figure 7 at feedrate V = 500 in/min with cross coupled P type controllers based on the exact contour error (upper) and an osculating circle contour error estimation (lower). Results for the relative gain values k cc = 0, 1, 2, 4, 8 are shown. Note that the case k cc = 0 corresponds to no cross coupling of the axes, while for k cc > 8 there are signs of incipient dynamic instability. 16

19 Figure 10 illustrates the feed errors corresponding to the contour errors in Figure 9. If p, r(ξ k ), and ε are the current machine position, reference point, and contour error, the magnitude of the feed error is r(ξ k ) p 2 ε 2, and it is considered to be positive/negative according to whether the dot product [r(ξ k ) p ] r (ξ k ) is positive/negative (the feed error and contour error are regarded as tangential and normal components of the position error vector). The two methods yield similar results, although the osculating circle estimation incurs a slightly more pronounced variation with k cc. A feed error of 0.3 in is apparent, consistent with the fact that P control does not ensure steady state error suppression. This is relatively unimportant with the exact contour error method, but for the osculating circle method it implies that the circle of curvature at a point r(ξ k ) differing substantially from the machine position p will be used to approximate the curve r(ξ), yielding an inaccurate estimate of contour error when the curvature is varying rapidly. Figure 11 illustrates, for s 0.3 in, the ratio (6) that characterizes the deviation of r(ξ) from its osculating circle. In the vicinity of the sharp turn in the curve (see Figure 7), this ratio is evidently not small compared to unity, indicating that the osculating circle becomes a rather poor approximation to the path r(ξ) for the steady state feed error s 0.3 in incurred by the P type controller. This largely explains the poorer contour error performance of the controller that employs the osculating circle estimation, rather than the exact contour error computation, evident in Figure 9. The feedrate achieved by the machine can be determined by differencing the data obtained from real time position encoders. The measured feedrate for the two methods is illustrated in Figure 12. Both controllers evidently encounter some difficulty maintaining the commanded 500 in/min feedrate near the sharp turn in the path (see Figure 7), but the performance of the exact contour error method is clearly superior to that of the osculating circle estimation, especially as the relative gain k cc is increased. As noted above, when using a P type controller, the observed difference in performance between the methods based on exact contour error measurement and the osculating circle approximation is primarily due to the steady state position error, which causes the latter approach to use inappropriate circles of curvature when the curvature varies rapidly. The experiments were therefore repeated using a PI (K p = 5000, K i = 50) cross coupled controller, again with relative gains k cc = 0, 1, 2, 4, 8 and feedrate V = 500 in/min. Figure 13 shows the measured contour error for both methods. As distinct from the results for the P type controller in Figure 9, the PI type controller 17

20 k cc = feed error (in) k cc = k cc = feed error (in) k cc = time (sec) Figure 10: Measured feed error in executing the test curve at 500 in/min with cross coupled P type controllers based on exact contour error (upper) and the osculating circle approximation for the contour error (lower). Results for the relative gain values k cc = 0, 1, 2, 4, 8 are shown. 18

21 0.3 deviation from osculating circle arc length s (in) Figure 11: Variation of the ratio (6), that characterizes the deviation of the osculating circle from the commanded path r(ξ), for the steady state feed error s 0.3 in observed in Figure 10. The fact that this ratio is not small compared to unity near the sharp turn of the curve in Figure 7 indicates that the osculating circle becomes a poor approximation to r(ξ) in this region. yields measured contour errors for the two methods that (at each k cc value) are virtually indistinguishable. Comparing Figures 9 and 13, we see that for the P and PI controllers the maximum magnitude contour error is negative and positive i.e., to the right and the left of the curve respectively. Also, the maximum contour error in the k cc = 0 (no cross-coupling) case with the PI controller is in, compared to in for the P controller. For k cc = 1, 2, 4, 8 the peak contour error magnitudes with the PI controller are in, in, in, in these values are slightly higher than for the P controller (using the exact contour error method). Figure 14 shows the feed errors obtained with the PI controller. As with the contour errors in Figure 13, the PI controller gives measured feed errors for the two methods that are remarkably similar. The steady state feed error is nearly zero except for transient effects associated with the initial start up and the sharp turn in the curve (see Figure 7), as expected for a PI controller. The large initial feed error at start up is unimportant, because of the low initial curvature. Figure 15 shows, for the peak feed error s 0.1 in near the sharp turn in the curve, the ratio (6) characterizing the deviation of r(ξ) from its osculating circle. It is seen that this ratio is always small compared to unity, even near the sharp turn of the curve (compare with Figure 11), 19

22 500 k cc = k cc = 4 feedrate (in/min) k cc = 2 k cc = k cc = time (sec) time (sec) Figure 12: Comparison of measured feedrate, for a specified constant value V = 500 in/min, with cross coupled P type controllers using exact contour error measurement (left) and the osculating circle contour error estimation (right). Results for the values k cc = 0, 1, 2, 4, 8 of the relative gain are shown, with successive plots displaced vertically for clarity. The feedrate is obtained by differencing of the real time encoder position data, incurring some noise. 20

23 k cc = contour error (in) k cc = k cc = contour error (in) k cc = time (sec) Figure 13: Measured contour error in executing the test curve at 500 in/min with cross coupled PI type controllers based on exact contour error (upper) and the osculating circle contour error estimation of contour error (lower). Results for the relative gain values k cc = 0, 1, 2, 4, 8 are shown. 21

24 feed error (in) k cc = k cc = feed error (in) k cc = 0 k cc = time (sec) Figure 14: Measured feed error in executing the test curve at 500 in/min with cross coupled PI type controllers based on exact contour error (upper) and the osculating circle approximation for the contour error (lower). Results for the relative gain values k cc = 0, 1, 2, 4, 8 are shown. 22

25 and hence the osculating circle remains a good approximation to r(ξ). This largely explains the disparate observed behavior of the cross coupled P and PI controllers, using the two methods of contour error computation. 0.3 deviation from osculating circle arc length s (in) Figure 15: Variation of the ratio (6) describing the deviation of the osculating circle from the commanded path r(ξ), for the feed error s 0.1 in observed in Figure 14, corresponding to the sharp turn of the curve in Figure 7. The fact that this ratio is always small compared to unity guarantees that the osculating circle remains a good approximation to r(ξ) in this region. Finally, Figure 16 shows the feedrates obtained with the PI controller. Again, only minor differences between the two methods, based on the exact and approximate contour error computation, are apparent. Note that the PI controller incurs large feedrate fluctuations at start up and on encountering the sharp turn, compared to the P controller (see Figure 12). For a constant spindle speed, these feedrate fluctuations incur corresponding variations in chip load, which may cause chatter or breakage of the cutting tool. Hence, the P type controller combined with the exact contour error measurement is in many respects preferable for optimum smooth tracking performance. 5 Closure Cross coupled controllers for CNC machines use the contour error (the actual instantaneous distance of the machine position from the commanded path), in addition to individual axis errors, to optimize the machine tracking accuracy. 23

26 500 k cc = k cc = 4 feedrate (in/min) k cc = 2 k cc = k cc = time (sec) time (sec) Figure 16: Comparison of measured feedrate, for a specified constant value V = 500 in/min, with cross coupled PI type controllers using exact contour error measurement (left) and the osculating circle contour error estimation (right). Results for the values k cc = 0, 1, 2, 4, 8 of the relative gain are shown. 24

27 Typically, the contour error for general curved paths has been estimated by using the osculating circle approximation about the current reference point. This study has demonstrated the feasibility of exact real time computation of contour error for general curved paths, through a polynomial root tracking algorithm, on CNC machines with representative servo sampling frequencies and relatively modest processor speeds. Experiments were conducted by an implementation of the cross coupled control schemes using both approximate and exact contour error meaures on an open architecture CNC machine, to assess their comparative performance under P and PI controllers, with increasing values for the cross coupling gain relative to that of the individual axis position errors. The exact contour error measure was found to offer substantially improved performance under one or more of the following circumstances (1) strong curvature of the path; (2) strong variation of path curvature; and (3) significant steady state position errors. In particular, circumstance (3) typically associated with a P type controller using relatively low gain at high feedrates causes the traditional osculating circle approach of contour error estimation to produce inaccurate results. With a PI controller, the steady state error is effectively suppressed and the difference between the approximate and exact contour error methods is much less pronounced. However, the PI control incurs much larger feedrate fluctuations in high curvature regions, and the actual contour error is slightly larger than with the P controller based on the exact measure. The choice of controller type and control gains for optimum performance without sacrificing control stability of cross coupled schemes deserves further attention, since the experiments reveal behavior that is quite sensitive to these choices. Moreover, real machines obviously have axes with different dynamic characteristics, which the cross coupled controller should take into account for optimum performance. Another important but difficult extension of the cross coupled control paradigm is to 5 axis motions of CNC machines, involving the characterization of deviations from spatial motions combining translational and rotational components. References [1] Y. Altintas (2000), Manufacturing Automation: Metal Cutting Mechanics, Machine Tool Vibrations, and CNC Design, Cambridge University Press. 25

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