Equivalent-plane cross-coupling control method

Abstract

The invented equivalent-plane cross-coupling control method belongs to high-precision and high-efficiency intelligent multi-axis CNC (Computer Numerical Control) machining filed, featured a three-axis cross-coupling controller based on the equivalent plane which can be used for improvement of the three-dimensional contour-following accuracy. This method first find the foot point from the actual motion position to the desired contour using a tangential back stepping based Newton method. Then, establish an equivalent plane which containing the spatial contouring-error vector by passing through the actual motion position and the tangential line at the foot point. After that, estimate the three-dimensional contouring error in a scalar form, thus controlling the signed error using a PID based two-axis cross-coupling controller. Finally, calculate the three-axis control signals according to the geometry of the equivalent plane, thus realizing the three-axis contouring-error control by using the well-studied two-axis contour controllers.

Claims

1. An equivalent-plane cross-coupling control method, approximating to a foot point from actual motion position to a desired contour using a tangential back stepping based Newton algorithm, and establishing an equivalent plane where a contouring-error scalar instead of a contouring error vector can be obtained so that a proportional-integral-differential (PID) controller based two-axis cross-coupling controller (CCC) can be utilized to constrain a spatial contouring error, thus improving contour-following accuracy of three-axis computer numerical control (CNC) systems; wherein the method is as follows: Step One: establishment of the equivalent plane; (i) denoting an equation of a desired contour as C=C(u), where u denotes curve parameter; (ii) denoting desired and actual motion positions R=[r.sub.x,r.sub.y,r.sub.z] and P=[p.sub.x, p.sub.y, p.sub.z], respectively; (iii) denoting u.sub.r as a parameter corresponding to the desired motion position; and (iv) defining a tangential error d.sub.t(u) on a point of the desired contour C(u) as a projection of vector C(u)P on the tangential direction at the position of C(u), and being computed as: $\begin{array}{cc}{d}_t\left(u\right)=\left(C\left(u\right)-P\right).Math.\frac{C^{}\left(u\right)}{.Math.C^{}\left(u\right).Math.}& \left(1\right)\end{array}$ where C(u) is the first-order derivative of C(u) with respect to u, and means the Euclidean norm; where contouring error is defined as the orthotropic distance from the actual motion position to the desired contour; therefore, the tangential error d.sub.t(u) must be zero when C(u) is the foot point from P to the desired contour; accordingly, foot-point parameter u.sub.f is obtained by solving d.sub.t(u)=0; where tangential back-stepping point parameter u.sub.b is first calculated by projecting the tangential error to the desired contour as: $\begin{array}{cc}{u}_b=u_r-\frac{\left(C\left(u_r\right)-P_p\right).Math.C^{}\left(u_r\right)}{{.Math.C^{}\left(u_r\right).Math.}^2}& \left(2\right)\end{array}$ then, the tangential back-stepping point parameter u.sub.b is taken as the initial value of the Newton method so as to find the solution u.sub.N of d.sub.t(u)=0 by: $\begin{array}{cc}{u}_N=u_b-\frac{\left({.Math.C^{}\left(u_b\right).Math.}^2.Math.\left(C\left(u_b\right)-P\right).Math.C^{}\left(u_b\right)\right)}{\left(\begin{array}{c}{.Math.C^{}\left(u_b\right).Math.}^4+{.Math.C^{}\left(u_b\right).Math.}^2.Math.\left(C\left(u_b\right)-P\right).Math.C^{}\left(u_b\right)-\\ \left(\left(C\left(u_b\right)-P\right).Math.C^{}\left(u_b\right)\right).Math.\left(C^{}\left(u_b\right).Math.C^{}\left(u_b\right)\right)\end{array}\right)}& \left(3\right)\end{array}$ and if |d.sub.t(u.sub.N)|<|d.sub.t(u.sub.b)|, thus indicating that the tangential back stepping based Newton algorithm is convergent, taking the solution as foot-point parameter u.sub.f where u.sub.f=u.sub.N; otherwise, applying the tangential back stepping based Newton algorithm again at u.sub.b to obtain the foot-point parameter u.sub.f; to summarize, the foot-point parameter u.sub.f is calculated by the following equation: $\begin{array}{cc}{u}_f=\{\begin{array}{c}{u}_N,.Math.d_t\left(u_N\right).Math.<.Math.d_t\left(u_b\right).Math.\\ u_b-\frac{\left(C\left(u_b\right)-P\right).Math.C^{}\left(u_b\right)}{{.Math.C^{}\left(u_b\right).Math.}^2},.Math.d_t\left(u_N\right).Math..Math.d_t\left(u_b\right).Math.\end{array}& \left(4\right)\end{array}$ establishing an equivalent plane by passing through the actual motion position P and a tangential line of the desired contour at approximated foot point C(u.sub.f); where normal vector n.sub.E of the equivalent plane is computed by: $\begin{array}{cc}{n}_E=\left(C\left(u_f\right)-P\right)\frac{C^{}\left(u_f\right)}{.Math.C^{}\left(u_f\right).Math.}& \left(5\right)\end{array}$ wherein means outer production; equivalent-plane horizontal axis, denoted as X.sub.E, is taken as an intersection direction of the equivalent plane and original plane XY; and equivalent-plane vertical axis, denoted as Y.sub.E, is taken as a direction that is perpendicular to X.sub.E and n.sub.E; where X.sub.E and Y.sub.E are determined by: $\begin{array}{cc}\{\begin{array}{c}{X}_E=\frac{n_E{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T}{.Math.n_E{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T.Math.}\\ Y_E=\frac{X_E{n}_E}{.Math.X_E{n}_E.Math.}\end{array}& \left(6\right)\end{array}$ Step Two: contouring-error calculation and cross-coupling control in the equivalent plane; calculating the contouring error scalar form in the equivalent plane; where X.sub.E-direction and Y.sub.E-direction tracking errors from P to C(u.sub.f) are denoted as e.sub.x,E and e.sub.y,E, respectively, and are computed as: $\begin{array}{cc}\{\begin{array}{c}{e}_{x,E}=\left(C\left(u_f\right)-P\right).Math.X_E\\ e_{y,E}=\left(C\left(u_f\right)-P\right).Math.Y_E\end{array}& \left(7\right)\end{array}$ such that estimated contouring error {circumflex over ()} is:
{circumflex over ()}=C.sub.x,E.Math.e.sub.x,E+C.sub.y,E.Math.e.sub.y,E(8) wherein C.sub.x,E and C.sub.y,E are X.sub.E-direction and Y.sub.E-direction cross-coupling gains, respectively, and are obtained by: $\begin{array}{cc}\{\begin{array}{c}{C}_{x,E}=-\sin \left(\right)\\ C_{y,E}=\cos \left(\right)\end{array}& \left(9\right)\end{array}$ where is the included angle of vectors C(u.sub.f) and X.sub.E, and $=\arctan \left(\frac{C^{}\left(u_f\right).Math.Y_E}{C^{}\left(u_f\right).Math.X_E}\right);$ when the PID controller based two-axis CCC is utilized to control the estimated contouring error {circumflex over ()} as a control object, the output control signal of the CCC at time of t is thus obtained as: $\begin{array}{cc}{U}_c\left(t\right)=k_p\widehat{\mathrm{.Math.}}+k_i{}_0^t\widehat{\mathrm{.Math.}}\mathrm{dt}+k_d\frac{d\widehat{\mathrm{.Math.}}}{\mathrm{dt}}& \left(10\right)\end{array}$ where k.sub.p, k.sub.i, and k.sub.d are proportional, integral, and differential gains, respectively; according to U.sub.c(t), X.sub.E-direction and Y.sub.E-direction control signals, denoted by .sub.x,E and .sub.y,E respectively, are computed as: $\begin{array}{cc}\{\begin{array}{c}{}_{x,E}=C_{x,E}.Math.U_c\\ {}_{y,E}=C_{y,E}.Math.U_c\end{array}& \left(11\right)\end{array}$ Step Three: calculation of three-axis control signals in real space; according to the geometry relationship between X.sub.E/Y.sub.E and the spatial X/Y/Z axes, calculating coupling gains from an equivalent plane's two axes to real three-dimensional space's three axes as: $\begin{array}{cc}\{\begin{array}{c}{k}_{x,x}=X_E.Math.{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T\\ k_{x,y}=X_E.Math.{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T\\ k_{y,x}=Y_E.Math.{\left[\begin{array}{ccc}1& 0& 0\end{array}\right]}^T\\ k_{y,y}=Y_E.Math.{\left[\begin{array}{ccc}0& 1& 0\end{array}\right]}^T\\ k_{y,z}=Y_E.Math.{\left[\begin{array}{ccc}0& 0& 1\end{array}\right]}^T\end{array}& \left(12\right)\end{array}$ where k.sub.x,x, k.sub.x,y are gains from X.sub.E axis to X and Y axes, respectively, and k.sub.y,x, k.sub.y,y, and k.sub.y,z are gains from Y.sub.E axis to X, Y, and Z axes, respectively; calculating X-axis, Y-axis, and Z-axis control signals, denoted by .sub.x, .sub.y, and .sub.z, respectively, as: $\begin{array}{cc}\{\begin{array}{c}{}_x=k_{x,x}{}_{x,E}+k_{y,x}{}_{y,E}\\ {}_y=k_{x,y}{}_{x,E}+k_{y,y}{}_{y,E}\\ {}_z=k_{y,z}{}_{y,E}\end{array}& \left(13\right)\end{array}$ and subsequently performing a curve-interpolation CNC motion in a three-axis CNC system by adding the obtained .sub.x, .sub.y, and .sub.z to control signals of X-axis, Y-axis, and Z-axis position loops within each interpolation period, the equivalent-plane cross-coupling control can hence be realized, which can reduce the three-axis spatial contouring error effectively.

Description

INSTRUCTION FIGURES

(1) FIG. 1: Overall flow diagram of the invented method;

(2) FIG. 2: Geometric model of curved toolpath in Cartesian coordinate system;

(3) FIG. 3: The contouring errors before and after utilization of the invented method, where A and B axes means the time with the unit of s and the contouring error with the unit of mm, respectively, and the curves 1 and 2 means the errors before and after utilization of the invented method, respectively.

SPECIFIC IMPLEMENTATION EXAMPLE

(4) The specific implementation procedure of the invention is described in detail with an example in combination with the technical scheme and attached figures.

(5) During curve-interpolation CNC motion, the contouring error will be formed due to the existence of single-axis tracking errors and multi-axis dynamic mismatch. To reduce the three-axis contouring error thus improving the CNC curved contour following accuracy, an equivalent-plane cross-coupling control method is invented.

(6) FIG. 1 shows the overall flow diagram of the invented method, and FIG. 2 illustrates the geometry of the testing toolpath. By taking the curved toolpath shown in FIG. 2 as an instance, the detail implementation procedures are illustrated as follows.

(7) According to the flow diagram shown in FIG. 1, execute the equivalent-plane cross-coupling control for the testing toolpath shown in FIG. 2.

(8) First, establish the equivalent plane. According to the method provided in Step One of the contents of the invention, the tangential back stepping based Newton method is used to search the foot point C(u.sub.f) from actual motion position P to the desired contour C(u). Then, utilize Eq. (5) to calculate the normal vector n.sub.E of the equivalent plane. Additionally, take Eq. (6) to calculate the horizon axis X.sub.E and vertical axis Y.sub.E.

(9) Second, contouring-error estimation and cross-coupling control in the equivalent plane. Take Eq. (8) to calculate the estimated signed contouring error {circumflex over ()}, then based on its PID control, calculate the X.sub.E-direction control signal .sub.x,E and Y.sub.E direction control signal .sub.y,E in the equivalent plane.

(10) Third, calculate the three-axis control signals. Determine the equivalent-plane axes to real three-dimensional axes coupling gains using Eq. (12), thus computing the X-axis control signal .sub.x, Y-axis control signal .sub.y, and Z-axis control signal .sub.z using Eq. (13). Add the control signals to the position loops of corresponding axes within each interpolation period, thus realizing the equivalent-plane cross-coupling control of the three-axis contouring error.

(11) FIG. 3 illustrates the contouring errors before and after utilization of the invented method, where A and B axes means the time with the unit of s and the contouring error with the unit of mm, respectively, and the curves 1 and 2 means the errors before and after utilization of the invented method, respectively. As can be seen from FIG. 3 that before utilization of the invented method, the maximum contouring error is about 0.4 mm, while after using the invented method, the maximum contouring error is reduced to about 0.07 mm. As a conclusion, the invented equivalent-plane cross-coupling control method decreases the contouring error by 82.5%, which demonstrates that the method can effectively reduce the spatial contouring error and improve the three-axis contour following accuracy.

(12) Aiming at controlling the three-axis contouring error induced by reasons such as servo lag and external disturbances, an equivalent-plane cross-coupling control method is invented. This method can estimated the spatial contouring error in a scalar form by establishment of the equivalent plane, which not only is beneficial to flexible design of the contour controller, but also can control the three-axis contouring error using the well-studied two-axis CCCs.