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=[rx,ry,rz] and P=[px, py, pz], respectively; (iii) denoting ur as a parameter corresponding to the desired motion position; and (iv) defining a tangential error dt(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:

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 dt(u) must be zero when C(u) is the foot point from P to the desired contour; accordingly, foot-point parameter uf is obtained by solving dt(u)=0; where tangential back-stepping point parameter ub is first calculated by projecting the tangential error to the desired contour as:

then, the tangential back-stepping point parameter ub is taken as the initial value of the Newton method so as to find the solution uN of dt(u)=0 by:

and if |dt(uN)|<|dt(ub)|, thus indicating that the tangential back stepping based Newton algorithm is convergent, taking the solution as foot-point parameter uf where uf=uN; otherwise, applying the tangential back stepping based Newton algorithm again at ub to obtain the foot-point parameter uf; to summarize, the foot-point parameter uf is calculated by the following equation:

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(uf); where normal vector nE of the equivalent plane is computed by:

wherein × means outer production;

equivalent-plane horizontal axis, denoted as XE, is taken as an intersection direction of the equivalent plane and original plane XY; and equivalent-plane vertical axis, denoted as YE, is taken as a direction that is perpendicular to XE and nE; where XE and YE are determined by:

Step Two: contouring-error calculation and cross-coupling control in the equivalent plane;

calculating the contouring error scalar form in the equivalent plane; where XE-direction and YE-direction tracking errors from P to C(uf) are denoted as ex,E and ey,E, respectively, and are computed as:

such that estimated contouring error ? is:

?=Cx,E·ex,E+Cy,E·ey,E (8)

wherein Cx,E and Cy,E are XE-direction and YE-direction cross-coupling gains, respectively, and are obtained by:

where ? is the included angle of vectors C?(uf) and XE, and

when the PID controller based two-axis CCC is utilized to control the estimated contouring error ? as a control object, the output control signal of the CCC at time of t is thus obtained as:

where kp, ki, and kd are proportional, integral, and differential gains, respectively;

according to Uc(t), XE-direction and YE-direction control signals, denoted by ?x,E and ?y,E respectively, are computed as:

Step Three: calculation of three-axis control signals in real space;

according to the geometry relationship between XE/YE 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:

where kx,x, kx,y are gains from XE axis to X and Y axes, respectively, and ky,x, ky,y, and ky,z are gains from YE axis to X, Y, and Z axes, respectively;

calculating X-axis, Y-axis, and Z-axis control signals, denoted by ?x, ?y, and ?z, respectively, as:

and subsequently performing a curve-interpolation CNC motion in a three-axis CNC system by adding the obtained ?x, ?y, and ?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.