Patent Document ID: 4794540
Application ID: 06876995
Patent Flag: 1

Claim One:
1. Apparatus for driving a movable element in space about N axes of movement, as a function of an independent variable t having a first value t.sub.1 and a second value t.sub.2, between a first position P.sub.1 and a second position P.sub.2 along a first path section represented in parametric form as a function of the independent variable t by a first set of M-order polynomial equations p.sub.n (t)=a.sub.n,0 +a.sub.n,1 t+a.sub.n,2 t.sup.2 +. .. a.sub.n,M t.sup.M for each axis n=1. .. N by computing I intermediate positions PI.sub.i =[p.sub.1,i, p.sub.2,i,. .. p.sub.N,i ] for each i=1, 2,. .. I, separated by subintervals .DELTA.t and approximating positions along the path intermediate the first position p.sub.1 =[p.sub.1 (t.sub.1), p.sub.2 (t.sub.1),. .. p.sub.N (t.sub.1)] and the second position P.sub.2 =[p.sub.1 (t.sub.2), p.sub.2 (t.sub.2),. .. p.sub.N (t.sub.2)] during an interval t.sub.1 .ltoreq.t.ltoreq.t.sub.2, and moving the element from the first position through the intermediate positions to the second position comprising: input means for receiving: data representative of the first position, p.sub.n (t.sub.1), and the second position, p.sub.n (t.sub.2), of the element on each of the N axes; data representative of at least M-1 derivatives of the polynomial equations for each of the N axes at the first and second element positions, the data representative of the at least M-1 derivatives and the first and second positions characterizing at least M+1 initial condition values of element position and motion along each axis; data representative of the interval (t.sub.2 -t.sub.1); and data representative of the number, I, of intermediate positions PI.sub.i ; computer means responsive to the input means for computing and producing signals representative of: polynomial coefficients a.sub.n,0, a.sub.n,1, a.sub.n,2,. .. a.sub.n,M for each of the N axes by simultaneously solving, for each axis, a set of at least M+1 equations describing one of the at least M+1 initial condition values in terms of the polynomial coefficients; initial difference values Dm.sub.n,0 for each of the M orders of each of the N axes, the difference values approximating an average value of the derivatives of the polynomial equations over a first subinterval .DELTA.t, wherein the difference values Dm.sub.n,0 can be determined and evaluated in terms of the polynomial coefficients by recursively evaluating an equation: EQU Dm.sub.n,s.DELTA.t =D(m-1).sub.n,(s+1).DELTA.t -D(m-1).sub.n,s.DELTA.t, where: s is a number; EQU DO.sub.n,s.DELTA.t =p.sub.n (s.DELTA.t); and .DELTA.t=(t.sub.2 -t.sub.1)/I; first position P.sub.1 and second position P.sub.2 ; and intermediate positions PI.sub.i where for each of the N axes; EQU p.sub.n,i =p.sub.n,(i-1) +Dl.sub.n,(i-1) ; Dm.sub.n,i =Dm.sub.n,(i-1) +D(m+1).sub.n,(i-1) for each order m=1, 2,. .. M-2; and EQU D(M-1).sub.n,i =D(M-1).sub.n,(i-1) +DM.sub.n,0 ; and actuator means responsive to the computer means for continuously moving the element from the first position, through the intermediate positions, and to the second position, as a function of the signals representative thereof.