Patent GB-925393-A (GB)
Title: Improvements relating to computing devices
Inventor: ARROWSMITH JOHN RICHARD
IPC: G05B19/4105
Filed: 1958-06-30
Abstract: 925,393. Electric analogue calculating. ELECTRIC & MUSICAL INDUSTRIES Ltd. June 24, 1959 [June 30, 1958], No. 23825/58. Class 37. An electric analogue computer comprises means responsive to a group of successive input signals representing respective dimensional co-ordinates, means for generating a modified co-ordinate signal when a co-ordinate represented by one such input signal is displaced from a smooth curve intersecting a majority of the represented co-ordinates such that the modified signal represents a co-ordinate of the smooth curve; wherein such generating means comprise plural transformers having primary turns across which the input signals are applied, or signals representing the difference between successive input signals; the ratios and secondary interconnections being such that required modified signal represents the solution of an equation representing the required co-ordinate in terms of the successive co-ordinates represented by the input signal group. In a machine tool control system, a profile is defined by five equally-spaced co-ordinates x -2 , x -1 , x 0 , x 1 , and x 2 , for corresponding values - 2, - 1, 0, 1, 2 of independent variable t and a smooth cubic curve x=a 0  + a 1 t + a 2 t<SP>2</SP> + a 3 t<SP>3</SP>=#(t)(1) is fitted to the defined points of the profile to minimize the sum of the squares of the displacements from the curve to the defined ordinal points (Fig. 1) where r is a given value of t and x r  is the corresponding x ordinate. It is shown by mathematics that 9 0  =1/35 [ - 3x- 2 + 12x- 1 + 17x- + 12x 1  - 3x 2 ] (2) 9 1  = 1/12[x -2  - 8x -1  + 8x 1  - x 2 ] (3) 9 2  = 1/14[2x -2  - x -1  - 2x 0  - x 1  + 2x 2 ] (4)   9 3  = 1/12[-x -2  + 2x -1  - 2x 1  + x 2 ] (5)   and that x-<SP>1</SP>=a 0 , so that for a group of five ordinates a modified value of x 0  may be produced (=x-<SP>1</SP>) and substituted whereby all five ordinates lie on a cubic curve. In Fig. 2, alternating voltages analogous to the input ordinal values x- 2 , x- 1 , x 0 , x 1 , and x 2  are applied to the primaries of transformers 6 to 10 wherein 6 and 10 are of ratio 3/35, 7 and 9 are of ratio 12/35 and 8 is of ratio 17/35, and the secondaries are connected in such senses as to develop a total output voltage representing equation (2), which is balanced in centre-tapped transformer 11 against voltage from a potentiometer 14, adjusted by a servomotor 13 responsive to an error voltage developed by the secondary of 11, so that the potentiometer voltage represents a 0  and the modified ordinal value, x 0 <SP>1</SP>, of x 0 . Modified ordinal voltage values of x- 1 , x- 2 , x 1 , and x 2  may similarly be produced using equations (3), (4), (5) above to more closely fit the required cubic curve. In a modification a cubic curve B (Fig. 3) is drawn to intersect points ( - 2, x- 2 ), (-1, x- 1 ), (1, x 1 ), and (2, x 2 ) and a point x 0 <SP>1</SP> on the zero ordinate which is shown by mathematics to have the ordinal value x<SP>1</SP> 0  = 9 0  = 1/6 [-x 2 +4x 1  + 4x 1  - x -2 ] (6)   determinable by the circuit of Fig. 2 with transformer ratios 1/6 and 4/6 respectively (and utilizing the method of successive differences). Alternatively, analogue voltage of ordinal values x -2 , x -1 , x 0 , x 1  and x 2  are applied to terminals 21, 22, 23, 24, 25 (Fig. 4). Primaries of transformers 26 to 29 are series-connected and terminals 21, 22 are connected to the primary of 29 and terminals 24, 25 to the primary of 26 while the secondaries are connected respectively differentially in groups of two as shown to energize the series-connected primaries of transformers 30, 31, 32. The respectively differentially-connected secondaries of the latter energize the series-connected primaries of transformers 33, 34 whose secondaries are opposedly series-connected to amplifier 36 which energizes transformer 35 connected between the junction of the primaries of 27, 28 and the input terminal 23 receiving input x 0 . All transformers have unity ratios, and it is shown that first, second, and third difference signals appear across the secondaries of transformers 26, 27, 28, 29; second difference voltages across the secondaries of 30, 31, 32 and third differences across the secondaries of 33, 34 which are combined in opposition to the x 0  signal at terminal 23 to a fourth difference voltage. If the values x- 2 , x- 1 , x 0 , x 1 , x 2  do not lie on a cubic curve, the fourth difference will deviate from zero and the error voltage across the secondary of 35 will develop a differing ordinal value x1 0  at 37 lying on the curve, since the fourth difference is linearly dependent on any one ordinal value. Corrected values of x 0  may be obtained for sequentially selected groups of five ordinal values.