GB2080523A

GB2080523A - Improved Instrumented Remote Center Compliance Device

Info

Publication number: GB2080523A
Application number: GB8127834A
Authority: GB
Original assignee: Charles Stark Draper Laboratory Inc
Current assignee: Charles Stark Draper Laboratory Inc
Priority date: 1979-09-19
Filing date: 1980-09-16
Publication date: 1982-02-03
Also published as: GB2080522B; GB2060167A; IT8024777A0; JPH0323842B2; JPS5696206A; DE3035838A1; GB2080522A; GB2060167B; CA1161925A; IT1132981B; GB2080523B; FR2465996B1; FR2465996A1

Abstract

A remote center compliance device (10), having an operator member (16) and a remote compliance center (70) near the end of the operator member, and including one or more translational displacement sensors (58, 60) proximate the operator member of for sensing displacement in one or more degrees of freedom of the operator member, is first calibrated by (a) applying known displacements (X) to the operator member; (b) measuring sensor signals (X1); and (c) calculating a transfer matrix D defined by X1=DX, where X1 is a vector whose elements are the signals X1; and X is a vector whose elements are the displacements X. The device is then used by (d) applying an unknown displacement (Y); (e) measuring sensor signals (Y1); and (f) calculating Y=AY1, where Y is a vector whose elements are the displacements Y; Y1 is a vector whose elements are the signals Y1; and A is a transfer matrix which is the inverse of D. <IMAGE>

Description

1 GB 2 080 523 A 1

SPECIFICATION Improved Instrumented Remote Center Compliance Device

This invention relates to an improved instrumented remote center compliance device, and more particularly to an instrumented system including one or more translational displacement sensors located about the operator member of the device for sensing displacement in one or more degrees of 5 freedom of the operator member.

A remote center compliance device (RCC) is a passive device for aiding insertion and mating maneuvers in robot machines and assembly equipment. RCC's typically include a structure which supports an operator member and establishes a remote compliance center near the functioning end of the operator member. See U.S. Patent Specifications No. 4,098,001; 4,155, 169. In some robot and 10 assembly applications there is a need for feedback from the FICC; however, force measurement is not always ideal for this purpose. For example, force sensors generally cannot withstand the large forces that occur when the RCC is driven to the limit against mechanical stops. In addition, force sensors generally cannot resolve the very small forces on the RCC when it is operating in its more normal range, not at its limits. Further, the mounting of the force sensors, usually between the RCC and its 15 support from the host machine, interferes with the compliance of the RCC; the compliance of the remote center is not simply that of the RCC but the combination of the compliance of the RCC and the compliance of the force sensor apparatus.

Typical angular deflections of an RCC are of the order of 50 and such angles are very difficult to measure using angular rotation sensors. In addition, the kinematic rotation canter of the RCC is not a 20 fixed point but rather a varying point. The general angular rotational sensor enforces a fixed axis of rotation within the sensor. In addition, the operator means of the RCC issubject to various translation.

In consequence, angular rotational sensors need be connected to the operator means of the RCC by means of the kinematic equivalent of splined shafts accommodating either axial or lateral relative motion. This is a significant difficulty which is avoided by the use of translational displacement sensors 25 exclusively.

It is therefore an object of this invention to provide an improved instrumented RCC which uses only translational-displacement measuring sensors to measure position and angular displacement in one or more degrees of freedom.

The invention is featured in an improved instrumented remote center compliance device which 30 has an operator member and a remote compliance center near the end of the operator member. More specifically, the invention features one or more translational displacement sensors located proximate the operator member for sensing displacement in one or more degrees of freedom of the operator member. By translational displacement sensor is meant one which measures motion of the operator member in terms of translations which can then be resolved into 35 actual translational and rotational displacements. The sensors are arranged so as to produce a change in output from at least one of the sensors in response to changes in position relative to the radial axes of the operator member. In a preferred embodiment, there are first and second translational displacement sensors spaced from the operator member for sensing displacement thereof, and the first and second sensors are disposed to one another at a first angle about the axis of the operator member. 40 There are third and fourth translational diaplacement sensors spaced from the operator member for sensing displacement thereof. The third and fourth sensors are spaced from the first and second sensors along the axis of the operator member and are disposed to each other at a second angle about the axis of the operator member. The sensors are disposed so as to produce an output from at least one sensor for changes of position of the operator member relative to the radial axes. By displacement 45 herein is meant both angular and translational movement.

Typically, the first and second sensors are in one plane and the third and fourth are in a second, parallel plane, and the first and second angles are equal. In a simple case the first and second angles may both be equal to 901 and the sensor in each pair may be aligned with a sensor in the other pair. If necessary or desirable, fewer or more translational displacement sensors may be used, for example a 50 fifth sensor may be used to sense a fifth degree of freedom or motion of the operator member.

The invention also features means for solving the equation R=AX,, where A is a transfer matrix relating the translational displacement sensor output signals to displacement applied to the remote center compliance device which produces those output signals; R, is the vector whose elements are the outputs of the sensors for determining A, which is a vector whose elements are the components of 55 the displacement supplied to the device. Although the sensors illustrated in the specification are of the photoelectric type, this is not a necessary limitation of the invention, for other types of translational tranducers may be used and are sufficient, for example LVDT's, linear- displacement potentiometers, or any other translational displacement- sensitive transducer. 60 An embodiment of the invention will now be described by way of example with reference to the 60 accompanying drawings in which: Fig. 1 is an axonometric view of a remote center compliance device, RCC, of the type shown in U.S. Patent Specification No. 4,155,169 which may be instrumented according to this invention;

GB 2 080 523 A 2 Fig. 2 is a more detailed sectional view of an RCC device such as shown in Fig. 1 with instrumentation according to this invention taken along line 2-2 of Fig. 3; Fig. 2; Fig. 3 is a plan view of the improved instrumented RCC device of Fig. 2 taken along line 3-3 of Fig.4 is a block diagram of a computer circuit for resolving measured translational displacement 5 signals into resolved translational and rotational displacements; Fig. 5 is an axonometric schematic diagram illustrating basic parameters used in resolving measured displacement signals into resolved translational and rotational displacements; Fig. 6 is a more simplified schematic of the instrumentation of an RCC where the angle between the sensors in each pair of sensors is at an angle other than 900; Fig. 7 is a diagram illustrating a basis for conversion from the axes of the sensors in Fig. 6 to the X, Y axes in Fig. 6; Fig. 8 is a block diagram of a computer circuit for calculating the resolved translational and rotational displacements from the measured displacement outputs of the sensors; Fig. 9 is a simplified schematic diagram of the placement of sensors on an instrumented RCC when each of the sensors is in a different plane and at a different angle to the X, Y axes and there is no vertical alignment between any of the sensors; Fig. 10 is a schematic diagram showing one placement of a fifth sensor, which enables monitoring of all five degrees of freedom of motion of an RCC; Fig. 11 is a block diagram of a computer circuit for resolving the measured displacement signals 20 into the resolved translational and rotational displacements; Fig. 12 is a block diagram of a circuit which may be constructed using calibration techniques to resolve measured displacement signals into resolved translational and rotational displacements; Fig. 13 is a schematic diagram of an RCC of the type shown in U.S. Patent Specification No.

4,098,001; Fig. 14 is a simple block diagram of a circuit for resolving the measured displacement signals into the resolved translational and rotational displacements for the RCC in Fig. 13; and Fig. 15 shows an alternative circuit similar to that in Fig. 14 for resolving measured displacement signals into resolved translational and rotational displacements for the FICC's in Fig. 13.

26 The invention may be accomplished by disposing a number of translational displacement sensors 30 on an RCC so that the relative angular and translational displacement between the moveable or operating member and the rest of the RCC can be detected. FICC's are disclosed in U.S. Patent Specifications No. 4,098,001 and 4,155,169, which are incorporated here by reference. Any number of sensors may be used, but they should be disposed so that an output is produced from at least one sensor for changes of position of the operator member relative to the remainder of the FICC. The number of sensors used may be commensurate with the number of degrees of freedom that it is desired to monitor; for example, one sensor may be used to monitor one degree of freedom, four sensors to measure four degrees of freedom, five sensors to measure five degrees of freedom. However, more sensors may of course be used. RCC's typically do not have more than five degrees of freedom; they are constrained to permit no displacement in the axial direction.

Once the translational displacement signals are obtained from the sensor or sensors, they must be converted or resolved into an acceptable coordinate form to provide useful information. Typically in an X and Y coordinate system the X and Y translations, X, Y, and the X and Y rotations 0, OV, of the operator member are desired to describe the movement of the operator member with respect to the remote compliance center. Computing circuits are necessary to resolve the measured displacement 45 signals into the actual displacements of the operator member. The soundness of the block diagrams of the computing circuits disclosed herein may be verified and defined analytically through the use of geometry and algebra, or empirically by a calibration technique also taught herein.

A simple arrangement which uses a relatively simple computing circuit results from using four translational displacement sensors two in the plane perpendicular, i.e. radial, to the axis, of the operating member and two in a second plane parallel to the first plane. The sensors in each plane are at 901 to each other, and each is aligned with the sensor in the plane above. However, this is not a necessary limitation on the invention. For example the angle between the sensors in each pair need not be 901, and in fact the angle between one pair of sensors need not be equal to the angle between the other pair of sensors. The sensors need not be aligned each with another one, and in fact each of the 55 sensors may be in a different plane. Of course, the number of sensors need not be fixed at four, but may be any number equal to or greater than the number of degrees of freedom that it is desired to monitor.

For proper results, however, the sensors should be disposed so that there is an output produced from at least one of the sensors for changes of position of the operator member. In contrast the use of rotationally sensitive sensors for any one or more of the sensors results in the difficulty of connecting 60 such sensors, with their fixed axes of rotation, to the operator member, with its five degrees of freedom, two of translation and three of rotation, by means of the kinematic equivalent of a splined shaft with universal joints at the ends, with all attendant disadvantages such as backlash, friction and inertia.

There is shown in Fig. 1 a remote center compliance device 10 which includes a deformable 65 Y 2.

5d 1V 3 GB 2 080 523 A 3 structure 22, from the central portion 24 of which is suspended operator member 16 having longitudinal or axial axis 17. Deformable structure 22 may also include three or more radially extending beams 26, 28, and 30, which are equally spaced and terminate in an intermediate rigid annular member 32. Beams 26, 28, and 30 lie along radial axes, i.e. axes perpendicular to the axial axis.

Member 32 is carried by a second deformable structure 34 which includes three longitudinal beams 5 36, 38, and 40, which extend to a fixed portion such as housing 12, as illustrated in Fig. 2.

Attached to member 16, Figs. 2 and 3, is a stop member 42 which limits the extent of motion of operator member 16 to prevent damage to the RCC. Also mounted to operator member 16 is a shade 44 which has two shade elements 46 and 48, whose outer edges 50, 52 sharply delineate the shadow area from that illuminated by light sources 54, 56 on translational displacement sensors 58, 60 carried 10 by supports 62, 64 on housing 12. Light sources 54, 56 are carried by support 66, also mounted on housing 12. Light source 54 and 56 may be Monsanto Electronic Special Products MVII OB light emitting diodes, for example, and translational displacement sensors 58 and 60 may be Reticon RL 256G solidstate line scanners with associated timing and counting circuitry, for example. Sensors 58 and 60 may be considered the X axis sensors. A second set of translational displacement sensors 76 15 and 78 (partly obscured) is typically provided, Fig. 3, with a second pair of light sources 72 and 74 (partly obscured). Shade 44 includes a second pair of shade elements 80 and 82 (not shown). The signals from sensors 58 and 60 are referred to as X2 and Xl, respectively, and those from sensors 76 and 78 as Y2 and Y1, respectively.

The measured translational displacement signals XV X21 Yll Y2 may be resolved into the actual 20 translational X, Y, and angular or rotational 0, 0, displacements of operator member 16 by the computing circuit shown in Fig. 4. There, X, is multiplied by the factor (L/S) in multiplier circuit 100, and by the factor (]/S) in multiplier circuit 102, whire L is the distance from remote centre 70 to the X21 Y2 signal sensing position and S is the distance between X,, Y1 and X2, Y1 signal sensing positions, shown in Fig. 5. X2 measured displacement signal is multiplied by the factor (I-L/S) in multiplier 104, and by the factor -(1/S) in multiplier 106. Measured signal Y, is multiplied by the factor (L/S) in multiplier 108 and -(1/S) in multiplier 110. Measured displacement signal Y, is multiplied by the factor O-L/S) in multiplier circuit 112, and by the factor 0/S) in multiplier circuit 114. These multiplier factors are relatively simple since the translational displacement sensors are arranged in pairs in parallel planes aligned with each other and with each pair at 901 to each other. The X,, X2 derived outputs from circuits 100 and 104 are combined in summer 116 to provide the actual displacement X.

The Xl, X2 derived outputs from circuits 102 and 106 are combined in summer circuit 118 to provide the displacement Oy. The Y1, Y2 derived outputs from circuits 108 and 112 are combined in summer circuit 120 to provide the actual Y displacement, and the Y1, Y, derived outputs from circuits 110 and 114 are combined in summer 122 provide the 0,, output. That this approach is sound may be seen from Fig. 5, where A1 and A2 are both equal to 900. The relationship of the measured displacement signals xl, X21 Y1, Y2 to the actual displacement X, Y, Ox, OV in terms of L and S is:

Xl=X-0 v (L-S) X2 =X-0 v (L) Y1=Y+O,jL-S) Y2=Y+O,JIL) These equations may be expressed in matrix form for easier solution:

Xl X 2 Y 1 Y 2 which may be simply expressed as:

(1) (2) (3) 40 (4) L- X v 0 1 (L-S) 0 ex 0 1 (L) 0 Y 1 0 0 (S4) 1 0 0 (_0 Ri=A-13Z (6) 45 Thus, 3Z1 is a vector whose elements are the outputs of the sensors.

These equations of course are not exact; they are standard approximations known and used in geometry. In order to solve for the actual displacementsX, the matrix may be inverted, X=AX1 to the state:

(7) 4 GB 2 080 523 A 4.

1 X v ex k 8 Y) (44 (ItL) 0 0 S S (1) WL) S S 0 0 (-"% 1 _) (i-) S S 0 Xl X2 Y1 Y2 (8) cl In the general case, wherein X, may be of higher dimension than X, equations (6) and (7) can be written _Al=CR- and R=AX, respectively, where A+D equals 11 and 11 is the identity matrix. 1 the useful special case previously described in detail wherein X and X1 are of the same dimension, A and D are 5 square matrices and simply D equals A.

Alternatively, when the sensors are not at 901 to each other, that is when A1 and A2 are not equal to 901 as shown in Fig. 6, the measured signals are represented by UJ, U21 V, and V2, which can be resolved in terms of the X, X, Y, and Y2 coordinates shown in Fig. 6, where U, and X, are separated by the angle a,; U2 and X, by the angle %; V, and Y1 by the angle P,; and V2 and Y2 by the angle P2. This 10 conversion maybe accomplished as shown in Fig. 7 using equations:

1 x= a cos a a=u - h b= e sin a y=e + c c= a sin a f coso = v j coso = X j sinp = 9 + f = Y (9) which result in the matrix expression:

cos a -sin a cos a cos 0 (10) sin a --GS2 a cos 0 For the specific case of X,, Y1 the matrix expression appears as U1 sin al cos 2 al V1 cos 01 cos a, sinal cosal cos P1 ( 11) 15 i and for X2, Y2 the matrix expression appears as:

0 1 i X2 V2 X / COS02 - sin 1a2 cOsa2 cos 02 (12) sin 02 cos2 a2 cOsP2 GB 2 080 523 A 5 These two expressions (11) and (12) are combined to produce:

X cosal sin al cos al 0 cos P1 Y1 X2 V2) sin a, C052 al COSP1 0 0 0 0 cos Q2 - sina2 cosa2 cos 02 0 S'nG2c OS2 Q2 COS02 which when rearranged to present Xl, X2, Y1, Y2 in the desired order, appears as:

1 Xl X2 V1 V2) cosal - sinal cosal COSP1 0 0 Sinai cOS2a, COSPI 0 Expression (14) may now be simply stated:

1 U1 V1 1 U2 V1 0 COS02 - 5'na2COS112V1 COS02 0 0 U2 0 Sin02 c o$2 2 cos P2 0 X1=BU1 Since we know from expression (7) that 5=A7Xl, we can substitute in expression (15) to arrive at expression (16), which fully expanded appears as expression (17).

-A=AE-Ul X v 8 X \ 11) ) sin al cos al 5 yin(I2cO G2 (1)cosai -(-5 (1-.L) cOsa2 cOS02 cos 01 1)sin a, _1 C0s2a, AL) sin a2 (1 -.L) S 5 cos 01 S S COS02 1 - 1 cOS201 )sinal (Isin Q2 S cos pi 5 5 cOsP 2 (!)C OS a, j.)s in a, cos a 1 A-4c0S02 (1) sinQ2cOsa2 S S COSP, S 5 cos P2 (13) (14) (15) (16) U1 V1 U v (17) jo 6 GB 2 080 523 A 6 A computer circuit which implements this statement includes a plurality of multiplier circuits 130 through 160, Fig. 8, which multiply the measured displacement signals U, V, U2, and V2, by the various factors shown in terms of the dimensions L, S, and the angles a and P, and are then combined in the combinations shown in summer circuits 162, 164, 166, and 168, to provide the actual displacements X, Y, 0, and 0..

In a similar manner, in the case where there are four translational displacement sensors providing four measured signals UJ, U21 V1, and V2, but none of the sensors are in the same plane nor aligned with each other, as shown in Fig. 9, the measured signals UJ, U2, V1, V2 may be resolved into the desired X, Y, OX, 0Y components, as shown by the following expressions:

Ulf--f(cos al)X+(sin al)Y-(S1 sin al)0x+(S1 cos al)O, (18) 10 'i U2--(cos P1)X+(sin P1W-(L, sin P1)0x+(L1 cos P,)0, (19) It Vi--(cos a2)X+(sin %)Y_(S2 sin al)ox+(S2 COS 401 (20) V2--f (cos P,)X+(si n NY-(L, sin P2) Ox+(L, COS PAY These expressions may be placed in matrix form, (21) Ul U2 m V1 V2) and then inverted in the usual way, X V 1 h BX ey) cos al sin al -Slsinal S1Cosal cos P1 sinOl -11sinfil 11COSP1 cOsa2 sinG2 S2sina2 S2COSIL2 cos 02 sin02 -12 Sin P 2 12 cOsP2 M11 M12- M13 M14 mn M22 M23 M24 M31 M32 M33 M34 M41 M42 M43 M44 X V ex a Y 1 (22) 15 Ul U2 V1 V2 (23) to obtain the terms which define the multiplier factors that are implemented in multiplier circuits 180-210 in the computer circuit, Fig. 11. 20 The outputs of circuits 180-210 are combined in the groups shown in summer circuits 212, 214, 216, and 218, to provide directly the X, Y, and 0, OV displacements. Although thus far the illustrations have used four translational displacement sensors to sense four degrees of freedom, this is not a necessary limitation of the invention, as fewer or more sensors may be used to monitor fewer or more degrees of freedom. For example, a fifth sensor 220, Fig. 10, may be added to produce a 0. displacement signal which senses rotation about the Z axis, axis 17, of the operator member. Senser 220 is placed parallel to sensor 78 in order to sense the rotation about Z axis 17. To effect this, a non-circular member with some sort of camming surface 222 is used in conjunction with sensor 220 to detect the rotation. Alternatively, sensor 220 might be placed elsewhere.

With five degrees of freedom being sensed by five sensors, the four expressions (18)-(2 1) would be expanded to include a fifth equation, and there would be a fifth column and fifth row added to the matrices in expressions (22) and (23), while the implementation shown in Fig. 11 would be expanded by the addition of one more multiplier circuit associated with each of the measured input signals UP U21 V1, V2, and also the addition of a fifth input, for example 0 c 2W z It c 7 GB 2 080 523 A 7 Alternatively, a calibration technique may be used to verify the multiplier factors in the computer circuit which resolves the measured displacement signals obtained from an instrumented RCC into the actual displacements of the operator member and body of the FICC relative to each other. First the instrumented RCC is fixed so that each degree of freedom, for example, X, Y, Ox, OV, and 0., can be varied independently while all the others remain fixed or at zero displacement. This may b e stated in matrix form as..

ul U 2 U 3 U4 \ US 1 1 Nil M12 N13 H14 H15 N21 N22 N23 N24 N25 931 M32 N33 N34 N35 M41 H42 N43 N44 N45 N 51 N52 N53 N54 N55 X v 4 OX a Y 9Z \ 1 (24) The U terms are the sensor outputs. If X is displaced a known amount Xl, not equal to zero, and all the remaining possible displacements, Y, 0,, Oy, 0, are held at zero, the result may be expressed:

Ul=l\111xl U2 =N"Ilxl U3 =N31xl U4 =N41Xl U5 =N51xl (26) 5 (26) (27) (28) (29) and since the measured values U,-, are known and the displacement X, is known, this set of equations may be transformed into:

1 Nil- U (30) (31) (32) xl U2 N21 X, U3 xl Nal 4 N41 U (33) 10 xl 1 N51us xl (34) In a similar fashion, with Y set equal to Y1, not equal to zero, and the remaining terms X, 0,, OV, 0Z all set at zero, the same action may be taken to obtain the numerical values for the second column of the matrix of expression 24. When this is done, with all the numerical values in place in the matrix of 25 expression (24), a simple matrix inversion results in:

8 GB 2 080 523 A 8 X v ex 8 Y 9 z 1 M11 M12 m 13 m 14 M15 M21 M22 M23 M24 M25 M31 M32 M33 M34 M35 M41 M42 M43 M44 M45 M51 M52 M53 M54 M55 U1 U2 U 3 U 4 U 5) (35) 2 where each of the M values in each row and column of the matrix is a numerical value and may be directly inserted in the multiplier circuits 250-298, Fig. 12, which are combined as shown in summing circuits 300, 302, 304, 306 and 308 to provide the actual displacements X, Y, 0, OY, oz. An even simpler implementation occurs with the RCC of U.S. Patent Specification No. 4,098,001, as shown in Fig. 13. In such a device, operating member 16'typicaliy is supported by a member 310, which in turn is supported by an intermediate device 312 by means of a number, typically three, flexures 314, 316, only two of which are shown, which converge toward each other and meet at a point 70' which generally establishes the remote compliance center along the axis 17' of operator member 16', Flexures 314, 316 are in turn fastened to intermediate member 312, which is 10 attached to a support 318 by means of typically three additional flexures 320, 322, only two of which are shown. Typically flexures 320, 322 control only translational motion, while flexures 314, 316 independently provide the rotational flexibility for the instrument. In this case a single translational displacement sensor 330, located as shown between support 318 and intermediate member 312, provides the signal X'2, which is indicative of motion along the X axis. Motion about the Y axis is indicated by the measured signal X'i obtainable from translational displacement sensors 332, or from XY1, obtainable from sensor 334. Similar signals Y'l, Yul, and Y'2 are obtained in the same way with respect to the Y axis translational and X axis rotational displacements. Because of this independent action in the translational and rotational motion modes in RCC 10', the actual displacement along the X axis, X, is equal to the measured value X'2, X=X,2 and similarly the displacement along the Y axis is equal to the measured displacement Y2:

Y=Y'2 Rotational motion about the X axis, 0, is either:

- 1 1 (36) (37) OX= (38) 25 L or:

Similarly, 0. is either:

or:

-Yll 1 0 X = S 0 v- X1 1 L OV= X11 1 (41) S (39) (40) Thus the computing circuit implementation to resolve the X'l, Y'll X'2, and Y'2 measured displacements into the actual disp'lacements OV, Ox, X, and Y, may be simple: direct connections 350, 352, to resolve the X' 2 and Y'2 signal into X and Y displacements. Multiplier circuit 354 multiplies X', signal by a factor of (I/L) to obtain 61., and multiplier circuit 356 multiplies V, displacement signal by a factorof (-I/L) to obtain 0, 1 k T tl Ir 9 GB 2 080 523 A 9 Similarly, using the combination X"2. Y1 2,andX" Y' only multiplier circuits 358, providing a Ito%f (-1/S), are necessary to complete the factor of fl/S), and multiplier circuit 360, providing a fa r computing circuit 10. As in Fig. 14, X2 and V, directly provide the X and Y displacements.

Claims

1. A method of measuring an unknown displacement on the operator member of a remote center 5 compliance device having a remote compliance center near the end of the operator member comprising:

disposing one or more translational displacement sensors about said device for producing output signals representative of displacements of said operator member; applying known displacements to said operator member, so that there exists no sensed displacement which cannot be expressed in terms of one or more applied displacements; measuring each of the output signals produced by each of the sensors in response to the applied displacements; calculating a transfer matrix D defined by the equation R11=ER, as determined by the applied displacements and the corresponding sensor output signals, where 3Z is a vector whose elements are 15 the components of the applied displacements and 3Z1 is a vector whose elements are the outputs of the sensors; applying an unknown displacement to the operator member; measuring each of the output signals from each sensor in response to the unknown displacement; calculating the unknown displacement by solving the equation X= AX,l, where X1 is a vector whose elements are the sensor outputs in response to the unknown displacement and X is a vector whose elements are the components of the unknown displacement.

2. A method of measuring an unknown displacement on the operator member of a remote centre compliance device having a remote compliance centre near the end of the operator member substantially as herein described with reference to and as shown in the accompanying drawings.

Printed for Her Majesty's Stationery Office by the Courier Press, Leamington Spa, 1982. Published by the Patent Office, 25 Southampton Buildings, London, WC2A 1 AY, from which copies may be obtained.

1 i