Acta Astrmautica Vol. 2, pp. 1031-1034. Pergamon Press 1975. Printed in Great Britain

Communication

A novel technique for estimating

VICTOR LARSON?

relative motion

Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91103, U.S.A.

(Received 3 December 1974; revised 26 June 1975; communicated by Professor George Leitmann, U.S.A., Corresponding Member, Section 2)

Abstract-This paper provides an interesting potential scheme for estimating the relative motion between two vehicles. It represents an application of modem filtering and estimation concepts.

Introduction

A POTENTIAL method for determining the information needed for a successful docking between two vehicles (e.g. an Orbit-to-Orbit Shuttle (00s) and a malfunctioned satellite) is described herein.

The proposed technique is novel in that a TV alone is used to obtain all the necessary information required for docking and stationkeeping. Four passive reflectors, designed so that the electronically scanned images are distinguishable, are mounted on the satellite to provide the only means of cooperation required between the satellite and 00s.

TV calibration

The conversion of the apparent lengths of the vectors dii (the vector dii originates on the i th reflector and terminates in the j th reflector) from TV units to feet is accomplished by a technique similar to the familiar method of triangulation. First, the images of the vectors Ri9 on the TV screen (having coordinates BP) are recorded for one position of the 00s. Then, the 00s is moved a known distance D away from the initial position while either maintaining the same TV screen orientation or recording the transformation which relates the TV screen coordinates for the two positions.

The estimate for the k (the sought scale factor) can be obtained from a generalized least-square technique and is given by

E = (H,TR -‘&)-‘HcTR -‘z (1)

where R is the covariance matrix of the white noise process ~1 and 2% is given by

tMember Technical Staff; Member AIAA.

1031

1032 V. Larson

(21

The measurement z is given by z = H,x + u where x is the vector having components k,, kz and u represents the sensor noise.

Initial estimates of direction cosines

By relating the resolution error of the TV to a white noise process U, the measurement equation for the technique associated with the determination of the initial estimates of the direction cosines becomes

Z = HAX +2, (3)

where z represents the twelve scalars dii . ek and x represents the six direction cosines b - BP. The 12 x 6 matrix IL is given by

14)

where E,, k; are the estimates of the scale factors determined above. The estimate of x is obtained in the same way that the estimate of k was obtained and is given by

i = (HATI? -‘Ha)-‘HA% lz (5)

and H,, is the 6 x 3 matrix containing the components of the six vectors dij in the satellite coordinate system .L. Equation (5) provides six of the nine direction cosines relating the s^, and fi= bases.

The remaining three direction cosines (the second column of C,,“) is given by

C? = c-,x (‘1 (6)

where cj represents the jth column of CD‘. The matrix formed from the columns ci is then orthonormalized so that is satisfies the properties of a direction cosine matrix.

Initial relative angular velocity The initial estimate of the angular velocity of the satellite relative to the TV

coordinate frame (and expressed in satellite coordinates) is given by

A novel technique for estimating relative motion 1033

(5= C,“(i - l)C,“(i)-E At - (7)

Relative attitude and attitude rate The linearized rotational equations for a symmetric dual-spin vehicle (Larson

and Likins, 1974) are given by (A is the rotor spin axis inertia, II is the transverse moment of inertia of the satellite, and aRS is the rotor angular velocity relative to the despion portion of the satellite)

and the linearized rotational equations of motion for a symmetric spinning vehicle are given by (P is the ratio (I, -&)/I, and ON is the nominal velocity of the satellite)

6=&d-WNxe; &,=--bh.,x&+w. (9)

The direction cosine matrix C,” is updated according to (E is an identity matrix)

&i(i) = [E - 8x]((?i (i)]. (10)

The relative angular velocity o is updated according to

4(i) = k’(i) + S;(i) (11)

where G’(i) represents the previous estimate &(i - 1) propagated forward to the current time ti. The initial estimates for the direction cosine matrix C,” and the relative angular velocity w are obtained in the manner discussed previously.

Determination of relative range and range rate The equations governing the behavior of the relative range and range rate are

particularly simple if it is assumed that the satellite is merely rotating relative to the 00s and not translating. If the relative range is denoted by x1 and the relative range rate by x2, the differential equations become

il= x2; &= w (12)

where w is a scalar white noise process used to represent the disturbance in acceleration.

Measurement process The discrete measurement process associated with the filter used to obtain e^

and 86 is described below. The process z’ is defined in terms of the scaled projections of the vectors du (known in satellite coordinates) on the screen

1034 L’. Larson

coordinates; that is, for a particular vector dij, 7’ is defined as

z;= [“o’ p, /!I C,“(k)dij + ~‘k = AC,"(k)dii + ~)k. (13)

By stacking up the observations corresponding to the six vectors dti to 4 and i # j, ij # ji) the complete measurement process becomes

(with i, j = 1

(14)

where Q is a 12 x 18 diagonal matrix containing as its elements the 2 x 3 matrix ACsP(k - 1) and d is the 18 x 3 matrix formed by stacking up the six 3 x 3 matrices dj.

The measurement model associated with the technique for determining the range and range rate is determined from

d XI 1 1 - -xi d’=x;; d’-dx; (15)

where x, is the desired range, d is the length of the projection of a particular vector dij along the e3 axis, and d’ is the measured or observed length along the J?~ axis of the same vector dtj. The measurement equation is given by

z=Hx+c (16)

where x is the 2 x 1 vector having components x, (the range) and xz (the range rate). The 6 x 1 vectors d and d’ are given by

with

d’ = i&d;; . $2 d = di, . $3 = H,,ci

i,j=1,4; i+j; ij#ji

where k; is the previously determined scale factor associated with axis Lj3, cz is the third column of the previously determined direction cosine matrix C,“, and H,, is the 6 x 3 matrix containing the components of the six vectors dij in the satellite coordinates system s^,. The 6 x 1 vector z is made up of the reciprocals of the elements of d’, the first column of the 6 x 2 measurement matrix H is made up of the reciprocals of the product of the elements of d and the scalar x’ and the second column of H is the 6 x 1 null vector.

References Kalman, R. E. and Bucy, R. (1961) New Results in Linear Filtering and Prediction, Tmns. ASME 83D,

35. Larson, V. (1971) Docking Dynamics Associated with the OOS-C and A Malfunctioned Satellite,

Aerospace Report No. TOR-0059(6531-04)-2. Larson, V. and Likins, P. (1974) Closed-Form Solution for the State Equation for Dual-Spin and

Spinning Spacecraft, Journal of Astronautical Sciences XXI (5 & 6), 244-251.