Introduction

Tools of Kinematics:A Crital Review

ASIM K. SEN, Senior Member, IEEESynchrosat Limited, Canada

Abstract

A review of the basic tools of kinematics widely used by aerospace

engineers is presented. Two different approaches are discussed. One

is the familiar "vectorial mechanics," while the other is called the

"variational method." It is indicated that either approach should

produce the same set of motion equations for the dynamical system

considered.

Since the time Newton first formulated the laws ofmotion, the science of mechanics has developed along twomain directions. One branch is called "vectorial mechanics,"which originates directly from Newton's second law ofmotion. The other branch is called "variational theory ofmechanics," as founded originally by Euler and Lagrange.While both of these approaches consider ultimately the samephysical phenomena, namely the motion and equilibrium ofa dynamic system, they differ basically in that the formerbegins with the consideration of two fundamental vectors,"momentum" and "force," whereas the latter bases every-thing on the consideration of two scalar quantities, "kineticenergy" and "work function." Although many well writtentexts [1 -6] are now available on the subject which containelaborate descriptions of these two basic analytical ap-proaches, it is felt that a good deal of elaboration and clari-fication is still needed to make these tools readily applicableto solve real world problems. The purpose of this reviewpaper is to fil this deficiency in these original presentations.

Vectorial NMechanics

As mentioned above, the vectorial approach of mechanicshad its origin in Newton's second law of motion. Accordingto this law

IFF=mr

Manuscript received July 10, 1978;revised July 31, 1978.

Author's address: Synchrosat Limited, P.O. Box 1259, Stn. B,Ottawa, Ont., Canada, K1P 5R3.

0018-9251/79/0100-0040 $00.75 IEEE 1979

where EF is the vector sum of all the forces acting on a

particle or a system of particles of mass m, and r~is the vec-

tor acceleration of the mass measured relative to an inertialreference frame which Newton postulated as being fixed inthe absolute space.

In solving problems of mechanics, however, a straight-forward application of the Newton's law is not generallyfeasible because the kinematic variables involved such as thevelocity and acceleration of the mass particle are often onlyavailable as measured relative to a moving reference frame.Clearly, a simple artifice to overcome this difficulty is touse a kinematic law which would translate the variables inthe moving frame into appropriate variables in the inertialreference frame. The required kinematic law may be derivedas follows.

Consider a vector r, say, the position vector of a particleof mass m, defined as a function of time in a referencecoordinate system (x, y, z). Let x y5, z denote the unit vec-

tors associated with the reference coordinate system (x, y, z)which is moving with respect to a fixed inertial system(X, Y, Z) at an angular velocity co. Let X, Y, Z representthe unit vectors associated with the inertial reference sys-

tem (X, Y, Z). By expressing the vector r as componentsin either of the two reference frames, one can write

r = rxx+ ryY + rzZ = rx + rJy +rrZ (2)

which, on differentiation with respect to the inertial refer-ence frame, yields

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 1 JANUARY 1979

(1)

40

rr X+r Y+r2ZX y z

=rx +rYA+rZ +r++r +rz.

Note that the unit vectors X, Y, Z associated with the fixedinertial coordinate system are constants both in magnitudeand direction and, therefore, no time-rate-of-change in theseunit vectors are taken in (3). However, the unit vectors x, 9,

z, associated with the moving coordinate system, are con-

stants only in magnitude, but their directions do change inthe inertial space giving rise to terms containing the deriva-tives x, y, z. By noting further that the unit vectors x, y, z

can change directions in the inertial space only due to thecomponents of C3 which are orthogonal to the respectiveunit vectors, it follows, by reference to Fig. 1, that

x=WZY - WyZ

=h) Xx.

z

(A) (B)Fig. 1. (A) Motion of x.due to wz. (B) Motion of x due to wy.

Fig. 2. Basic dual-spin configuration.

(4)

Likewise, one can also write for the time-rate-of-changeof the unit vectors and z, as follows:

y=cO Xy (5)

z=c^X 2. (6)

Now, by using the notations (df/dt)N and (dP/dt)B forthe time-rates-of-change of r in the fixed inertial systemand the moving coordinate system, respectively, and byusing the preceding expressions for x, y, and z, (3) can berewritten as

(df/dt)N = (df/dt)B + cCX X(ri+r+y z

which finally takes the form

(df/dt)N = (dP/dt)B + X X r. (8)

This is the basic kinematic law which relates the vectorderivatives in the moving frame to the vector derivatives inthe inertial reference frame, and it offers a powerful tool tosolve problems related to space vehicle dynamics. Althoughapplication of the law is rather straightforward to the case

of a simple spinner, difficulty arises when two or more mov-

ing frames are involved as in the case of a space vehicle con-

taining two or more moving parts. The following examplemay be helpful to further illustrate this point.

Consider the dual-spin satellite configuration as shown inFig. 2. In the inertial space, let body B have the angularvelocity cw with its spin component designated as wo, where-as co' denotes the angular velocity of body B' with its spin-axis component labeled as cw4. Assuming that the twobodies B and B' are constrained to have a common axis ofrotation along the z (or z') body-axis direction, one can

write(A)r= W^ +2Q(9)

where Q is the relative angular velocity between the twobodies of the dual-spin system considered. Let x, y, zrepresent the unit vectors associated with the reference co-

tl A, ifordinate system, xyz, defined for body B, whereas x^, y, z

denote the corresponding unit vectors associated with thereference coordinate system, x'y'z', defined for body B'.

For the sake of simplicity, let it also be assumed that themass centers of the two bodies B and B' are coincident andare located at the origin 0 of the inertial reference frame,XYZ.

Now, in order to write the motion equations for the dual-spin system of Fig. 2 with the help of the basic kinematiclaw, one should note that the total angular momentum ofthe system is actually contributed by two different parts,each having a different angular velocity in the inertial space.

Thus, one can write

(dHT/dt)N = (dH/dt)N + (dHr/dt)N

= [(dH!dt)B +c X hr] + [(dH'/dt)B±+ CZ'X H'] (10)

whlere HT is the total angular momentum of the dual-spinsystem considered, and H and H' are the two momentumcomponents ofHT contributed by the bodies B and B',respectively. It might be important to point out here that,by using the basic kinematic law, the second term on theright-hand side of (10) which involves vector derivative ofH' in the B'-body frame, can be rewritten as

SEN: TOOLS OF KINEMATICS: A CRITICAL REWIEW

A

(3)

41

[(AH'/dt) + & xfj(A[(dH/Idt)B + X H']

thus yielding

(dHT/dt)N = [(dT/dt)B + CZ X HT] (12)

where HT =H + H'.However, a word of caution is in order. For correctly

evaluating the term (dHTldt)B in the preceding expression,it is necessary that one first have the vector quantity HTexpressed in the form

H = H +H +H x+H'X + H'Y + Hz (13)T X y z X

and then time derivatives of the unit vectors x', Y' and z^' betaken in the reference coordinate frame of body B, withthe result

[(dH'/dt)B + (D X I'[j = [(dH'/dt)B + Q X H' + & X H']

= [(dH'/dt)B' + 'X ] (14)

where co '= X^ + Q, Q being the angular velocity of thexpy'z'-frame of reference measured with respect to the refer-ence coordinate frame of body B. More specifically, onemay note that, whereas

H =H +H'T

=(H +H+)x+(H + ( +HZ)Z (15)XX y y z z

is a kinematically valid expression and is widely used inavailable texts to represent the total angular momentum ofthe two-body system considered, the expression given by(13) is the more appropriate form to use in situations wherebody frame differentiations are involved. In the event thepreceding form for HT is to use, one should rewrite (1 1) inthe forrn

[(dH'/dt)B + c.' X H'] (16)

with the result

(dHT/dt)N = [(dHT/dt)B + X X HT Q X ] (17)

Basic Kinematic Law and Coordinate Transformation

The basic kinematic law and the principle of coordinatetransformation are two different but closely related kine-matic tools often confused by an analyst dealing withmultiple-body dynamics. The cause of the confusion canbe attributed to the fact that both these analytical toolspermit a kinematic variable, such as the acceleration orvelocity of a mass particle or a system of mass particles orig-inally measured with respect to one coordinate system, tobe expressed as components in a different coordinate sys-tem. However, to avoid any problem in their application,a rule of thumb will be to recognize that the kinematic law

approach should be concerned with a pair of coordinatesystems which are embedded in two different inertial bodieseach moving with a different angular velocity in the inertialspace. On the other hand, the transformed coordinate sys-tem is applied simply by a change of base in the same iner-tial body, and, consequently, the two coordinate framesinvolved in the transformation process should have the sameangular velocity in the inertial space. The dual-spin exampleconsidered earlier in this discussion can again be used tofurther illustrate this point.

For the dual-spin system of Fig. 2, a coordinate transfor-mation matrix, CT, often used to express a kinematic vari-able associated with the B'-body frame (x'y'z') as compo-nents in an auxiliary coordinate system, xjyizi, can bewritten as

cos e sin e 0 1CT = -sine cos e 0

OT 0 1

Therefore, in terms of the unit vectors of the two frames,one can write

(19)CT ].

By referring to Fig. 2, one may note that the auxiliarycoordinate system, xjyizi, is one which has been chosen tocoincide with the reference coordinate frame, xyz, for bodyB at an arbitrary instant of time, t. Thus, for the instantan-eous time, t, one may write

x x

_' = CT _

(20)

where the angular parameter e appearing in CT now becomesa time varying parameter. This time varying property of thecoordinate transformation matrix is a very significant pointoften confused by the analysts. For the dual-spin exampleconsidered, the problem arises when an analyst attempts totake a time derivative of the augular momentum vector H'by the application of the basic law of kinematics while, atthe same time, treats the coordinate transformation matrixC as time varying. In the following it will be demonstratedthat coordinate transformation should always be treated asbeing a time-invariant process because the cross-productterm of the kinematic law is actually there to take care ofthe required differentiation of the coordinate transforma-tion matrix when moving frames are involved.

Consider the angular momentum vector, H', which, byusing (18) and (20), can be expressed as components in thetwo reference frames x'y'z' and xyz as follows:

H' =X'H +y 'H, + z 'X y z

=X(H', cose - H',sin e)x y

(18)

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 1 JANUARY 197942

+ y(x, sin e +H', cos e) + zHzxy z

= xH' + yHy + zHz.X Yy z

From (21) one can write

H' =(H' ,cos c- H', sine)x x y

H' =(H', sin c + H',cose)y X Y

H' = Hl,.z z

2) The angular momentum associated with each spinningpart must be considered separately when the kine-

(21) matic law is applied.3) Under no circumstances should differentiation of the

elements of any coordinate transformation matrix becarried out.

Thus, by applying the guidelines 1) and 2) to the case ofthe dual-spin system of Fig. 2, one can write

(22)

By using (21) and (22) and by noting that the angularparameter e is time variant, the time derivative of H' in thereference frame xyz can now be obtained as follows:

(dH'Idt)B =HX' +yH + z

=X[(I, cos c- H' sin e) + e(-H>, sin e- H'f, cos e)]

+ Y4(2x sine+ Hyt cos e) + e(H' cos c - H' ,sin e)]YVIIX y xy

+ z [Hz]

=(xcose+9sine)H, +(-xsine+9 coseW', + ^2f,y1X y ~~~y z

+ (-x sin c + i cos e) eH',- (i cos e + 9 sin c) cH',

=x'+ H+)y x y

=(dH'/dt)B,+ 2XH' (23)

where e = 2, Q2 being the relative angular rate between thetwo reference frames xyz and x'y'z'. Note that the preced-ing expression represents the basic kinematic law which re-lates the vector derivative of H' in the xyz frame to the de-rivative taken in the x'y'z' frame of reference. Note also thatthe cross-product term of the kinematic law is actually gen-erated by differentiating the elements of the coordinatetransformation matrix. Clearly, this differentiation wasneeded because the angle e appearing in the coordinatetransformation matrix was a time-varying parameter. So, itmay be concluded that, when the basic kinematic law isused to obtain the time derivative of the angular momentumvector H', the elements of the coordinate transformationmatrix C should never be differentiated because the cross-

product term of the kinematic law is actually there to takecare of the required differentiation of the coordinate trans-formation matrix when moving frames are involved.

Application of the Basic Kinematic Law

From the preceding discussions it is now clear that, inapplying the basic kinematic law to obtain the motion equa-tions for a dual-spin or multiple-spin body in space, the fol-lowing guidelines must be observed:

1) A separate body axes coordinate system must be usedfor each spinning part.

(dHT/dt)N = [(dH/dt)B + W^ X H] + [(dH'/dt)B + &) X Hp]

=x(h +Ho -Hw')X z y y z

+y(ht +Hcw-H )y X Z Z X

+2(f +H w -H )z y X X y

+^'(A", + H' w',)+X ~~y zX Z y yZ

+y (H +H u H', w',)y X z z X

+ z'(1, + Hp Wgi - If,cDy,) (24)z y X X y

where

W xwx y Z wz

H =5xH +YH + z HX y z

CO X()CO + (A) ,-Z^ W,.X y z

H' py1-H=x'H, + y f¢, + zH,.(25)X y z

Now, by noting that for the dual-spin system of Fig. 2the unit vector 2 equals 2', and also noting that the unitvectors x, j, x', 5' and 2 are linearly independent, as theysimply represent the basis vectors for the system considered[7], one may readily obtain, for a torque-free environment,the following set of motion equations:

H +Hco -Hcw =0 (26)x z y y z

H +HHc -Hcw =0 (27)y x z z x

I¢, + Ht ,x',-H' ,x', = O (28)x z y y z

H', + H' ,x', - Ht ,x',' = O(29)y x z z x

H +HHw -Hcw +H',+H',c ',-H',c',=O. (30)z y x x y z y x x y

Note that in the preceding derivation of the motion equa-tions which uses a five degrees-of-freedom dynamic modelfor the dual-spin system considered, the guideline 3) hasnot been used at all. This, of course, does not imply thatthe guideline 3) is insignificant. It should be pointed outthat in situations where coordinate transformations areused by the analysts to reduce the total number of degreesof freedom of the original dynamic model to only three,it does represent a very significant guideline. It can be

SEN: TOOLS OF KINEMATICS: A CRITICAL REWIEW 43

shown that violation of this guideline would result in im-proper cancellations of two important dynamic terms orig-inally produced by the kinematic law [81.

Variational Theory of Mechanics

While the vectorial approach of mechanics as discussedin the preceding section uses Newton's laws of motion asthe basic postulate, in the, variational approach it usuallybegins from Hamilton's Principle. For an n degrees-of-free-dom dynamic system which is conservative, the principlestates that the motion of the system from time tl to timet2 is such that the time integral

I= ft L(qi, q2- qnp ql,,q n' t)1~~~~~(31)

is an extremum for the path of motion. To put it in anotherway, the principle states that the motion is such that thevariation of the line integral I for fixed times t1 and t2 iszero. That is,

2, ..., n are now denoted as bq1, one can write

I = E 5I.i I

t2= f E [(L/6q.) - d/dt(L/61j.)] bq. dt

which, by using Hamilton's Principle, finally yields

(38)

(39)E [(LISqi)-dldt(8L/bqj)] bqI = O.

For a dynamic system which is nonconservative, (39)takes the form

(40)i

) + Q1] 1q, = O

where Q, represents the frictional forces. In terms of theRayleigh Dissipation Function D given by

t2

U=8Itl L(ql, q2.qqn l, q2 , ..., 4n, t)dt=O

where L is a scalar function of the generalized coordina(qcl, q2, ..., qn) and (41, q2, ..*,qn), called the Lagrangiaand is given by

L=T- V

(32) D=.Bq2

tes the frictional forces Q1 can be written asQn

Q. = -(6D/6qj)

(33) thus giving

where T represents the kinetic energy and V is the potentialenergy for the dynamic system considered.

For a dynamic system with only one degree of freedom,the variation of the line integral can be expressed as

&I, = bfti L(q, , t) dt

= ft [L(ql + dql, q1 + dq4, t)-L(q1,Pq1, t)] dt

= ft [(UL6q1 ) dql + (SL/6ql1) dq4 ] dt. (34)

By the method of integration by parts, the second termof (34) can be evaluated as follows:

(8L/6q1) dq1 dt = [(L/6141) dq 1]tt2

- ft [dldt(SLIS41l) dql dt- (35)

By noting that the flrst term of (35) is zero because ofthe boundary conditions

(dq1)t=t =(dql)t= =0 (36)

one finally obtains from (34)

8I, =|t [(SLI8q1) - ddt(6L1841)1 dql dt. (37)

Similarly, by considering a dynamic system with n de-grees of freedom in which the variations of all q1 for j = 1,

z [(6L/6qj) - dldt(6L/64') - (SD/64')] 6q. = 0.

i(43)

This is a general equation which applies to all dynamicsystems with any arbitrary number of degrees of freedom,and from which the motion equations for any specific sys-

tem can be readily found. Clearly, the procedure would beto first define a set of generalized coordinates to specifythe state of the system at any given instant of time t andthen to invoke the condition that the virtual variations ofthese coordinates, bqj at the given time t are independentof each other. When the condition is fulfilled, then one

gets a set of n simultaneous equations given by

[d/dt(&L/b,) - (L/6q.) + (8D/lb')] = 0. (44)

These are called Lagrange's equations of motion. But, insituations where not all the 6q1 are found to be independentof each other, then the analysis becomes complicated since,in that case, not all the coefficients of (43) can be equatedto zero. The dual-spin system of Fig. 2 may be consideredto further illustrate this point.

By assuming that the mass centers of the two bodies Band B' are coincident and are located at the point of origin0 of the inertial reference system XYZ and also assuming,for the moment, that the two bodies B and B' are not con-

strained to have a common axis of rotation along the z (orz') body axis direction, one can define two independentsets of Euler angles-one set 4, 0,y' for the body B, whilethe other set 4', 0', 4' is for the body B'-and then use them

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO.1 JANUARY 1979

(41)

(42)

44

as the generalized coordinates for the two-body system con-sidered. Thus, for the dual-spin system of Fig. 2 with noconstraints and no dissipative forces present (Q, = 0), onecan write the following from (43):

[d/dt(8L/6 4)- (L/64,)]680 + [d/dt(6L/6) - (6L/60)]680

+ [d/dt(6L/64;) - (6L/6t)]64 + [d/dt(6L/6 ')

-(6L/64')]6' + [d/dt(6L/60') - (6L/60')]60'

+ [d/dt(6L/6q/) - (8L/6 7')16 4 ' = 0Fig. 3. Definition of Euler angles and their rates.

where the Euler angle sequence used for body B is illustratedin Fig. 3. A similar Euler angle sequence (not shown) is alsoused for body B'.

Since in the above hypothetical system the virtual varia-tions of all the coordinates, viz., 6 ,, 60, 64i, 64,', 60', and64p, are independent of each other, it follows that all thecoefficients in (45) should be equated to zero, thus giving

d/dt(6L/6I) - (6L/6) = 0 (46)

d/dt(6L/6) - (6L/60) = 0 (47)

d/dt(6L/6) -(6L/64) = 0 (48)

d/dt(6L/64 ')-(6L/6 o') = 0 (49)

d/dt(6L/6 ') - (6L/60') = 0 (50)

d/dt(6L/ 4) (L/6 4") = 0. (51)

Note that, of the preceding set of Lagrange's equations,only (48) and (51) represent the torque equations corre-

sponding to the z and z' body axis coordinates, respectively.By noting further that in the actual system of Fig. 2 thesebody-axis coordinates (z and z') are in fact the same be-cause of the constraint that actually exists, one may readilyconclude that the virtual variations 6 and 6 4' in (45) mustbe identical and not independent as previously assumed.This yields

d/dt(6L/6) - (6L/6 4) = 0 (52)

ci/dt(6L/60) - (6L/60) = 0 (53)

d/dt6L/6 4')-(6L/6 4') = 0 (54)

d/dt(6L/6 6') - (6L/60') = 0 (55)

[d/dt(6L/6 4) - (6L/6 )] + [d/dt(6L/6l 4)

available in terms of the generalized coordinates. Note that,for the system of Fig. 2 with no impressed torque, theLagrangian L will contain only the kinetic energy term Tgiven by

T= W(I2 +I W2 +I 2 +I' +I2i+Ir 2 +I ,W2X X y y z z X X y y z Z

(57)

where the I represent the principal moments of inertia,while the Cw represent the body-axis angular rates for thesystem. So, in order to obtain T in terms of the generalizedcoordinates, one may use the following:

xx =4sin0 sin +O cos

W = sin 0 cos 4-O sin

Cz= f cos 0 +;

= ;' sin G' sin 4' + 6' cos 4'X

c = 4'sinG' cos 4'-0' sin 4,'y

CW', = cos 0' + 4;'.z

(58)

(59)

(60)

(61)

(62)

(63)

These relationships follow directly from the transformationscorresponding to the three Euler angle sequence used foreach body (Fig. 3).

Often, for the purpose of analysis, it is found more con-

venient to have the equations of motion expressed in termsof the co variables and as torque equations correspondingonly to body-axis coordinates. Thus, by noting from theabove that

(6coW/6P) = 1, (6W /6P) = W' (6CO /64) =-c

z X y ~~~~yX(64)

-(6L/6 )] = 0. (56) (6 co1' /64') = 1, (6o'16 ,) = co', (6J'/64' ) = -Xx

These are the true Lagrange's equations of motion for thedual-spin system of Fig. 2. Clearly, to solve these equationsin their present form, one requires the Lagrangian L to be

(65)

the torque equation corresponding to z (or z) body-axis

SEN: TOOLS Ol KINEMATICS: A CRITICAL REWIEW

z

(45)

45

coordinate may be found from (56), as follows:

[d/dt( T/6Iwz) + wX(6T/6wy) -wy(6T/15&X)j+ [d/dt(6 T/6cz',) + w' ,(6T/6Iw',)-Sy yo X

- Wo' (6,TIet'A) = 0.

It should be pointed out that, while the equation invol-ving the generalized coordinates ;' and P' given by (56)directly provides the torque equation corresponding to z(or z') body-axis coordinate, the remaining equations in-volving the other generalized coordinates and given by (52)-(55), do not likewise directly provide the torque equationscorresponding to the other body-axis coordinates, viz., x, y,x' andy'. llowever, by observing that choosing the z axisas the body axis in Fig. 3 is completely arbitrary, one mayeasily permute the indices in the development of the equa-tions (58)-(65), with the result

dI/dt(6T/ISw) + wy (6T/6ci) - w(6T/lw ) = O

d/dt(bT/l5wy) + wz(6T/ltc%) - to (6 TIeStz) = 0

H +Hwj -H w =0x z y y z

h +H w -Ho =0y x z z X

H', + H' ,c', - H' ,C', = 0x z y y z

(66) H',+H',&', H',w',=y X z z X

H +H X -H w +H H'+', w',z y x x y z y x

-H'Y,o', = 0.x y

(67)

(68)

d/dt( T/6w' ,) + w' ,(6T/l5o',)-to' T(eT/eo' ,) = ° (69)X y z z y

dldt('Tl'y,) + w'z,(6T/6o',)- w',(T/eo',)=0. (70)

The set of motion equations given by (66){70) can nowbe readily evaluated with the help of (57) and, followingsubstitutions ot Hx = Ixox, HY = Iycjy, Hz = Icoz, Hx=IJ'xto, H,l = I4tcoy and H'z = Iz'z', the evaluation finallyyields

(71)

(72)

(73)

(74)

(75)

It can be seen that the above equations correspond iden-tically to the set of motion equations (26){30) obtainedearlier for the dual-spin system of Fig. 2 by using the vec-torial approach of mechanics.

References

[I] ET. Whittaker, A Treatise on theAnalyticalDynamics ofParticles and Rigid Bodies. New York: Cambridge Univ.Press, 1959.

[21 W.T. Thomson, Introduction to Space Dynamics. New York:Wiley, 1963.

[31 D.T. Greenwood, Principles ofDvnamics. Englewood Cliffs,N.J.: Prentice-Hall, 1965

[41 H. Goldstein, Classical Mechanics. Reading, Mass.: Addison-Wesley, 1965.

[51 C. Lanezos, The VariationalPrinciplesofMechanics.Toronto: Univ. Toronto Press, 1970.

[61 L. Meirovitch, Methods of A nalytical Dynamics. New York:McGraw-llill, 1970.

[71 D.R. Ostbery and F.W. Perkins,An Introduction to LinearAnalysis. Reading, Mass.: Addison-Wesley, 1966, pp. 21-22.

[81 A.K. Sen, "On a recent breakthrough on the dual-spinstabilization theory," IEEE Trans. A erosp. Electron. Syst.,vol. AES-9, pp. 960-964, 1973.

Asim K. Sen (M'63-SM'72) was born in Calcutta, India, on January 2, 1939. He re-ceived the B.Sc. degree in physics in 1958, the M.Sc. (Tech.) degree in radio physics andelectronics in 1961, and the Ph.D. (Sc.) degree for work on control system analysis in 1967,all from the University of Calcutta.

From 1962 to 1965 he was a Research Fellow at the Institute of Radio Physics andElectronics, University of Calcutta, and from 1965 to 1967 he was a Lecturer in physicsat the University of Kalyani, West Bengal, India. From 1967 to 1969 he was at theUniversity of Manitoba, Winnipeg, Man., Canada, on a postdoctoral fellowship. From 1969to 1971 he was a National Research Council Postdoctoral Resident Research Associate ontenure at the NASA-Goddard Space Flight Center, Greenbelt, Md., and, after spending ayear with the Communications Research Centre, Ottawa, Canada, on contract employment,he founded his own research firm, Synchrosat Limited, in 1973 where he is presently thePresident and Head of the Satellite Research Division. He has published many papers inthe areas of spacecraft dynamics and stability theory for control systems. He is the au-thor of a forthcorming research monograph entitled "The GYROLITE Stabilizer: aBraakthrough of the Seventies."

Dr. Sen is a registered Professional Engineer in the Province of Ontario, Canada.

IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. AES-15, NO. 1 JANUARY 197946