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Annual Report for 1965 on Research Contract NAS 8-2619 NASA CR71308
APPLICATIONS OF CALCULUS OF VARIATIONS TO TFWECTORY ANALYSIS
Submitted to National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama
by M. G. Boyce, Principal Investigator J. L. Linnstaedter and G. E. Tyler, Assistants p *ai; ..%-a -k a*. Department of Matnema tics set t Lt Vanderbilt University a Annual Report for 1965 on Research Contract NAS 8-2619
APPLICATIONS OF WCULUS OF VARIATIONS TO TRAJECTORY ANAz;YSIS
Submitted to National Aeronautics and Space Administration Marshall Space Flight Center Huntsville, Alabama
by M. G. Boyce, Principal Investigator J. L. Linnstaedter and G. E. Tyler, Assistants
Department of Math ema t ic s Vanderbilt University Nashville, Tennessee
January, 1966 INTROD3CTION AND SUMMARY
During the year 1965 two reports were made at the two contractor conferences on guidance and space flight theory held at Huntsville in February and August. They were entitled "The Multistage Weierstrass and Clebsch Conditions with Some Applications to Trajectory Optimization," by M. G. Boyce, and "Applications of Multistage Calculus of VEriaticns Theory to Two and Three Stage Rocket Trajectory Problems," by G. E. Tyler. Also a paper was contributed to the Progress Report No. 7 on Studies in the Fields of Space Flight and Guidance Theory, NASA TM X - 53292, pp. 7 - 31, on "Necessary Conditions for a Multistage Bolza - Mayer Problem Involving Control Variables and Having Inequality and Finite Equation Constraints," by M. G. Boyce and J. L. Linnstaedter. A summary of the results in these reports, omitting proofs of theorems, is given in this annual report. Additional work has been done recently on gradient methods and on sufficient conditions, but results are as yet incomplete. The first section of this report contains a modification of the Denbow multistage calculus of variations theory, allowing for discon- tinuities at stage boundaries. This modification is an intermediate step in passing from the classical theory to a form readily applicable to trajectory optimization. The second section summarizes extensions of the multistage theory to problems involving control variables and having inequality and finite equation constraints. The Mayer formulation is used, and differential constraints are taken in normal form since in trajectory problems the equations of motion are in such form. A three stage re-entry problem is treated in Section 111. This example serves to partially illustrate the theory of Section 11. To avoid computational complexity, simple intermediate point constraints are assumed and first order approximation to gravitational attraction is used. 4. 2
Discontinuities will be allowed in the functions appearing in the differential equation constraints and in the dependent variable coordinates defining admissible paths Let t be the independent
variable. For fixed p, define a set of variables(to, tl, eO., to be a partition set if and only if t < tlcu. < t Let I - 0 P denote the interval t < t < t and I the subinterval ta,l <- t < ta 0- - p a for a = 1, .*., p - 1 and taWl--< t < ta for a = p. Let z(t> denote the set of functions (z,(t), '.., zN(t)), where each za(t), a = 1, *.., N, is continuous on I except possibly at partition points
tl, e.. t At these points right and left limits z (t-),z,(t,), + +'-!'are assumed to exist and we let za(tb) = az,(tb), 1+ b = 1, 0.. z a(t p-1 ..., p - 1. The problem will be to find in a class of admissible arcs
t 0- satisfying differential equations t in I~, f3 = l,*.*,M< N, (1) $aB (t,z,i) = 0, and end and intermediate point conditions one that will minimize Let R a be an open connected set in the 2N+1 dimensional The (t,.,;) space whose projection on the t-axis contains Ia a functions e' B are required to have continuous third partial deriva- I 1 Is assumed of rank M in R tives in Ra and each matrix a 3 I Let S denote an open connected set in the 2Np+p+l dimensional space of points (to, ,t z (to),z ,z *. ,z (t,)) in which the functions P , (ti) it,),+ f , p = 0, 1, e. , K have continuous third partial derivatives and the P matrix is of rank K+1. An admissible -set is a set (t,z,;) in R for some a=l,oo.,po I a An admissible subarc C is a set of functions z(t), t on Ia, with each a (t,z,?) an admissible set and such that z(t> is continuous and k(t) I is piecewise continuous on An admissible arc E is a parti- ‘a * I tion set (to, t ) together with a set of admissible subarcs ..., P + -. ‘a’ a = 1, ..., p, such that the set (to,..., t ,z(t ),z(t;),z(tl) ,...,z(tp)) I PO is in S e Multiplier Rule. An admissible arc E’ that satisfies equations (l), (2), (3) is said to satisfy the multiplier rule if there exist con- stants e not all zero and a function P F(t,z,i,h) = h BB$a(t,z,i), t in Ia, with multipliers h,(t) continuous except possibly at corners or dis- continuities of E’, where left and right limits exist, such that the fol- lowing equations hold: + Every minimizing --arc must satisfy -the multiplier -rule. An extremal is defined to be an admissible arc and set of muitipiiers satisfying equations (1) and (5) and such that the functions ia(t), (t) have continuous first derivatives except possibly at par- B tition points, where finite left and right limits exist. An extremal is -non-singular in case the determinant is different from zero along it. An admissible arc with a set of multi- pliers satisfying the multiplier rule is called normal if e = 1. With 0 this value of eo the Set Of multipliers is unique. Weierstrass Condition. An admissible arc E’ with a set of multi- liers (t) is said to satisfy the Weierstrass condition if -B 1 holds at every element (t,z,i,h) -of E’ for all admissible sets (t,z,i) satisfying the equations fl; = 0. Every normal minimizing arc must satisfy the Weierstrass condition. ,?‘Lebsch Condition. An admissible arc E’ with a set of multipliers is said to satisfy the Clebsch condition if B (t) F. (t,z,i,h) flu% 2 0 za% *. 5 holds at every element (t,z,i,X) -of E' for all sets (nl,, a e ,%) satisfying the equations a .- r\ ,z ) 31a '- J. @Pia (t ,i Every normal minimizing -arc -must sati s f y -the Clebsch condition. 6 By the introduction of new variables and by notational transformations the theory of Section I can be utilized to es- tablish necessary conditions for the more general formulation of this section. As before, let t be the independent variable and define a set of variables (t ..., t ) contained in the range of 0' P t to be a partition set if and only if t < tl < < t e Let I 0 P denote the interval to -< t -< t p' and let Ia denote the sub-interval t < t < ta for a = 1, p 1 and ta,l < t < ta for a = pa a.-1 - ..., - - - Let x(t) denote the set of functions (xl (t) , ..*, x n (t)). For each i, i = 1, n, assume x (t) to be continuous on I ..., i except possibly at partition points tb, b = 1, p - 1, where finite left and right limits exist; denote these limits by xi(tb) + and xi(tb), respectively. The amount of discontinuity of each member of x(t) at each partition point will be assumed known, and we write + with each dib a known constant. Also let xi(tb) = xi(tb)* Thus x.(t) is continuous at t, if and only if dib = 0. 1 Let y(t) denote the set (yl(t), ym(t)), where y.(t) is J piecewise continuous on I, .j = 1, .*., m, finite discontinuities being allowed between, as well as at, partition points. In the formulation of the problem the y.(tj will occur only as undifferentiated variables J and will not occur in the function to be minimized nor in the end and intermediate point constraints. Such variables are called control variables, while the x. (t) are called state variables. 1 The problem is to find in a class of admissible arcs which satisfy differential equations a 2 = Li(t,x,y,), t in I~, a = 1, p, i = 1, n, i *.., 7 finite equations Mi(t,x,y) = 0, g = 1, 0.0, q, inequalities N&X,Y) ,> 0, h = 1, ..*, r, q+r5m, and end and intermediate point conditions Jk(to, t x(t ), x(t;), ~(~1,+ x(tp)) = 0, ..., P' 0 ..., k = 1, ..., s C (n + 1) (p + l), + xi($)- - dib = 0, b = 1, ..., p - 1, one that will minimize In order to state precisely the properties of the functions in- be an open connected set in the m + n + 1 volved in the problem, let Ra dimensional. (t,x,y) space whose projection on the t-axis contains the interval Ia, and let S be an open connected set in the 2np + p + 1 dimensional space of points The functions L:, Mi, Nt are assumed continuous with continuous partial derivatives through those of third order in R a.nd a'. Jok J are to have such continuity properties in S. For each a, the matrix is assumed of rank q + r in R a,' where Da1 is an r by r diagonal a matrix with Nl, , Na as diagonal elements e The matrix . . , r is assumed of rank s -t 1 in S. s An admissible set is a set (t,x,y) in Ra for some a = 1, ..., p. An admissible sub-arc Ca is a set of functions x(t), y(t), t on Ia' with each (t,x,y) admissible, and such that x(t) is continuous and ?(t), y(t) are piecewise continuous on Ia . An admissible arc is a partition set (to, ..., t ) together with a set of admissible sub-arcs P + C , a = 1, a.. , p, such that the set (to, , t x(to), x(t;),X(ti),. ,x(tp)) a ... P' . is in S. On introducing a generalized Hamiltonian function H as defined below and utilizing the normal form of the differential equation constraints, one can now apply the theory of Section I to obtain the following multiplier rule. The Multiplier Rule An admissible arc E for which is said to satisfy the multiplier rule if there exists a function with multipliers hi(t) ' %3 (t),v h (t) continuous except possibly at partition points or corners of E, where finite left and right limits exist, such that for each t in In,- a = 1, ...) pL a a I@= 0, NE > 0, HX dt + ci, H = 0, ki = Li, g - a-1 i yJ and such that the -- transversality matrix The multipliers V are zero when N > 0. Everx is of rank s + 1. h h minimizing arc E must satisfy the multiplier rule. 9 Between corners of a minimizing arc E the equations 2 =H hi=-H = 0, V H = 0 (not summed) i xi , X H i Yj 'h hold and hence also Tranavzrsality Conditions for Normal Arcs Under the usual normality assumptions, the transversality matrix can be put into a form having one fewer rows. This leads to the following statement of transversality conditions. For a normal minimizing arc the transversality matrix is of rank s. Since the matrix is of order s + 1 by (n+l) (p+l), the requirement that the rark be a iaiposes (ii+;> (p+;> - s conditions. This is one more condition than was imposed by (2), which was sufficient to determine the multipliers up to an arbitrary proportionality factor. Weierstrm ss Cnaditdon For a normal minimizing arc E the inequality XiLi (t,X,Y) 2 XiLi (t,x,Y) must hold at each element (t,x,y,X,p,v) -of E for all admissible sets (t,x,Y) satisfying; Mg(t,x,Y) = 0 -and Nh(t,x,Y) 2 0. Clebsch Condition For a normal minimizing arc E the inequality must hold at each element (t,x,y,X,p,v) of E for all sets satisfying M (t,x,y)flj = 0 and N (t,x,y)7[ = 0, where in the last - hYJ j gyj equation h ranges only over the subset of' 1, r for which Nh(t,x,y) = OQ a normal minimizing arc the multipliers v are all non- For h negative a 11 In this section the theory of Section 11 is applied to a three stage re-entry problem. Since it is primarily an illustrative ex- ample, certain simplifying assumptions are made. In particular, the vehicle is assumed to be a particle of variable mass, with thru-st mag- nitude proportional to mass flow rate and thrust direction subject to instantaneous change. Moreover, external forces are required to be functions of position only, while the earth is assumed spherically symmetrical and nonrotating with respect to the coordinate system of the vehicle. Finally, motion is restricted to two dimensions, gravitational acceleration is approximated by first order terms, and air resistance is neglected, The foregoing conditions allow the motion of the vehicle to be described by the following equations : -a2x + CB m-1 cos e, t < t < tl, ( 1 0- ;= { -a2x t 1-< t < t2, -a2x + e3 m -1 cos e, [ 3 tg 5 t 5 t3, -1 + 2a2y -+ CB m sin e, t < t < tl, (- -go 1 0- (-go + 2a2y + e3 m-l sin e, t2 5 t 5 t3, 3 g= u, t Jl to = 9, J2 E u (to)- uo =o, = v (to)= 0, J3 - J4 H x (to)= 0, J5 = Y (to)- Yo = 0, J6 X (t,) - X1 = 0, J y (t,) y2 = 0, 7 E - J8 E x (t,) - x3 = 0, Jlo E m (t,) - m3 = 0, a nd m it,,-\ - m \tljI.+\ = dl, The function to be minimized is taken to be the sum of the times of the powered stages, that is, J 0 tl - to + t3 - t2. If B1. = B this is equivalent to requiring that the fuel used be 3’ minimized, or Jo - m (to)*The conditions J and J insure the 6 7 existence of three stages. The Multiplier Rule of Section I1 allows the following Hamiltonian to be written: 13 11 (-a2x + cBlrn-lcos e) + X, (-go + 2a2y + cBlrnllsin 6) + u -I- + C-B1), to < t < tl, X 3 X~V X 5 H= (-a2x) + (-g + 2a2y) + u + x4v, t < t < t2, x1 l2 x 3 1- Ca2x + CB m-lcos e) + X, (-go + 2a2y + CB m-'sin e) x1 3 3 -!- u -I- + (-B3), t2 <-- t < t30 13 X~V X 5 The Euler equations for this Hamiltonian are: il + 13 OJ A2 i-X4 = 0, t3 - a2A1 = 0, i4+ 2a2h = 0, 2 cBlrn-2 (A, cos 8 + sin e) = 0, I5 - 12 CB rn-l sin e COS e) = 0, 1 (9 - x2 for t in ctoJtl); il + l3 = 0, + 2a21 = 0, l4 2 -2 cB m cos 8 f X2 sin = 0, $.5 - 3 (Il e) @B rn-l (1, sin e 1 cos e> = 0, 3 - 2 for t in It2, t31 14 Simple techniques for integration allow these equations to be expressed in integrated form as follows: = A~ sin a(t -t 11 c,), x2 = A2 sinh &(t + C2), x3 = -a% cos a(t + c~), for to -< t < tl; l4 = -aA;& cosh E& (t f Ci), for t -< t < t2; and 11 11 sin a(t + C1 ), hl = A1 h2 = A; sinh &(t + Cd ), 11 11 x3 = -Altos a(t + c~), h4 = -ad & cosh &(t f CL ) , for t.2- < t -< t-.5 It, is clear in expressing xi; a.2J ),4 as functions of time with two constants of integration, that the last two Euler equations in stage 1 and stage 3 have been ignored. These equations together with the Weierstrass condition will be used to expressed the control angle as a function of the multipliers X1 and h From the 2' last Euler equation of stage 1 and stage 3 we have (for hl # 0, cos 8 # 0) tan = l2/hl and hence and From the Weierstrass condition of section 11, cB m -1 1 (1, cos 8 + x2 sin 8 - x1 cos a - x2 sin a) -> 0 for to -C t C t. Here 8 is the control angle that actually optimizes, and a! ranges over all possible control angles for which the original equations of motion are satisfied. This expression being non-negative is equivalent to maximizing the following function (with respect to a): 2% < 0 which gives Thus -=Ja?W 0 and - - 3s sin a + cos a = 0 -A1 12 and -Al cos a - X2 sin a -< 0. thus tan a = A2/hl and I , -.--..- I which implies that cos a = /J- 1;' + and similarly that 1I sin a = ~2/~~~e Hence the control angle 8 is expressed as follows : for stage 1. The same expressions for control angle 8 hold for stage 3. The fifth Euler equation on stage 1 and stage 3 becomes The transversality mat$ix which is given at the end of this section has eleven rows and twenty-four columns and is of rank ten, From this matrix fourteen end and intermediate conditions are found. These conditions imply that all multipliers, except possibly at tl and h4 at t2, are continuous across staging times. h3 16 Also the following condition holds at tl: CB mml (llcose + A2sine) + i5 (B~)+ (A+3 - 1;) u + 1 = o 1 -!- + where 1 = X (tl). 33 A similar condition that holds at t is: 2 + CB m-l (X1cosB + 1 sine) - 1 (B ) + (Aq - 1;) v - 1 = 0. 3 2 53 The other four conditions implied by the transversality condition are: h5(to) = 0, ~i(t3)= 0, ~2(t3) 0, - H(t ) + 1 = 0 3 An optimal trajectory for this problem requires the finding of fifteen constants of integration from the equations of motion, a like number from the Euler equations, and the four times t tl, t2, and 0 , t Fourteen transversality conditions, ten end and intermediate 3' conditions, and ten requirements on state variables at staging points provide the necessary number of conditions for the determination of these constants. It is possible to start at the last stage to determine the integration constants for the Euler equations in terms of multiplier values. The constants for the third stage are: 11 is the final value of X ) , 4=-133/a, (X33 3 A, = 2 I1 c:= c = -t 1 2 3" Because of the continuity of X 1 and 13 at t2, I1 A= 1 11 I c: := 1 e 11 The values for A2 and Cr only hold for the third stage. To proceed 2 from the third stage back into the second stage we need the value of the difference This can be found from the transversality condition above which holds at tp, Supposing this equation solved, the deter- mination of constants A' C' for the second stage can proceed, and 2' 2 these values also hold for the first stage for 1 and A4' An 2 analogous procedure is applied to 1 and h for the first stage. 1 3 18 TRANSCTERSAZ;ITY MATRIX FOR THm STAGE PROBLEM H (to) -1 H (tl I-'t-1 H ( t2 1I -+-1 -H (t3 )+1 -I1 (to1 -I2 (to1 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 i 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -14(to) -Ig (to1 +J; X,(t,)l; 13 (ti)I ; 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 hg(tl)l; x1 (t& 12(t,) I: 13(t2) 1; 1dt2)I0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 c! A, (t31 I2 (t, 1 I3 (t31 I4(t3 1 (t, 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 0 1 0 0 0 0 0 1 0 0 0 0 0 1 SECTION IV. CONCLUSI9NS AND RFICOMMENDATIONS The multistage theory swmnarized in Section I1 and illustrated in Section 111 involves multipoint boundary value problems. Closed form solutions are to be expected only in problems with simple con- straints. However, considerable practical information is obtainable from the general theorems. In particular, the transversality matrix may yield usable switching functions for determining coasting and powered stages, and the Weierstrass condition can be interpreted as the Maximum Principle of Hestenes and Pontryagin, which is especially important for optimal paths lying partially along region boundaries. The procedure in Section 111 of beginning with the final stage and successively determining the constants of integration for the several stages in terms of the Lagrange multipliers would seem applicable to more complicated problems. Trial estimates of the multipliers at the final point would determine an optimal trajectory which in general would not satisfy all intermediate and initial conditions. New estimates could then be made and the process re- peated. This suggests attempting to obtain a sequence of optimal trajectories converging to one that would satisfy all the multipoint boundary conditions, Another type of sequential process consists of using non-optimal solutions of the equations of motion, each satisfying the multipoint conditions, with the sequence converging to an optimal trajectory satisfying all conditions. Gradient, or steepest descent, methods would apply to this procedure, and further study is recommended. Some study has been made under this contract of analogues to the Jacobi conjugate point conditions for multistage problems. Further effort toward obtaining such conditions in computationally useful form is recommended. Also the combination of necessary conditions, suitably strengthened, into a set of sufficient conditions is desirable. Application of multistage calculus of variations theory to the optimization of six stage earth-moon trajectories is in progress under this contract. Special attention to the reduction of computational difficulties is needed. BIBLIOGRAPHY Balakrishnan, A. V., and Neustadt, L. W. (editors). Computinq Methods in Optimization Problems. New York: Academic Press, 1964 Bliss, Gilbert Ames. Lectures on the Calculus of Variations. Chicago: University of Chicago Press, 1946. Boyce, M. G. and Linnstaedter, J. L. "Necessary Conditions for a Multistage Bolsa-Mayer Problem Involving Control Variables and Having Inequality and Finite Equation Constraints, I' Promess Report No. 7 on Studies in the Fields of Space Flight and Guidance Theory. Huntsville, Alabama: NASA-MSFC, Aero-Astrodynamics Laboratory, 1965. Denbow, C. H. "A Generalized Form of the Problem of Bolza," Contributions to the Calculus of Variations 1933-1937. Chicago : University of Chicago Press, (1937), pp.449 - 484. Hesteness, Magnus R. A General Problem in the Calculus of Variations with Application to Paths of Least Time. U. So Air Force Project Rand Research Memorandum RM - 100. Santa Monica, Calif.,: Rand Corporation, March 1, 1950 Leitmann, G. "Variational Problems with Bounded Control Variables," Optimization Techniques e New York: Academic Press, 1962, pp. 171 - 204, Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V., and Mishchenko, E. F. The Mathematical Theory of Optimal Control Processes. New York: Interscience Publishers, 1962 Tyler, G. E. Applications of Multistage Calculus of Variations Theory to Rocket Trajectory Problems. Master's thesis, Vanderbilt University, Nashville, Tennessee, 1965.