GENERALIZATIONS AND APPLICATIONS OF THE LAGRANGE IMPLICIT FUNCTION THEOREM

Transcription

1 AAS 8-3 GENERALIZATIONS AND APPLICATIONS OF THE LAGRANGE IMPLICIT FUNCTION THEOREM John L. Junkins *, James D. Turner, Manoranjan Majji Dedicated to F. Landis Markley INTRODUCTION The Implicit unction theorem due to Lagrange is generalized to enable high order implicit rate calculations o general implicit unctions about a nominal solution o interest. The sensitivities thus calculated are subsequently used in determining neighboring solutions about a nominal point, or in the case o a dynamical system, a trajectory. The generalization to dynamical systems, as a special case, enables the calculation o high order time varying sensitivities and the sensitivity o the solutions o two point boundary value problems subject to system parameter and boundary condition variations. The generalizations thus realized are applied to various problems arising in trajectory optimization. It was ound that useul inormation relating the neighboring extremal paths can be deduced rom the implicit rates characterizing the behavior in signiicant inite neighborhoods centered along the nominal motion. The accuracy o the solutions obtained is subsequently enhanced using a Global Local Orthogonal Polynomial (GLO-MAP) weight unctions developed by the irst author to blend many local approximations in a continuous ashion. Example problems illustrate the wide applicability o the presented generalizations o Lagrange s classical results to static and dynamic optimization problems. The Implicit unction theorem originally due to Lagrange is an important result in analysis[1, ]. Related methods or dierentiation o implicit unctions are even more important. The convergence o most versions o the Newton s method to solve nonlinear equations and the theoretical oundations o virtually all algorithms or solving nonlinear dierential equations (local existence and uniqueness o solutions) are closely related to these classical results. Other than serving as an important theoretical tool, the authors recognize that the implicit unction theorem and related concepts can be generalized to arrive at high order sensitivity equations or algebraic and dierential equations about a given pre-computed solution. This perspective enables the conception o methods that construct amilies o neighboring solutions and blend them together in a ashion that simultaneously improves the accuracy o the local approximations and guarantees global piecewise * Regents Proessor, Distinguished Proessor, Royce E. Wisenbaker 39 Chair in Engineering, Department o Aerospace Engineering, Texas A&M University, College Station, TX, junkins@aeromail.tamu.edu. Research Proessor, Director o Operations, Consortium o Autonomous Space Systems, Aerospace Engineering Department, Texas A&M University. turner@aeromail.tamu.edu. PhD Candidate, Department o Aerospace Engineering, Texas A&M University. majji@tamu.edu

2 continuity to a prescribed degree o partial dierentiation. Although the most elementary implicit sensitivities presented here are used widely in existing methods, the high order sensitivity calculation and the generalizations to dynamical systems yield new results and algorithms that signiicantly extend existing approaches. However there are several important results in the literature that are special cases o (or are closely related to) the implicit unction concepts that are developed in the present paper. In particular, the Classical Davidenko Homotopy Method [3, 4] is an important application o Lagrange s implicit unction theorem, used to obtain solutions o nonlinear optimization problems and to solve multi-dimensional root solving problems. A second order sensitivity calculation based on the implicit unction theorem was used by the irst author to derive second order sensitivities o distinct eigenvalues and eigenvectors o a matrix unction which has a locally smooth dependence on a parameter set [5-7]. A related irst order sensitivity method was developed by Malanowski and Maurer [8, 9] to investigate the parametric variation o solutions to constrained optimal control problems. Theoretical issues such as regularity are also discussed using the irst order sensitivity analysis developed by the same authors in companion papers [8, 9]. A recent result by Pinho and Rosenblueth [1] utilizes the implicit unction theorem to transorm a constrained optimal control problem to an unconstrained orm and proposes a solution approach. In the light o these theoretical and computational advances, it is elt that it is important to place these methods in a common theoretical ramework, investigate the high order sensitivities, and to develop more general methods, and in particular, develop methods that enable computation o extremal ield maps or neighboring optimal trajectories. Section II derives the high order implicit rates central to the paper. Ater presenting the scalar and vector versions o the high order sensitivities o the solution o the implicit unction equations, analogous solution sensitivities or linear and nonlinear dynamical systems are developed. The implicit rates developed are subsequently applied to several example problems in the next section. Section IV summarizes the current and uture eorts along with a discussion o the potential o the approach to improve the solution o large amilies o important dynamics and control problems. We conclude the paper with a summary o the results. GENERALIZED IMPLICIT FUNCTION THEOREM The Implicit unction theorem due to Lagrange is stated as : Lagrange's Implicit Function Inversion Theorem Given the equation: ( x) = y, where is analytic at x = a with d / dx, then the inverse unction is x = g( y) where g is analytic at y = b = ( x), Lagrange ound the general result: 1 n n n d x a ( y b) n 1 n 1 dx ( x) b n! x a y( a) b x = g( y) =, convergence depends on ( x). = = = Several versions o this classical theorem abound in the literature and the theorem has oundationally enabled the existence o solutions to ordinary dierential and equations (or theoretical considerations reer [, 11, 1]). The power o the theorem lies entirely in its generality and the applications spanning several spheres o engineering are a testimony to this act. The theorem and its proo are intimately connected to the idea o dierentiation o implicit unctions. In act, many o the needed tools or generalization are developed rom implicit unction dierentiation. To develop the oundational elements o the paper, let us consider a scalar algebraic root solving problem. Given p, we are required to solve ( ( ) ) y = ψ x p, p = (1)

3 The solution or a root x( p ) typically requires an iteration process (e.g., Newton s Method). Having arrived at a solution, Lagrange s implicit unction theorem enables us to analytically compute the x p sensitivity o this root, say ( ) * = x, with respect to the parameter, dx smooth and dierentiable, this is simply given by the chain rule applied to Eq. (1). dy ψ ψ dx = + = dp p x dp 1 dp. Assuming ( x ( p ), p ) ψ is dx ψ ψ = dp x p Now consider the ollowing generalization o the irst order implicit unction dierentiation by simply taking successive derivatives and recursively applying the chain rule. This leads us to the cascade o equations where the subscript ( ) * dy ψ ψ dx = + = dp p x dp d y ψ ψ dx ψ x ψ d x = = dp p x p dp x p x dp k k k d y ψ ψ d x = + + = k k k dp p x dp x= x has been dropped above and hereater or simplicity and it has been assumed that the partial derivatives are being evaluated at the nominal solution. Given that ψ ( x( p), p) vanishes at ( x( p), p ), then the vanishing o the derivatives as in Eq. (3) ollows rom variational arguments i x( p ) is a distinct root. Notice the kth derivatives o the x variable with respect to p are contained linearly in Eqs (3) and can thereore also be written as the ollowing cascade orm () (3) 1 dx ψ ψ = dp x p 1 d x ψ ψ ψ dx ψ x = + + dp x p x p dp x p ψ = + k 1 k d x ψ k k dp x p k 1 dx d x terms depending on... dp k 1 dp (4) Notice urther that the main condition or the derivatives above to exist is that the scalar (in this case) Jacobian o the nonlinear unction should be invertible, which is the requirement or the application o classical implicit unction theorem. In essence the high order derivatives explicitly constructed above, enable us to write the Taylor series or the answer x( p ) or neighboring solutions:

4 k n n d x p k+ 1 x( p) x( p) + + O n ( p ) (5) dp n! n= 1 p The high order derivatives presented above can be derived in the vector case (i.e., ( ( ) ) n m xψ, x p, p R, p R ), where the equations are given by 1 x ψ ψ =, i = 1,, r p x p i i 1 x ψ ψ ψ x ψ x = + + pi x pi x pi pi x p i 1 x ψ ψ ψ x ψ x = + + pi x pi x pi pi x p i (6) 1 x ψ ψ ψ x ψ x ψ x x = p p i1 i x p p x p p x p p x x p p k 1 k x ψ ψ = + p... p p... p i1 i x m i1 i m k k i1 i i1 i i i1 k1 k i1 i Example Demonstrations and Additional Tools: Example 1: Kepler s Equation As a irst illustration o the utility o the implicit derivatives, let us generate local Taylor series associated with a grid o roots o Kepler s equation, or various values o mean anomaly M and eccentricity e, and show that these local Taylor series approximations can be simply blended to provide an accurate interpolation that solves this equation or all practical purposes or all admissible (M, e). This is merely an illustration o one utility o implicit unction derivatives; we do not claim that this is the best way to solve Kepler s equation. This example, in addition to serving as a demonstration problem, is also used to show yet another tool we use to mask (and reduce!) the truncation errors caused by using the Taylor series (o equation (5)) in combination with the computed implicit rates. The tool is the partition o unity (PU) weight unctions developed by Junkins and Singla [13], briely summarized in the next section. The classical Kepler s equation ( M = E esin E) relating the mean anomaly M and eccentric anomaly E or a given orbit eccentricity e is given or elliptic orbits by Applying the ormula () using the nominal solution or implicit derivative M [ E( e, M ) esin E( e, M )] = (7) e = (note that ( ) e M = E ), leads to the = de sin E = = sin M de 1 ecos E e= e= E= M E = M Second and third order implicit derivatives can similarly be taken to yield (8)

5 de d E de 3 d E 3 e= E= M e= E= M = cos M sin M, = 3sin M cos M The high order terms agree with the classical series expansion solution derived by Battin [14] and the solution using Lagrange Inversion Theorem and can be expressed as as 3 e d e d 3 E ( e) = M + esin M + (sin M ) + (sin M ) +... (1)! dm 3! dm In the general case when expanding about some point where e, we must include the implicit partials o d E the cross terms (like dedm = = * * e e, M M ) which are generally non zero. Having computed the terms and using the Taylor series o the orm in Eq. (5), we can readily approximate the solution o the Kepler s equation about the nominal (with the usual convergence caveats). Within the domain e < , or the classical expansion about e =, convergence is reliable and outside this region, divergence is also reliable. Many other non-iterative methods (or instance [15, 16]) have been proposed to solve Kepler s equation over the entire domain. We demonstrate that a simple blending o Taylor series based on 3 rd order implicit derivatives accurately approximates the solution o Kepler s Equation by constructing the Taylor series approximants at a grid o (e, M ) points. However it is well known that Taylor series approximations diverge approximately radially and hence as we move away rom a nominal grid point (high order implicit derivative evaluation point in ( e, M ) space), the error introduced by such a Taylor series inversion is obviously ampliied as a unction o the distance rom the expansion point. So i adjacent nodes are placed such that the domains o convergence overlap and the convergence is to a suitable tolerance, then we introduce a judicious average o the overlapping approximations that can yield both a reduction in the truncation errors and accomplish rigorous piecewise continuity. This Global/Local (GLO-MAP) approximation process [13] yields an attractive piecewise continuous approximation that matches exactly the local Taylor approximations at the node where each is exact and osculates to a prescribed degree o continuity. The weights are a partition o unity and can be chosen to guarantee piecewise continuity through any order o partial dierentiation desired, and urthermore, it is logical to adopt weight unctions that ensure continuity consistent with the degree o the local Taylor approximations. It has been shown that the maximum approximation errors are very signiicantly reduced, and the approximation errors are made more uniorm by this averaging process. Figure 1 plots the second order implicit unction derivative approximated preliminary surace and the weighted PU averaged surace. Not surprisingly, the accuracy obtained depends on the spacing o the approximation nodes and the degree o the local Taylor series expansions. Similarly, Figure plots the e,.95, M, π ) by the third order preliminary approximant maximum error incurred (over all, [ ] [ ] (rom the local Taylor series approximations using implicit unction derivatives) and errors incurred by the PU averaging process applied to adjacent and overlapping Taylor series approximations. Note the odd unction symmetry about M =, so only one hal o the M range is required. As is evident the inal approximation errors (ater applying the averaging process) are approximately one order o magnitude more accurately, as measured by the maximum errors, than the preliminary Taylor series approximants. This is because the weight unctions have the eect o believing most the Taylor expansion associated with a node nearest the evaluation point, and systematically de-weighting the expansions as the distance increases rom the expansion point. The theory underlying these weight unctions oer a number o guarantees, and the numerical results o this paper indicate that the PU averaging process is highly eective in the amily o applications studied. (9)

6 Figure 1. Maximum Error Suraces : Implicit Derivative Preliminary Taylor Series Approximantions and GLO-MAP Averaged Functions Figure. Maximum Error (step size variation): Error Suraces or Implicit Approximations and GLO-MAP Averaged Functions

7 The averaged error distribution or a step size o ( he.1, hm.3) = = is shown in Figure 3. The nonuniorm errors evident in Figure 3 indicate that this simple approach yields a non iterative global solution or eccentric anomaly o suitable or sub cm computation o Earth orbiting satellites. Figure 3. GLO-MAP Approximation Errors o E vs (M, e), or 3rd Order Case (Step size >.1 both variables) Example : Linear Quadratic Regulator : (Fixed and Free time Problems) As a second illustration o the utility o the implicit unction derivatives, we consider one o the most important equations in linear control systems (the Riccati Dierential Equation). This equation must be solved i the perormance criterion involves the minimization o the L norm squared error o the control eort and the state variables with a ixed terminal time. However, the design is highly dependent on the plant and controller parameters, since the nonlinear dierential equation or the optimal state eedback gain history is a unction o the parameters appearing in the plant model. I these parameters change, conventional thinking would lead one to the necessity o re-solving the Riccati equation. For demonstration o the concepts, let us consider a scalar system together with ( ) t 1 1 min J = s x ( t ) + qx ( τ ) + ru ( τ ) dτ (11) subject to: t x t a x t bu t ( ) = ( ) + ( ) x t given, t ixed. The solution to this problem is a linear state eedback control b signal, u ( t) = s( a, t) x ( t), where the time varying state eedback gain is calculated by solving the r Riccati dierential equation backwards in time. (1)

8 b s = a s + s q (13) r with inal conditions s( a, t ) = s. So even or a scalar plant, the optimal gains satisy a nonlinear dierential equation. The gains thus calculated are typically stored or use by the controller to operate the plant in orward time. Notice however that, even or this simple problem, with a small change in plant parameters (say a = a + δ a ), the state eedback gains need to be recomputed i the perormance o the system cannot be sacriiced. Alternatively consider an implicit unction approach to the problem, that the state-eedback gain is an implicit unction o the varying plant parameters and so is the optimal state evolution, and thereore can be expanded in a Taylor series about the nominal trajectory (in this case the optimal path or the nominal parameter value) as s 1 s s ( a, t) = s( a, t ) + δ a + ( δ a) +... a! a a= a a= a x 1 x x( a, t ) = x ( a, t ) + δ a + ( δ a) +... a! a a= a a= a To determine the coeicients required in the above expansion, the dierential equations governing the closed loop together with the gain dierential equations need to be considered. We demonstrate the utility o such a calculation with an example problem with the nominal parameters as a b r q t = 1, = 1, =, = = 3sec, s = Experiencing a possible variation in the plant parameters in the range, a [.4,.5] δ about (14) (15) a. The gains predicted rom the sensitivity calculations presented are plotted in Figure 4 along with the exact solutions calculated or the corresponding parameters. It can be noted that the predictions and the exact calculations agree to plotting accuracy. Figure 4. State Feedback Gain Variation Predictions Using Implicit Function Theorem (Fixed Time Problem)

9 I we let the terminal time t, we are let to the ree time problem. It is well known that the steadystate state eedback gain in this case satisies the algebraic Riccati equation ( s ). b s a s q = (16) r Now let us consider this problem with both plant and controller parameters varying and compute implicit derivatives similar to the Kepler s equation example. Choosing the nominal parameters to be a = 1.5, b = 1.5, r = 1, q = (17) experiencing a possible variation in the plant parameters in the range, δ a [.5,.5 ], δ b [.5,.5] about a, b, we ound the ollowing. Errors incurred by the implicit unction theorem preliminary approximants and the GLO-MAP (Junkins Singla PU averaging method) averaged unction are plotted in Figure 5. Figure 5. Errors: GLOMAP Averaged and Preliminary Taylor Approximations in Log- Log Scale (Various nominal solution grid sizes considered as plotted in x axis) The two examples discussed so ar demonstrate the added advantages realized due to averaging, let us now look at a brie summary o the theory that makes this work out. GLO-MAP PU Averaging Method To illustrate the methodology, let us consider the above method applied on a grid in the parameter space (Consider the previous problem). About each node on this grid, a second order approximant o the gain as deined rom Eq (14). For the interior points our preliminary approximations to the gain can be constructed by using each corner node point as the nominal about which the coeicients o the expansion(14) are determined. To obtain a more reined approximation or the gains, a special (Global/Local Orthogonal Polynomial) GLO-MAP weight unction is used or blending the adjacent preliminary approximants. This interpolant or the D problem has the ollowing orm

10 Φ ( x, x ) = F J + F J + F J + F J 1 j1 j j1 j j1 + 1 j j1 + 1 j = i= j = 1 1 j 1 1 j1 1 J ( x, x ) F ( x, x ) i, j 1 j i, j + j or all the points in the region x x x, x x x + + unction approximant and J : J ( x 1, x ) i1 i i1 i, with F : F ( x 1, x ) j j 1 The GLO-MAP weight unctions orm a partition o unity j1 j j1 j (18) = as the preliminary = as the weight unction, valid within the region o interest. 1 1 i = j= J ( x, x ) = 1 and have been ound by the i, j 1 irst author to possess impressive properties, applicable to solution o partial dierential equations, scattered data interpolation and improvement o preliminary approximations by the said blending. The recent book by Singla and Junkins[13], documents these derivation and gives other applications o these results. While the earlier examples used local Taylor series to construct the preliminary local approximations, in the generalized setting, these approximations are quite general and may be any desired smooth unctions that are appropriate in a particular problem, and the nature o these unctions may be tailored to the local irregularity o the multidimensional unction being represented. Thus rom one perspective, the averaging process can be viewed as a way to blend arbitrary local approximations and thereby establish a piecewise continuous model rom virtually any desired amily o local approximations. GENERALIZATIONS AND APPLICATIONS TO DYNAMICAL SYSTEMS Having looked at the generalization to multivariate vector unctions and the applications o the implicit unction theorem to demonstration problems, we now generalize our developments to dynamical systems. There is no assumption in the theorem that would preclude its application to solutions o dierential equations. In act some proos o existence and uniqueness o solutions to dierential equations use the implicit unction paradigm as has been observed in the introduction. To aid this investigation consider a unction being given by ( t t ) y = ψ x(, x( ), c), p = (19) n where the unction o time x( t, x( t ), c) R is the solution o the system o dierential equations ( t) = ( t,, ) p1 x x c with initial conditions x = x ( t ) and a parameter c R. This parameter participating in the dynamics o the problem o interest can be a means to parameterize the control variable or can simply be some constant in the model o the system dynamics. Notice that there is another parameter vector p p R present in the terminal maniold constraint deinition equation (19), that might be used to represent a level surace o the terminal maniold o interest, or be any parameter in the terminal boundary condition constraint surace. This pervasive presence o parameters in nonlinear dierential equations (be their presence in dierential equations or in the boundary conditions), motivates us to apply the derived sensitivities to problems in dynamical systems and obtain an ininite set o neighboring solutions, thereby avoiding re-solving these neighboring problems at the cost o a loss in perormance, determinable a priori. Furthermore, we will see that the GLO-MAP ideas can be invoked to establish extremal ield maps o neighboring optimal solutions or a large amily o values o the system parameters. The idea o using partition o unity unctions to orm global solutions rom local solutions in a more abstract setting is presented by Abraham, Ratiu and Marsden[17]. The current approach o developing extremal ield maps using neighboring optimal solutions using a partition o unity approach is an instance o the abstract result presented therein. To motivate the thinking, assume that all boundary conditions and other parameters are speciied and p1 we have solved or c R to satisy the condition o Eq. (17). It is evident that c = c( p, x) is a unction

11 o the remaining parameters, e.g., ( p, x ) and thereore the issue is how to obtain an approximation o c = c( p, x ). We now demonstrate the sensitivity computations o implicit unctions involving a dynamical system solution with an example problem. Consider the implicit unction (19) and let us assume we are required to c c evaluate,. These ollow by starting with the chain rule derivatives o the implicit unction in x p question. y ψ x ψ c = = +, similar equation or x p x x x c x Using equation (), we see that we require the ollowing implicit partials () 1 c ψ ψ x ψ x ψ x = = x c x x x c x x 1 1 c ψ x ψ = p x c p x x where the sensitivities, = Φ( t, t ), = χ ( t, t ) are the classical irst order state and parameter x c transition tensors obtained by solving the dierential equations (1) Φ = x xnom χ = χ + x c xnom Φ xnom () Φ = I χ =. The kth order generalization o the state sensitivities with initial conditions, ( t, t ), ( t, t ) ollows exactly as in the scalar case, with the qualiication that the high order state and parameter transition tensors need to be solved compounding the notational requirements and will also dictate a computational challenge or the solution. This implementation challenges are worked around by automating the process by eiciently using symbolic manipulators working with a vector matrix representation o tensors similar to the methods used in a recent paper by the authors [18]. In this paper, the authors propose the use o Automatic dierentiation tool Object oriented Coordinate Embedding Algorithm (OCEA) [19] along with a vector matrix representation o tensors that utilizes Kronecker products and sparse matrix algebra tools, allows us to perorm the needed computation eiciently. Although still computationally intensive, the methodology proposed gives much more inormation and such inormation is oten unavailable by classical gradient based iterative schemes. The investigation o methods to acquire and utilize this inormation is a central aim in the current study. As will be evident automatic dierentiation and implicit dierentiation is a marriage made in heaven. Let us now look at the application o the generalization presented above to some problems o optimization. Dynamical System Demonstrations Example 3: Application to Terminal Maniold Parameter Optimization

12 Consider a problem with terminal constraint. The requirement o the controller is to minimize the given perormance index while satisying a given terminal condition exactly. Classical open loop solution involves a shooting method where one has to use optimizers to solve or the initial conditions o the costate vector. In such a situation, when the level set o the terminal maniold changes even a little, we need to solve the nonlinear optimization problem all over again []. The problem statement is as ollows: 1 min J = ( x1 ( τ ) + x ( τ ) + u ( τ )) dτ subject to: 1 t x ( t) = x ( t) x ( t) = u( t) and t ( ) Ψ x ( t ), x ( t ), p = x ( t ) + x ( t ) p = 1 1 with given initial conditions x1 ( t ), x ( t ) and a ixed terminal time, t, while the states are ree to lie on the constraint surace. The question we ask (and answer) is the ollowing. Let us assume we have a nominal * solution or p = p. Now i the level set o constraint surace changes by a little bit, can I use this nominal solution to produce an acceptable answer? Notice that the implicit unction theorem does not apply in a straight orward manner. We get around this by augmenting the given terminal constraint with an additional condition that comes out o the necessary conditions or an extremal. λ ( t ) = υx ( t ), λ ( t ) = υx ( t ) (4) where ( t ), ( t ) λ λ are the elements o the costate vector and υ is the terminal maniold constraint Lagrange multiplier. Notice that eliminating this υ, we have an additional condition that needs to be satisied at the terminal time, φ = λ ( t ) x ( t ) λ ( t ) x ( t ) = (5) 1 1 Thus taking the implicit derivatives, we can use the nominal solution and extend it to accommodate the terminal constraint variations. Simply including the irst order variations, we get agreement o plotting accuracy. These results are plotted in Figure 6 and Figure 7. (3)

13 Figure 6. Family o Terminal Controllers : Implicit Derivatives in Phase Space Figure 7. Family o Terminal Controllers : Implicit Derivative Based Realization in Phase Space (Blown Up)

14 Example 4: Orbit Transer Problem : Controller Parameter Variations (Thrust and Final Time) Consider the ollowing maximum radius transer optimal control problem, a classical problem introduced by Bryson and Ho[]: max subject to: ( ) υ r = u, θ = r T sin u = + ( φ ) ( φ ) υ µ r r m m t uυ T cos υ = + r m m t (6) and r t ( ), ( ), υ ( ) µ Ψ 1 = u( t ) = ; Ψ = υ( t ) = r t with initial conditions r = r u = = where the terminal time t is assumed to be ixed. Notice that these equations are the simpliied equations governing the three body problem (simpliication involves considering only the planar motion or more details reer [1], three dimensional maneuvers are more complicated and methods or such problems are discussed by [] and the reerences there-in). We list here Bryson and Ho s necessary conditions or optimal control as Hamiltonian ( t) 1 u = tan λυ µ r ( ) ( ) ( φ ) υ µ T sinφ uυ T cos H = λr u + λu + + λ υ + r r m m t r m m t Co-state Equations υ µ uυ λr = λu + 3 λυ r r r υ λu = λr + λυ r υ u λυ = λu + λυ r r Optimal Control φ λ (7)

15 Together with conditions at terminal time, t ( t ) λ υ µ λ ( t ) = 1 + λ ( t ) = v, λυ ( t ) = v r ( t ) r 3 u 1 The conditions at the terminal time which have to be satisied by the optimizer in a shooting type method were treated as the basic implicit unctions to derive the derivatives. Figure 8 plots a nominal solution to this problem (this solution happens to be a minimum time solution or the given parameters), * * with simulation parameters, m = 1, T =.145, t = TU, m =.7487 giving us a solution o max ( ) r t = 1.54 AU. The canonical units (see Bate[3]) have been employed in the non - dimensionalization o the problem and the solutions. The implicit unction theorem predicted variations and T * + δt = T * +.,. the true variations o the optimal solution with changes in the thrust parameter [ ] are plotted in Figure 9. We obtain plotting accuracy with irst order sensitivities. In the case when we calculate the sensitivities o the BVP with respect to two parameters, Thrust, T and the inal time, * * t = t + δ t = t +.1,.1,we obtained the results plotted in Figure 1. [ ] (8) Figure 8. Nominal Solution o the Orbit Transer Problem

16 Figure 9. Optimal Solution Variation with a single plant parameter, Thrust (True and Implicit Approximated trajectories) Blown Up View Figure 1. Optimal Solution Variation with two parameters o BVP, Thrust T and inal time t (True and Implicit Approximated trajectories)

17 The situation analyzed in this problem is very simplistic and problems o interest in trajectory optimization are oten more challenging. For instance, the departure and arrival orbits were assumed to be circular and this consideration allows the analyst to disregard the phasing inormation. In particular, the phase is vital and is the mechanism by which initial and inal time are associated with, or example, the Earth and the motion o the target celestial body. More realistically, one has to make sure the target object at least has a position matching condition at the terminal time and phasing typically dictates the launch window or which a mission is easible. Such a problem is setup in the next example. With a sequence o these simple problems, it is aimed that we obtain signiicant insight on the undamental physics o the problem, which allows us to add complexity one at a time and obtain open loop solutions to complex problems in trajectory optimization (ollowing similar process as outlined in []). We also mention that in Figure 1, we show only one solution and it s neighborhood, but we can clearly use GLO-MAP to continuously blend such local approximations blend to span a amily o thrust and inal times as illustrated elsewhere in this paper Example 5: Orbit Transer Problem: Transer to a Coplanar Eccentric Orbit We now consider a problem where the phasing inormation and the departure and arrival times are important to account or. Preliminary results obtained are discussed as ollows. The target orbit was obtained by rotating the orbit plane o the asteroid Apophis [4, 5] to be co-planar with the Eccliptic (or simplistic illustration in the current discussion). The time rame o interest was assumed to be in January o 11. To gain initial insight in to the physics o the problem and to ascertain convergence uniormly with a single initial guess on the initial costate, we report the results obtained by considering a launch during the irst ten days, starting January 1, 11, and times o light to Apophis ranging rom 145 through 195 days. The problem considered can be ormally stated as, 1 min subject to: ( ( u( t ) uasteroid ) + ( υ ( t ) υasteroid ) ) υ r = u, θ = r T sin u = + ( φ ) ( φ ) υ µ r r m m t uυ T cos υ = + r m m t and ( ), θ ( ) Ψ = θ ( t ) = θ ; Ψ = r( t ) = r 1 Asteroid Asteroid with initial conditions r t = r t = θ Earth at t Earth at t The necessary conditions or optimality yield: ( ) υ ( t ), u t =, = µ ( ) r t (9)

18 Hamiltonian ( t) ( φ ) υ υ µ T sinφ uυ T cos H = λr u + λθ + λu + + λ υ + r r r m m t r m m t Co-state Equations υ µ uυ υ λr = λu + λ,, 3 υ λ + θ λ θ = r r r r υ υ u λθ λu = λr + λυ, λυ = λu + λυ r r r r Optimal Control φ = tan 1 λu λ υ along with the boundary conditions, r ( t ) rasteroid, θ ( t ) time λu ( t ) u ( t ) uasteroid, λυ ( t ) υ ( t ) υasteroid Asteroid (3) = = θ and the conditions or costates at inal = =. A nominal solution to the problem with parameters o the spacecrat chosen as m * 1, T.45, = =. t * =.473 TU, m =.7487 lead to the nominal trajectory plotted in Figure 11. Note that rom the speciication o the problem, we can iner that the problem does not have a hard constraint on the target velocity. This was intentional, as the purpose o the problem was to analyze a mission setup or an initial continuous thrust phase to reach the target, ollowed by an impulse to correct or the velocity. As an impulsive maneuver is expensive and a heavy thruster needs to be carried, an investigation was initiated to explore the entire times o light and departure times to isolate regions with smallest correction required. The solution thus determined would lead to identiication o mission critical times (launch window) similar to the two impulse Lambert problem solution mission design (using Pork-Chop plots to locate easible launch windows).

19 Figure 11. Orbit Transer Example: A Nominal Solution Initial simulations reveal that there is a threshold minimum time below which no initial guess can lead to a convergent solution. Physically this reachability can be interpreted as the least amount o time possible or a given thrust level to accomplish the transer by the given inal time. So ar, only a small raction o region o interest has been explored. Figure 1 plot the converged solutions to the initial costates or the solution o the problem as a unction o the initial departure time and arrival time. The details o the suraces indicate the nonlinearity o the problem. Figure 13 plots the impulse requirements or the region under study. Note that the impulse requirements go to zero as we approach transer time o 17 days with a departure on 1 st o January, 11. Notice, however, we generated the amily over +/- 5 day variation in the launch date and +/- days variation in the time o light. The insight gained rom these computations indicates that the approach is easible and, or example, allows one to initiation insightul exploration o the neighboring extrema in the vicinity o the nominal extremal trajectory (or the nominal departure date and time o light constraint). However, we have not yet attempted to globalize these local solutions, and based on prior experience, we anticipate holes in the space o possibilities were no easible solutions exist (exactly analogous to the impulsive Lambert type transers with constraints on the magnitude o the initial and inal velocity impulse).

20 Figure 1. Initial Costates (All 4 Components: plotted as unction o initial and inal time) Figure 13. Magnitude o Impulse required or the Rendezvous to the Target (Continuous Thrust ollowed by an impulse)

21 Applying the implicit unction theorem, we can calculate the irst order sensitivity o the initial co state vectors with respect the plant parameter, thrust (T). These are plotted in Figure 14. Figure 14. Initial Costate Sensitivity wrt Thrust Value (All 4 Components: plotted or dierent initial and terminal times) CONCLUSIONS In this paper, we present generalizations o the classical implicit unction theorem and derive high order sensitivities o solutions to systems o nonlinear equations. More speciically, the idea o implicit unction dierentiation was moved into new settings, interpreting the underlying implict unction relationship as any nonlinear continuous and dierential mapping. It was observed that the generalizations span is extended to include implicit unctions involving solutions o systems o ordinary dierential equations, both initial value problems and two-point boundary value problems where the parameters that are the argument list or implicit dierentiation can include boundary condition parameters as well as parameters contained in the dierential equation model. Explicit ormulas or the high order implicit derivatives were obtained leading to the ormal inversions that allow the computation o high order sensitivities o implicit unctions in terms o parameters in either the dierential equations or the boundary conditions. The resulting equations were arranged in cascade orm that is attractive or eicient recursive computation. It was also ound that recent progress in the utilization o operator overloading to automate the simultaneous derivation and computation o partial derivatives played a signiicant role in mitigating the curse o dimensionality. The high order sensitivities thus derived were applied to a range o problems, ranging rom

22 approximating the inverse solutions o transcendental equations; important in both static (Kepler s equation and Riccati equation) and dynamic optimization problems (orbit transer). Several application problems o immediate interest primarily in the exploration o extremal ield maps or optimal trajectories o dynamical systems. We augment the construction o local implicit derivative approximations with the GLO-MAP averaging o adjacent approximations to construct local/global approximations. The results obtained on several idealized applications orm a basis or optimism on the avorable impact these generalizations o the implicit unction theorem is likely to have on a wide class o problems in engineering and applied mathematics. REFERENCES 1. Hale, J., K., Ordinary Dierential Equations. Pure and Applied Mathematics, ed. R. Courant, Bers, L., Stoker, J. J. Vol. XXI. 1969, New York, NY: Wiley-Interscience.. Lagrange, J.L., Nouvelle Methode Pour Resoudre Les Equations Litterales (Pare le moten des series), in Euvres de Lagrange. 177, Gauthier - Villars, Imperium - Libraire: Paris. 3. Dunyak, J.P., Junkins, J., L., and Watson, Layne, T., Robust Nonlinear Least Squares Estimation Using the Chow-Yorke Homotopy Method. Journal o Guidance, Control and Dynamics, (6): p Krantz, S., G., and Parks, Harold, R., The Implicit Function Theorem: History, Theory and Applications., Boston, MA: Birkhauser. 5. Lim, K.B., Junkins, J. L., and Wang, B. P.,, Re-Examination o Eigenvector Derivatives. Journal o Guidance, Control and Dynamics, (6): p Junkins, J.L., and Kim, Youdan, First and Second Order Sensitivity o the Singular Value Decomposition. Journal o Astronautical Sciences, (1): p Junkins, J.L., and Kim, Youdan, Introduction to Dynamics and Control o Flexible Structures. AIAA Education Series, ed. J.S. Przemieniecki. 199, Washington, D.C.: AIAA, Inc. 8. Malanowski, K., and Maurer, H.,, Sensitivity Analysis or Parametric Control Problems with Control State Constraints. Computational Optimization and Applications, : p Malanowski, K., and Maurer, H.,, Sensitivity Analysis or Optimal Control Problems Subject to Higher Order State Constraints. Annals o Operations Research, 1. 11: p De Pinho, M.D.R., and Rosenblueth, Javier.,, Mixed Constraints in Optimal Control: An Implicit Function Theorem Approach. IMA Journal o Mathematical Control and Inormation, 7. 4: p Chicone, C., Ordinary Dierential Equations with Applications. Texts in Applied Mathematics, ed. J.E. Marsden, Sirovich, L., and Antman, S. S.,. 6, New York, NY: Springer Publishing Company. 1. Moulton, F.R., Dierential Equations. 193, New York: Macmillan Company. 13. Singla, P., and Junkins, J. L.,, Multi-Resolution Methods or Modeling and Control o Dynamical Systems. Applied Mathematics and Nonlinear Science. Vol. In Press. 8, Boca-Raton, FL: CRC Press/ Chapman Hall. 14. Battin, R.H., An Introduction to Mathematics and Methods o Astrodynamics. AIAA Education Series, ed. J.S. Przemieniecki. 1999, Reston, VA: American Institute o Aeronautics and Astronautics Markley, F.L., Kepler Equation Solver. Celestial Mechanics and Dynamical Astronomy, : p Mortari, D., and Clocchiatti, A.,, Solving Kepler's Equation using Bezier Curves. Celestial Mechanics and Dynamical Astronomy, 7. 99(1): p Abraham, R., Ratiu, T., and Marsden, J. E., Maniolds, Tensor Analysis and Applications. ed. Applied Mathematical Sciences. 1, New York: Springer - Verlag. 18. Majji, M., Turner, J. D., and Junkins, J. L., High Order Methods or Estimation o Dynamic Systems Part 1: Theory, in AAS - AIAA Spacelight Mechanics Meeting. 8, Advances in Astronautical Sciences, American Astronautical Society: Galveston, TX. p. (In Print). 19. Turner, J.D., Automatic Generation o High-Order Partial Derivative Models. AIAA Journal, 3. 41(8).. Bryson, A.E., and Ho, Y. C., Applied Optimal Control : Optimization, Estimation and Control. Revised Printing ed. 1975, Washington, DC: Hemisphere Publishing Corporation. 1. Szebehely, Theory o Orbits, the Restricted Problem o Three Bodies. 1st ed. 1967, New York: Academic Publishers.. Nah, R.S., Vadali, S. R., and Braden, E., Fuel-Optimal, Low Thrust, Three-Dimensional Earth - Mars Trajectories. Journal o Guidance, Control and Dynamics, 1. 4(6): p Bate, R.R., Mueller, D. D., and White, J. E.,, Fundamentals o Astrodynamics. Dover Books on Astronomy. 1971, New York, NY: Dover Publications. 4. Junkins, J.L., Singla, P., Mortari, D., Bottke, W., and Durda, D.,, A Study o Six Near Earth Asteroids, in International Conerence on Computational and Experimental Engineering and Sciences. 5: Chennai, India.

23 5. Davis, J., Singla, P., and Junkins, J. L.,. Identiying Near-Term Missions and Impact Keyholes or Asteroid 9994 Apophis. in 7th Cranield Conerence on Dynamics and Control o Systems and Structures in Space. 6. Old Royal Naval College, Greenwich, UK: Cranield University Press.