AAS 18-282

THE THEORY OF CONNECTIONS APPLIED TO PERTURBEDLAMBERT’S PROBLEM

Hunter Johnston∗ and Daniele Mortari†

Lambert’s problem remains important in celestial mechanics for application suchas rendezvous, targeting, guidance, and orbit determination and has many robustalgorithms for solving this type of problem using Kepler’s equation of motion.Solutions to the unperturbed case have been studied extensively and are leverageand built upon by developing an algorithm that allows for the solution of the per-turbed Lambert problem by the addition of all perturbations at once. By doingthis, both analytical and numerical perturbation models to be handled in a uni-fied manner. Since at its core, Lambert’s problem is simply a two point boundaryvalue problem for the differential equation of motion, extension of the Theoryof Connections (ToC) for this case is immediate. In this paper, a new solutionto the perturbed Lambert Problem is developed using the Theory of Connection(ToC) where first Lambert’s problem is solved for the unperturbed case, then thissolution is adjusted for all incorporated perturbations simultaneously utilizing aconstrained expression. From this, a solution is produced with sub-meter accu-racy.

INTRODUCTION

Although the solution dates back to the 1700s, the rise of the space-age found specific interest inunderstanding and applying Lambert’s problem to a vast array of problems in celestial mechanics.With the straightforward application of this including rendezvous, targeting, guidance, and orbitdetermination, much research has been conducted to optimize the speed and efficiency of solvers.Of all of the fundamental work produced by Euler, Lambert, Lagrange and Gauss, a need for a robustalgorithm surfaced during this space age where Lancaster and Blanchard [1] revisited Lambert’sproblem and developed a solution that was valid for elliptic, hyperbolic, and parabolic orbits. Thisalgorithm reduced solving Lambert’s problem to computing an inverse of either a trigonometric orhyperbolic function. Not long after, an algorithm was developed by Gooding [2] which increasedthe speed of convergence while achieving high precision. Gooding’s algorithm, through extensivetesting, claimed primacy over all other method. Yet, this problem was revisited by Izzo [3], who,building upon the fundamental work of Lancaster and Blanchard, proposed a new transformationof time of flights curves which further simplified the problem. The algorithm developed by thismethod provided accurate solutions in a shorter time when compared to the Gooding algorithm. Inthis paper, the algorithm developed by Izzo is used as the initial input rk to the perturbed Lambertproblem solution.

Other studies have looked to incorporate perturbations such as the method of Particular Solutionswhich uses Modified Chebyshev-Picard (MCPI) Iteration [4, 5]. In this, the perturbed solution is∗PhD student, Aerospace Engineering, Texas A&M University, College Station, TX. E-mail: hunterjohnston@tamu.edu†Professor, Aerospace Engineering, Texas A&M University, College Station, TX. IEEE and AAS Fellow. E-mail:mortari@tamu.edu

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considered as,r(t) = rref + ∆r,

where rref defines a reference trajectory and ∆r is the departure motion, or the adjustment due tothe perturbations. With this, the departure motion differential equation defined by,

∆r = g(t, rref + ∆r, rref + ∆r) + u+ ∆r,

was then linearized to,∆r = A∆r +B∆r + O(∆2),

and is solved for using the MCPI method. In all, this method hold a major benefit because it doesn’trequire the computation of the state transition matrix.

Furthermore, recent work conducted by Armellin et al. [6] developed two unique solvers basedon on higher order expansion of the flow enabled by Differential Algebra. The first solver was de-veloped for when a Kepler solution was available and utilized the application of homotopy on theperturbations. While convergence of this method only takes a few iterations, it does not guaranteethe minimum ∆v solution. From this, a second method was proposed that removes the need for aninitial Kepler solution by using Differential Algebra techniques that allow for the high order expan-sion of the problem residuals. This method, however slower computationally than the homotopymethod benefits from not requiring an “initial guess” to the problem.

The technique proposed in this paper relies heavily on the development of the Theory of Connec-tions [7], the method to derive a constrained expression, and the process to apply this to solving adifferential equation [8, 9]. The next section provides background to ToC and a detailed explain ofits application to boundary-value problems.

BACKGROUND ON THE THEORY OF CONNECTIONS

At its heart, the solution of the perturbed Lambert problem remains to be a boundary-value prob-lem since the initial and final position are still desired to be the same as the initial Kepler solution.For this, much can be learned from the process of solving a boundary-value problem for a generaldifferential equation using the Theory of Connections (ToC), which can then be specified to theperturbed Lambert problem. ToC is an analytical procedure to derive general expressions, calledconstrained expressions, that always satisfy a set of k linear constraints. The general form of theconstrained expression assumes the form,

y(t) = g(t) +k∑

i=1

ηipi(t), (1)

where k is the number of constraints, g(t) is a freely chosen function, and the ηi coefficients arederived from the k linear constraints. Since g(t) is completely free, it spans all possible functionsfor y(t) which always satisfy the given constraints. This technique has already been applied to least-squares solutions of initial-value (IVP), boundary-value (BVP), and multi-value (MVP) problemson linear [8] and nonlinear [9] differential equations with machine error accuracy. In this specificcase, the main interest lies in the boundary-value problem. Let us consider solving a simplified caseof a general nonlinear second-order differential equation subject to boundary-value constraints

N (t, y, y, y) = 0

{y(t0) = y0y(tf ) = yf

. (2)

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From this, the constrained expression can be written from Eq. (1),

y(t) = g(t) + η1 + η2 t,

where η1 and η2 can be solved for by setting up a linear system of equations based on the twoconstraint conditions:

y(t0) = g(t0) + η1 + η2 t0

y(tf ) = g(tf ) + η1 + η2 tf .

The linear system becomes, {y0 − g0yf − gf

}=

[1 t01 tf

]{η1η2

},

which can be written in terms of the two unknowns,{η1η2

}=

1

tf − t0

[tf −t0−1 1

]{y0 − g0yf − gf

},

leading to the solutions:

η1 =tf

tf − t0(y0 − g0)−

t0tf − t0

(yf − gf )

η2 = − 1

tf − t0(y0 − g0) +

1

tf − t0(yf − gf ).

From this, the solution can be plugged into Eq. (2),

y(t) = g(t) +tf

tf − t0(y0 − g0)−

t0tf − t0

(yf − gf )− t

tf − t0(y0 − g0) +

t

tf − t0(yf − gf ),

and can be simplified to,

y(t) = g(t) +tf − ttf − t0

(y0 − g0) +t− t0tf − t0

(yf − gf ) . (3)

By analyzing Eq. (3), it can be seen that if t = t0 that the equation in fact simplifies to y(t0) = y0and if t = tf it becomes y(tf ) = yf , thus satisfying both constraints regardless of the functiong(t). Now, in order to solve the differential equation, the function g(t) must be “searched.” In priorwork, g(t) was represented by either Chebyshev or Legendre orthogonal polynomials since afterderiving the constrained expression polynomial basis allow for easy analytical differentiation andintegration. The free function then becomes,

g(t) = ξ Th(x)

where ξ is the unknown vector of coefficients to be solved and h(x) represents a vector of m basisfunctions and can be added to Eq. (3) leading to,

y(t) = ξ Th(x) +tf − ttf − t0

(y0 − ξ Th0) +t− t0tf − t0

(yf − ξ Thf ) . (4)

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Yet, in general the independent variable of t ∈ [t0, tf ] does not coincide with the range of bothChebyshev or Legendre polynomials which are strictly defined on the interval x ∈ [−1,+1]. There-fore, a change of variable is needed in order to correctly incorporate these basis functions where xis set linearly to t,

x = x0 +xf − x0tf − t0

(t− t0) ⇐⇒ t = t0 +tf − t0xf − x0

(x− x0).

With this, the first derivative of g(t) becomes,

dg

dt=dξ Th(x)

dt= ξ Tdh(x)

dx

dx

dt,

where it can be seen that dxdt =

xf−x0

tf−t0 ≡ c from the variable mapping and in general becomes,

dkg

dtk= ckξ Td

kh(x)

dxk

Using this method, all subsequent derivatives can be calculated. From this y(t) and its derivativescan be solved for analytically and plugged into the differential equation which yields a nonlinearexpression in the unknown vector ξ. For the resulting equation a discrete set ofN different values ofx (and from this also t) can be determined. In expressing the solution of a differential equations asa expansion of Chebyshev/Legendre orthogonal polynomial sets, the optimal distribution of pointsis provided by collocation points [10, 11], defined as,

xi = − cos

(iπ

N

)for i = 0, 1, · · · , N, (5)

where N is the number of point in the interval. As compared to the uniform distribution point, thecollocation point distribution allows a much slower increase of the condition number as the num-ber of basis functions, m, increases. Although for the solution of linear differential equations theconstant term ξ remains linear and can automatically be solved through least-squares, the nonlinearcase requires an extra step with a nonlinear least-squares technique to iteratively solve for ξ.

With an understanding of this development, the solution of the perturbed Lambert problem isonly a slight extension. First, Lambert’s problem is solved for the unperturbed case providing theKepler solution to the equation of motion. Then using ToC and techniques developed above a con-strained expression is written for the perturbed part of the solution which captures all perturbationssimultaneously.

PERTURBED LAMBERT PROBLEM

In the perturbed Lambert problem, it is desired to solve for the trajectory such that the body startswith position r0 and after a given ∆t arrives at the final position rf . Yet, if only the unperturbedLambert solution was used, the incorrect values of v0 and vf would be determined and the orbitpropagated subject to the perturbations would place the orbiting body some ε distance away fromthe desired final position. The solution of the for the true perturbed trajectory using ToC takes intoaccount the perturbations and adjusts the Kepler solution rk and provides the corrected trajectoryr = rk + δr. From this, the correct v0 and vf can be calculated to reach the desired position rf inthe given time. A visual representation of the problem can be seen in Fig. 1. In this, the dotted lines

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represent the unperturbed Lambert solution propagated under a perturbed environment, the dashedline represents the unperturbed Lambert solution propagated under no perturbations, and finally, thesolid line (although exaggerated) represents the desired solution. From this it can be seen that theToC solution provides linearly adjustments to the Kepler solution and that will be the basis for thedevelopment of the perturbed theory.

Fig. 1. Description of problem (the distances are exaggerated for illustration purposes)

Additionally, since the derived constrained expression can be easily differentiated, the adjustmentto velocity is simply δr = δv and is illustrated in Fig. 2.

Fig. 2. Velocity correction as a part of the perturbed solution

To start, let us consider the perturbed two-body problem, assuming that all the perturbations arecaptured in the term ap,

r = − µr3r + ap(r). (6)

Although many perturbations admit an analytical equation (planet harmonics and third-body effects)the ap does not need to be define analytically. Since the solution reduces to an iterative least-

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squares solution, ap can also handle numerical models such that the perturbation are given by alook-up table based on position (r), or even time. Additionally, ap can also be a combinationof analytical and numerical models allowing for increased flexibility and robustness of solution inpractical applications. The solution of this problem is searched as a sum of the Keplerian solution(unperturbed part) and the contribution generated by the ap(r) term (perturbed part), that is,

r(t) = rk(t) + δr(t) such that:{r(t0) = rk(t0) = r0r(tf ) = rk(tf ) = rf

. (7)

We assume rk(t) be known as, for instance, obtained using one of the existing approaches [2,3,12].The δr(t) term satisfies the boundary conditions, δr(t0) = δr(tf ) = 0. Using the ToC method [7],the term δr(t) can be written as the constrained expression,

δr(t) = δg(t) + η1 + η2 t, (8)

Where δg(t) is defined as a “free-function” and the variables η1 and η2 are solved subject to theboundary conditions. For this formulation the δr is zero on the boundaries and therefore the con-strained expression can be written as a system of linear equations as follows,

[−δg0 −δgf

]=[η1 η2

] [1 1t0 tf

],

and by matrix inversion, this system can be solved for in terms of η1 and η2,

[η1 η2

]=[−δg0 −δgf

] [1 1t0 tf

]−1.

The solution of this equation can be plugged back into Eq. (8) which leads to the constrainedexpression for δr(t),

δr(t) = δg(t)−t− tft0 − tf

δg0 −t− t0tf − t0

δgf , (9)

where δg(t) is a freely chosen function. Substituting in Eq. (7) we obtain,

r(t) = rk(t) + δg(t)−t− tft0 − tf

δg0 −t− t0tf − t0

δgf . (10)

This is an expression that always satisfies the constraints, regardless of the value of δg(t). The freefunction, δg(t), can now expressed as a linear combination of a set of basis functions. Chebyshev orLegendre orthogonal polynomials are excellent selections as basis functions. However, since theyare defined in v ∈ [−1,+1] a linear mapping from time, t, and the new variable, v, is applied. Thismapping is,

v = 2t− t0tf − t0

+ 1 ←→ t = t0 +(tf − t0)(v + 1)

2.

The free function can be expressed now in terms of the new variable, v, as a linear combination ofthe selected basis functions,

δg(v) =[ξx, ξy, ξz

]Th(v) = Ξh(v), (11)

where Ξ is a matrix of unknown coefficients to be solved for and h(v) is the vector containingthe selected basis functions. The unknown matrix, Ξ, which actually constitutes the solution of

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our problem, can be then estimated by least-squares for linear [8] and nonlinear [9] differentialequations. Using Eq. (10) and Eq. (11) we can rewrite the constraint expression in terms of the newvariable v which leads to,

r(t) = rk(t) + Ξh(v)− t− t0tf − t0

Ξhf −t− tft0 − tf

Ξh0,

where h0 = h(−1) and hf = h(+1). Using the relationship between t and v we obtain

t− t0tf − t0

=v + 1

2and

t− tft0 − tf

= −v − 1

2.

Therefore,

r(t) = rk(t) + Ξ

[h(v)− 1

2(hf + h0)−

v

2(hf − h0)

]. (12)

The second derivative of this equation leads to,

r(t) = rk(t) + Ξd2hdv2·(

dvdt

)2

= rk(t) + Ξh′′ · 4

(tf − t0)2.

(13)

where h′′ indicates the second derivative with respect to v. Using the derived constraint expressionand its derivative, Eq. (12) and Eq. (13), this assumed form of r can be plugged into the governingequation of motion, Eq. (6), and the unknown coefficients Ξ can be solved for using a nonlinearleast-squares approach [9]. From this, a solution is produced which absolutely satisfies the boundaryconstraints and produces a highly accurate solution for the overall trajectory.

A step-by-step road map of the algorithm is provided in Fig. 3 which summarizes the algorithmand key equations used in solving the perturbed Lambert problem. Following the process of thischart, the inputs to the algorithm are r0, rf , and ∆t. These input values are then used in a Lambertsolver (this study leveraged the algorithm developed by Izzo [3]), where only the the unperturbedequation of motion is used for the solution. Output from this is the Kepler solution, rk to theproblem. This becomes the input to the ToC method along with r0, rf , and ∆t. From this, followingthe process described above, all perturbations are captured in ap(r) and an initial estimation of thecoefficient vector ξ0 and the perturbations ap0 are computed. With this initial estimation, a iterativeleast-squares method is used to converge on the final values for ξ and ap. From this, ξ representsthe the correct function for g(t) and when plugged into the constrained expression represents thesolution to the differential equation absolutely satisfying the given constraints (or the inputs). Tofurther explain the ToC box in Fig. 3, a summary of the transformations of the equations used isprovided in Fig. 4. The left column represents the transformation of the constrained expression andthe incorporation of Chebyshev polynomials. The right column describes the change of variableand adjusted on the constrained expression with subsequent derivatives. Finally, the bottom rowrepresents both the constrained expression and its second derivative.

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Fig. 3. Detailed flowchart for the solution of the perturbed Lambert problem.

Fig. 4. Summary of transformations for use in the ToC Solution

NUMERICAL TESTS

In order to analyze the accuracy of this technique a method was used to create a “true solution”as a point of comparison. In order to establish this “true solution” a high accuracy solver (ode45

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with strict relative and absolute tolerances) was given values for r0, v0, and tf and used to solvefor a perturbed trajectory. The boundary conditions r0 and rf plus the time of flight, ∆t, were usedas the input to the algorithm which is the starting point of the flowchart shown in Fig. 3. Once theToC method produced a solution to the perturbed problem, a comparison was made between thissolution and the one produce by the unperturbed case.

Validation Test

As an initial test of the method, a simple perturbation model was used where ap(r) was consid-ered to be acting in the constant radial direction over the entire course of the flight. The magnitudeof this perturbation was consider to be 10−6 km/s2 (or the order of J2 perturbations) to closely mir-ror physical orders of magnitude. The initial conditions resembled that of an equatorial orbit and itwas expected most of the ToC correction would be exhibited in the radial direction along the flightpath. The inputs to the algorithm were, r0 = [7533.8, 0, 0] T km, rf = [−8855.4, −2386.5, 0] T

km, and ∆t = 70 minutes and the results of this test are displayed in Fig. 5.

Fig. 5. Magnitude of position error for the Kepler Solution and the ToC predictedtrajectory under constant radial perturbations.

Looking at this plot it can be seen that the ToC method converged with sub-meter precisionover the entire course of the flight. Having solved for r, the initial and final velocities were easilydetermined by differentiating the analytical constrained expression which can be easily calculatedsince it is described by a polynomial expansion. Doing this, Table 1 was produced displaying theerror in initial and final velocity for both the Kepler solution and the ToC solution. The velocityprediction accuracy closely mirrored the accuracy in the trajectory with both being on the order of10−11 km/s2.

Furthermore, to complete the analysis, components of the error are plotted in Fig. 6. As expected,most of the error (and thus most of the correction) lies in the radial direction of the orbit. This test,however, is limited because it does not represent a physical perturbations experienced during orbit.Therefore, the next section considers both J2 perturbation and third-body effects from the Moon.

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Table 1. Error in initial and final velocity prediction for Kepler and ToC solutions under constantradial perturbation.

Solution V0 [m/s] Vf [m/s]

Kepler 0.1127 1.9809

ToC 0.1332−10 0.0266−10

Fig. 6. Radial, tangential, and normal error of Kepler and ToC solution under con-stant radial perturbations.

Perturbation Models

As described in Montenbruck and Gill [13] the largest perturbations experience in Earth orbit aredue to oblateness contributions and third body effects from the sun and moon. Since for this methodall perturbations are captured in the single function, ap, one of each was incorporate to demonstratethe method as a linear combination,

ap = aJ2 + a$.

From Schaub and Junkins [14] an analytical expression for Earth J2 perturbations can modeled as,

aJ2 = −3

2J2

( µr2

)(reqr

)2(1− 5( zr )2

)xr(

1− 5( zr )2) y

r(3− 5( zr )2

)zr

, (14)

where J2 is the harmonic coefficient (1.082629×10−3 km/s2), µ is the Earth gravitational parameter,r is the position of the satellite in the Earth-centered inertial (ECI) frame, req is the Earth equatorialradius, and x, y, and z correspond to the component directions in ECI frame. From Vallado [15]third-body perturbations can be modeled as,

a⊕ = −µ⊕r3⊕

[r −

{3r · r⊕r2⊕

− 15

2

(r · r⊕r2⊕

)2}r⊕

], (15)

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where µ⊕ is the gravitational parameter for third-body, and r⊕ is the position vector of third-body(with magnitude r⊕) as measured in the ECI frame. For this test, the inputs to the algorithm were,r0 = [7533.8, 0, 0] T km, rf = [−8793.3, −0.1, −2477.4] T km, and ∆t = 70 minutes and theresults are displayed in Fig. 7 and Fig. 8. It can be seen that for this test, the ToC method producesa solution with sub-meter accuracy while absolutely satisfying both boundary points.

Fig. 7. Magnitude of position error for the Kepler Solution and the ToC predictedtrajectory under J2 and Moon perturbations.

Looking at the component error, it can be seen that most of the correction from the ToC meth-ods in the radial and normal component of the trajectory which is to be expected since the targettrajectory is very close to a polar orbit.

Fig. 8. Radial, tangential, and normal error of Kepler and ToC solution under J2 andMoon perturbations.

Additionally, Table 2 presents the error in predicted velocity at the initial and final points. Forthe ToC solution the error is larger than that of the first test, however, it is well below the accuracyneeded. Although not visible in the plot, the boundary values are exactly met from the constrainedexpression.

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Table 2. Error in initial and final velocity prediction for Kepler and ToC solutions J2 and Moonperturbations.

Solution V0 [m/s] Vf [m/s]

Kepler 5.3245 2.4489

ToC 0.0319−6 0.1096−6

FUTURE WORK

Although the tests successfully solved the perturbed Lambert problem, all tests were conductedfor short time of flights (under one orbit). Future work will focus on a robust method to solve forproblems with multiple revolutions. For longer times of flights the current method diverges dras-tically from the initial Kepler solution and converges on another trajectory. For example, considerFig. 9, where the red trajectory is the initial Kepler solution which completes just over one orbit.The blue trajectory represents the ToC solution, which has diverged from the rk initial estimate.

Fig. 9. Trajectory error in solution for ToC Lambert solution for long time of flights.

However, by checking the residuals on the differential equation shown in Fig. 10, it seems that thisis still a valid solution to the problem.

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Fig. 10. Time history of the differential equation residuals.

This divergence can be explained by the Lambert problem having 1 + 2N unique solutions.Currently the algorithm converges to another equally valid solution, however, in the development itis desired that the perturbed solution be the one closest to rk. Therefore, future work will look toincorporate a method to penalize the algorithm if it diverges from the baseline solution (rk). Forexample the penalty function could be expressed as,

p =

{∑Ni=1 δr

Tδr if k ≤ kmax

0 if k > kmax

, (16)

where kmax is a define parameter used to switch off the penalty function after a specified number ofiterations such that the algorithm will converge on the solution with the lowest δr value.

CONCLUSIONS

In this paper the perturbed Lambert’s problem is solved using the Theory of Connections and aniterative least-squares method. Building off of prior algorithms for the unperturbed case [2,3], thismethod adds the perturbation adjustment through a constrained expression which is at the heart ofthe ToC method. Initial results of this test short times of flight (less than one orbit) produce sub-meter accuracy of the trajectory, while the constrained expression, which is an analytical equations,easily produced the velocity of the spacecraft over the course of the trajectory. With this, V0 andVf were corrected for with equal precision to the trajectory. The test in this study covered a fewparticular case and in future work this technique will be generalize for multiple revolutions and amore extensive test of the algorithm will be conduct for both speed and accuracy with a comparisonto the current state-of-the-art.

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