Examining Geometric Integration for Propagating Orbit Trajectories with Non-Conservative Forcing Course Project for CDS 05 - Geometric Mechanics John M. Carson III California Institute of Technology June 1, 005 Abstract The precision of numerical integration greatly affects the ability to track an navigate a spacecraft throughout the solar system. In fact, precision of up to fifteen significant igits is require for accurate navigation of spacecraft missions heaing towar the outer planets. This paper explores the use of numerical integration techniques base symplectic an variational integrators to propagate orbit trajectories, plus an examination is mae of how to incorporate into these methos the non-conservative forces affecting the trajectories. These numerical techniques are of interest ue to their intrinsic ability to conserve quantites such as energy, momentum, the Hamiltonian, etc. from the inherent ynamics. Research Objective The ability to successfully navigate the solar system is irectly relate to how well a spacecraft state can be preicte. State etermination is accomplishe through both estimation from Earth-base observations, as well as through numeric propagation of governing equations of motion. Traitional techniques for numerical integration are RK (Runge-Kutta) an similar avance iscretization techniques that typically require small step sizes to ensure precision in the trajectory propagation of large solar istances. A benefit of these techniques is in their ability to incorporate non-conservative forces, along with starts an stops of the propagation in orer to inclue ifferent perturbations. The objective of this research is to implement symplectic an variational integrators for propagating orbit trajectories an to etermine the feasibility an methoology for incorporating non-conservative forces an perturbations into the evelope techniques. The approach to this research will be a step-wise increase in complexity of the orbit problem. The first step will be to formulate the Hamiltonian for a simple two-boy problem (involving a spacecraft in the gravity well of a single planet) with no external forcing other than the gravitational potential of the host planet (See simple schematic in figure 1). In aition, the integration scheme for a basic symplectic integrator will be evelope an formulate for this initial problem; the chosen symplectic integrator will be base on the mipoint rule. This first step will facilitate comparisons with classic RK methos (such as variable an fixe time step methos) since the two-boy problem has analytic solutions. Further, this step will enable an assessment of step size on numerical accuracy. The secon stage of this research will emonstrate that the chosen mipoint rule is not only a symplectic integrator but also a variational integrator. Through this extension, which comes 1
through the iscrete Euler-Lagrange equations (DEL) an the iscrete Legenre transform, a possible metho of incorporating non-conservative forces becomes apparant. The thir stage of the research will consier how non-conservative forcings enter variational integration schemes through the use of the Alembert Principle. Either the DEL-like approach just mentione will be consiere, or perhaps a return to the Hamiltonian-sie of mechanics will be necessary. Figure 1: Representation of Two-Boy Problem The secon step of the research will introuce impulsive forces at preset times in orer to etermine how symplectic versus RK propagation hanles the change in trajectory irection. Again, the analytic solutions for two-boy orbit mechanics are known, so meaningful comparisons can be mae. To implement the impulsive forces, both integrators will be stoppe, an a step change in velocity will be mae to current velocity values (thus imparting an impulsive force). The impulsive force is essentially bumping the energy (Hamiltonian) of the orbit to a new, constant level. The thir step of the research will be to etermine how to incorporate non-conservative forces into the equations of motion such that symplectic integration remains vali. The technique for oing this is not yet known, however, there are known classes of systems that conserve a Hamiltonian but not energy. Further, the notion of being locally Hamiltonian may play an important role in this implementation. This all remains to be etermine through the research. Equations of Motion The ynamics of an orbiting spacecraft uner the influence of only gravity obey a conservation of energy principal from which the equations of motion can be erive [1]. The energy (an Hamiltonian) for a spacecraft orbiting a planet is given by H(q, p) = 1 m p mµ q where q = q is the raius from the planet c.m. (center of mass), m is the spacecraft mass, µ is the gravitational parameter for a particular planets or boy being orbite, an p is the spacecraft momentum. The spacecraft mass is assume insignificant to the planet mass, so the spacecraft exerts no influence on the planet s orbit. Thus, the center of gravity of the two-boy system remains fixe at the planet c.m. The Hamiltonian in equation 1 governs the resulting equations of motion: q t = H p = 1 m p (1) ż = J H(z) () p t = H q = mµ q q 3 [ ] where z = (q, p) T 0 I an J =. Note also the secon-orer equation of motion that I 0 arises from the left han sie of equation : m q = mµ q q. 3
For spacecraft orbiting multi-boy systems (i.e.. systems with one spacecraft an multiple planets), a conservation of energy principal exists as well. However, the presence of nonconservative forces causes energy not to be conserve. Symplectic Integration For a n-imensional symplectic vector space (Z, Ω), let z = (q, p) Z enote canonical coorinates such that symplectic matrix J represents Ω. The evolution equations of a Hamiltonian system on phase space Z are given by [ where J = Proposition 1. 0 I I 0 ]. ż = J H(z), with z t=0 = z 0 The mapping φ : z n Z z n+1 Z base on the mipoint rule is symplectic, such that ż = z n+1 z n z = z n+1 + z n ( ) zn+1 + z n z n+1 = z n + J H (3) Proof. For this mapping to be symplectic, the Jacobian of the upate scheme must be symplectic: z n+1 = I + J ( ) zn+1 + z n H z n z n = I + 1 ( ) ( zn+1 + z n J H I + z ) n+1 (4) z n From equation 4, efine the following: where H is evaluate at A = 1 J H (5) ( ) zn+1+z n. Then, equation 4 can be re-written as (I A) z n+1 z n = I + A z n+1 z n = (I A) 1 (I + A) (6) where an assumption has been mae on the invertibility of I A. In other wors, the Hessian of H an the size of are such that I A is invertible. Assuming provie I A is invertible, the right-han sie of equation 6 is the Cayley transform, which is symplectic if an only if A is Hamiltonian [, p. 80]. Since A is a linear mapping, then A is Hamiltonian if an only if A is Ω-skew [, Prop..5.1, p. 77]. An, Ω- skewness of A is equivalent to the matrix JA being symmetric (i.e.. JA + A T J = 0) [, p. 78]. Thus, evaluate A from equation 5 as follows: 3
( ) ( ) T 1 1 JA + A T J = J J H + J H J = 1 (JJ H + ( H ) ) T J T J = 1 ( I H + H I ) = 0 since H is symmetric. Thus, A is Ω-skew, implying A is Hamiltonian. Finally, by the Cayley transform, equation 6 is symplectic, inicating equation 3 efines a symplectic mapping. Implementing Symplectic Integration The ifficultly with implementing the symplectic integration scheme of equation 3 is that the metho is implicit. An initial metho for ealing with this problem is to utilize a Newton solve for fining the roots of the equation. Equation 3 can be viewe as ( ) zn+1 + z n f(z n+1 ) = z n+1 z n J H = 0 (7) where the roots of function f are the solutions z n+1 to equation 3. The Newton solve is initiate with a guess, such as guess = z n, such that f(z n+1 ) = 0 = f(guess + δ) f(guess) + f (guess)δ (8) with f (guess) being the Jacobian of f, as given in equation 6. The value of δ is etermine from equation 8: δ = (f (guess)) 1 f(guess) which is use to iterate on the initial guess (guess = guess + δ) until f(guess) < ɛ for some esire, small value of ɛ. Variational Algorithms an Variational Integration [3] The action integral on a Lagrangian L(q, q) is efine as S = b a L(q(t), q(t))t (9) where the Euler-Lagrange equations extremize S given fixe en points. The evelopment of a iscrete version of the Euler-Lagrange equations involves iscretizing the action integral S over a time step such that L (q k, q k+1 ) = tk+1 t k L(q(t), q(t))t L( q, q) (10) where L( q, q) is constant over the interval t (t k, t k+1 ) with q an q a iscrete approximation for q an q an t k+1 t k =. This efines a iscrete Lagrangian such that L : Q Q R. The associate iscrete analog to the action integral is then given by S = N 1 k=0 L (q k, q k+1 ), (11) 4
where S : Q N+1 R, an similar to the continuous case, extremizing S prouces the iscrete version of the Euler-Lagrange equations or DEL (Discrete Euler-Lagrange) equations: D 1 L (q k+1, q k+ ) + D L (q k, q k+1 ) = 0 (1) for k = 1,..., N 1. Note, equation 1 efines a variational algorithm. The Legenre transform has an analogous iscrete version, as efine in [3], FL : Q Q T Q; (q k, q k+1 ) (q k+1, D L (q k, q k+1 )) (13) which further implies p k+1 = D L (q k, q k+1 ). For the research conucte herein, the iscrete approximation to the continuous Lagrangian is chosen to be the symmetric form L sym,α (q k, q k+1 ) = L + L ( (1 α)q k + αq k+1, q ) k+1 q k ( αq k + (1 α)q k+1, q ) k+1 q k Theorem 1. [3] The mipoint rule in equation 3 is a variational algorithm through the iscrete Legenre tranformation with the iscrete Lagrangian L sym,α for α = 1. Proof. (in the reverse irection) Let q m = q k+1+q k an q = q k+1 q k, then for α = 1, ( L sym, 1 qk + q k+1 (q k, q k+1 ) = L, q ) k+1 q k = L (q m, q m ) (15) an (14) D 1 L sym, 1 (q k, q k+1 ) = D 1L (q m, q m ) D L (q m, q m ) (16) D L sym, 1 (q k, q k+1 ) = D 1L (q m, q m ) + D L (q m, q m ) (17) Through the DLT (iscrete Legenre transform) in equation 13 an the DEL equations (equation 1), p k DLT DEL = D L (q k 1, q k ) = D 1 L (q k, q k+1 ) (18) p k+1 = D L (q k, q k+1 ) (19) which efines an implicit algorithm such that (q k, p k ) (q k+1, p k+1 ). For L sym, 1, the variables p k an p k+1 are equivalent to equations 16 an 17, respectively. Aing an subtracting equations 18 an 19, plus incorporating the results of equations 16 an 17 for α = 1 provies the following algorithms: p k+1 p k p k+1 + p k Restrict the class of Lagrangian to those of canonical form = D l L (q m, q m ) (0) = D L (q m, q m ) (1) L(q, q) = 1 qt M q V (q) () 5
where M is a symmetric matrix, an V is a potential function. Note, this loses the generality in the proof, but this class of Lagrangian is vali for most mechanical systems, incluing those in the research within this paper. The continuous Legenre transform provies the following Hamiltonian for this class of canonical system: H(q, p) = p T q L(q, q) = 1 M 1 p T p + V (q) (3) with p = M q For the Lagrangian of equation, the first an secon slot erivatives are D 1 L(q, q) = V (q) an D L(q, q) = M q (4) which when combine with the algorithms in equations 0 an 1 result in the following equations p k+1 p k = V (q m ) = V ( q k+1 + q k ) p k+1 p k = H ( ) qk+1 + q k (5) q p k+1 + p k = M q m = M q k+1 q k q k+1 q k = H ( ) pk+1 + p k (6) q which are the exactly the form of equation 3 for z = (q, p) an the Hamiltonian of equation 3. Thus, the mipoint rule efines a variational algorithm with L sym,α when α = 1. Incorporating Non-Conservative Forces The results of Theorem 1 are quite powerful an provie a framework from which non-conservative forces might be incorporate into variational-like algorithms. No results will be shown from this brief section but it serves as a basis for how future work will continue. As iscusse in [3], the integral Lagrange- Alembert principal is given by δ L(q(t), q(t))t + F (q(t), q(t))t = 0 (7) which appears similar to the action integral in equation 9. In fact, a iscrete version of equation 7 exists from which force DEL equations can be erive: D 1 L (q k+1, q k+ ) + D L (q k, q k+1 ) + F (q k+1, q k+ ) + F + (q k, q k+1 ) = 0 (8) From these force DEL equations, an implicit algorithm is likely erivable in a manner analogous to that from the proof of Theorem 1. A key to the evelopment of this algorithm will require appropriate iscretization of the non-conservative forces, likely in a symmetric fashion as with L sym,α. Application of Symplectic/Variational Integration to Circular Orbits The first-orer symplectic integration scheme from equation 3 was implemente in Matlab for the Hamiltonian ynamics of equation 1 for a spacecraft orbiting a planet (two orbit types are analyze, one circular an one elliptic). The parameters chosen for the spacecraft an planet are given in table 1. Consequently, the initial conitions also set the value of the conserve Hamiltonian from equation 1 for the two orbit types; the Hamiltonian values for each orbit type are inclue in the table (they are negative, which correspons to circular an elliptic orbits). 6
Table 1: Simulation Parameter Data an Initial Conitions µ (km 3 /s ) 300.9998 m (kg) 1 q 0 (km) (0, 1664.09, 0) T p 0 (circular) (km/s) (1.386955, 0, 0) T H(q, p)(circular) (kg km /s ) 0.961811 p 0 (elliptic) (km/s) (1.550663, 0, 0) T H(q, p)(elliptic) (kg km /s ) 0.7136658 The Matlab coe also implements three aitional schemes for comparing numerical performance over roughly one ay (86400 secons) of orbit ata. The comparison schemes were the Matlab intrinsic RK45 scheme ODE45 (which has a variable time step) an two Mathworks fixe-step RK schemes calle ODE an ODE4 (RK an RK4, respectively), available from the company web site. Two orbit types were simulate with the numerical integrators: a circular orbit an an elliptical orbit. With both simulations, which are two-boy conic orbits, analytic solutions for the orbits are available, which facilitates comparison of numerical schemes an etermination of their accuracy. Circular Orbit Simulation Figure juxtaposes results from the symplectic, RK45, an RK4 integration schemes, along with an analytic solution for the two-boy circular orbit. Due to the choice of initial conitions, the orbit is planar. Figure : Comparison of Numerical Techniques Versus the Analytic Solution From figure, the variable-step RK45 scheme is seen to win towar the planet; the metho is essentially issipating energy from the orbit trajectory. The symplectic an RK4 (fixe-step) schemes track the exact orbit more closely an cannot be separate in figure. To istinguish these two schemes, figure 3 zooms in on a segment in the upper-left portion of figure. 7
Figure 3: Close-Up of Circular Orbit from Figure The close up of figure 3 reveals that both the RK4 an symplectic integrators track the analytic solution well, though both with an offset. Interestingly the symplectic integrator has a smaller offset than the RK4 metho, which is interesting given that RK4 is fourth orer an this particular symplectic scheme is secon orer. Utilizing an integrator that intrinsically accounts for the system geometry (conserving the Hamiltonian) oes have benefit over a more naive technique, at least with a simple circular orbit. The conservation of energy by the symplectic integrator is reveale in figure 4, which also clearly shows how RK4 introuces false issipation, something that woul cause a loss in accuracy over long-term integration. The issipation is in the 10 th igit only uring this time interval, so the plot scale is cryptic. Figure 4: Energy of Circular Orbit shoul be Conserve 8
Elliptical Orbit Simulation Figure 5 juxtaposes results from all four integration schemes (symplectic, RK, RK4, an RK45), along with an analytic solution for the two-boy elliptic orbit. The orbit is again planar ue to the choice of initial conitions; for the elliptic. Figure 5: Comparison of Numerical Techniques Versus the Analytic Solution From figure 5, the variable-step RK45 scheme is again seen to win towar the planet, artificially issipating energy as was the case with the circular orbit. The remaining schemes track the exact orbit more closely, which can be seen in the zoom of the orbit shown in figure 6. Figure 6: Close-Up of Ellipse Orbit from Figure 5 The close up of figure 6 shows that the RK4 integrator is significantly more accurate than the symplectic or RK technique. This is slightly iffering from the circular case but seems reasonable since the RK4 technique is 4th-orer, whereas the RK an chosen symplectic techniques are both secon orer. Outsie of the circular case, these plots woul appear to make the RK4 9
technique the integrator of choice. However, an examination of the energy of the system brings aitional unerstaning of the ability of the ifferent schemes to prouce longterm accuracy (See figures 7 an 8). Figure 7: Elliptic Orbit Energy Showing RK Decay an Symplectic Conservation Figure 8: Close-Up of Elliptic Orbit Energy Revealing Slow Decay in RK4 The results of figures 7 an 8 inicate that the symplectic integrator, though not constant in energy, oscillates about a mean value, offset from the actual energy. A correction factor is likely realizable for the numerical implementation of the symplectic integrator that woul make it oscillate about the exact value. The RK4 technique appears to conserve energy too in figure 7; however, a closer examination in figure 8 again shows that a very long-term integration woul exhibit energy ecay with RK4. Note, the number of significant igits isplaye in the MATLAB generate plots is cryptic as it oes not reflect the issipation occurring in the 10th igit for the RK4 energy, though the tren in the plot is unmistakable. 10
Future Steps One immeiate question to answer is how versatile is the symplectic/variational integrator approach? The RK methos an other similar techniques can incorporate iscretization of non-conservative forces an prouce certain known accuracies. However, can the force DEL equations prouce their own variational-like algorithms that preserve some element of the force ynamics, thus proviing an even higher accuracy than traitional techniques? This question is of immeiate interest to the research in this paper. The subsequent focus of this research will be to incorporate non-conservative forces, such as thruster firings an solar raiation pressure. From the aition of these forces, performance an precision comparisons can be mae with existing integration techniques. To assess precision an accuracy, simple two-boy orbits with impulsive thruster firings can be propagate to etermine how well ifferent numerical integration schemes track the known, analytic solutions. For an integration scheme to be useful in the real-worl spacecraft navigation, the scheme must have high precision an have timely performance. Future work can focus on the integration scheme itself; the mipoint rule is the lowest-orer symplectic integrator, so higher-orer schemes shoul be more accurate. Accuracy requires not only reproucing proper position an velocity solutions but also arriving at the solutions on a similar time scale; for instance, the perio of a conic orbit shoul be recoverable from the integration scheme. Regaring performance, explicit formulations, or goo approximations to implicit forms coul likely improve performance, so such ieas will be explore. References [1] R.R. Bate, D.D. Mueller, an J.E. White, Funamentals of Astroynamics, Dover Publications, 1971. [] J.E. Marsen an T.S. Ratiu, Introuction to Mechanics an Symmetry, Springer, n eition, 00. [3] C. Kane, J.E. Marsen, M. Ortiz, an M. West, Variational Integrators an the Newmark Algorithm for Conservative an Dissipative Mechanical Systems, Int. J. Num. Math. Eng., 49, pp. 195-135, 000. 11