Submitted to the Journal of the Astronautical Sciences
AAS 05-471 Fundamental Limits on Spacecraft Orbit Uncertainty and Distribution Propagation D.J. Scheeres∗, F.-Y. Hsiao†, R.S. Park ‡, B.F. Villac §, J.M. Maruskin
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Abstract In this paper we present and review a number of fundamental constraints that exist on the propagation of orbit uncertainty and phase volume flows in astrodynamics. These constraints arise due to the Hamiltonian nature of spacecraft dynamics. First we review the role of integral invariants and their connection to orbit uncertainty, and show how they can be used to formally solve the diffusion-less Fokker-Plank equation for a spacecraft probability density function. Then, we apply Gromov’s Non-Squeezing Theorem, a recent advance in symplectic topology, to find a previously unrecognized fundamental constraint that exists on general, nonlinear mappings of orbit distributions. Specifically, for a given orbit distribution, it can be shown that the projection of future orbit uncertainties in each coordinate-momentum pair describing the system must be greater than or equal to a fundamental limit, called the symplectic width. This implies that there is always a fundamental limit to which we can know a spacecraft’s future location in its coordinate and conjugate momentum space when mapped forward in time from an initial covariance distribution. This serves as an “uncertainty” principle for spacecraft uncertainty distributions.
1
Introduction
The topic of orbit uncertainty and its mapping is both a practical and a theoretically interesting problem. The practical issues that it covers include the prediction and reconstruction of spacecraft trajectories, the prediction of solar system bodies such as Near-Earth asteroids, the tracking of orbit debris, and the correlation and identification of seemingly uncorrelated spacecraft observations. While there has been an enormous amount of research devoted to all aspects of this practical problem, there has been relatively little attention paid to the more theoretical aspects of the dynamics of uncertainty propagation. Standard texts on spacecraft navigation [12] provide clear analytical descriptions of Gaussian orbit statistics and linear mappings, but generally resort to numerically intensive approaches when non-linearities are to be accounted for. There has been much analysis for practical processing and estimation techniques when dealing with non-linearities, extensively reviewed in [2], but these usually do not take advantage of deeper constraints that arise from spacecraft dynamics being described by Hamiltonian systems. In a series of recent investigations, however, the implications of specific system properties such as stability and instability for the propagation of orbit uncertainty and for the orbit determination and control of a spacecraft have been explored [15, 16, 18]. These studies also relied on the use of linearized orbit mappings. In a different vein of research, analytical insights on the non-linear ∗ Associate
Professor, Department of Aerospace Engineering, The University of Michigan,
[email protected] Professor, Department of Aerospace Engineering, Tamkang University, Taiwan ‡ PhD Candidate, Department of Aerospace Engineering, The University of Michigan § Member of the Technical Staff, Jet Propulsion Laboratory/California Institute of Technology ¶ Graduate Student, Department of Mathematics, The University of Michigan † Assistant
1
propagation of orbit uncertainty was investigated in [6] to develop analytic insights into the propagation of initially Gaussian distributions. In [17] a discussion of useful “measures” of the effect of instabilities on an orbit distribution were discussed. Recently, in [13] a different approach to these non-linear propagations was taken which relies on explicit expansions of a trajectory in terms of its initial conditions. Using this approach it is possible to model highly non-linear orbit propagations and to analytically compute shifts in the statistical mean and covariance due to these non-linearities. Absent from all these investigations, however, were any deeper constraints on the evolution of an uncertainty distribution. To date, the most profound constraint identified for an evolving uncertainty distribution is that its total phase volume is constant, due to Liouville’s Theorem, and that the probability density function of the spacecraft is an integral invariant, meaning that probability is conserved along the flow [18]. These are somewhat obvious results, however, and are generally accepted without proof by spacecraft navigators. Recently, we have embarked on a more formal reinvestigation of possible constraints that may exist on orbit distributions. This has been driven by two basic observations. First is that there have been no in-depth investigations on the implications of the absolute integral invariants for the propagation of spacecraft uncertainty in systems that can be modeled using a Hamiltonian formalism. As discussed in this paper, there are situations that can be cast into a form where an integral invariant constraint exists. Although integral invariants have been used previously in celestial mechanics [8], they have not been studied in the context of orbit uncertainty distributions. In the current paper we introduce this idea and discuss some immediate implications of these constraints. We plan to further expound on these results in future research. Second is the existence of a relatively new result in symplectic topology known as Gromov’s NonSqueezing Theorem [11]. This specific result is of special interest to uncertainty distributions as it specifically applies to distributions of positive volume (unlike the lower-order integral invariants), and supplies clearly different constraints than those that arise from Liouville’s Theorem. In [5] the basic connections between this theorem and spacecraft uncertainty dynamics have been made at a more formal level. In the current paper our intent is to introduce this result to the astrodynamics community and identify its elementary implications. Again, our future research will focus on further investigating and applying this result to spacecraft navigation in general.
2
Hamiltonian Dynamics
We first review a number of results for Hamiltonian Dynamical Systems. We do not attempt a rigorous statement or proof of these items, as there are ample texts where these topics are covered in detail [3, 7]. Rather, our intent is to introduce new ideas and connections that can be made between spacecraft dynamics, orbit uncertainty propagation, and Hamiltonian dynamics.
2.1
Form of the Equations of Motion
Consider an even-dimensional vector space with elements that conform to a canonical form of equations of motion for their evolution. Let the state of the system be designated as x ∈ R2n , and divided into two sets, each of dimension n, the coordinates q = (qi ; i = 1, n) and the momenta p = (pi ; i = 1, n). Define the Hamiltonian as a scalar function of the coordinates, momenta, and possible time: H(q, p, t). The Hamiltonian is assumed to be analytic in all of its arguments. The equations of motion for a Hamiltonian dynamical system are then stated as: x˙ J
∂H = J ∂x On = −Idn
Idn On
(1) (2)
where On and Idn are n × n dimensional zero and identity matrices, respectively. In this form of the equations we assume the standard form for the vector x = [q, p].
2
We can also define an alternate version of the equations of motion, which in some cases is more natural to use, where we group the conjugate coordinates and momenta together. In this case we ˜ = (qi , pi ; i = 1, n) and the equations of motion are: define x ∂H ˜˙ = J˜ x ˜ ∂x J22 O J˜ = ... ... O
O J22 ... O ...
... O ... J22 O
O ... ... O J22
(3)
(4)
where J22 is just the 2 × 2 version of the symplectic matrix from Eq. 2. Given such a dynamical system, there are many properties that are invariant under any transformation that preserves the same form of the equations of motion. Specifically, consider a transformation from the set (q, p) → (Q, P), or let Q = Q(q, p, t) and P = P (q, p, t). Then, in this new system, if the equations of motion of Q and P can be obtained from a properly transformed Hamiltonian function K(Q, P, t) using Eq. 1, then the transformation is canonical. For a time invariant transformation the Hamiltonian K is found by directly substituting the inverse expressions q = q(Q, P) and p = p(Q, P) into H. For a time-varying transformation an additional term must be added [3] to properly find the right form of K. Such a transformation is called a “canonical” transformation, and may be refered to as a “symplectomorphism”, a term we will (non-rigorously) use interchangeably with canonical transformation. We finally note that Lagrangian dynamical systems that arise from potential force fields, such as nearly all astrodynamics problems, can usually be cast into a Hamiltonian form by a Legendre Transformation. Exceptions include when significant non-gravitational forces such as atmospheric drag act on the spacecraft. It is important to note that solar radiation pressure, a common nongravitational force, may often be treated as if arising from a potential. Thus, in the following we will assume that a standard Hamiltonian form of the equations of motion can be found for our generic astrodynamics system.
2.2
Solutions and their properties
From the standard theory of ordinary differential equations, Eqs. 1 have well defined solutions as a function of their initial conditions. We denote a solution by a smooth function φ ∈ R2n that depends on time and the state at a specified epoch. x(t)
= φ(t; xo , to )
and which satisfies the following two constraints φt
∂H = J ∂x x=φ
φ(to ; xo , to ) = xo
(5)
(6) (7)
where φt denotes the partial derivative with respect to time. The solution φ is usually thought of as a point travelling in phase space. We note, however, that it can also represent a general solution flow of a region of phase space. Indeed, given a general set Bo ⊂ R2n specified at an epoch to in phase space, we can denote the solution flow B(t) = φ(t; Bo , to ), defined as: B(t) = {x(t) = φ(t; xo , to )|xo ∈ Bo }
(8)
The solution flow is a one-to-one mapping between the “initial conditions” and the current state, which allows us to define the inverse mapping xo = φ(to ; x, t), and which leads to the identity: xo
= φ(to ; φ(t; xo , to ), t) 3
(9)
This represents both an identity map and a defining relation for the integrals of motion for a general solution x(t), as the initial conditions of a trajectory are integrals of motion of that trajectory [3]. We define the integrals of motion for a specific trajectory as xo
= ψ(t, x; to )
(10)
In this notation, the function ψ is considered to be a function of time t and state x, and the epoch to is a parameter. This function takes the current state of a dynamical system and maps it back to its state at a specified epoch to . Integrable problems will in general have such a set of relations available as an analytical function of the xo . For non-integrable problems, such analytic functions do not exist or, at best, may only be found locally in the phase space about a solution. Recent research has focused on using such local solutions to develop non-linear descriptions of motion in non-integrable problems [4, 13]. From Eq. 9, and from the fact that the xo are integrals of motion, we note that dψ dt
=
∂ψ ∂ψ ∂H + ·J ≡0 ∂t ∂x ∂x
(11)
We will use this abstract property of the solution flow later in our analysis. Finally, we note that the epoch conditions xo are also the state of the system. Thus, we can view the solution flow φ as a coordinate transformation from xo to x with the time t as a free parameter. A fundamental result from Hamiltonian dynamics is that the solution flow is a canonical transformation [3]. Thus, the solution flow must satisfy the usual conditions on canonical transformations. Here, we mainly rely on one specific property for canonical transformations, related to the Jacobian of the transformation. Define the gradient of the solution flow transformation as Φ(t, xo , to )
=
∂x(t) ∂xo
(12)
This is just the usual “state transition matrix” from linear systems theory. Since φ is a canonical transformation the matrix Φ must be symplectic. Thus, Φ satisfies the following relations: ΦT JΦ
= J |Φ| = 1
Φ(to , to )
=
Id2n
(13) (14) (15)
Most notably, we note that the Jacobian of the transformation, |Φ|, is unity.
2.3
Distributions in Phase Space
In the following we consider distributions of initial conditions on manifolds of varying dimensionality. In general, we consider an n-dimensional distribution in phase space as:
Bn = x|x ∈ M n ⊂ R2n (16) where the set M n is a manifold of dimension n and is closed and connected. If we assign an epoch to the set, Bn (to ), then we can define the evolution of the set as a function of time: Bn (t) = φ(t; Bn (to ), to ). In terms of simple geometric objects, we see that the set B0 is a point in R2n , B2 is a 2-dimensional surface embedded in R2n , and B2n is a 2n-dimensional subset of R2n with non-zero volume.
2.4
Integral Invariants
A standard result in Hamiltonian dynamics is the existence of absolute integral invariants. Simply stated, if a quantity is an integral invariant it means that the integrations of an arbitrary distribution B2k is conserved when summed over all possible symplectic combinations of degree 2k. By 4
a “symplectic combination” or “symplectic pair” we mean the pair of coordinate and momentum variables, (qi , pi ), that correspond to each other. Specifically, define the integral over a set B2k taken at a time t:
I2k (t) = dqi1 ∧ dpi1 ∧ · · · ∧ dqik ∧ dpik (17) 1≤i1
