sfm 2012 abstract

7/28/2019 SFM 2012 Abstract

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AAS XX-XXX

ITERATIVE MODEL REFINEMENT FOR ORBITAL TRAJECTORY

OPTIMIZATION

Jennifer Hudson and Ilya Kolmanovsky

An iterative trajectory optimization method is developed for optimal control ofa low-thrust spacecraft. A high-fidelity model and a low-fidelity model are usedto iteratively refine solutions. The high-fidelity model accurately represents thesystem but has high computational complexity, such that numerical optimizationis prohibitively time-consuming. The low-fidelity model can be used for numericaloptimization, but approximates the system dynamics with an unknown error. Theiterative model refinement method systematically reduces the difference betweenthe two models and converges on a solution with efficient execution time.

INTRODUCTION

An iterative optimization method is developed for a system represented by two models: a high-

fidelity model and a low-fidelity model. The high-fidelity model accurately represents the system

but has high computational complexity, such that iterative simulations necessary to compute the op-

timal control are prohibitively time-consuming. The low-fidelity model approximates the behavior

of the system with a small but unknown error, and can be evaluated at a much faster rate.

The iterative model refinement method is applied to the problem of optimal control of a low-thrust

spacecraft about the L4 Lagrange point in the Earth-Moon system. The objective is to optimally

control the spacecraft from a perturbed initial state to the L4 point. Several cases with different

boundary conditions and constraints will be analyzed.

In addition to demonstrating and validating the iterative model refinement method on the orbital

control problem, this paper will analyze the convergence properties of the method. Conditions on

the system models and on the control to ensure convergence will be discussed.

METHODS

The method assumes a system described by two models: a high-fidelity model,

x = f(x(t), u(t)), (1)

where x is the state, u is the manipulated input; and a low-fidelity model,

x = Ax(t) + Bu(t) + d(t), (2)

where d represents unmeasured disturbances.

Postdoctoral Researcher, Department of Aerospace Engineering, University of Michigan, 1320 Beal Ave., Ann Arbor,

MI 48109-2140.Professor, Department of Aerospace Engineering, University of Michigan, 1320 Beal Ave., Ann Arbor, MI 48109-2140.

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The iterative model refinement algorithm is shown in Table 1. First, d0(t) is set to zero and anumerical optimizer is used to calculate the optimal control, u0(t), of the low-fidelity model. Thisresults in the trajectory xl

0(t).

Next, the high-fidelity model is simulated with the control u0(t), resulting in the trajectory xh0

(t).

Finally, xl0

(t) and xh0

(t) are used to determine the disturbance d1(t) that minimizes the differencebetween the low-fidelity model and the high-fidelity model under u0(t). These disturbances arethen used in the next iteration. Each iteration of this algorithm requires many evaluations of the

low-fidelity model but only one evaluation of the high-fidelity model.

Table 1. IMR Algorithm

1. Optimize low fidelity model, find optimal control u(t)2. Simulate high fidelity model with u(t) (and possibly with excitation added), find trajectory xh(t)

3. Compare xh(t) to low fidelity model output, estimate disturbance d(t)4. Repeat

ORBITAL TRAJECTORY OPTIMIZATION PROBLEM

The iterative model refinement approach will be demonstrated on the problem of optimally con-trolling a spacecraft near the L4 Lagrange point. The nonlinear equations of the planar circular

restricted three-body model are the high-fidelity model of the system,

x 20y 2

0x =

1(xD1)

r31

2(x + D2)

r32

+ x + ax (3)

y + 20x 2

0y =1y

r31

2y

r32

+ y + ay, (4)

where the coordinate frame is rotating with constant velocity 0 about the center of mass of theEarth-Moon system, D1 is the distance of the Earth from the center along the positive x axis,D2 is the distance of the Moon from the center along the negative x axis, and 1 and 2 are the

constant gravity parameters of the Earth and Moon respectively. The perturbations x and y areadded to represent unkown system behavior, such as control disturbances, solar radiation pressure

perturbations, and gravitational perturbations.

The linearized restricted 3-body orbital dynamics for perturbations from L4 are the low-fidelity

model of the system,

x 20y c12

0x = ax (5)

y + 20x c22

0y = ay, (6)

where the Cartesian coordinate components x and y describe the perturbation of the positionvector from the Lagrange point.

Comparing the two models, the high-fidelity model alone is more difficult and computationally

expensive to optimize due to its nonlinear structure. The low-fidelity model is easily optimized, but

the solution does not solve the high-fidelity optimal control problem, as the linear model does not

fully capture the nonlinear system dynamics. The iterative approach using both models converges

on a solution to the nonlinear problem with computational efficiency.

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Figures 1 - 5 show an example of the iterative method. The objective is to control the space-

craft from a perturbed initial state to the L4 point within a five hour timespan. The initial state

perturbation from L4 is

xyxy

=

100km100km

10km/s10km/s

. (7)

The perturbations are x = 0.0016km/s2 and y = 5.6861e005km/s2. Figures 1 and 2 show thetrajectories calculated by the high-fidelity model with the optimal control found by the low-fidelity

model after each iteration. Figures 3 and 4 show the two elements of the optimal control vector

found in each iteration. All plot dimensions are normalized by the Earths radius RE and a time

constant

R3E

1.

Figure 5 shows the cost,

J=

tf0

xTQx + uTRu, (8)

calculated by each iteration of the high-fidelity model, where Q = I4 and R = I2. This exampleshows convergence to a minimum cost after a small number of iterations.

Figure 1. Trajectories in Rotating Frame

CONCLUSIONS

The iterative model refinement method offers an efficient, robust approach for trajectory optimiza-

tion. It is applicable to many problems in spaceflight mechanics, including low-thrust spacecraft

trajectory design. Future space missions will involve complex trajectories with many constraints.

Appropriate and efficient use of models of varying fidelity will be useful in many mission design

scenarios.

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Figure 2. Trajectories in Inertial Frame

Figure 3. Optimal Control u(1)

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Figure 4. Optimal Control u(2)

Figure 5. Cost

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sfm 2012 abstract

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