Analytic Solution for Fuel-Optimal Reconfiguration in Relative Motion

In this paper, we consider a linear quadratic regulator control problem for spacecraft rendezvous in an elliptical orbit. A new spacecraft rendezvous model is established. On the basis of this model, a linear quadratic regulator control problem is formulated. A parametric Lyapunov differential equation approach is used to design a state feedback controller such that the resulting closed-loop system is asymptotically stable, and the performance index is minimized. By an appropriate choice of the value of a parameter, an approximate state feedback controller is obtained from a solution to the periodic Lyapunov differential equation, where the periodic Lyapunov differential equation is solved on the basis of a new numerical algorithm. The spacecraft rendezvous mission under the controller obtained will be accomplished successfully. Several illustrative examples are provided to show the effectiveness of the proposed control design method.OPTIMAL CONTROL APPROACH TO SPACECRAFT RENDEZVOUS 159 As it is well known, the propellant storage capacity of a spacecraft rendezvous system is very small. Thus, it is important to reduce fuel consumption in the spacecraft rendezvous process. Once the chaser spacecraft runs out of fuel, the spacecraft rendezvous mission will no longer be possible to continue. Thus, the minimization of the fuel consumption during the spacecraft rendezvous mission is critically important. Consequently, the minimum fuel consumption problems in spacecraft rendezvous have been actively studied among the control community. For instance, the minimum fuel rendezvous problem is studied in [8], where the spacecraft is driven by normal power; a minimum fuel consumption for spacecraft rendezvous problem in a general central gravity force field is investigated in [9]; a minimum fuel consumption problem subject to constraint on the inverse square field is considered in [10]; low-thrust optimal rendezvous maneuver in the vicinity of an elliptical orbit is studied in [11]; differential game theory is applied in [12] to achieve the optimum lowacceleration rendezvous maneuver for spacecraft rendezvous; an exact penalty function method is used in [13] to solve a fuel optimization problem for continuous-thrust orbital rendezvous subject to collision avoidance constraint. However, the optimal control methods mentioned previously are for the design of open-loop controllers. It remains a challenge to design optimal closed-loop controllers, which are much preferred in engineering applications.In this paper, we consider a linear quadratic regulator (LQR) control problem. The nonlinear spacecraft rendezvous system is first linearized around the origin, resulting in a linear time-varying system. By considering the true anomaly as a new variable, the linear time-varying system is transformed into a linear periodic system. A parametric Lyapunov differential equation approach is proposed to design a state feedback controller such that the resulting closed-loop system is asymptotically stable, and the performance index is minimize...

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“…Theorem Consider system , where A ( θ ) and B ( θ ) are defined by and , respectively. Then,

MathClass-bin- P MathClass-rel' (θ) MathClass-rel= A T (θ) P (θ) MathClass-bin+ P (θ) A (θ) MathClass-bin- P (θ) B (θ) R MathClass-bin- 1 (θ) B T (θ) P (θ) MathClass-bin+ ϵP (θ)

has a unique positive definite 2 π ‐periodic solution, where R ( θ ) > 0 is a given 2 π ‐periodic matrix, ϵ is an adjustable parameter, and

ϵ MathClass-rel\in (0 MathClass-punc, scriptE]

with

scriptE MathClass-rel> 0

. Proof From Example 2.4 in , it can be shown that

Ψ (θ) MathClass-rel= ([], \begin{matrix} falsenone none none none nonefalsearray \\ arraycenter 0 & arraycenter - \cos θ - e \cos^{2} θ & arraycenter 0 & arraycenter - MathClass-open(1 + e \cos θ MathClass-close) \sin θ & arraycenter 3 e MathClass-open(1 + e \cos θ MathClass-close) \sin θφ MathClass-open(θ MathClass-close) - 2 & arraycenter 0 \\ arraycenter 1 & arraycenter MathClass-open(2 + e \cos ... \end{matrix})

…”

Section: Resultsmentioning

confidence: 99%

“…From the controllability of OEA.Â/; B.Â/; (16), and [21], it follows that (28) has the unique periodic nonnegative definite solution P 0 .Â/ D 0, that is,…”

Section: Proofmentioning

confidence: 99%

An optimal control approach to spacecraft rendezvous on elliptical orbit

Gao

Teo

Duan

2014

Optim Control Appl Methods

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“…They found complete solutions for elliptical orbits in terms of the eccentric anomaly. This advancement was followed by additional papers which present the complete analytical solution explicit in time, expanding the state transition matrix in terms of eccentricity (Yamanaka and Ankersen, 2002;Carter, 1998;Melton, 2000;Broucke, 2003;Inalhan et al, 2002;Sengupta and Vadali, 2007;Cho and Park, 2009). This form of solution is used to analyze the relative motion between the chaser and the target vehicles in the relative frame of motion more efficiently and rapidly than solving the exact nonlinear differential equations in the inertial coordinate system.…”

Section: Introductionmentioning

confidence: 99%

Relative Motion Guidance, Navigation and Control for Autonomous Orbital Rendezvous

Okasha

Newman²

2014

J.Aerosp. Technol. Manag.

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In this paper, the dynamics of the relative motion problem in a perturbed orbital environment are exploited based on Gauss' variational equations. The relative coordinate frame (Hill frame) is studied to describe the relative motion. A linear high fidelity model is developed to describe the relative motion. This model takes into account primary gravitational and atmospheric drag perturbations. In addition, this model is used in the design of a control, guidance, and navigation system of a chaser vehicle to approach towards and to depart from a target vehicle in proximity operations. Relative navigation uses an extended Kalman filter based on this relative model to estimate the relative position and velocity of the chaser vehicle with respect to the target vehicle and the chaser attitude and gyros biases. This filter uses the range and angle measurements of the target relative to the chaser from a simulated Light Detection and Ranging (LIDAR) system, along with the star tracker and gyro measurements of the chaser. The corresponding measurement models, process noise matrix and other filter parameters are provided. Numerical simulations are performed to assess the precision of this model with respect to the full nonlinear model.The analyses include the navigations errors, trajectory dispersions, and attitude dispersions.

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“…Palmer presented an analytic solution for the optimal reconfiguration problem based on the Fourier series expansion of thrust (Palmer 2006). Cho & Park derived a simple analytic solution for linear dynamics in the local vertical, local horizontal frame (Cho & Park 2009). Lee and Park derived approximated analytical solutions considering nonlinearities of the terrestrial gravity, eccentricities of a chief satellite, and J2 effects by perturbing the solution to the linear Hill equation (Lee & Park 2011).…”

Section: Introductionmentioning

confidence: 99%

Performance Analysis of Generating Function Approach for Optimal Reconfiguration of Formation Flying

Lee¹,

Park²,

Park³

2013

Journal of Astronomy and Space Sciences

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The use of generating functions for solving optimal rendezvous problems has an advantage in the sense that it does not require one to guess and iterate the initial costate. This paper presents how to apply generating functions to analyze spacecraft optimal reconfiguration between projected circular orbits. The series-based solution obtained by using generating functions demonstrates excellent convergence and approximation to the nonlinear reference solution obtained from a numerical shooting method. These favorable properties are expected to hold for analyzing optimal formation reconfiguration under perturbations and non-circular reference orbits.

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Analytic Solution for Fuel-Optimal Reconfiguration in Relative Motion

Cited by 37 publications

References 18 publications

An optimal control approach to spacecraft rendezvous on elliptical orbit

An optimal control approach to spacecraft rendezvous on elliptical orbit

Relative Motion Guidance, Navigation and Control for Autonomous Orbital Rendezvous

Performance Analysis of Generating Function Approach for Optimal Reconfiguration of Formation Flying

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