Comparison of Implicit Integration Methods for Solving Aerospace Trajectory Optimization Problems

Riehl, John P.; Paris, Stephen; Sjauw, Waldy K.

doi:10.2514/6.2006-6033

Cited by 15 publications

(18 citation statements)

References 9 publications

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“…Adding more nodes will only capture negligibly-higherfrequency components and will not result in any significant improvement in accuracy. This notion has been validated by Riehl et al [46] on several problems in aerospace trajectory optimization. Furthermore, as described by Ross et al [47], it is possible to capture even the high-frequency terms by anti-aliasing the solution through an application of Bellman's principle.…”

Section: A Spectral Coefficient Testmentioning

confidence: 68%

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Gong

Fahroo

Ross

2008

Journal of Guidance, Control, and Dynamics

159

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Recent convergence results with pseudospectral methods are exploited to design a robust, multigrid, spectral algorithm for computing optimal controls. The design of the algorithm is based on using the pseudospectral differentiation matrix to locate switches, kinks, corners, and other discontinuities that are typical when solving practical optimal control problems. The concept of pseudospectral knots and Gaussian quadrature rules are used to generate a natural spectral mesh that is dense near the points of interest. Several stopping criteria are developed based on new error-estimation formulas and Jackson's theorem. The sequence is terminated when all of the convergence criteria are satisfied. Numerical examples demonstrate the key concepts proposed in the design of the spectral algorithm. Although a vast number of theoretical and algorithmic issues still remain open, this paper advances pseudospectral methods along several new directions and outlines the current theoretical pitfalls in computation and control.

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Section: A Spectral Coefficient Testmentioning

confidence: 68%

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Gong

Fahroo

Ross

2008

Journal of Guidance, Control, and Dynamics

159

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“…The most obvious method is that of algebraic integration which however is not possible for most realistic trajectory optimization problems. In Riehl et al 19 a overview of possible implicit integration methods are given (including Pseudo-Spectral integration). The methods transform the problem of numerical integration by replacing the ODEs with a set of defect constraints.…”

Section: Example Temporal Decorrelation: Integration Blow-upmentioning

confidence: 99%

“…The constructed NLP problem is typically solved by available solver based on gradient methods 16 and genetic algorithms. Direct transcription methods has been successfully applied in many fields such as spacecraft trajectory optimization, 17,18 aircraft trajectory optimization 19,20 etc. As stated, in most cases the applied methods cannot provide guarantees on finding the global optimal solution (region of convergence may be very small) and moreover can have severe problems of finding any valid trajectories given the (in)equality constraints imposed on the problem.…”

mentioning

confidence: 99%

Trajectory Optimization Based on Interval Analysis

Weerdt

Kampen

Chu

et al. 2011

AIAA Guidance, Navigation, and Control Conference

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Trajectory optimization has been a large field of research for many years. The drawback is that for non-convex, constrained problems practically all available solvers cannot guarantee that the globally optimal trajectory is found. Interval analysis based solvers however can provide this guarantee. Interval analysis has been applied to trajectory optimization before, but the previously presented methods suffered from major drawbacks which limited their application to small scale problems. In this paper a new interval based method is introduced which incorporates state parameterization to prevent explicit integration. The performance of the proposed method is demonstrated by applying it to a spacecraft formation flying optimization problem. The results are compared with a gradient based solver and it is shown that the interval method is guaranteed to find the global optimal solution. Finally the first steps for another new trajectory optimization method based on interval analysis and direct collocation are presented.

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“…This limitation can be overcome by using the so-called "multiple shooting" algorithm in which the time is divided into segments and both control and state variables are discretized at these segment end points, or cardinal nodes [55], while only the control variables are discretized within each segment. This way the errors due to gradient approximations remain within the segments and the algorithm achieves better overall accuracy, although at the expense of a larger NLP.…”

Section: Some Theoretical Considerations and Limitations Of The Algormentioning

confidence: 99%

Dynamics and control of electromagnetic satellite formations

Ahsun

Miller

2006

2006 American Control Conference

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Satellite formation flying is an enabling technology for many space missions, especially for space-based telescopes. Usually there is a tight formation-keeping requirement that may need constant expenditure of fuel or at least fuel is required for formation reconfiguration. Electromagnetic Formation Flying (EMFF) is a novel concept that uses superconducting electromagnetic coils to provide forces and torques between different satellites in a formation which enables the control of all the relative degrees of freedom.With EMFF, the life-span of the mission becomes independent of the fuel available on board. Also the contamination of optics or sensitive formation instruments, due to thruster plumes, is avoided. This comes at the cost of coupled and nonlinear dynamics of the formation and makes the control problem a challenging one. In this thesis, the dynamics for a general N-satellite electromagnetic formation will be derived for both deep space missions and Low Earth Orbit (LEO) formations. Nonlinear control laws using adaptive techniques will be derived for general formations in LEO. Angular momentum management in LEO is a problem for EMFF due to interaction of the magnetic dipoles with the Earth's magnetic field. A solution of this problem for general Electromagnetic (EM) formations will be presented in the form of a dipole polarity switching control law. For EMFF, the formation reconfiguration problem is a nonlinear and constrained optimal time control problem as fuel cost for EMFF is zero. Two 3 different methods of trajectory generation, namely feedback motion planning using the Artificial Potential Function Method (APFM) and optimal trajectory generation using the Legendre Pseudospectral method, will be derived for general EM Formations. The results of these methods are compared for random EM Formations. This comparison shows that the artificial potential function method is a promising technique for solving the real-time motion planning problem of nonlinear and constrained systems, such as EMFF, with low computational cost. Specifically it is the purpose of this thesis to show that a fullyactuated N-satellite EM formation can be stabilized and controlled under fairly general assumptions, therefore showing the viability of this novel approach for satellite formation flying from a dynamics and controls perspective.

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Comparison of Implicit Integration Methods for Solving Aerospace Trajectory Optimization Problems

Cited by 15 publications

References 9 publications

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Spectral Algorithm for Pseudospectral Methods in Optimal Control

Trajectory Optimization Based on Interval Analysis

Dynamics and control of electromagnetic satellite formations

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