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John E. Prussing - One of the best experts on this subject based on the ideXlab platform.

  • Oxford Scholarship Online - Improving a Nonoptimal Impulsive Trajectory
    Oxford Scholarship Online, 2018
    Co-Authors: John E. Prussing
    Abstract:

    Improving a nonoptimal trajectory is analysed, including adding terminal coasts and midcourse impulses in fixed-time trajectories. Orbit transfer is also analysed. If the Primer Vector evaluated along an impulsive trajectory fails to satisfy the necessary conditions (NC) for an optimal solution, the way in which the NC are violated provides information that can lead to a solution that does satisfy the NC. The necessary gradients were first derived by Lion and Handelsman.

  • General Theory of Optimal Rocket Trajectories
    Optimal Control with Aerospace Applications, 2013
    Co-Authors: James M. Longuski, Jose J Guzman, John E. Prussing
    Abstract:

    In this chapter we develop a general theory of optimal spacecraft trajectories based on two pioneering works: Breakwell [1959] and Lawden [1963]. Lawden introduced the concept of the Primer Vector, which plays a dominant role in minimum-propellant trajectories and also in other types of optimal trajectories. A more complete discussion of the topics in this chapter, including several example trajectories, is in Prussing [2010].

  • spacecraft trajectory optimization Primer Vector theory and applications
    2010
    Co-Authors: John E. Prussing
    Abstract:

    Introduction In this chapter, the theory and a resulting indirect method of trajectory optimization are derived and illustrated. In an indirect method, an optimal trajectory is determined by satisfying a set of necessary conditions (NC), and sufficient conditions (SC) if available. By contrast, a direct method uses the cost itself to determine an optimal solution. Even when a direct method is used, these conditions are useful to determine whether the solution satisfies the NC for an optimal solution. If it does not, it is not an optimal solution. As an example, the best two-impulse solution obtained by a direct method is not the optimal solution if the NC indicate that three impulses are required. Thus, post-processing a direct solution using the NC (and SC if available) is essential to verify optimality. Optimal Control, a generalization of the calculus of variations, is used to derive a set of necessary conditions for an optimal trajectory. The Primer Vector is a term coined by D. F. Lawden in his pioneering work in optimal trajectories. [This terminology is explained after Equation (2.24).] First-order necessary conditions for both impulsive and continuous-thrust trajectories can be expressed in terms of the Primer Vector. For impulsive trajectories, the Primer Vector determines the times and positions of the thrust impulses that minimize the propellant cost. For continuous thrust trajectories, both the optimal thrust direction and the optimal thrust magnitude as functions of time are determined by the Primer Vector.

  • Optimal Two- and Three-Impulse Fixed-Time Rendezvous in the Vicinity of a Circular Orbit
    Journal of Spacecraft and Rockets, 2003
    Co-Authors: John E. Prussing
    Abstract:

    Minimum-fuel, multiple-impulse orbital rendezvous is investigated for the case in which the transfer time is specified (time-fixed case). A method for obtaining optimal solutions is employed which is applicable to rendezvous or orbit transfer between elliptical orbits of low eccentricity. In this method optimal solutions are constructed by satisfying the necessary conditions for the Primer Vector. It is assumed that the terminal orbits lie close enough to an intermediate circular reference orbit that the linearized equations of motion can be used to describe the transfer. The linear boundary value problem for the impulse magnitudes for rendezvous is then solved analytically. As an application of the method, optimal two-and three-impulse fixed-time rendezvous transfers between coplanar circular orbits are obtained for a range of transfer times. These linearized solutions combined with previously obtained four-impulse solutions provide a complete solution for fixed-time coplanar circle-to-circle rendezvous between close orbits for transfer times up to nearly two terminal orbit periods.

  • Preliminary study of optimal thrust-limited path-constrained maneuvers
    Journal of Guidance Control and Dynamics, 1996
    Co-Authors: Russel S. Wenzel, John E. Prussing
    Abstract:

    Fuel-optimal maneuvers of a constant-specific-impulse, thrust-limited spacecraft in field-free space are analyzed. The simple problem of an optimal maneuver from a state of rest at one location to a state of rest at another location becomes very complex when a path constraint is introduced. Solutions are obtained using a direct numerical optimization method that combines Hermite-Simpson transcription and nonlinear programming. The resulting Lagrange multipliers provide a discrete approximation to the Primer Vector. The necessary conditions for an optimal solution can then be checked to validate the solution. Fundamental concepts, such as the existence of boundary arcs or boundary points and the optimal number of coast arcs, are examined. A comprehensive solution is obtained for symmetric rest-to-rest maneuvers. More general maneuvers are also examined.

Denis Arzelier - One of the best experts on this subject based on the ideXlab platform.

  • fuel optimal impulsive fixed time trajectories in the linearized circular restricted 3 body problem
    IAC 2018 - 69th International Astronautical Congress; IAF Astrodynamics Symposium, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Florent Brehard, Mioara Joldes
    Abstract:

    The problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the Primer Vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the Primer Vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the Primer Vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane (OOP) rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the OOP elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the out-of-plane elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical optimal solutions of impulsive out-of-plane rendezvous around elliptic orbits
    IFAC Proceedings Volumes, 2014
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre, J.l. Calvet
    Abstract:

    Abstract This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting, and a Keplerian relative motion. It is shown that the out-of-plane Keplerian relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. The different optimal solutions, for different durations of the rendezvous, are obtained via the analysis of the optimal conditions expressed in terms of the Primer Vector. A numerical example illustrate sthese results.

  • Using Polynomial Optimization to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem
    Journal of Guidance Control and Dynamics, 2011
    Co-Authors: Denis Arzelier, Mounir Kara-zaitri, Christophe Louembet, Akın Delibaşı
    Abstract:

    Nomenclature a = semi-major axis ; e = eccentricity ; ν = true anomaly ; φ(ν) = fundamental matrix of relative motion ; B(ν) = input matrix in the dynamic model of relative motion ; R(ν) = φ(ν)B(ν) = φ(ν)B(ν) = Primer Vector evolution matrix ; uf = φ(νf )Xf − φ(ν1)X1 6= 0 = boundary conditions ; N = number of velocity increments ; νi, ∀ i = 1, · · · , N = impulses application times ; ∆vi = impulse modulus at νi ;

Christophe Louembet - One of the best experts on this subject based on the ideXlab platform.

  • A New Mixed Iterative Algorithm to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem
    Journal of Optimization Theory and Applications, 2013
    Co-Authors: D. Arzelier, Christophe Louembet, A. Rondepierre, Mounir Kara-zaitri
    Abstract:

    The optimal fuel impulsive time-fixed rendezvous problem is reviewed. In a linear setting, it may be reformulated as a non-convex polynomial optimization problem for a pre-specified fixed number of velocity increments. Relying on variational results previously published in the literature, an improved mixed iterative algorithm is defined to address the issue of optimization over the number of impulses. Revisiting the Primer Vector theory, it combines variational tests with sophisticated numerical tools from algebraic geometry to solve polynomial necessary and sufficient conditions of optimality. Numerical examples under circular and elliptic assumptions show that this algorithm is efficient and can be integrated into a rendezvous planning tool.

  • Using Polynomial Optimization to Solve the Fuel-Optimal Linear Impulsive Rendezvous Problem
    Journal of Guidance Control and Dynamics, 2011
    Co-Authors: Denis Arzelier, Mounir Kara-zaitri, Christophe Louembet, Akın Delibaşı
    Abstract:

    Nomenclature a = semi-major axis ; e = eccentricity ; ν = true anomaly ; φ(ν) = fundamental matrix of relative motion ; B(ν) = input matrix in the dynamic model of relative motion ; R(ν) = φ(ν)B(ν) = φ(ν)B(ν) = Primer Vector evolution matrix ; uf = φ(νf )Xf − φ(ν1)X1 6= 0 = boundary conditions ; N = number of velocity increments ; νi, ∀ i = 1, · · · , N = impulses application times ; ∆vi = impulse modulus at νi ;

  • polynomial optimization for the solution of fuel optimal impulsive rendezvous using Primer Vector theory
    Conference on Decision and Control, 2010
    Co-Authors: Mounir Karazaitri, Denis Arzelier, Akın Delibaşı, Christophe Louembet
    Abstract:

    In this paper, the optimal fuel impulsive time-fixed rendezvous problem is considered. Under some simplifying assumptions, this problem may be recast as a non convex polynomial optimization problem. A numerical solving algorithm using a convex relaxation based on sum-of-squares representation of positive polynomials is proposed. Numerical results are evaluated on the PRISMA technology in-orbit formation flying testbed mission.

  • Mixed Iterative Algorithm For Solving Optimal Impulsive Time-Fixed Rendezvous Problem
    AIAA Guidance Navigation and Control Conference, 2010
    Co-Authors: Mounir Kara Zaitri, Denis Arzelier, Christophe Louembet
    Abstract:

    In this paper, an iterative algorithm for solving impulsive fuel-optimal rendezvous in a linearized gravitational field is proposed. The developed algorithm combines the iterative approach presented by Lion and Handelsman with a polynomial solver based on homotopy continuation methods using algebraic formulation of optimality conditions introduced recently by Carter. This allows to overcome the problems of non-smoothness of the resulting trajectory of the Primer Vector faced by several authors, when applying the LionHandelsman approach. The algorithm is designed for a general keplerian ellipse-to-ellipse impulsive transfer, through the use of Yamanaka-Ankersen transition matrix. For each iteration, homotopy continuation technique is used to solve polynomial equations system induced by the optimality conditions for a fixed number of impulses. Numerical results obtained on academic as well as non academic test cases are analysed through simulations

  • CDC - Polynomial optimization for the solution of fuel-optimal impulsive rendezvous using Primer Vector theory
    49th IEEE Conference on Decision and Control (CDC), 2010
    Co-Authors: Mounir Kara-zaitri, Denis Arzelier, Akın Delibaşı, Christophe Louembet
    Abstract:

    In this paper, the optimal fuel impulsive time-fixed rendezvous problem is considered. Under some simplifying assumptions, this problem may be recast as a non convex polynomial optimization problem. A numerical solving algorithm using a convex relaxation based on sum-of-squares representation of positive polynomials is proposed. Numerical results are evaluated on the PRISMA technology in-orbit formation flying testbed mission.

Aude Rondepierre - One of the best experts on this subject based on the ideXlab platform.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane (OOP) rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the OOP elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the out-of-plane elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical optimal solutions of impulsive out-of-plane rendezvous around elliptic orbits
    IFAC Proceedings Volumes, 2014
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre, J.l. Calvet
    Abstract:

    Abstract This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting, and a Keplerian relative motion. It is shown that the out-of-plane Keplerian relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. The different optimal solutions, for different durations of the rendezvous, are obtained via the analysis of the optimal conditions expressed in terms of the Primer Vector. A numerical example illustrate sthese results.

Romain Serra - One of the best experts on this subject based on the ideXlab platform.

  • fuel optimal impulsive fixed time trajectories in the linearized circular restricted 3 body problem
    IAC 2018 - 69th International Astronautical Congress; IAF Astrodynamics Symposium, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Florent Brehard, Mioara Joldes
    Abstract:

    The problem of fixed-time fuel-optimal trajectories with high-thrust propulsion in the vicinity of a Lagrange point is tackled via the linear version of the Primer Vector theory. More precisely, the proximity to a Lagrange point i.e. any equilibrium point-stable or not-in the circular restricted three-body problem allows for a linearization of the dynamics. Furthermore, it is assumed that the spacecraft has ungimbaled thrusters, leading to a formulation of the cost function with the 1-norm for space coordinates, even though a generalization exists for steerable thrust and the 2-norm. In this context, the Primer Vector theory gives necessary and sufficient optimality conditions for admissible solutions to two-value boundary problems. Similarly to the case of rendezvous in the restricted two-body problem, the in-plane and out-of-plane trajectories being uncoupled, they can be treated independently. As a matter of fact, the out-of-plane dynamics is simple enough for the optimal control problem to be solved analytically via this indirect approach. As for the in-plane dynamics, the Primer Vector solution of the so-called primal problem is derived by solving a hierarchy of linear programs, as proposed recently for the aforementioned rendezvous. The optimal thrusting strategy is then numerically obtained from the necessary and sufficient conditions. Finally, in-plane and out-of-plane control laws are combined to form the complete 3-D fuel-optimal solution. Results are compared to the direct approach that consists in working on a discrete set of times in order to perform optimization in finite dimension. Examples are provided near various Lagrange points in the Sun-Earth and Earth-Moon systems, hinting at the extensive span of possible applications of this technique in station-keeping as well as mission analysis, for instance when connecting manifolds to achieve escape or capture.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane (OOP) rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the OOP elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory
    IEEE Transactions on Control Systems Technology, 2018
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre
    Abstract:

    This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting. It is shown that the out-of-plane elliptic relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. Indeed, the approach relies on the Primer Vector theory by writing down and directly solving the optimality necessary conditions. After analyzing the characteristics of the dynamics of the optimal Primer Vector candidates, the complete analytical optimal solution is obtained for arbitrary durations of the rendezvous and arbitrary boundary conditions.

  • Analytical optimal solutions of impulsive out-of-plane rendezvous around elliptic orbits
    IFAC Proceedings Volumes, 2014
    Co-Authors: Romain Serra, Denis Arzelier, Aude Rondepierre, J.l. Calvet
    Abstract:

    Abstract This paper focuses on the fixed-time minimum-fuel out-of-plane rendezvous between close elliptic orbits of an active spacecraft, with a passive target spacecraft, assuming a linear impulsive setting, and a Keplerian relative motion. It is shown that the out-of-plane Keplerian relative dynamics are simple enough to allow for an analytical solution of the problem reviewed. The different optimal solutions, for different durations of the rendezvous, are obtained via the analysis of the optimal conditions expressed in terms of the Primer Vector. A numerical example illustrate sthese results.