Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory

Serra, Romain; Arzelier, Denis; Rondepierre, Aude

doi:10.1109/tcst.2017.2656022

Cited by 12 publications

(5 citation statements)

References 16 publications

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“…and are defined for all u ∈ R m . Next, while g is defined on open intervals, the behavior of the cost at the boundary of these intervals satisfies the requirements in (9). As such, the sets T 1 and T 2 can be defined as…”

Section: Validation and Performance Assessmentmentioning

confidence: 99%

See 1 more Smart Citation

Fast Algorithm for Fuel-Optimal Impulsive Control of Linear Systems With Time-Varying Cost

Koenig

D’Amico

2021

IEEE Trans. Automat. Contr.

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This paper presents a new computationally efficient and robust algorithm that provides energy-optimal impulsive control input sequences that drive a linear timevariant system to a desired state at a specified time. This algorithm is applicable to a broad class of problems where the cost is expressed as a time-varying norm-like function of the control input. This degree of generality enables inclusion of complex, time-varying operational constraints in the control planning problem. First, optimality conditions for this problem are derived using reachable set theory, which provides a simple geometric interpretation of the meaning of the dual variables. Next, an optimal impulsive control algorithm is developed that iteratively refines an estimate of the dual variable until the optimality criteria are satisfied to within a user-specified tolerance. The iteration procedure simultaneously ensures global convergence and minimizes computation cost by discarding inactive constraints. The algorithm is validated through implementation in challenging example problems based on the recently proposed Miniaturized Distributed Occulter/Telescope small satellite mission, which requires periodic formation reconfigurations in a perturbed orbit with time-varying attitude constraints. Additionally, it is demonstrated that the algorithm can be implemented with run times more than an order of magnitude faster than other approaches.

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Section: Validation and Performance Assessmentmentioning

confidence: 99%

“…However, such solutions are inherently specific to the prescribed state representation, dynamics model, and cost function. Indeed, such solutions have only been found to date for a limited class of problems in spacecraft formation-flying [1], [2], [7]- [9].…”

Section: Introductionmentioning

confidence: 99%

Fast Algorithm for Fuel-Optimal Impulsive Control of Linear Systems With Time-Varying Cost

Koenig

D’Amico

2021

IEEE Trans. Automat. Contr.

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“…The two-impulse trajectory of the CS subject to the initial and final positions is determined by two parameters of t and ∆t i+1,i that is formulated in (8). The set R i,i+1 (t) contains all position vectors which can be reached at t ∈ T ∆ti+1,i by any ∆t i+1,i ∈ T π/κ .…”

Section: B Impulse Design Under Path Constraintsmentioning

confidence: 99%

“…Relative control of spacecraft is widely discussed in the literature and many different schemes are proposed. Optimal impulsive approaches based on the primer vector solutions are investigated in [6]- [8]. Gao et al discussed the robust H ∞ control of relative motion [9], while solutions to the matrix inequalities are proposed by Tian and Jia [10].…”

Section: Introductionmentioning

confidence: 99%

On the Impulsive Formation Control of Spacecraft Under Path Constraints

Shakouri

2019

IEEE Trans. Aerosp. Electron. Syst.

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This paper deals with the impulsive formation control of spacecraft in the presence of constraints on the position vector and time. Determining a set of path constraints can increase the safety and reliability in an impulsive relative motion of spacecraft. Specially, the feasibility problem of the position norm constraints is considered in this paper. Under assumptions, it is proved that if a position vector be reachable, then the reach time and the corresponding time of impulses are unique. The trajectory boundedness of the spacecraft between adjacent impulses are analyzed using the Gerschgorin and the Rayleigh-Ritz theorems as well as a finite form of the Jensen's inequality. Some boundaries are introduced regarding the Jordan-Brouwer separation theorem which are useful in checking the satisfaction of a constraint. Two numerical examples (approximate circular formation keeping and collision-free maneuver) are solved in order to show the applications and visualize the results.

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“…A complete analytic solution for the out-of-plane component of the primer vector is shown to exist, which is independent of the semi-major axis of the transfer orbit [66]. The problem of time-fixed fueloptimal out-of-plane elliptic rendezvous between spacecraft in a linear setting is studied with a complete analytical closed-form solution [67].…”

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confidence: 99%

How Many Impulses Redux

Taheri

Junkins

2019

J Astronaut Sci

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A central problem in orbit transfer optimization is to determine the number, time, direction and magnitude of velocity impulses that minimize the total impulse. This problem was posed in 1967 by T. N. Edelbaum, and while notable advances have been made, a rigorous means to answer Edelbaum's question for multiple-revolution maneuvers has remained elusive for over five decades. We revisit Edelbaum's question by taking a bottom-up approach to generate a minimum-fuel switching surface. Sweeping through time profiles of the minimum-fuel switching function for increasing admissible thrust magnitude, and in the high-thrust limit, we find that the continuous thrust switching surface reveals the N -impulse solution. It is also shown that a fundamental minimum-thrust solution plays a pivotal role in our process to determine the optimal minimum-fuel maneuver for all thrust levels. Remarkably, we find the answer to Edelbaum's question is not generally unique, but is frequently a set of equal-∆v extremals. We further find, when Edelbaum's question is refined to seek the number of finite-duration thrust arcs for a specific rocket engine, that a unique extremal is usually found. Numerical results demonstrate the ideas and their utility for several interplanetary and Earth-bound optimal transfers that consist of up to eleven impulses or, for finite thrust, short thrust arcs. Another significant contribution of the paper can be viewed as a unification in astrodynamics where the connection between impulsive and continuous-thrust trajectories are demonstrated through the notion of optimal switching surfaces. NOMENCLATURE f = vector function of unforced dynamics B = control influence matrix c = exhaust velocity, m/s g 0 = Earth gravitational acceleration, m/s 2 h = specific angular momentum vector, km 2 /s H = Hamiltonian I sp = specific impulse, s J = cost functional

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Analytical Solutions for Impulsive Elliptic Out-of-Plane Rendezvous Problem via Primer Vector Theory

Cited by 12 publications

References 16 publications

Fast Algorithm for Fuel-Optimal Impulsive Control of Linear Systems With Time-Varying Cost

Fast Algorithm for Fuel-Optimal Impulsive Control of Linear Systems With Time-Varying Cost

On the Impulsive Formation Control of Spacecraft Under Path Constraints

How Many Impulses Redux

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