Hodograph analysis in aircraft trajectory optimization

Cliff, Eugene M.; Seywald, Hans; Bless, Robert R.

doi:10.2514/6.1993-3742

Cited by 13 publications

(6 citation statements)

References 8 publications

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“…If the domain of maneuverability is convex, then the minimization results in a unique solution. This convexity property is satisfied for the problem studied in this paper, and we will not discuss the delicate issues of nonconvexity and singular optimal control (see e.g., [2,5,21]). …”

Section: The Relationmentioning

confidence: 95%

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Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

Techy

2011

Intel Serv Robotics

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This paper is concerned with time-optimal navigation for flight vehicles in a planar, time-varying flow-field, where the objective is to find the fastest trajectory between initial and final points. The primary contribution of the paper is the observation that in a point-symmetric flow, such as inside vortices or regions of eddie-driven upwelling/downwelling, the rate of the steering angle has to be equal to onehalf of the instantaneous vertical vorticity. Consequently, if the vorticity is zero, then the steering angle is constant. The result can be applied to find the time-optimal trajectories in practical control problems, by reducing the infinite-dimensional continuous problem to numerical optimization involving at most two unknown scalar parameters.

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Section: The Relationmentioning

confidence: 95%

“…One can attribute an intuitive geometric interpretation to the minimum principle using hodograph analysis [5]. The hodograph is a figure in the velocity-space with the states frozen at their current values:…”

Section: The Relationmentioning

confidence: 99%

Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

Techy

2011

Intel Serv Robotics

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“…It is well known that an optimal solution does not exist and numerical methods to find the optimal solution must break down as soon as there is one instant of time along a trajectory at I M I I I I I | I I I I I I I I I | I I I I I I I I I | I I I I I I I I I | I I I I U I I I which, loosely speaking, PMP "wants to select the control" such that the associated state rate x = f(x,u,t) lies in a nonconvex domain of the hodograph. 3 Unfortunately, for the ALS vehicle the dependence of aerodynamic drag on angle of attack a is such that the hodograph becomes nonconvex if a is allowed to vary within fixed bounds and the location of this nonconvexity is such that it is likely to play a role in the optimal control process (see Figs. [2][3][4].…”

Section: Phase 1: Optimal Control Applied On Point-mass Modelmentioning

confidence: 99%

“…3 Unfortunately, for the ALS vehicle the dependence of aerodynamic drag on angle of attack a is such that the hodograph becomes nonconvex if a is allowed to vary within fixed bounds and the location of this nonconvexity is such that it is likely to play a role in the optimal control process (see Figs. [2][3][4]. Hence it is expected that, with angle of attack a included in the set of control variables and the dependence of the aerodynamic forces on a modeled somewhat precisely, the optimal control problem does not have a solution.…”

Section: Phase 1: Optimal Control Applied On Point-mass Modelmentioning

confidence: 99%

Neighboring optimal control based feedback law for the advanced launch system

Seywald

Cliff

1994

Journal of Guidance, Control, and Dynamics

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In this paper a robust feedback algorithm is presented for a near-minimum-fuel ascent of a generic two-stage launch vehicle operating in the equatorial plane. The development of the algorithm is based on the ideas of neighboring optimal control and can be divided into three phases. In phase 1 the formalism of optimal control is employed to calculate fuel-optimal ascent trajectories for a simple point-mass model. In phase 2 these trajectories are used to numerically calculate gain functions of time for the control(s), for the total flight time, and possibly for other variables of interest. In phase 3 these gains are used to determine feedback expressions for the controls associated with a more realistic model of a launch vehicle. With the advanced launch system in mind, all calculations in this paper are performed on a two-stage vehicle with fixed thrust history, but this restriction is by no means important for the approach taken. Performance and robustness of the algorithm is found to be excellent. cg y cg * TB yxB JC TC J TC T B T c ft Nomenclature = x coordinate of center of gravity = y coordinate of center of gravity = x coordinate of point from where booster thrust emanates = y coordinate of point from where booster thrust emanates = x coordinate of point from where core thrust emanates = y coordinate of point from where booster thrust emanates = thrust magnitude of booster = thrust magnitude of core = gimbal angle for core thrust S re f = reference length J re f = reference area

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“…17 and 18. Assuming that the hodograph of the problem is strictly convex 19 (also referred to as a regular Hamiltonian), it has been shown that the control must be continuous across a junction point. In this case, for a scalar state constraint, Ref.…”

Section: Problem Definitionmentioning

confidence: 99%

Finite element method for the solution of state-constrained optimal control problems

Bless¹,

Hodges

Seywald

1995

Journal of Guidance, Control, and Dynamics

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This paper presents an extension of a finite element formulation based on a weak form of the necessary conditions to solve optimal control problems. First, a general formulation for handling internal boundary conditions and discontinuities in the state equations is presented. Then, the general formulation is modified for optimal control problems subject to state-variable inequality constraints. Solutions with touch points and solutions with stateconstrained arcs are considered. After the formulations are developed, suitable shape and test functions are chosen for a finite element discretization. It is shown that all element quadrature (equivalent to one-point Gaussian quadrature over each element) may be done in closed form, yielding a set of algebraic equations. To demonstrate and analyze the accuracy of the finite element method, a simple state-constrained problem is solved. Then, for a more practical application of the use of this method, a launch vehicle ascent problem subject to a dynamic pressure constraint is solved. The paper also demonstrates that the finite element results can be used to determine switching structures and initial guesses for a shooting code.

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Hodograph analysis in aircraft trajectory optimization

Cited by 13 publications

References 8 publications

Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

Optimal navigation in planar time-varying flow: Zermelo’s problem revisited

Neighboring optimal control based feedback law for the advanced launch system

Finite element method for the solution of state-constrained optimal control problems

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