Time optimal Zermelo's navigation problem with moving and fixed obstacles
TL;DR: This paper considers a time optimal Zermelo's navigation problem (ZNP) with moving and fixed obstacles that can be formulated as an optimal control problem with continuous inequality constraints and terminal state constraints using the control parametrization technique together with the time scaling transform.
About: This article is published in Applied Mathematics and Computation.The article was published on 2013-11-01. It has received 37 citations till now. The article focuses on the topics: Optimal control & Penalty method.
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TL;DR: The basic concepts underlying the MISER software, which include the control parameterization technique, a time scaling transform, a constraint transcription technique, and the co-state approach for gradient calculation, are described in this paper.
Abstract: The FORTRAN MISER software package has been used with great success over the past two decades to solve many practically important real world optimal control problems. However, MISER is written in FORTRAN and hence not user-friendly, requiring FORTRAN programming knowledge. To facilitate the practical application of powerful optimal control theory and techniques, this paper describes a Visual version of the MISER software, called Visual MISER. Visual MISER provides an easy-to-use interface, while retaining the computational efficiency of the original FORTRAN MISER software. The basic concepts underlying the MISER software, which include the control parameterization technique, a time scaling transform, a constraint transcription technique, and the co-state approach for gradient calculation, are described in this paper. The software structure is explained and instructions for its use are given. Finally, an example is solved using the new Visual MISER software to demonstrate its applicability.
38 citations
TL;DR: It is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer, which can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories.
Abstract: We consider the following differential game of pursuit and evasion involving two participating players: an evader, which has limited maneuverability, and an agile pursuer. The agents move on the Euclidean plane with different but constant speeds. Whereas the pursuer can change the orientation of its velocity vector arbitrarily fast, that is, he is a “pedestrian” a la Isaacs, the evader cannot make turns having a radius smaller than a specified minimum turning radius. This problem can be seen as a reversed Homicidal Chauffeur game, hence the name “Suicidal Pedestrian Differential Game.” The aim of this paper is to derive the optimal strategies of the two players and characterize the initial conditions that lead to capture if the pursuer acts optimally, and areas that guarantee evasion regardless of the pursuer’s strategy. Both proximity-capture and point-capture are considered. After applying the optimal strategy for the evader, it is shown that the case of point-capture reduces to a special version of Zermelo’s Navigation Problem (ZNP) for the pursuer. Therefore, the well-known ZNP solution can be used to validate the results obtained through the differential game framework, as well as to characterize the time-optimal trajectories. The results are directly applicable to collision avoidance in maritime and Air Traffic Control applications.
36 citations
Cites background from "Time optimal Zermelo's navigation p..."
...Zermelo’s Navigation Problem (ZNP) is a well-known result in optimal navigation, which has received a lot of attention in the literature (see for example [1,2,11])....
[...]
TL;DR: In this paper , a control-based UAV trajectory optimization problem for UAV aided wireless communication is studied, which takes into account both of the UAV's kinematic equations and the dynamic equations.
Abstract: This paper studies the three-dimensional (3D) trajectory optimization problem for unmanned aerial vehicle (UAV) aided wireless communication. Existing works mainly rely on the kinematic equations for UAV’s mobility modeling, while its dynamic equations are usually missing. As a result, the planned UAV trajectories are piece-wise line segments in general, which may be difficult to implement in practice. By leveraging the concept of state-space model, a control-based UAV trajectory design is proposed in this paper, which takes into account both of the UAV’s kinematic equations and the dynamic equations. Consequently, smooth trajectories that are amenable to practical implementation can be obtained. Moreover, the UAV’s controller design is achieved along with the trajectory optimization, where practical roll angle and pitch angle constraints are considered. Furthermore, a new energy consumption model is derived for quad-rotor UAVs, which is based on the voltage and current flows of the electric motors and thus captures both the consumed energy for motion and the energy conversion efficiency of the motors. Numerical results are provided to validate the derived energy consumption model and show the effectiveness of our proposed algorithms.
24 citations
TL;DR: In this paper, the authors considered a zero-sum differential game in which there were two players, a pursuing spacecraft that tried to minimize a payoff and an evading spacecraft that attempted to maximize the same payoff.
Abstract: This paper considers a spacecraft pursuit-evasion problem taking place
in
low earth
orbit. The problem is formulated as a zero-sum differential game in
which there are two players, a pursuing spacecraft that attempts to
minimize a payoff, and an evading spacecraft that attempts to maximize
the same payoff. We introduce two associated optimal control problems
and show that a saddle point for the differential game exists if and
only if the two optimal control problems have the same optimal
value. Then, on the basis of this result, we propose two
computational methods for
determining a saddle point solution: a
semi-direct control parameterization
method (SDCP method), which is based on a piecewise-constant control
approximation scheme, and a hybrid method,
which combines the new SDCP method with the multiple shooting
method. Simulation results show that the proposed SDCP and hybrid
methods are superior
to the semi-direct collocation nonlinear programming method
(SDCNLP method), which is widely used to solve pursuit-evasion
problems in the aerospace field.
24 citations
Cites background from "Time optimal Zermelo's navigation p..."
...The advantage of control parameterization is that it results in a finite-dimensional approximate problem of much smaller dimension [15, 17]....
[...]
01 Dec 2014
TL;DR: It is shown that the problem reduces to a special version of Zermelo's Navigation Problem (ZNP) for the pursuer, and the well-known ZNP solution can be used to validate the results obtained through the differential game framework as well as to characterize the time-optimal trajectories.
Abstract: In this paper we consider a differential game of pursuit and evasion involving two players with constant, but different, speeds, and different maneuverability constraints. Specifically, the evader has limited maneuverability, while the pursuer is completely agile. This problem is an asymmetric version of the well-known Game of Two Cars. The aim of this paper is to derive the optimal strategies of the two players and characterize areas of initial conditions that lead to capture if the pursuer acts optimally, and areas that guarantee evasion regardless of the pursuer's strategy. It is shown that the problem reduces to a special version of Zermelo's Navigation Problem (ZNP) for the pursuer. Therefore, the well-known ZNP solution can be used to validate the results obtained through the differential game framework as well as to characterize the time-optimal trajectories. The results are directly applicable to collision avoidance problems.
18 citations
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TL;DR: The relations between the different sets of optimality conditions arising in Pontryagin's maximum principle are shown and the application of these maximum principle conditions is demonstrated by solving some illustrative examples.
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937 citations
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272 citations
TL;DR: This work considers an optimal control problem with a nonlinear continuous inequality constraint and proposes an algorithm that computes a sequence of suboptimal controls for the original problem that converges to the minimum cost.
149 citations
TL;DR: This paper develops a computational method for a class of optimal control problems where the objective and constraint functionals depend on two or more discrete time points that is approximated by a sequence of approximate optimal parameter selection problems.
109 citations