Time optimal Zermelo’s navigation problem with moving and fixed obstacles

doi:10.1016/j.amc.2013.08.092

Applied Mathematics and Computation

Volume 224, 1 November 2013, Pages 866-875

https://doi.org/10.1016/j.amc.2013.08.092 Get rights and content

Abstract

In this paper, we consider a time optimal Zermelo’s navigation problem (ZNP) with moving and fixed obstacles. This problem can be formulated as an optimal control problem with continuous inequality constraints and terminal state constraints. By using the control parametrization technique together with the time scaling transform, the problem is transformed into a sequence of optimal parameters selection problems with continuous inequality constraints and terminal state constraints. For each problem, an exact penalty function method is used to append all the constraints to the objective function yielding a new unconstrained optimal parameters selection problem. It is solved as a nonlinear optimization problem. Different scenarios are considered in the simulation, and the results obtained show that the proposed method is effective.

Introduction

The information of the surrounding environment of a vessel, an airplane or a robot can be collected by using sensor and data processing technology. For example, the ocean wave-field surrounding a moving or stationary vessel can be measured by an innovative coherent (Doppler) X-band radar mounted on a ship. The optimal navigation can be enhanced by incorporating such information [1]. In [1], an efficient path finding algorithm based on discrete dynamic programming is presented for finding the fastest path from a given starting point to a predetermined point, where the visibility horizon and sharpest turning radius are taking into consideration.

Zermelo’s navigation problem is a classical optimal control problem, which was introduced by Zermelo in the early 1930s. In this problem, a vessel is required to traverse the river in the presence of the flow current in the minimum time. The problem was then studied with a number of extensions and variations in literature (see, for example, [2], [3]). In this paper, we investigate a Zermelo’s navigation problem in the presence of moving and fixed obstacles. To be more specific, a fast vessel wishes to cross the river with strong flow current so as to reach the harbor in the opposite shore, while needing to avoid n slow moving vessels and some fixed obstacles in the river. The fast vessel is regarded as an autonomous fast agent and the n slow moving vessels are regarded as n slow agents. It is assumed that the trajectories of the n slow agents are known to the fast agent in advance. The objective is to find a control such that the mission is accomplished with a minimum time. The problem is motivated by the ship-to-port cargo transfer problem subject to obstacle avoidance, including some moving vessels and/or some fixed obstacles near the port.

The natural formulation of this Zermelo’s navigation problem is in the form of a time optimal control problem subject to continuous inequality constraints and terminal state constraints. [4] is an excellent survey on necessary conditions for optimality of various types of constrained optimal control problems. Furthermore, there are several numerical methods available in the literature that can be applied to solve this constrained optimal control problem, such as the discretization method [5], [6], the nonsmooth Newton method [7], and the control parameterizations method [8], [9], [10], [11], [12], [13]. In this paper, we choose to solve this time optimal control problem as follows. The control parametrization approach and the time scaling transform [13] are applied to approximate the time optimal control problem as a sequence of optimal parameter selections problems subject to continuous inequality constraints and terminal state constraints. The exact penalty function method proposed in [14] is applied to construct a constraint violation function for the continuous inequality constraints and the terminal state constraints. It is then appended to the cost function, forming a new cost function. In this way, each of the optimal parameter selection problems is further approximated as an optimal parameter selection problem subject to a simple non-negativity constraint on a decision parameter. This problem can be solved as a nonlinear optimization problem by any effective gradient-based optimization technique, such as the sequential quadratic programming (SQP) method. This computational scheme is supported by rigorous convergence analysis. Numerical simulations are carried out, where two scenarios are considered. In the first case, it is assumed that there is no fixed obstacle in the river. In the second scenario, two fixed obstacles are placed along the trajectory obtained in the first scenario.

Section snippets

Statement of the problem

Given $n + 1$ agents in a 2-dimensional (2D) flow field, where two slow agents follow navigated trajectories, while the third one, which is faster, is autonomously controllable. Let the trajectories of the n slow agents be denoted as ${\vec{z}}_{i} (t) = {[x_{i} (t), y_{i} (t)]}^{⊤}, i = 1, 2, \dots, n$ and $t ⩾ 0$ . We assume that the velocity components at any point $(x, y)$ in the 2D flow field can be denoted by $G (x, y, t)$ and $H (x, y, t)$ , respectively. The flow dynamics ${[G (x, y, t), H (x, y, t)]}^{T}$ can be modeled by the famous Navier–Stokes equation [15].

Control parametrization and time scaling transform

Problem (P) is a nonlinear optimal control problem subject to continuous inequality constraints. It is well known that nonlinear optimal control problem is difficult to be solved by means of the classical optimal control theory. Furthermore, there is an inequality constraint in each time point, which implies that there are infinite number of constraints. For this, control parametrization and time scaling transform are applied to transform this problem into a nonlinear semi-infinite optimization

Simulation results

The trajectories of the two slow agents, $A_{1}$ and $A_{2}$ , are described as: $A_{1} : \{\begin{matrix} x_{1} (t) = 25 + (1 / 25) {(t - 25)}^{2}, \\ y_{1} (t) = t, \end{matrix}$ $A_{2} : \{\begin{matrix} x_{2} (t) = 50 + t / 50, \\ y_{2} (t) = t \end{matrix}$ and the safety region is 1. The equations of motion of the faster agent $A_{3}$ in the presence of the flow dynamic is described as: $A_{3} : \{\begin{matrix} {\dot{x}}_{3} (t) = V \cos (u (t)) - y_{3} (t) (1 - y_{3} (t)), \\ {\dot{y}}_{3} (t) = V \sin (u (t)) . \end{matrix}$

Note that, for the current $G [x (t), y (t), t] = y_{3} (1 - y_{3})$ and $G [x (t), y (t), t] = 0$ , it is as shown in Fig. 4, where $b = 1 / 2$ . It implies that the current is strongest in the middle of the river and its

Conclusions

This paper presents an effective computational method for solving a special Zermelo’s Navigation problem where there are moving and fixed obstacles. This method is then applied to two scenarios of the problem. The results obtained clearly demonstrate the effectiveness of the method proposed.

Acknowledgement

This work is supported by a grant from the Australia Research Council and partially supported by a Major State Basic Research Development Program 973 (No. 2012CB215202) and a National Natural Science Foundation of China (61134001). The first author would like to thank Professor Yong Hong Wu for his kindness help in this research.

References (18)

C. Buskens et al.
SQP-methods for solving optimal control problems with control and state constraints: adjonit varibles, sensitivity analysis and real time control
Journal of Computational and Applied Mathematics
(2000)
T.W.C. Chen et al.
Inequality path constraints in optimal control: a finite iteration $ε$ -convergent scheme based on pointwise discretization
Journal of Process Control
(2005)
R. Loxton et al.
Optimal control problems with a continuous inequality constraint on the state and control
Automatica
(2009)
Q. Lin et al.
Optimal control computation for nonlinear systems with state-dependent stopping criteria
Automatica
(2012)
Q. Chai et al.
A class of optimal state-delay control problems
Nonlinear Analysis: Real World Applications
(2013)
R. Loxton et al.
Optimal control problems with multiple characteristic time points in the objective and constraints
Automatica
(2008)
I.S. Dolinskaya
Optimal path finding in direction, location and time dependent environment
Naval Research Logistics
(2012)
E. Bakolas, P. Tsiotras, Time-optimal synthesis for the Zermelo–Markov–Dubins problem: the constant wind case, in:...
A. Enes, W. Book, Blended shared control of Zermelo’s navigation problem, in: American Control Conference, Woodruff...

There are more references available in the full text version of this article.

Cited by (45)

Enhanced moving finite element method based on error geometric estimation for simultaneous trajectory optimization
2023, Automatica
The direct method has been widely used in various fields of trajectory optimization due to the need to optimize the performance of a controlled dynamic system. An essential issue in the development of the direct method is the selection of the finite element mesh. Although there have been many pieces of research on adaptive mesh refinement, few works focused on finite element layout optimization before the refinement. This paper proposes an enhanced moving finite element method (EMFE), taking the length of each finite element as a variable, based on the geometric estimation of non-collocation-point error (NCPE) and generalized finite element length (GFEL). The features of EMFE include flexible problem reformulation, stable numerical calculation and the ability to locate breakpoints for singular control problems (the control variables appear linearly in the problems) and finding the local minimum numerical error. The theoretical basis of the ability to locate the breakpoints for singular control problems and the feature of NCPE estimate has also been investigated. Three numerical examples, two of which are singular control problems for lunar landing and the other is a multi-phase nonlinear problem for launch vehicles, demonstrated the performance and features of the proposed EMFE method.
Time-optimal Dubins trajectory for moving obstacle avoidance
2022, Automatica
Citation Excerpt :
In Fathi, Bidabad, and Najafpour (2021) the use of direct method for a 3-D Dubins model (Chitsaz & LaValle, 2007) while incorporating collision avoidance as a soft constraint to the cost was proposed. Such approaches also lead to only approximate optimal control and are prone to converge at local minima as indicated in Li, Xu, Teo, and Chu (2013). The indirect method coupled with multiple shooting method for solving the resulting MPBVP is effective in computing highly accurate solutions.
In this paper, the problem of computing the global time-optimal Dubins trajectory between two configurations (two points with prescribed heading angles) while avoiding a moving obstacle by a specified safe distance is discussed. The obstacle is allowed to move on any arbitrary trajectory as long as its trajectory satisfies some regularity constraints. By Pontryagin’s maximum principle, the fundamental segments of the time-optimal trajectory are proposed. Along each segment, the control input is either constant or is expressed in state-feedback form. Thereafter, using the necessary conditions for state-inequality constraints, the underlying geometric and analytical properties that govern the concatenation of these fundamental segments are revealed. All these properties exhibit solely state-dependent relations. This has two-fold applications: (1) such relations are essential in ruling out spurious extremals obtained by numerical optimal control methods, and (2) they enable the formulation of a set of nonlinear equations whose solution gives rise to the exact time-optimal trajectory. Numerical examples are presented, validating and highlighting the application of these properties.
Deep neural networks-based real-time optimal navigation for an automatic guided vehicle with static and dynamic obstacles
2021, Neurocomputing
Citation Excerpt :
From the perspective of optimal control methods, the path planning of AGVs can be regarded as various optimal trajectory planning problems according to the given starting and ending points with path constraints. Contributions to these related areas can be found in the literature [9–15]. For instance, in [9], a numerical optimal control approach based on interior-point method is proposed to solve a time-optimal maneuver planning problem in automatic parallel parking of a wheeled vehicle.
In this paper, a hybrid-intelligent real-time optimal control approach based on deep neural networks (DNNs) is proposed to improve the autonomy and intelligence of automatic guided vehicles (AGVs) navigation control. We first formulate the motion planning problem of an AGV with static and dynamic obstacles as a nonlinear optimal control problem (OCP). Subsequently, a direct method incorporating a smooth transformation technique is utilized to obtain the high-fidelity optimal solutions off-line. The optimal state-action samples are then extracted from the optimal trajectories generated by the OCP from perturbed initial states and stored as a large optimal trajectory data set. Thereafter, the DNNs architecture is designed and trained off-line through the collected data set to learn the optimal state-action relationship. As a result, the well-trained DNNs are used to produce the corresponding to optimal feedback actions on-board. The main advantage of our approach is that the time-consuming computation process is only carried out off-line and thus the traditional repeated off-line planning for the AGV due to changes such as the perturbed initial conditions in the system can be effectively avoided. Numerical experimental results are carried out to demonstrate our designed DNNs-based optimal control approach can generate the optimal control instructions on-board to steer the AGV to the desired location with high robustness to initial conditions as well as satisfying different obstacle constraints.
Dynamic optimization of nonlinear systems with guaranteed feasibility of inequality-path-constraints
2021, Automatica
An algorithm is proposed for locating an optimal solution satisfying the KKT conditions to a specified tolerance to inequality-path-constrained dynamic optimization (PCDO) problem within finite iterations. This algorithm solves the optimization problem by iteratively approximating the original optimization problem through adaptive convexification of its lower level programs. In the process of convexification of the lower level programs, $α$ BB technique is used adaptively to construct convex relaxations of a path constraint in each time subinterval. Compared to the result in (Fu, Faust, Chachuat, & Mitsos, 2015), the distinguishing feature is that the proposed algorithm avoids numerically solving the non-convex lower level program to global optimality at each iteration. Two numerical examples are shown to demonstrate the performance of the algorithm.
Time optimal control of triple integrator with input saturation and full state constraints
2020, Automatica
Citation Excerpt :
There are several methods to deal with the state constraints, such as the dynamic programming scheme (Bellman, 1952; Fu, Faust, Chachuat, et al., 2015) and the control parametrization scheme (Lin, Loxton, & Teo, 2014; Teo, Goh, & Wong, 1991). The control parametrization scheme includes the exact penalty function method (Li, Xu, Teo, et al., 2013; Li et al., 2011) and the constraint transcription method (Liu, Gong, Teo, et al., 2017; Liu, Loxton, Lin, et al., 2018), while these two methods are not easy to generate analytical solutions describing the detailed optimal control process. Yuan, Chen, Yao, et al. (2019) and Yuan et al. (2018) developed a sliding surface based nonlinear filter for double integrator with partial state constraints and input constraint.
Multiple integrator systems with input saturation and state constraints ubiquitously exist in practical problems such as trajectory planning in CNC (computer numerical control) systems, robots, automotive systems and industrial processes. Without state constraints, the optimal control of the multiple integrator with only input saturation is well solved in classic optimal control theory. However, if state constraints are considered, it is still challenging to obtain a global time optimal analytical solution for multiple integrator systems with even higher than second order, e.g., a double integrator plant. In this paper, the global time optimal control law for triple integrator with input saturation and full state constraints is considered. The system has a serial structure of three integral elements, while the control input and system states are within box constraints. A bang-singular-bang time optimal control law is synthesized according to Pontryagin minimum principle. Then, costates and jump conditions are analyzed in detail. Based on Bellman’s principle of optimality, switching surfaces and curves in phase space are obtained through dynamic programming method. The time optimal control law is then obtained in analytical state feedback form, and its global convergence property is proved. Through the comparison with numerical solutions, the effectiveness of the proposed control strategy is verified. The proposed scheme will be useful for trajectory planning under input saturation and full state constraints in practical applications.
Online inverse optimal control for control-constrained discrete-time systems on finite and infinite horizons
2020, Automatica
In this paper, we consider the problem of computing parameters of an objective function for a discrete-time optimal control problem from state and control trajectories with active control constraints. We propose a novel method of inverse optimal control that has a computationally efficient online form in which pairs of states and controls from given state and control trajectories are processed sequentially without being stored or processed in batches. We establish conditions guaranteeing the uniqueness of the objective-function parameters computed by our proposed method from trajectories with active control constraints. We illustrate our proposed method in simulation.

View all citing articles on Scopus

View full text

Time optimal Zermelo’s navigation problem with moving and fixed obstacles

Abstract

Introduction

Section snippets

Statement of the problem

Control parametrization and time scaling transform

Simulation results

Conclusions

Acknowledgement

Journal of Computational and Applied Mathematics

Journal of Process Control

Automatica

Automatica

Nonlinear Analysis: Real World Applications

Automatica

Optimal path finding in direction, location and time dependent environment

Naval Research Logistics