Automated vehicles are increasingly getting main-streamed and this has pushed development of systems for autonomous manoeuvring (e.g., lane-change, merge, and overtake) to the forefront. A novel framework for situational awareness and trajectory planning to perform autonomous overtaking in high-speed structured environments (e.g., highway and motorway) is presented in this paper. A combination of a potential field like function and reachability sets of a vehicle are used to identify safe zones on a road that the vehicle can navigate towards. These safe zones are provided to a tube-based robust model predictive controller as reference to generate feasible trajectories for combined lateral and longitudinal motion of a vehicle. The strengths of the proposed framework are: 1) it is free from non-convex collision avoidance constraints; 2) it ensures feasibility of trajectory even if decelerating or accelerating while performing lateral motion; and 3) it is real-time implementable. The ability of the proposed framework to plan feasible trajectories for high-speed overtaking is validated in a high-fidelity IPG CarMaker and Simulink co-simulation environment.

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Trajectory Planning for Autonomous High-Speed Overtaking in Structured Environments Using Robust MPC

On Dubins paths to a circle

Model-based path planning for autonomous vehicles may incorporate knowledge of the dynamics, the environment, the planning objective, and available resources. In this chapter, we first review the most commonly used dynamic models for autonomous ground , surface, underwater, and air vehicles. We then discuss five common approaches to path planning—optimal control , level set methods, coarse planning with path smoothing , motion primitives, and random sampling—along with a qualitative comparison. The chapter includes a brief interlude on optimal path planning for kinematic car models. The aim of this chapter is to provide a high-level introduction to the field and to suggest relevant topics for further reading.

Model-Based Path Planning

In this paper we present a Dubins Path based framework to find emergency landing paths for fixed-wing airplanes in case of total loss of thrust. Using a simple motion model under the assumption of best glide, the aircraft can be treated as a Dubins Vehicle in the horizontal plane. Under these assumptions the optimal landing path can be described with a small set of generalized schemes. The resulting nonlinear systems of equations are solved using a modified Newton’s method, which enables fast solutions with low calculation effort. This enables the proposed algorithms for on-board real-time applications.

Fast Generation of Landing Paths for Fixed-Wing Aircraft with Thrust Failure

This paper is concerned with characterizing the shortest path of a Dubins vehicle from a position with a prescribed heading angle to a target circle with the final heading tangential to the target circle. Such a shortest path is of significant importance as it is usually required in real-world scenarios, such as taking a snapshot of a forbidden region or loitering above a ground sensor to collect data by a fixed-wing unmanned aerial vehicle in a minimum time. By applying Pontryagin's maximum principle, some geometric properties for the shortest path are established {without considering any assumption on the relationship between the minimum turning radius and radius of the target circle}, showing that the shortest path must lie in a sufficient family of 12 types. By employing those geometric properties, the analytical solution to each type is devised so that the length of each type can be computed in a constant time. In addition, some properties depending on problem parameters are found so that the shortest path can be computed without checking all the 12 types. Finally, some numerical simulations are presented, illustrating and validating the developments of the paper.

On Dubins Paths to a Circle

In 1889, Andrey Andreevich Markov published a paper in “Soobscenija Charkovskogo Matematiceskogo Obscestva” where he considered four mathematical problems related to the design of railways. The simplest among these problems (and the first one in course of the presentation) is described as follows. Find a minimum length curve between two points in the plane provided that the curvature radius of the curve should not be less than a given quantity and the tangent to the curve should have a given direction at the initial point. In 1951, Rufus Philip Isaacs submitted his first Rand Corporation Report on differential game theory where he stated and lined out a solution to the “homicidal chauffeur” problem. In that problem, a “car” with a bounded turning radius and a constant magnitude of the linear velocity tries as soon as possible to approach an avoiding the encounter “pedestrian”. At the initial time, the direction of the car velocity is given. In 1957, in American Journal of Mathematics, Lester Eli Dubins considered a problem in the plane on finding among smooth curves of bounded curvature a minimum length curve connecting two given points provided that the outgoing direction at the first point and incoming direction at the second point are specified. Obviously, if one takes a particular case of Isaacs’ problem with the immovable “pedestrian”, then the “car” will minimize the length of the curve with the bounded turning radius. The arising task coincides with the problem considered by A. A. Markov. The difference from the problem by L. E. Dubins is in the absence of a specified direction at the incoming point. The fixation of incoming and outgoing directions presents in the other three problems by A. A. Markov. However, they contain additional conditions inherent to the railway construction. In such a way the notion of a “car” which moves only forward and has bounded turning radius appeared. In 1990, in Pacific Journal of Mathematics, James Alexander Reeds and Lawrence Alan Shepp considered an optimization problem where the object with bounded turning radius and constant magnitude of the linear velocity can instantaneously change the direction of motion to the opposite one. In a similar way, carts move around storage rooms. Thus, the model of the car that can move forward and backward has appeared. 8

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Reachable Sets for Simple Models of Car Motion

An approach to systematization of types of formal cognitive maps

A three-dimensional reachable set “at the instant” for a controlled object called the Dubins car is considered. The turn angular velocity of the linear velocity vector is a control variable. Along with the case when, according to the statement of the problem, rotation is possible in both directions, the cases of one-sided rotation are studied. Three-dimensional images of reachable sets are presented. Sufficiency of the Pontryagin maximum principle conditions for the control leading to the boundary of the reachable set is analyzed. Possibility of using reachable sets in the problem of motion observation under conditions of inaccurate measurements of the geometric position is considered.

Reachable set for Dubins car and its application to observation problem with incomplete information

A strapdown inertial navigation system as part of a control object is under consideration. The influence of the spatial closed angular motion of the control object on the structure of its angular position errors in the SINS instrument axes is studied. It is assumed that these errors are proportional to the measured signal. Two types of closed angular motion are under consideration, characterized by the same orientations at the initial and final instants of time. It is shown that such types of motion enable to concentrate angular position errors in one of the instrument axes or redistribute them between the instrument axes. Due to this, the parameter observability of the SINS error model increases, the target characteristics of the control object can be improved. This approach can be particularly effective when using multi-precision gyroscopic sensors (including for a pre-adjusted parameters of the SINS error model).

Determination of Significant Characteristics of Strapdown Inertial Navigation System as Part of a Control Object Using Typical Motion Sections

For the Dubins car model, the paper highlights the cases when the Pontryagin Maximum Principle (PMP) is both necessary and sufficient condition for the motions leading onto the boundary of the reachable set at a given instant. Some relation is revealed between the PMP sufficient property and convexity of the reachable set sections on the angular coordinate. Uniqueness of the extremal controls leading onto the boundary is analyzed in the class of piecewise-constant controls.

Andrey A. Fedotov

Papers

Reachable Sets for Simple Models of Car Motion

An approach to systematization of types of formal cognitive maps

Reachable set for Dubins car and its application to observation problem with incomplete information

Determination of Significant Characteristics of Strapdown Inertial Navigation System as Part of a Control Object Using Typical Motion Sections

Three-dimensional reachable set at instant for the dubins car: Properties of extremal motions