Sikha Hota

Optimal path planning for an aerial vehicle in 3D space

We address optimal path planning in three-dimensional space for an aerial vehicle with bounded turn radius. The objective is to calculate a feasible path of minimum length when the initial and final positions and orientations of the vehicle are given. The proposed method is based on 3D geometry. Unlike the existing iterative methods, which yield suboptimal path and are computationally more intensive, this geometrical method generates an optimal path in much lesser time. Due to its simplicity and low computational requirements this approach can be implemented on an aerial vehicle with constrained turn radius to reach a final point with a prescribed orientation. But, if the path demands very high pitch angle (as in the case of steep climbs), then the generated path may not be flyable for an aerial vehicle with limited range of flight path angle and we need to use a numerical method such as multiple shooting to obtain the optimal solution.

Optimal geometrical path in 3D with curvature constraint

This paper presents a path planning approach for achieving an optimal feasible path satisfying a maximum curvature bound in three dimensional space, given initial and final configurations specified by position and orientation vectors. Based on Dubins strategy two types of solution approaches will be discussed, the first is a numerical technique which is computationally intensive and the second is based on 3D geometry from which we will derive an analytical solution. In the second approach, the computational time is very low and the strategy can be implemented for real-time path planning problems. Unlike the existing iterative methods which yield suboptimal paths and are computationally more intensive, this geometrical method generates an optimal path in lesser time. Due to its simplicity and low computational requirements this approach can be implemented on fixed wing aerial vehicles with constrained turn radius.

Optimal Trajectory Generation for Convergence to a Rectilinear Path

This paper presents a strategy to determine the shortest path of a fixed-wing Miniature Air Vehicle (MAV), constrained by a bounded turning rate, to eventually fly along a given straight line, starting from an arbitrary but known initial position and orientation. Unlike the work available in the literature that solves the problem using the Pontryagin’s Minimum Principle (PMP) the trajectory generation algorithm presented here considers a geometrical approach which is intuitive and easy to understand. This also computes the explicit solution for the length of the optimal path as a function of the initial configuration. Further, using a 6-DOF model of a MAV the generated optimal path is tracked by an autopilot consisting of proportional-integral-derivative (PID) controllers. The simulation results show the path generation and tracking for different cases.

Optimal Trajectory Planning for Unmanned Aerial Vehicles in Three-Dimensional Space

T HIS paper generates an optimal path from an initial configuration (position, heading, and flight-path angle) to a final configuration in three-dimensional (3-D) space for a turn-rateconstrained fixed-wing unmanned aerial vehicle (UAV) flying both in the absence and presence of wind. Because the problem is fundamental to 3-D waypoint-following problems in complex environments, it is a challenge to generate an optimal trajectory that is computationally fast for online implementation. Dubins [1] showed that, in a two-dimensional (2-D) plane, for fixed initial and final positions and orientations, the shortest path for a constant speed and turn-radius-constrained vehicle consists of three consecutive path segments, each of which is a circle ofminimum turn radius (C) or a straight line (S). In 3-D space, Sussmann [2] used a maximum principle on manifolds to show that every minimizer in 3D is either a helicoidal arc or a concatenation of three pieces, each of which is a circle or a straight line. In coordinated path planning of multiple UAVs in 3-D space for sufficiently far-apart points, Shanmugavel et al. [3] computedCCSC-type paths for a 3-D Dubins problem. Babaei andMortazavi [4] used two planes in which the 3-D path obtained is also ofCCSC type. The solutions of [3,4], according to Sussmann [2], are not optimal. A suboptimal trajectory was also generated for the 3-DDubins problem in [5] based on the assumption of independent bounded control over the altitude velocity and turn rate in the plane. In [6], using an iterative algorithm, a suboptimal path consisting of a maximum of five different segments, each of which is either a circle or a straight line, was generated for the 3-D Dubins problem with the flight-path angle constraint. The algorithm is stated to be suboptimal only when the distance between the two points projected onto the X-Y plane is large; otherwise, this solution is very far from the optimal one. In [7], using Pythagorean hodograph curves, flyable paths were designed between 3-D waypoints satisfying maximum curvature and torsion bound. The 3-D waypoint-following problem was discussed in a cluttered environment in [8], where collision-free waypoints were generated by rapidly exploring random trees. Then, the unnecessary waypoints were removed by a simple path-pruning algorithm, and piecewise linear paths were obtained. To develop a smooth continuous curvature path from these linear segments, cubic Bezier spiral curves was used, and the generated smooth path also satisfied the curvature constraint of the vehicle. The presence ofwind has been considered in some papers [9–18], but mostly in the 2-D plane. In [19,20], considering the steady-wind effect at high altitude, a 3-D path was generated numerically based on the Pontryagin minimum principle. With the same assumption as in [3,4], that is, when the points are situated sufficiently far away, a 3-D CSC path is generated using geometry. As per [2], thisCSC path is only a candidate optimal path. To establish the optimality of the CSC path for the points situated sufficiently far apart, the problem is formulated in the optimal control framework and solved using the multiple shooting method coupled with nonlinear programming (MS-NP) [21]. The optimal path generated by the MS-NP technique completely matches with the CSC path generated by 3-D geometry. If the generated CSC path demands steep climb or descent (when the vertical separation between the points is significantly higher than their horizontal separation), it may not be flyable due to flight-path angle constraints. This issue is addressed by including the flight-path angle constraint and solved numerically using the MS-NP technique to obtain the optimal path. Some preliminary work on this problem is available in [22] for the absence of wind condition. The present paper also extends the idea of generating a 3-D optimal path in the presence of steady and time-varying wind, considering wind flow in horizontal plane [23]. In summary, the paper develops the time-optimal path in 3-D space both in the absence and presence of wind for a constant-speed, fixedwing UAV for given initial and final configurations. Unlike existing iterative methods, which yield suboptimal paths and are computationally more intensive, the geometrical method proposed here is computationally simple and fast and hence implementable in real time when the final configuration changes en route. This is useful for 3-D waypoint-following problems in dynamically changing environments. Finally, using a six-degree-of-freedom (DOF) UAV model, this path is tracked by an autopilot consisting of proportional– integral–derivative controllers.

Rectilinear Path Following in 3D Space

This paper addresses the problem of determining an optimal (shortest) path in three dimensional space for a constant speed and turn-rate constrained aerial vehicle, that would enable the vehicle to converge to a rectilinear path, starting from any arbitrary initial position and orientation. Based on 3D geometry, we propose an optimal and also a suboptimal path planning approach. Unlike the existing numerical methods which are computationally intensive, this optimal geometrical method generates an optimal solution in lesser time. The suboptimal solution approach is comparatively more efficient and gives a solution that is very close to the optimal one. Due to its simplicity and low computational requirements this approach can be implemented on an aerial vehicle with constrained turn radius to reach a straight line with a prescribed orientation as required in several applications. But, if the distance between the initial point and the straight line to be followed along the vertical axis is high, then the generated path may not be flyable for an aerial vehicle with limited range of flight path angle and we resort to a numerical method for obtaining the optimal solution. The numerical method used here for simulation is based on multiple shooting and is found to be comparatively more efficient than other methods for solving such two point boundary value problem.

Optimal trajectory planning for path convergence in three-dimensional space

This article addresses the problem of determining the shortest path that connects a given initial configuration (position, heading angle, and flight path angle) to a given rectilinear or a circular path in three-dimensional space for a constant speed and turn-rate constrained aerial vehicle. The final path is assumed to be located relatively far from the starting point. Due to its simplicity and low computational requirements the algorithm can be implemented on a fixed-wing type unmanned air vehicle in real time in missions where the final path may change dynamically. As wind has a very significant effect on the flight of small aerial vehicles, the method of optimal path planning is extended to meet the same objective in the presence of wind comparable to the speed of the aerial vehicles. But, if the path to be followed is closer to the initial point, an off-line method based on multiple shooting, in combination with a direct transcription technique, is used to obtain the optimal solution. Optimal paths are generated for a variety of cases to show the efficiency of the algorithm. Simulations are presented to demonstrate tracking results using a 6-degrees-of-freedom model of an unmanned air vehicle.

Curvature-constrained trajectory generation for waypoint following for miniature air vehicle

This paper addresses trajectory generation problem of a fixed-wing miniature air vehicle, constrained by bounded turn rate, to follow a given sequence of waypoints. An extremal path, named as γ-trajectory, that transitions between two consecutive waypoint segments (obtained by joining two waypoints in sequence) in a time-optimal fashion is obtained. This algorithm is also used to track the maximum portion of waypoint segments with the desired shortest distance between the trajectory and the associated waypoint. Subsequently, the proposed trajectory is compared with the existing transition trajectory in the literature to show better performance in several aspects. Another optimal path, named as loop trajectory, is developed for the purpose of tracking the waypoints as well as the entire waypoint segments. This paper also proposes algorithms to generate trajectories in the presence of steady wind to meet the same objective as that of no-wind case. Due to low computational burden and simplicity in the design procedure, these trajectory generation approaches are implementable in real time for miniature air vehicles.

Time-Optimal Convergence to a Rectilinear Path in the Presence of Wind

This paper considers the problem of determining the time-optimal path of a fixed-wing Miniature Air Vehicle (MAV), in the presence of wind. The MAV, which is subject to a bounded turn rate, is required to eventually converge to a straight line starting from a known initial position and orientation. Earlier work in the literature uses Pontryagin’s Minimum Principle (PMP) to solve this problem only for the no-wind case. In contrast, the present work uses a geometric approach to solve the problem completely in the presence of wind. In addition, it also shows how PMP can be used to partially solve the problem. Using a 6-DOF model of a MAV the generated optimal path is tracked by an autopilot consisting of proportional-integral-derivative (PID) controllers. The simulation results show the path generation and tracking for cases with steady and time-varying wind. Some issues on real-time path planning are also addressed.

Optimal transition trajectory for waypoint following

This paper discusses the problem of a turn-rate constrained vehicle, such as a fixed-wing MAV (miniature air vehicle) required to fly along the path defined by a series of waypoints. An extremal path, named as γ-trajectory, that transitions between two consecutive waypoint segments (obtained by joining two waypoints in sequence) in a timeoptimal fashion, is obtained. This algorithm is also used to track the maximum portion of waypoint segments with the desired shortest distance between the trajectory and the associated waypoint. Subsequently, the proposed trajectory is compared with existing transition trajectory in the literature to show better performance in several aspects.