Reactive Collision Avoidance of Using NonlinearGeometric and Differential Geometric Guidance
U NMANNED Aerial Vehicles (UAVs) hold good promise for autonomously carrying out complex civilian and military operations. However, many of these missions require them to fly at low altitudes, making them vulnerable to collision with both stationary as well as moving obstacles. Hence, it is vital that UAVs are equipped with autonomous capability to sense and avoid collisions, especially for the pop-up threats. When such a threat is sensed and a collision is predictedwithin a short time ahead, theUAV should be able to react and maneuver away quickly so that the collision is avoided. An algorithmwhich can assure such amaneuver is called a “reactive collision avoidance algorithm.” Since the available reaction time in such a scenario is usually is small and UAVs are usually limited by computational resources, such an algorithm should also be computationally very efficient (it should preferably be noniterative). It is also required that whilemaneuvering away, it should notmaneuver toomuch away from the obstacle either. This is both to avoid collision from other nearby obstacles and not to compromise on the overall mission objective. There are various attempts in the literature to develop algorithms for collision avoidance purpose, many of which are inspired from global path planning algorithms. The artificial potential field method is such an approach where the motion of the vehicle is guided under the influence of a potential field. The potential field (which is essentially a cost function) is designed in such a way that obstacles have repulsive fields while the destination has an attractive field. The safe path of the UAV is then found by optimizing the carefully selected cost function. To tune this basic philosophy for reactive collision avoidance, a model predictive control-based algorithm has been proposed in the literature. This algorithm essentially assures path following under safe conditions (i.e. if no collision is predicted in the near future) and invokes the potential field function when new collisions are sensed. However, in potential field based techniques the associated optimization process is typically done in an iterative manner. Because of this they are usually computationally intensive and hence are not suitable for reactive collision avoidance of airborne UAVs in general. A promising algorithm in collision avoidance and global path planning is the philosophy of rapidly-exploring random tree (RRT) [1], which has also been used for reactive collision avoidance. However, there are many concerns about the RRT approach, which can largely be attributed to the random nature of the algorithm. For example, the path predicted by RRT is usually a sting of connected straight lines that does not reflect the path followed by a vehicle with nonholonomic constraints. More important, it is a probabilistic approach and hence there is no guarantee of finding a feasible path within a limited finite time. Other graph search algorithms such as best-first search are also implemented for reactive collision avoidance [2]. However, this is not systematic approach and could result in the algorithm searching far too many nodes under some conditions. Moreover, precomputing motion primitives and saving them in a lookup table is infeasible for UAVs, which are usually resource-limited. An interesting perspective to collision avoidance problem is the minimum effort guidance [3], where an optimal control-based approach has been proposed after applying the collision cone philosophy to detect collisions. This method is computationally nonintensive as a closed form solution has been proposed. Even though this is an interesting idea, by minimizing the lateral acceleration, perhaps it imposes unwanted extra constraint on the problem formulation as reactive collision avoidance problems do not necessarily have to be carried out with minimum lateral acceleration. More important, one can observe that this formulation only assures position guarantee and no constraint is imposed on the velocity vector. Hence, even though it guides the vehicle to a carefully selected target point on the safety boundary (we call it the “aiming point”), it causes the vehicle to maneuver until this point. This can be risky as the vehicle may enter the safety ball before reaching the aiming point. Even though the collision cone based aiming point philosophy is a very good idea, the authors of this Note strongly believe that instead of only position guarantee, rather the velocity vector should be aligned towards the aiming point as soon as a collision is detected (which will automatically lead to position guarantee as well). Towards this objective, two new nonlinear guidance laws are proposed in this Note, which are named as nonlinear geometric guidance (NGG) and differential geometric guidance (DGG). These guidance laws are inspired by the philosophy of “aiming point guidance” (APG) [4], which has been proposed in missile guidance literature. It turns out that the APG is a simplified case of the NGG where the associated since function is replaced by its linear approximation (hence, for a systematic discussion, it is renamed as linear geometric guidance (LGG) in this Note). Both of the guidance algorithms proposed in this Note quickly align the velocity vector of the UAValong the aiming point within a part of the available time-togo, which ensures quick reaction and hence safety of the vehicle. The main feature of this philosophy is that they effect high maneuvering at the beginning, causing the velocity vector of the UAV to align with the aiming point direction quickly and then settling along it. Therefore there is no need to maneuver all the way until the aiming point is reached and hence the chance of the UAV entering into the safety ball is minimized. Using the point of closest approach (PCA) [5], the proposed NGG andDGGalgorithms have also been extended for collision avoidance with moving obstacles in both cooperative as well as ignorant scenarios. Mathematical correlations between the guidance laws have also been established, which show that the NGG and DGG are exactly correlated to each other with appropriate gain selections, while the LGG is an approximation of DGG. A “sphere-tracking algorithm” is also proposed in this Note where the UAV is guided to track the surface of the safety spherewhenever a brief violation of the safety boundary occurs after reaching the aiming point because of the location of the next aiming point (which may include the target in Presented as Paper 2010-8315 at the AIAA Guidance, Navigation and Control, Toronto, 2–5August 2010; received 26May 2010; revision received 4 October 2010; accepted for publication 6 October 2010. Copyright © 2010 by Radhakant Padhi. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/11 and $10.00 in correspondence with the CCC. ∗Former Project Assistant, Department of Aerospace Engineering; anushamujumdar87@gmail.com. Associate Professor, Department of Aerospace Engineering; padhi@ aero.iisc.ernet.in. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 34, No. 1, January–February 2011
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