# Three-Dimensional Obstacle Avoidance Strategy for Fixed-Wing UAVs Based on Quaternion Method
## 1 Introduction
### Related Works  
Substantial research on motion planning for UAVs has been conducted, including sampling-based methods [5] and search-based methods [6]. For UAVs, the methods mentioned above are not directly applicable since they heavily rely on target prior information and large-scale iteration.  
In view of the aforementioned considerations, reactive collision avoidance methods have been utilized to tackle the motion planning problem of systems with limited computing resources. The VO method was first proposed in [7], and it was originally developedfor moving obstacles, considering the velocity of each obstacle in the environment. Since the VO method computes the avoidance velocity received based on the instantaneous geometric relationship of collisions, target prior information becomes needless. Compared with other reactive collision avoidance methods, such as artificial potential field (APF) methods [8; 9; 10], the VO method guarantees a lower computational cost and is less prone to problems with local minimums. Refs. [11; 12; 13] extend VO methods to heterogeneous agents. Ref. [11] relies on specific implicit coordination, Ref. [12] assumes that agents have the same type of control input, and Ref. [13] depends on an abstraction kinodynamic model of the robot; the latter makes the VO algorithm truly independent of the model. VO methods were further extended to the system with non-holonomic constraints by testing the optimality of sampling control [14], by mapping between the holonomic speed and the nonholonomic control input [15]. The latter guarantees a lower computational cost. Following the concept proposed in Ref. [15], we map the avoidance velocity to the nonholonomic control input. Thanks to the contributions of Refs. [16; 17] on the three-dimensional velocity obstacle (3D-VO) method, the range of feasible solutions (obstacle avoidance velocity or acceleration) has been significantly expanded. Our method builds on the concept proposed in [17] which was successfully verified by experiments for collision avoidance with static obstacles [18] and extended to the dynamic system with nonholonomic constraints. The approaches mentioned above directly update velocities to avoid moving obstacles, and thus, it is summarized as the first-order method. Actually, compared with the quadcopter, the fixed-wing UAV cannot instantly change the direction of the velocity vector. Instead of computing an avoidance velocity command, it is possible to produce an avoidance acceleration that ensures that the collision avoidance can always be maintained.  
A systematical extension of VO methods to second-order or higher-order systems has been conducted [12; 19; 20; 13; 21]. However, the aforementioned approaches contain some assumptions that limit their application to UAVs, such as only considering the simplified linear robot model, calculating the solution in a two-dimensional plane, and supposing that the robot can implement omni-directional acceleration. In [22], a 3D collision cone was utilized to examine collision condition and rigid quadrotor dynamics were considered. However, the obstacle avoidance only considers static obstacles in the environment. The fixed-wing UAV involved in our research has more complex dynamics. To the best of our knowledge, the extension of the 3D-VO method to a second-order nonlinear dynamic system of an underactuated fixed-wing UAV cannot be found in the literature. Mainly, there are the following difficulties: (1) a fixed-wing UAV is not only a nonholonomic system, but also a kinodynamic constrained nonholonomic system. It must take explicitly into account this nature of UAVs in planning strategies. However, collision avoidance may then no longer be guaranteed. (2) The dynamics of a fixed-wing UAV is represented as a second-order nonlinear differential equation with variable coefficients, which make it extremely difficult to directly derive a feasible velocity set or acceleration set, as described in [12; 19; 20].