Vector

Vector Path Following

Assuming that the UAV is flying at a constant altitude and airspeed(V), held constant by the control of the longitudinal dynamics. The following is a simple first order model of the navigational dynamics to be used to study the path following behavior of the UAV.

x = VcosΨ + Wx y = VsinΨ + Wy where Wx,Wy represent x and y components of the wind velocity. Heading (Ψ) of the UAV will not be controlled directly in this method. Instead, we shall focus on the ground track heading, ( χ ). An alternative representation of the above equations is x = VcosΨ + Wx = Vx + Wx = Sx y = VsinΨ + Wy = Vy + Wy = Sy where S = Groundspeed V = Airspeed W = Wind Speed

Or,as seen in figure x = S cos χ y = S sin χ The key difference from the DTS is that the equations of motion are expressed in terms of groundspeed and ground track heading and are independent of the wind velocity.This dramatically improves performance in situations where wind is a factor, which is true in the case of a small fixed wing UAV. Secondly, since we are not generating a trajectory, time is not taken into account thereby significantly reducing computational complexity. It is assumed that the UAV is equipped with an autopilot that implements a ground track heading hold loop and that the resulting dynamics are represented by the following first order system. χ = α( χc – χ ) where χc is the commanded ground track heading, and α is a known positive constant.

A ) Straight Path following

1. When the UAV is far away from the line(lateral distance greater than 2 to 3 times the minimum turn radius) , the objective is to fly toward the path. 2. The transition region around the path is indicated by dashed lines which lie at a distance ( τ ) on each side of the path. Outside the transition region, the desired heading or entry angle, χc is constant. 3. Once inside the transition region, the desired heading begins to transition from χc to the heading along the desired path, χf . The rate of transition is controlled by a gain k >=1 .

A complete list of the variables used for the straight line algorithm is as follows. Variables χf s* Ε τ w1, w1x , w1y Ρ z = (x ,y)T χc χe k

Description Heading from waypoint 1 to 2 Progress along path, s* lies between 0 and 1 Lateral tracking error Transition region boundary distance Waypoint 1 and it’s north and east components The side of the path that the UAV is on, having a value of 1 or -1 Current location of the UAV Commanded heading Entry heading angle( 0 < χe < 90 ) Transition gain, k >=1

The parameters (τ, k and χ e ) can be tuned based on the capabilities of the UAV to achieve the desired performance. The following algorithm maneuvers the UAV to follow straight line paths with asymptotically decaying error provided it can generate enough thrust to yield a positive ground speed. Lyapunov arguments are used to justify these claims.

Note : MAV- Micro Aerial Vehicle

** Notes 1. Before Step 2, current position and heading is received as input , z(x,y) and χ. 2. In step 4, ρ = sign( axby – aybx )