Chapter 10 Path following

For small UAVs, a major issue is wind

• Always present to some degree
• Usually significant with respect to commanded airspeed

Wind makes traditional trajectory tracking approaches difficult, if not infeasible.

• Have to know the wind precisely at every instant to determine the desire airspeed

Better approach: path following Rather than “follow this trajectory”, we control the UAV to “stay on this path”.

Path types

Straight lines between two points in 3-D

• Inclination of path within climb capabilities of UAV
• Circular orbits or arcs in the horizontal plane

Paths for common applications can be built from these path primitives

• Methods for following other types of paths found in literature

We need an algorithm for each of the path primitives.

Straight line path description

Lateral tracking problem

Relative error dynamics in path frame:

$$\left[\begin{array}{c}{\dot{e}}_{px}\\ {\dot{e}}_{py}\end{array}\right]=\left[\begin{array}{cc}\cos {\chi }_q& \sin {\chi }_q\\ -\sin {\chi }_q& \cos {\chi }_q\end{array}\right]\left[\begin{array}{c}{V}_g\cos \chi \\ V_g\sin \chi \end{array}\right]=V_g\left[\begin{array}{c}\cos (\chi -{\chi }_q)\\ \sin (\chi -{\chi }_q)\end{array}\right]$$

Regulate the cross-track error $e_{py}$ to zero by commanding the course angle:

$${\dot{e}}_{py}=V_g\sin (\chi -{\chi }_q)$$ $$\ddot{\chi }=b_{\dot{\chi }}({\dot{\chi }}^c-\dot{\chi })+b_{\chi }({\chi }^c-\chi )$$

Select ${\chi }^c$ so that $e_{py}\to 0$

$\chi ={\chi }_d$ course autopilot.

Longitudinal tracking problem

By similar triangles

$$\frac{(h_d+r_d)}{\sqrt{s_n^2+s_e^2}}=\frac{-q_d}{\sqrt{q_n^2+q_e^2}}$$

Desired altitude based on current location

$$h_d(r,p,q)=-r_d-\sqrt{s_n^2+s_e^2}\left(\frac{q_d}{\sqrt{q_n^2+q_e^2}}\right)$$

Select $h^c$ so that $h\to h_d(r,p,q)$

Longitudinal guidance strategy

Lateral tracking- Vector field concept

Vector field tuning

$k_{path}$ is a positive constant that affects the rate of transition of the desire course

• $k_{path}$ large → short, abrupt transition
• $k_{path}$ small → long, gradual transition

Lyapunovs 2nd method

For a system having a state vector $x$, consider an energy like function $V(x)$: $R^n\to R$ such that

$V(x)\ge \text{(positive definite)}$ $V(x)=0$ for $x=0$ and $\dot{V}(x)\le 0\text{(negative definite)}$ $\dot{V}(x)=0$ for $x=0$

If such a function $V(x)$ can be defined, then $x$ goes to zero asymptotically and the system is stable.

Lateral tracking stability analysis

Smallest angle turn logic

Orbit following

Algorithm

Orbit tracking stability analysis