Discrete vector fields for 2-D navigation under minimum turning radius constraints

Abstract

To greatly extend the capabilities of unmanned aerial vehicles, automation systems for the various piloting functions are being developed. However, the development of these auto-systems is not without challenges. One of these challenges is the ability to prove the robustness of an autonomous system against unpredicted events. For the navigation task, this means to prove that the system can recover safely after a deviation from the original plan. In a decoupled scheme where the planner generates a plan online and the controller executes the plan, it is difficult to prove the safety of a system exhaustively. Conversely, using a feedback motion plan generated off-line, it is possible to pre-verify and pre-approve a large number of cases simultaneously. Therefore, it is easier to prove system safety. In this work, this approach is utilized to integrate the guidance toward a target location or a target path with geofence avoidance and recovery for a fully autonomous fixed-wing aircraft. Our formulation extends the wavefront expansion to the case of vehicles having minimum turning radius constraints. The solution is suitable for both single goal missions and path following problems in presence of geofences that can have both convex and non-convex shape. The main novelties introduced in the proposed method are the followings: (1) path following in presence of obstacles having an arbitrary shape, (2) definition of a transition function for the rotation of the flow around the predefined path, (3) use of a Gaussian filter to smooth the vector field. Simulation and experimental results demonstrate the effectiveness of the proposed method.

Appendices

Appendix A: Derivation of the control law

In this appendix, we derive the control law in (22). The autopilot used in our setup generates lateral acceleration commands. Such commands are generated using the nonlinear control law derived in [51] with a small variation. In [51] the required centripetal acceleration is computed as follows:

$$\begin{aligned} a_\mathrm{c}\left( x\right)= & {} 2\frac{v^2}{L_{1}}\sin \left( x\right) \end{aligned}$$

(24)

$$\begin{aligned} L_{1}= & {} \frac{1}{\pi }\xi Tv \end{aligned}$$

(25)

where $\xi$ and $\text {T}$ are tuning parameters defined, respectively, damping and period. In the autopilot [52] used in our setup, the (24) is modified by replacing the constant 2 with the term $K_{L1}$ as follows:

$$\begin{aligned} a_\mathrm{c}'\left( x\right)= & {} {\left\{ \begin{array}{ll} K_{L1}\frac{v^2}{L_{1}}\sin \left( x\right) , &{}\vert x\vert <\frac{\pi }{2}\\ {{\,\mathrm{sign}\,}}\left( x\right) K_{L1}\frac{v^2}{L_{1}}, &{}\vert x\vert \ge \frac{\pi }{2} \end{array}\right. } \end{aligned}$$

(26)

$$\begin{aligned} K_{L1}= & {} 4\xi ^{2} \end{aligned}$$

(27)

Exploiting that the turn rate can be obtained from the centripetal acceleration and the speed:

$$\begin{aligned} {\dot{\theta }}=\frac{a_\mathrm{c}'}{v} \end{aligned}$$

(28)

we can obtain the control law in (22) from (26) as follows:

$$\begin{aligned} \begin{aligned} z\left( x\right) =\frac{a_\mathrm{c}'\left( x\right) }{v}&={\left\{ \begin{array}{ll} K_{L1}\frac{v}{L_{1}}\sin \left( x\right) , &{}\vert x\vert<\frac{\pi }{2}\\ {{\,\mathrm{sign}\,}}\left( x\right) K_{L1}\frac{v}{L_{1}}, &{}\vert x\vert \ge \frac{\pi }{2} \end{array}\right. }\\&= {\left\{ \begin{array}{ll} \frac{4\pi \xi }{T}\sin \left( x\right) , &{}\vert x\vert <\frac{\pi }{2}\\ {{\,\mathrm{sign}\,}}\left( x\right) \frac{4\pi \xi }{T}, &{}\vert x\vert \ge \frac{\pi }{2} \end{array}\right. } \end{aligned} \end{aligned}$$

(29)

Appendix B: Additional flight data

See Figs. 23, 24.

Fig. 23

Data obtained in flight tests with cell size of 10 m and maps generated with the following parameters: $\alpha =2,\, \mu _{p}=\mu _{b}=\sigma _{p}=\sigma _{p}=1, \, \sigma =4$. b and e Show the UAV’s course versus the vector field, while c and f show their derivatives

Fig. 24

Data obtained in flight tests with cell size of 20 m and maps generated with the following parameters: $\alpha =2,\, \mu _{p}=\mu _{b}=\sigma _{p}=\sigma _{p}=1, \, \sigma =2$. b and e Show the UAV’s course versus the vector field, while c and f show their derivatives