Adaptive sampling of cumulus clouds with UAVs

Abstract

This paper presents an approach to guide a fleet of Unmanned Aerial Vehicles (UAVs) to actively gather data in low-altitude cumulus clouds with the aim of mapping atmospheric variables. Building on-line maps based on very sparse local measurements is the first challenge to overcome, for which an approach based on Gaussian Processes is proposed. A particular attention is given to the on-line hyperparameters optimization, since atmospheric phenomena are strongly dynamical processes. The obtained local map is then exploited by a trajectory planner based on a stochastic optimization algorithm. The goal is to generate feasible trajectories which exploit air flows to perform energy-efficient flights, while maximizing the information collected along the mission. The system is then tested in simulations carried out using realistic models of cumulus clouds and of the UAVs flight dynamics. Results on mapping achieved by multiple UAVs and an extensive analysis on the evolution of Gaussian processes hyperparameters is proposed.

Appendices

Appendix 1: Aircraft model

In this Appendix, we provide the details of the flight dynamics model adopted for this work. We consider a simplified aircraft model to enable fast trajectory computations, but still able to capture the essential characteristics of the flight mechanics for a realistic trajectory optimization simulation. The key parameters and coefficients used for the analytical calculations are estimated from a modified vortex-lattice analysis (Bronz et al. 2013) of the aircraft.

In particular, we consider the Mako aircraft shown in Fig. 7, which will be employed for future experiments within the SkyScanner project. Table 1 shows its general aerodynamic and geometrical specifications.

Table 1 General aerodynamic and geometrical specifications of the Mako aircraft

The trajectory computation is based on two control inputs: the power input $P_{in}$ and the turn radius R. The airspeed V is considered constant, therefore the angle of attack is kept fixed for simplification. The trajectory optimization mainly requires the drag force evaluation, which has to be compensated by the input power. Note the air density is kept constant during the simulations, again for the sake of simplicity.

In order to calculate the performance within the flight envelope, the flight phases are isolated as steady banked turn and constant climb. The climb calculations derived from energy equations, the pull-up and down transition phases are approximated according to the maximum lift capability limit.

Fig. 20

Forces applied during the steady banked turn phase (left) during the steady climb phase (right)

1.1 Steady banked turn phase

During the steady banked turn phase, the vertical component of the lift force $L_v$ is equal to the weight of the aircraft and its lateral component $L_h$ compensates the centrifugal force as shown in Fig. 20.

$$\begin{aligned} \sum F_z = 0 \Rightarrow W = L_v = L \cos \phi \end{aligned}$$

(16)

For a given bank angle $\phi$, the load factor n, given by

$$\begin{aligned} n&= \frac{L_v}{W} = \frac{L}{L \cos \phi } = \frac{1}{\cos \phi }, \end{aligned}$$

(17)

has to be accounted for in the lift force, which can then be expressed as:

$$\begin{aligned} L = \frac{1}{2} \rho V^2S_{ref}(nC_L) \; [C_{L_{max}} > nC_L], \end{aligned}$$

(18)

with drag coefficient and resultant drag force being

$$\begin{aligned}&C_D = C_{D_0} + k(nC_L)^2 \quad \quad \text {where} \quad \quad k = \frac{1}{\pi AR e} \end{aligned}$$

(19)

$$\begin{aligned}&D =\frac{1}{2}V^2S_{ref}C_D \end{aligned}$$

(20)

in oder to take into account the additional drag contribution coming from the induced lift force.

1.2 Rate of Climb (ROC) and power consumption

To maintain level flight, the required aerodynamic power is

$$\begin{aligned} P_{aero} = DV \end{aligned}$$

(21)

Incorporating the thrust, we can calculate the climb rate of the aircraft as:

$$\begin{aligned} ROC = V_{climb} = V\frac{(T-D)}{W} = \frac{P_{prop} - P_{aero}}{W} \end{aligned}$$

(22)

The maximum propulsive power is limited according to the specifications of the propulsion system used, whose efficiency $\eta _p$ results in a higher power drawn from battery: ${P_{prop} = \eta _p P_{in}}$, where $P_{in}$ is the input power drawn from the battery. The resulting $V_{climb}$ is then:

$$\begin{aligned} V_{climb} = \frac{\eta _p \times P_{in} - P_{aero}}{W}. \end{aligned}$$

(23)

The total propulsion system efficiency varies, as it is related to the flight speed and generated thrust force: a fine modeling of these variations would be required for a precise propulsion model. However, comparing electrical power input $P_{in}$ versus aerodynamic power output $P_{aero}$ shows that a linear relation is a fairly accurate model for the considered flight speeds range, as shown in Fig. 21. This fact is exploited to define the total propulsion efficiency $\eta _p$ in our simulation model.

Fig. 21

Resultant propeller aerodynamic power from battery input power for three different flight speeds

1.3 Pull-up and pull-down

The transition from level flight to steady climb is achieved by a short pull-up flight maneuver. Likewise, a pull-down flight maneuver is used to transition from level flight to steady descend phase. In our simplified aircraft model, as the flight angle of attack is assumed constant at all times, the distinction between climb and descent transitions is defined by the given power input $P_{in}$: if $P_{in} \times \eta _{p}$ is higher than the required level flight aerodynamic power $P_{aero}$, then the aircraft climbs.

The pitch turn radius can be calculated as:

$$\begin{aligned} R_{up} = \frac{V^2}{g (n-1)}, \quad R_{down} = \frac{V^2}{g (n+1)} \end{aligned}$$

(24)

where the contribution of the total aircraft weight is in the $(n-1)$ and $(n+1)$ terms. As the angle of attack is constant, the pitch rate $\dot{\gamma }$ is given by:

$$\begin{aligned} \dot{\gamma }= \pm \frac{V}{ R_{up/down}} \end{aligned}$$

(25)

Appendix 3: Trajectory computation

Assuming a fixed airspeed V, a steady turn radius R and input power $P_{in}$, the trajectory can be computed by separating it in a pull-up (or pull-down) phase and a steady phase. Pull-up and pull-down phases are executed assuming a maximal allowed bank angle, thus assuring that the maximal allowable load factor will not be exceeded. First we compute the maximum allowed load factor $n_{max}$:

$$\begin{aligned} C_L&= \frac{2 * W}{\rho V^2 S_{ref} } \end{aligned}$$

(26)

$$\begin{aligned} n_{max}&= \frac{C_{Lmax}}{C_L} \end{aligned}$$

(27)

Using Eq. (23), $V_{climb}$ can be obtained from $P_{aero}$ and $P_{in}$. This value represents the target value for vertical velocity for given $P_{in}$ and R. The climb rate $\gamma (t)$ can then be computed as:

$$\begin{aligned}&\varDelta _\gamma = \text {arcsin}(V_{climb}/V) - \gamma (0) \end{aligned}$$

(28)

$$\begin{aligned}&u = \text {sign}(\varDelta _\gamma )\end{aligned}$$

(29)

$$\begin{aligned}&R_{up/down} = \frac{V^2}{g*(n_{max}-u)} \end{aligned}$$

(30)

$$\begin{aligned}&\dot{\gamma }= \frac{uV}{R_{up/down}} \end{aligned}$$

(31)

$$\begin{aligned}&\varDelta _{tpull}(t) = min(t, \frac{ \varDelta _\gamma }{ \dot{\gamma }}) \end{aligned}$$

(32)

$$\begin{aligned}&\gamma (t) = \gamma (0) + \dot{\gamma }\varDelta _{tpull}(t) \, \end{aligned}$$

(33)

where $\varDelta _{tpull}(t)$ represents the duration of the pull up or pull down phase. Finally we can compute the projection on the z axis of the path on the $R_{up/down}$ circle during the pull phase, and assume a constant vertical velocity during the remaining time:

$$\begin{aligned}&\varDelta _{zpull}(t) = u R_{up/down}(\cos \gamma (0) - \cos \gamma (\varDelta _{tpull}(t))) \end{aligned}$$

(34)

$$\begin{aligned}&z(t)= \varDelta _{zpull}(t) + V_{climb}(t-\varDelta _{tpull}(t)) \end{aligned}$$

(35)

Knowing the vertical velocity, and assuming a constant total velocity V and turn radius R, we finally compute x(t) and y(t) using the change in the heading $\psi$ to project the position on a circle of radius R tangent to the horizontal velocity vector. We first compute the heading $\psi (t)$:

$$\begin{aligned} \dot{\psi }(t)&= \frac{V_H(t)}{R} \end{aligned}$$

(36)

$$\begin{aligned}&= \frac{V}{R}\cos \gamma (t) \end{aligned}$$

(37)

$$\begin{aligned} \varDelta _{\psi }(t)&= \int _0^{t} \dot{\psi }(x) dx \end{aligned}$$

(38)

$$\begin{aligned}&= \frac{V}{R}(\sin \gamma (\varDelta _{tpull}) - \sin \gamma (0)) + \gamma (\varDelta _t)(t-\varDelta _{tpull}) \end{aligned}$$

(39)

$$\begin{aligned} \psi (t)&= \psi _0 + \varDelta _\psi (t) \end{aligned}$$

(40)

Using $\psi (t)$ we deduce the xy position of the aircraft:

$$\begin{aligned} \begin{pmatrix} x \\ y \end{pmatrix} (t)&= \begin{pmatrix} x_0 \\ y_0 \end{pmatrix} + R\begin{pmatrix} -\sin (\psi _0) + \sin (\psi (t))\\ +\cos (\psi _0) - \cos (\psi (t)) \end{pmatrix} \end{aligned}$$

(41)

Finally, the remaining capacity of the battery J(t) (in joules) is:

$$\begin{aligned} J(t)&= J(0) - \int _0^t P_{in}(x)dx = J(0) - P_{in}t. \end{aligned}$$

(42)