# Estimating the Vertical Structure of Weather-Induced Mission Costs for Small UAS

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## Abstract

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## 1. Introduction

## 2. Basis Splines for Environmental Modeling

#### 2.1. Time Update

#### 2.2. Process Noise

#### 2.3. Observation Models

#### 2.4. Measurement Noise

#### 2.5. Sample Noise

#### 2.5.1. Wind Sampling Error

#### 2.5.2. Solar Sampling Error

#### 2.6. Summary of Kalman Filter Equations

## 3. Computing Cost Profile

## 4. Simulation Test

#### 4.1. Comparison between Aircraft

#### 4.2. Comparison to True Atmospheric State

## 5. Flight Test

#### 5.1. Test Description

#### 5.2. Process and Measurement Noise

#### 5.3. Results

#### 5.3.1. Comparison with Radiosonde Observations

#### 5.3.2. Comparison with a Numerical Weather Model

#### 5.4. Travel Cost

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The aircraft environmental model should be able to be constructed onboard the aircraft from in situ observations but be able to incorporate observations from other platforms if available. It should model the atmospheric structure in a way that permits transformation into other quantities of interest.

**Figure 2.**Basis splines can be used to approximate functions by a linear combination of the bases at each point. The coefficients for this example model were determined by a least-square fit of the data sampled from the equation $f\left(z\right)=-3{\left(\frac{z}{250}-400\right)}^{3}+2{\left(\frac{z}{250}-400\right)}^{2}+exp\left(\frac{z}{250}-400\right)+sin\left(\frac{z}{250}-400\right)$ with noise $\nu \sim \mathcal{N}(0,\phantom{\rule{3.33333pt}{0ex}}1.0)$. Spline bases are shown scaled by their respective coefficients.

**Figure 3.**The forecast error at each hour is computed using a triangular scheme and the analysis from subsequent forecast times as “truth.” For instance, the one-hour forecast error for the 12:00 UTC forecast is the value of the desired fields from the 12:00 UTC T1 step minus the analysis value from the 13:00 UTC forecast.

**Figure 4.**Mean absolute error profiles for the HRRR wind forecast at a point in central Pennsylvania.

**Figure 5.**Flight paths taken by the two simulated aircraft for a simulation beginning at 16:00 UTC on 2018-12-03. One aircraft remains between 500 and 1000 m while the other stays between 1000 and 1500 m so neither can independently sample the region of interest.

**Figure 6.**Mean absolute difference between the coefficients of the wind and solar models computed by the two aircraft as a function of the travel distance for a simulation beginning at 16:00 UTC on 2018-12-03. During time updates, slight differences in position and inertial velocity introduce small differences between the models maintained by each aircraft.

**Figure 7.**Estimated and true atmospheric profiles and $1\sigma $ bounds at 10 km into the flight for a simulation beginning at 16:00 UTC on 2018-12-03. Error bounds on the true profiles indicate the HRRR skill–the best that the atmospheric state can be known a priori.

**Figure 8.**Estimated and true atmospheric profiles and $1\sigma $ bounds at 40 km into the flight for a simulation beginning at 16:00 UTC on 2018-12-03. Error bounds on the true profiles indicate the HRRR skill–the best that the atmospheric state can be known a priori.

**Figure 9.**The Vulture, a small fixed-wing UAS based on a 2.5 m span model sailplane. It is equipped with an autopilot, onboard computer, and dual power systems permitting 30 min of flight time and more than 10 h of avionics run time. This allows multiple consecutive flights without disrupting the modeling system.

**Figure 10.**Wind and temperature at 925 mbar (approximately 750 m MSL) at 12:00 EST on December 6, 2018, approximately the midpoint of the test. The test site is located at the red dot, the approaching front can be seen as an abrupt wind shift along the eastern border of Michigan.

**Figure 11.**Aerial view of the test area showing the balloon trajectories and aircraft orbit location.

**Figure 12.**Time history of aircraft and balloon altitude (2018-12-06). The aircraft appears to be below ground between flights because its state estimator is incorporating airspeed measurements which are not informative when on the ground.

**Figure 13.**Time-altitude depiction of the profile of aircraft-estimated horizontal wind depicting its evolution throughout the day (2018-12-06).

**Figure 14.**Aircraft and radiosonde profiles from 08:00 local time (2018-12-06). Solid lines represent the mean profile estimated by the aircraft and radiosonde while dashed lines represent the $1\sigma $ error bounds. The raw aircraft and radiosonde observations are depicted with scattered points.

**Figure 22.**Evolution of the forecast and aircraft-estimated cost of traveling southwest at a groundspeed of 15 $\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$ (2018-12-06). Cost profiles are scaled such that one hour represents 4 $\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$ of specific power required to fly this mission. Systematic, slowly varying differences between the forecast and estimated cost profiles are seen.

**Figure 23.**Cost for the Vulture to travel in any direction at an average groundspeed of 15 $\phantom{\rule{3.33333pt}{0ex}}\mathrm{m}\phantom{\rule{3.33333pt}{0ex}}{\mathrm{s}}^{-1}$. Cost is expressed in specific power and evaluated at 08:00 local time (2018-12-06).

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**MDPI and ACS Style**

Bird, J.J.; Richardson, S.J.; Langelaan, J.W. Estimating the Vertical Structure of Weather-Induced Mission Costs for Small UAS. *Sensors* **2019**, *19*, 2770.
https://doi.org/10.3390/s19122770

**AMA Style**

Bird JJ, Richardson SJ, Langelaan JW. Estimating the Vertical Structure of Weather-Induced Mission Costs for Small UAS. *Sensors*. 2019; 19(12):2770.
https://doi.org/10.3390/s19122770

**Chicago/Turabian Style**

Bird, John J., Scott J. Richardson, and Jack W. Langelaan. 2019. "Estimating the Vertical Structure of Weather-Induced Mission Costs for Small UAS" *Sensors* 19, no. 12: 2770.
https://doi.org/10.3390/s19122770