# On the Self-Calibration of Aerodynamic Coefficients in Vehicle Dynamic Model-Based Navigation

^{*}

## Abstract

**:**

## 1. Vehicle Dynamic Model

#### 1.1. Frames Definition

**i**)-frame is an Earth Centered Inertial (ECI) frame, the Earth (i)-frame is an Earth Centered Earth Fixed (ECEF) frame, the local-level (l)-frame is oriented in north, east, and down directions with respect to the tangent plane on a reference ellipsoid [9]. Definitions of body (b)-frame and wind (w)-frame with respect to l-frame can be perceived from Figure 1.

#### 1.2. Motion States

#### 1.3. Dynamic Model

#### 1.4. Estimation Scheme

## 2. Simulation Setup

#### 2.1. Trajectory Definition

#### 2.2. Flight Simulation

#### 2.3. Sensor and VDM Parameter Errors

## 3. Analysis and Discussion

#### 3.1. Parameter Estimation

#### 3.2. Parameter Correlation

#### 3.3. Initial VDM Parameter Uncertainty

#### 3.4. Sequence of Maneuvers

#### 3.5. Brief Discussion on the Observability of VDM Parameters

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

CDF | Computational Fluid Dynamic |

ECEF | Earth Centered Earth fixed |

ECI | Earth Centered Inertial |

EKF | Extended Kalman Filter |

ENU | East North Up |

GNSS | Global Navigation Satellite System |

IMU | Inertial Measurement Unit |

INS | Inertial Navigation System |

UAV | Unmanned Aerial Vehicle |

VDM | Vehicle Dynamic Model |

## References

- Gleason, S.; Gebre-Egziabher, D. GNSS Applications and Methods; Artech House: Norwood, MA, USA, 2009. [Google Scholar]
- Jain, M.; Choi, J.I.; Kim, T.; Bharadia, D.; Seth, S.; Srinivasan, K.; Levis, P.; Katti, S.; Sinha, P. Practical, Real-time, Full Duplex Wireless. In Proceedings of the 17th Annual International Conference on Mobile Computing and Networking, Las Vegas, NV, USA, 19–23 September 2011; ACM: New York, NY, USA, 2011; pp. 301–312. [Google Scholar]
- Khaghani, M.; Skaloud, J. Autonomous Vehicle Dynamic Model-Based Navigation for Small UAVs. Navigation
**2016**, 63, 345–358. [Google Scholar] [CrossRef] - Khaghani, M.; Skaloud, J. Autonomous and Non-Autonomous Dynamic Model Based Navigation System for Unmanned Vehicles. US Patent 151,762,83, 8 June 2016. [Google Scholar]
- Khaghani, M.; Skaloud, J. Assessment of VDM-based Autonomous Navigation of a UAV under Operational Conditions. Robot. Auton. Syst.
**2018**, 106, 152–164. [Google Scholar] [CrossRef] - Khaghani, M.; Skaloud, J. Application of Vehicle Dynamic Modeling in UAVs for Precise Determination of Exterior Orientation. Int. Arch. Photogramm. Remote. Sens. Spat. Inf. Sci.
**2016**, 41, 827–831. [Google Scholar] [CrossRef] - Fitzgerald, R. Divergence of the Kalman filter. IEEE Trans. Autom. Control.
**1971**, 16, 736–747. [Google Scholar] [CrossRef] - Ducard, G. Fault-Tolerant Flight Control and Guidance Systems: Practical Methods for Small Unmanned Aerial Vehicles; Springer: London, UK, 2009. [Google Scholar]
- NIMA WGS84 Update Committee. Department of Defense.World Geodetic System 1984, Its Definition and Relationships with Local Geodetic Systems, 3rd ed.; Technical Report; National Imagery and Mapping Agency: Springfield, VA, USA, 2000.
- Rösch, N. Rotations, Quaternions and Double Groups; Altmann, S.L., ed.; Clarendon Press: Oxford, UK, 1986; 317p. Int. J. Quantum Chem.
**1987**, 32, 401. [Google Scholar] [CrossRef] - Gelb, A. Applied Optimal Estimation; MIT Press: Cambridge, MA, USA, 1974. [Google Scholar]
- Laupré, G.; Khaghani, M.; Skaloud, J. Sensitivity to Time Delays in VDM-Based Navigation. Drones
**2019**, 3, 11. [Google Scholar] [CrossRef][Green Version] - Intersense. Intersense Navchip. 2015. Available online: http://www.intersense.com/pages/16/246/ (accessed on 10 December 2015).
- Clausen, P.; Skaloud, J.; Orso, S.; Guerrier, S. Construction of dynamically-dependent stochastic error models. In Proceedings of the 2018 IEEE/ION Position, Location and Navigation Symposium (PLANS), Monterey, CA, USA, 23–26 April 2018. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**Local level, body, and wind frames with airspeed $\mathbf{V}$, wind velocity $\mathbf{w}$, and UAV velocity $\mathbf{v}$ [5].

**Figure 2.**Generalized vehicle dynamic model (VDM)-based navigation filter [5].

**Figure 4.**Normalized correlation matrix P between state-vector elements at the end of a maneuver: (

**a**)—Level flight, (

**b**)—Controlled Orbit (

**c**)—Infinity loop, (

**d**)—Combination of segments

**Figure 5.**VDM parameter error in percentage, averaged among all successful runs at the end of the different trajectories for several initial parameter uncertainty, for the whole range (

**a**) and zoomed (

**b**).

**Figure 7.**Error estimation evolution versus 1$\sigma $ during a specific trajectory: (

**a**)—Level flight, (

**b**)—Combination of maneuvers

Straight Line | Level | Level Descending | Level Ascending | |||

Orbit | Controlled | Auto pilot | ||||

Infinity Loop | Level | Descending/Ascending |

Error Type | Notation | Value | [Units] |
---|---|---|---|

Gyro. bias | ${b}_{G}$ | 720 | [deg/h] |

Gyro. correlated noise | ${\sigma}_{{G}_{GM1}^{PSD}}$ | 0.0028 | [deg/s/$\sqrt{Hz}$] |

$1/{\beta}_{G}$ | 200 | [s] | |

Gyro. white noise | ${\sigma}_{{G}_{WN}^{PSD}}$ | 0.18 | [deg/$\sqrt{Hz}$] |

Acc. bias | ${b}_{A}$ | 8 | [mg] |

Acc. correlated noise | ${\sigma}_{{A}_{GM1}^{PSD}}$ | 0.05 | [mg] |

$1/{\beta}_{A}$ | 200 | [s] | |

Acc. white noise | ${\sigma}_{{A}_{WN}^{PSD}}$ | 50 | [mg/$\sqrt{Hz}$] |

GNSS white noise (each direction) | ${\sigma}_{GNSS}$ | 1 | [m] |

Navigation States | Values | Units | |
---|---|---|---|

Position | - all axis | 1 | [m] |

Velocity | - horizontal | 1 | [m] |

- vertical | 0.5 | [m] | |

Attitude | - roll/pitch | 3 | [deg] |

- yaw | 5 | [deg] | |

Angular rate | - all axis | 1 | [deg/s] |

Trajectory | per VDM Category (Lowest to Largest) | Total | ||
---|---|---|---|---|

Descend. Straight L. | ${F}_{y}:$ 15.3% | $<{M}_{y}<{M}_{z}<{F}_{T}<{F}_{x}<{F}_{z}<$ | ${M}_{x}:$ 26.4% | 18.4% |

Straight line | ${M}_{y}:$ 10.0% | $<{F}_{y}<{F}_{T}<{F}_{z}<{M}_{x}<{M}_{z}<$ | ${F}_{x}:$ 17.6% | 13.8% |

Climbing Straight L. | ${M}_{y}:$ 9.1% | $<{F}_{y}<{F}_{T}<{F}_{z}<{M}_{x}<{M}_{z}<$ | ${F}_{x}:$ 17.4% | 13.7% |

Controlled Orbit | ${M}_{y}:$ 11.2% | $<{F}_{y}<{M}_{z}<{M}_{x}<{F}_{z}<{F}_{x}<$ | ${F}_{T}:$ 20.0% | 13.5% |

Orbit w. AutoPilot | ${M}_{y}:$ 10.8% | $<{F}_{y}<{M}_{x}<{M}_{z}<{F}_{z}<{F}_{x}<$ | ${F}_{T}:$ 17.9% | 13.2% |

8 Loop w. Alt. fix | ${M}_{z}:$ 8.3% | $<{M}_{x}<{F}_{y}<{M}_{y}<{F}_{z}<{F}_{x}<$ | ${F}_{T}:$ 17.5% | 11.1% |

8 Loop w. Alt. var. | ${M}_{z}:$ 8.3% | $<{M}_{x}<{M}_{y}<{F}_{y}<{F}_{z}<{F}_{x}<$ | ${F}_{T}:$ 16.8% | 10.9% |

Level Flight | Asc. Straight Flight | ‘8 loops’ with alt. Changes | Combination | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

${C}_{{M}_{y}1}$ | - | ${C}_{{M}_{y}\alpha}$ | 93.6% | ${{C}_{{M}_{y}1}}$ | - | ${{C}_{{M}_{y}e}}$ | 90.4% | ${C}_{{M}_{y}e}$ | - | ${C}_{{M}_{y}\alpha}$ | 90.2% | ${C}_{{F}_{z}\alpha}$ | - | ${C}_{{F}_{x}\alpha 2}$ | 90.9% |

$\overline{c}$ | - | ${C}_{{M}_{y}{\tilde{\omega}}_{y}}$ | 96.7% | D | - | ${C}_{{F}_{T}1}$ | 90.6% | ${C}_{{M}_{x}{\tilde{\omega}}_{x}}$ | - | ${C}_{{M}_{x}{\tilde{\omega}}_{z}}$ | 90.9% | ||||

${C}_{{M}_{y}1}$ | - | ${C}_{{M}_{y}\alpha}$ | 97.8% | ${C}_{{M}_{y}1}$ | - | ${C}_{{M}_{y}\alpha}$ | 92.2% | ${C}_{{M}_{z}{\delta}_{r}}$ | - | ${C}_{{M}_{z}\beta}$ | 95.2% | ||||

${C}_{{M}_{z}{\delta}_{r}}$ | - | ${C}_{{M}_{z}\beta}$ | 95.4% | S | - | ${C}_{{F}_{z}\alpha}$ | 97.1% | ||||||||

${\overline{c}}$ | - | ${{C}_{{M}_{y}{\tilde{\omega}}_{y}}}$ | 96.1% | ${C}_{{M}_{y}1}$ | - | ${C}_{{M}_{y}\alpha}$ | 97.5% | ||||||||

S | - | ${C}_{{F}_{z}\alpha}$ | 96.4% | ${C}_{{M}_{y}e}$ | - | ${C}_{{M}_{y}\alpha}$ | 97.5% | ||||||||

${C}_{{M}_{y}e}$ | - | ${C}_{{M}_{y}{\tilde{\omega}}_{y}}$ | 96.7% | ${{C}_{{M}_{y}1}}$ | - | ${{C}_{{M}_{y}e}}$ | 97.9% | ||||||||

${C}_{{M}_{x}\alpha}$ | - | ${C}_{{M}_{x}\beta}$ | 98.2% | ||||||||||||

D | - | ${C}_{{F}_{T}1}$ | 99.4% |

**Table 6.**Number of successful navigation runs per initial error in VDM parameters per trajectories (Monte-Carlo 100 runs).

Initial VDM Parameter Error $1\mathit{\sigma}$ | |||||||
---|---|---|---|---|---|---|---|

Trajectory | 10% | 20% | 30% | 40% | 50% | 75% | 100% |

Descend. Straight L. | 100 | 92 | 74 | 59 | 55 | 36 | 28 |

Straight line | 100 | 98 | 71 | 56 | 53 | 36 | 26 |

Climbing Straight L. | 100 | 98 | 87 | 70 | 60 | 40 | 31 |

Controlled Orbit | 100 | 100 | 100 | 95 | 83 | 61 | 40 |

Orbit w. AutoPilot | 100 | 99 | 95 | 79 | 57 | 31 | 19 |

8 Loop w. Alt. fix | 100 | 100 | 99 | 86 | 73 | 43 | 31 |

8 Loop w. Alt. var. | 100 | 100 | 98 | 79 | 66 | 41 | 21 |

Combinations | 100 | 91 | 74 | 56 | 43 | 22 | 13 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Laupré, G.; Skaloud, J. On the Self-Calibration of Aerodynamic Coefficients in Vehicle Dynamic Model-Based Navigation. *Drones* **2020**, *4*, 32.
https://doi.org/10.3390/drones4030032

**AMA Style**

Laupré G, Skaloud J. On the Self-Calibration of Aerodynamic Coefficients in Vehicle Dynamic Model-Based Navigation. *Drones*. 2020; 4(3):32.
https://doi.org/10.3390/drones4030032

**Chicago/Turabian Style**

Laupré, Gabriel, and Jan Skaloud. 2020. "On the Self-Calibration of Aerodynamic Coefficients in Vehicle Dynamic Model-Based Navigation" *Drones* 4, no. 3: 32.
https://doi.org/10.3390/drones4030032