#
Attitude Sensor from Ellipsoid Observations: A Numerical and Experimental Validation^{ †}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Theory of Attitude Determination from Imaged Ellipsoids

**t**

_{w}is the translation vector from the camera center to the origin of the target ellipsoid expressed in world frame (assumed known); ${C}^{*}$ is the inverse (or the adjugate) of the conic matrix $C$ computed from the coefficients of the ellipse quadratic equation:

_{x}and f

_{y}is the focal vector, p

_{x}and p

_{y}are the coordinates of the principal point, and $\alpha $ is the skew-angle (equal zero for orthogonal x-y axes).

- Find $\tilde{\rho}$ such that $\tilde{B}{}^{*}$ and ${\mathcal{C}}^{*}$ are orthogonally similar matrices apart from an unknown scaling.
- Once $\tilde{\rho}$ and thus $\tilde{B}{}^{*}$ are known, compute the attitude matrix as the solution of a modified orthogonal Procrustes problem.

#### Covariance Analysis

^{®}3D scene control. All images are created assuming the same relative position and attitude between the camera and the target, and differ one from each other only by some additive random noise. For each image, the limb is detected and fitted to an ellipse. Then, the attitude determination algorithm is run, and the error $\delta {\mathcal{C}}^{*}$ between the true and estimated ${\mathcal{C}}^{*}$ matrices is recorded, along with the estimated attitude error, $\delta R$. Once all images are processed, we numerically estimated the covariance matrix ${P}_{\mathcal{C}\mathcal{C}}$ from the recorded 1000 samples of $\delta {\mathcal{C}}^{*}$. Having ${P}_{\mathcal{C}\mathcal{C}}$ available, we computed for each test case the analytical covariance ${P}_{RR}$ according to Equation (19) to be compared to the actual estimation errors.

_{i,j}(i ≠ j). Since the assumed nominal R matrix is the identity (null attitude) r

_{i,j}is equal, to first order, to the error angle about k-axis, so that r

_{1,2}r

_{1,3}and r

_{2,3}correspond approximately to the error angles about the third (yaw), second (pitch), and first (roll) axis, respectively. Superimposed to each histogram, the curve of a Gaussian distribution probability function is shown, having variance equal to the corresponding element in the main diagonal of the analytic ${P}_{RR}$.

## 3. Sensor Architecture

^{®}and a FLIR Lepton

^{®}3 were chosen as the reference for configuration (a) and (b), respectively. Their main characteristics are collected in Table 1 and Table 2 [26,27].

#### Measurement Principle: Limb Detection and Fitting

- (a)
- Edge detection;
- (b)
- Edge fitting to an ellipse.

## 4. Numerical Simulator

^{®}-based simulation environment, which generates synthetic images of an ellipsoid target with the same flattening of the Earth. The simulator accounts for the main error source affecting the attitude determination from infrared Earth images, namely, the presence of a diffuse, inhomogeneous limb shift, due to the atmosphere. The consequence of a diffuse limb is that of resembling a slightly larger Earth, or a slightly smaller camera distance to the target. A non-homogeneous limb shift, instead, will result to a detected target shape, which differs from the solid Earth ellipsoid. While the former effect is rejected by the proposed algorithm, the latter is expected to induce some errors. The simplified atmospheric model implemented in our simulator aims at assessing, at least approximately, the magnitude of such errors, rather than providing a high-fidelity imaging simulation tool.

- -
- The normalized atmospheric radiance profile decreases with the tangent height roughly following an inverse S-shaped curve.
- -
- The spatial variation of the apparent limb shift has a systematic component, depending mainly on the latitude, plus a stochastic component. The two are almost equally important in magnitude.
- -
- Overall, the variability of the atmospheric radiance induces changes in the detected infrared limb height of about ±10 km.

_{k}, were tuned for leading to dw variations of up to about ±10 km.

## 5. Experimental Testbed

^{®}3 camera (see Table 2), thus representative of a lower resolution sensor implementation targeted to nano-satellites. Three rotational degrees of freedom are provided by micrometric stages, having 10 arcmin of resolution. The target consists of a spheroid made of PVC plastic having nominal radii equal to 68.75 and 69.00 mm, respectively and mounted with the axis of symmetry orthogonal to the test-bed plane. The remaining two translational degrees of freedom allows for fine alignment between the camera and the target center. Rotations are implemented as a 3-1-2 (yaw-roll-pitch) rotation sequence, with angular excursions of 360° about yaw (ψ), ±5° and ±10° for the roll (φ, inner) and pitch (ϑ, outer) axes. Yaw axis is directed towards the spheroid center, pitch axis is orthogonal to the test-bed plane (positive downward), and roll axis directed to create a right-handed triad. The camera itself is then mounted on a PVC support tilted by 45° about pitch axis, for having a portion of the spheroid limb in view at nominal orientation.

_{b/i}with respect to a fixed body-to-world R attitude, rather than assuming a fixed camera-to-body orientation R

_{b/i}w.r.t. a rotating body-to-world attitude, as in an actual operative scenario. However, implementing this last option requires the conical scan to be performed after the pitch-roll rotations, as in a 1-2-3 sequence, which, in turn, is not achievable with the mounting options offered by the three rotational stages in our testbed.

## 6. Results

#### 6.1. Results of Numerical Validation

_{E}= 0.1, 0.2, and 0.3 (note that in the short conference version of this paper [9], the dimensionless orbit altitudes were incorrectly reported as 0.01, 0.02, and 0.03, due to typos). For each altitude, vantage points were generated at latitudes ranging from 0 to 85°, with 5° of angular step. For each vantage point, a set of three images was generated (as if they were captured by the three sensor heads), prescribing a nominally nadir-pointing spacecraft attitude, with some randomly generated, normally distributed off-nadir angles ($\phi ,\vartheta $ having mean and standard deviation μ = 0°, σ = 1°, respectively). Images resolution corresponds to the one of a micro-satellite targeted camera, as found in Table 1.

_{E}= 0.1, 0.2, and 0.3, respectively.

^{−3}to 10

^{−2}degrees for pitch and roll angles. On the other hand, the angle about nadir (yaw) could be retrieved only very coarsely, with increasing error as the vantage point approaches the symmetry axis of the Earth spheroid (i.e., for latitudes close to 90°). This is an expected outcome, according to the discussion at the end of Section 2. Overall, the rms yaw error remained below 10° only up to a latitude of 60°. On the other hand, pitch and roll angles accuracies were only marginally degraded when approaching Earth symmetry axis.

#### 6.2. Results of Experimental Validation

_{b/i}according to:

_{j}j = 1, 2, 3 denotes the elementary rotation matrix around the j-th coordinate axis. Finally, an ellipse was fitted to the stacked vector of transformed points from the three views, using the method in [31]. Having the ellipse matrix available, the attitude determination algorithm was run applying the steps from Equations (11)–(14). The attitude matrix R output of this process was compared to the zero-reference attitude matrix ${R}_{0}$ computed running the algorithm for φ = ϑ = 0°. We defined the estimation errors as the angles of the 3-1-2 rotation sequence computed from the error attitude matrix $R{R}_{0}^{T}$. The process was repeated for four test-cases, corresponding to different pitch-roll combinations: the outcome is summarized in Table 4. A schematic representation of the overall measurement workflow, from image capturing to attitude matrix estimation, is given in Figure 9.

^{−2}deg, i.e., well below the accuracy limit of the experimental platform. This is evidence that the foreseen measurement procedure leads to results that are consistent with the previous theoretical and numerical analyses, providing high accuracy nadir direction determination. This is a relevant outcome, especially when considering the additional error sources affecting the experimental set-up, such as the imperfect camera distortion compensation, and imperfect knowledge of the relative orientation of the camera when capturing the three images. On the other hand, the estimation of the yaw angle was very poor, and worse than what was predicted by the numerical simulator, which is, nevertheless, not surprising, given the lower image resolution of the FLIR Lepton

^{®}3 camera.

## 7. Discussion

^{−2}deg (or less) for the off-nadir angles in most operating conditions. The capability of measuring the orientation about nadir was instead very coarse, and strongly dependent on the observation latitude and camera resolution.

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**Comparison between the numerically retrieved error distributions and analytic variances for three off-diagonal elements of the attitude matrix. (

**a**) r

_{1,2}; (

**b**) r

_{1,3}; and (

**c**) r

_{2,3}.

**Figure 3.**(

**a**) Cluster of three simulated limb images and (

**b**) detail of the edge region highlighting the blur induced by the atmosphere model [9].

**Figure 5.**Sample image checkerboard image used for infrared camera calibration. (

**a**) Raw and (

**b**) undistorted using the estimated radial distortion coefficients.

**Figure 6.**Attitude errors as a function of latitude for an Earth-like spheroid with atmosphere, h/R

_{E}= 0.1 [9].

**Figure 7.**Attitude errors as a function of latitude for an Earth-like spheroid with atmosphere, h/R

_{E}= 0.2 [9].

**Figure 8.**Attitude errors as a function of latitude for an Earth-like spheroid with atmosphere, h/R

_{E}= 0.3 [9].

**Table 1.**Physical characteristics of the IR camera core, FLIR Boson 320, taken as a baseline for a microsatellite targeted implementation of our horizon sensor prototype.

Image Size (px) | HFoV (deg) | Power (mW) | Weight (g) | Envelope (mm^{3}) |
---|---|---|---|---|

320 × 256 | 50 | 500 (operating) + 330 (shutter) | 20 g | 21 × 21 × 30 |

**Table 2.**Physical characteristics of the IR camera core, FLIR Lepton

^{®}3, taken as a baseline for a nanosatellite targeted implementation of our horizon sensor prototype.

Image Size (px) | HFoV (deg) | Power (mW) | Weight (g) | Envelope (mm^{3}) |
---|---|---|---|---|

120 × 160 | 57 | 150 (operating) + 500 (shutter) | 0.9 g | 10.5 × 12.7 × 7.14 |

Focal Lengths f_{x}, f_{y} | Principal Point Coordinates | Radial Distortion Coefficients |
---|---|---|

160.42 ± 1.22 | 80.70 ± 0.57 | −0.333 ± 0.012 |

160.35 ± 1.22 | 63.29 ± 0.41 | 0.212 ± 0.051 |

Test Case | ψ Error (°) | φ Error (°) | ϑ Error (°) |
---|---|---|---|

1: φ = 5°, ϑ = 0° | −3.29 | 0.03 | 0.04 |

2: φ = 0°, ϑ = 5° | 14.20 | −0.01 | −0.006 |

3: φ = 5°, ϑ = 5° | 8.05 | −0.02 | 0.02 |

4: φ = 10°, ϑ = 5° | −6.88 | 0.07 | −0.001 |

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**MDPI and ACS Style**

Modenini, D.; Locarini, A.; Zannoni, M.
Attitude Sensor from Ellipsoid Observations: A Numerical and Experimental Validation. *Sensors* **2020**, *20*, 433.
https://doi.org/10.3390/s20020433

**AMA Style**

Modenini D, Locarini A, Zannoni M.
Attitude Sensor from Ellipsoid Observations: A Numerical and Experimental Validation. *Sensors*. 2020; 20(2):433.
https://doi.org/10.3390/s20020433

**Chicago/Turabian Style**

Modenini, Dario, Alfredo Locarini, and Marco Zannoni.
2020. "Attitude Sensor from Ellipsoid Observations: A Numerical and Experimental Validation" *Sensors* 20, no. 2: 433.
https://doi.org/10.3390/s20020433