Surface Reconstruction from Distributed Angle Measurements

Citation

Talon, Thibaud (2020) Surface Reconstruction from Distributed Angle Measurements. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/ZG2D-2K77. https://resolver.caltech.edu/CaltechTHESIS:02282020-192725947

Abstract

This thesis presents an innovative solution to the shape measurement of large structures for space applications. The current state-of-the-art heavily relies on optical solutions such as cameras or lasers to recover the shape of a surface. Because of the impracticality of placing a system in front of a large structure flying in space, new solutions need to be developed. The proposed solution is to embed angular sensors (such as sun sensors) directly on the surface. The sensors provide a collection of distributed measurements that form a discrete map of the angular orientation of the structure. An integration scheme can then estimate the 3D shape of the surface.

A mathematical model to perform the integration from angle measurements to the shape of a 3D surface is presented first. This model is purely geometric and serves as a basis for similar concepts. The surface is known in a reference configuration and is assumed to have deformed inextensibly to its current shape. Inextensibility conditions are enforced through a discretization of the metric tensor generating a finite number of constraints. This model parameterizes the shape of the surface using a small number of unknowns, and thus requires a small number of sensors. We study the singularities of the equations and derive necessary conditions for the problem to be well-posed. The limitations of the algorithm are highlighted. Simulations are performed on developable surfaces to analyze the performance of the method and to show the influence of the parameters used in the algorithm. Optimal schemes which lower the RMS error between the reconstructed shape and the actual one are presented.

An experimental validation of the proposed solution and algorithm is performed on a 1.3 x 0.25 m structure with 14 embedded sun sensors. The sensors measure the two local angles of the surface from a light source placed in front of the surface. A small, lightweight and expandable design of the sensors is shown in this thesis. A calibration procedure accurately correlates the output of the sensor with a 0.5° precision. The procedure also highlights the limitations of the design. The structure was deformed in bending and torsion with amplitudes of a few centimeters, and its shape was reconstructed to an accuracy on the order of a millimeter.

The accuracy of the initial algorithm is found to be limited by local shape deformations caused by the mechanical response of the structure. A new algorithm, replacing the discrete inextensibility conditions with the equilibrium equations derived from a finite-element model, is shown. This new algorithm is tested on the experimental structure and the accuracy of the reconstruction is increased by a factor of 2. The RMS error is under a millimeter on average over the different applied shapes and goes as low as 0.3 mm.

To understand how this solution can apply to large space structures, simulations are performed on a model of a large planar spacecraft. A 25 x 25 m structure representing the current concept for the Caltech Space Solar Power Project satellite is used as an example. Sensors with similar noise properties as the ones built for the experiment are placed on the spacecraft. A finite-element model combining the vibration of the spacecraft with large rigid body rotations is presented. This model is used in a Kalman filter that estimates the shape of the structure by iterative prediction from the dynamic finite-element model and correction from the angle measurements. Simulations are performed around the thruster actuation applied at the corner of the structure to follow a specific guidance scheme that is optimal for space solar power satellites. The actuation creates both vibrations of the structure with amplitudes of few centimeters and large rotations of the spacecraft. The designed Kalman filter can accurately estimate both effects and it is shown that millimeter accuracy is achievable. The relationship between the number of sensors, the reconstructed shape error, as well as potential stiffness deviations in the FE model is studied. The results provide first order estimates of the performance of this measurement system, in order to enable the design of future space missions.

Thesis Files