The Dirichlet problem

Thesis (M.A.)--Boston UniversityThe problem of finding the solution to a general eliptic type partial differential equation, when the boundary values are given, is generally referred to as the Dirichlet Problem. In this paper I consider the special eliptic equation of ∇2 J=0 which is Laplace's equation, and I limit myself to the case of two dimensions. Subject to these limitations I discuss five proofs for the existence of a solution to Laplace's equation for arbitrary regions where the boundary values are given. [TRUNCATED

Sensor Characterization and Signal Fusion for InstantEye

The practicality and effectiveness of using a TerraRanger Duo—a parallel sonar and infrared time-of-flight distance sensor—payload for obstacle detection is investigated for use with Physical Science Inc.’s InstantEye drone. A Python program was developed to interface with the serial data output before comparing the sensor’s empirical performance against its data sheet. The two signals from the distinct sensor modules, each with their characterized strengths and weaknesses, were then fused with a Kalman filter. This was further refined by imposing conditional weighting based on the known sensor characteristics. The filter output, with conditional corrections, was able to accurately track a single object’s position and velocity within a maximum range of 14 meters