No Force Necessary: Turning on the Targeting Computer with Polynomial Trajectory Estimation and Quadcopter Control

Abstract

Reconstructing a three-dimensional trajectory from a series of two-dimensional images is a nontrivial problem with a wide range of applications in surveillance, communication, and defense. This thesis describes a method of successfully recovering the time-dependent polynomial trajectory of a tracked feature using a linear system of equations. This method is then expanded upon, with the same solution method applied to an oscillatory system composed of sines and cosines with an unknown frequency. The purpose of this was to develop a method of linearizing a system of equations that was not well-modeled by a polynomial. This was accomplished, with the system successfully generating predictive trajectories that drastically reduced pre- diction error when compared to the original polynomial method. The second half of this thesis involves application of this trajectory estimation method with a quadrotor helicopter (or quadcopter) base. The underlying dynamics of a quadcopter is described, and a linear state-space system was found for these (normally coupled) dynamics. Because of axial symmetry, the system was found to be neutrally stable about the yaw axis of rotation. A proportional-integral-derivative (PID) controller was implemented and manually tuned for attitude control, which demonstrated rapid rise time with relatively little overshoot. Altitude control was accomplished using a simpler proportional-derivative (PD) with gains derived from an linear quadratic regulator (LQR) optimization scheme. A method for positional control with simplified attitude control considerations is also proposed. Positional control was shown to be achievable in simulation, although the gains must be carefully selected to avoid driving the system unstable. Lastly, some of the barriers to practical implementation of the trajectory estimation algorithm and controller system were discussed; these include accurate calibration of the camera being used and mitigation of sensor noise.