Error-State LQR Formulation for Quadrotor UAV Trajectory Tracking

The control of quadrotor Unmanned Aerial Vehicles (UAVs) presents unique challenges due to their nonlinear dynamics, underactuation, and the need for precise trajectory tracking in dynamic environments. Traditional control techniques often struggle to handle these challenges efficiently while maintaining computational tractability for real-time applications. To address these issues, this work outlines an error-state Linear Quadratic Regulator (LQR) approach, leveraging the compact and singularity-free representation of orientation errors using exponential coordinates. Exponential coordinates provide a robust way to represent orientation errors without the singularities inherent in other parameterizations such as Euler angles. By formulating the controller in terms of error-state dynamics, this approach avoids the complexity of directly controlling the nonlinear dynamics, focusing instead on minimizing deviations from a nominal trajectory. This is achieved by driving the error-state--which includes position, velocity, and orientation errors--toward zero. The proposed controller uses an LQR formulation, a well-established concept in classical control theory for linear systems, to minimize a quadratic cost function balancing state deviations and control effort. Although the quadrotor dynamics are nonlinear, the error-state dynamics can be re-linearized about the current tracking error at a sufficiently high frequency, allowing the LQR controller to operate effectively in real time. This iterative re-linearization ensures that the controller remains responsive to changes in the tracking error while maintaining computational efficiency.