Angular and Linear Velocity Estimation for a Re-Entry Vehicle Using Six Distributed Accelerometers: Theory, Simulation and Feasibility
This report describes a feasibility study. We are interested in calculating the angular and linear velocities of a re-entry vehicle using six acceleration signals from a distributed accelerometer inertial measurement unit (DAIMU) [8, 10, 11, 9, 12, 13, 14] . Earlier work  showed that angular and linear velocity calculation using classic nonlinear ordinary differential equation (ODE) solvers is not practically feasible, due to mathematical and numerical difficulties. This report demonstrates
... the theoretical feasibility of using model-based nonlinear state estimation techniques to obtain the angular and linear velocities in this problem. Practical numerical and calibration issues require additional work to resolve. We show that the six accelerometers in the DAIMU are not sufficient to provide observability, so additional measurements of the system states are required (e.g. from a Global Positioning System (GPS) unit). Given the constraint that our system cannot use GPS, we propose using the existing on-board 3-axis magnetometer to measure angular velocity. We further show that the six nonlinear ODE's for the vehicle kinematics can be decoupled into three ODE's in the angular velocity and three ODE's in the linear velocity. This allows us to formulate a three-state Gauss-Markov system model for the angular velocities, using the magnetometer signals in the measurement model. This re-formulated model is observable, allowing us to build an Extended Kalman Filter (EKF) for estimating the angular velocities. Given the angular velocity estimates from the EKF, the three ODE's for the linear velocity become algebraic, and the linear velocity can be calculated by numerical integration. Thus, we do not need direct measurements of the linear velocity to provide observability, and the technique is mathematically feasible. Using a simulation example, we show that the estimator adds value over the numerical ODE solver in the presence of measurement noise. Calculating the velocities in the presence of significant measurement noise is not feasible with a classic ODE solver. The EKF is able to deal effectively with the noise and provide useful angular velocity estimates. The linear velocity estimates for this simulation show numerical difficulties associated with the nonlinear ODE's and the quadrature operation. Future work will focus on dealing with practical numerical issues and the issue of calibrating the DAIMU to deal with uncertainties in the accelerometer positions and locations.