Covariance Analysis of Attitude and Angular Rate Estimation using Accelerometers

TL;DR

This work tackles gyro-free spacecraft attitude estimation by leveraging an array of body-fixed accelerometers to recover angular velocity $\boldsymbol{\omega}$ and angular acceleration $\dot{\boldsymbol{\omega}}$ without gyros. It develops a linear least-squares framework that estimates the nine components of the kinematic matrix $\mathbf{A}(\boldsymbol{\omega},\dot{\boldsymbol{\omega}})$ from accelerometer readings $\tilde{\boldsymbol{g}}_i$, using relative sensor positions $\boldsymbol{a}_{i0}$. From the estimated $\mathbf{A}$, it derives $\boldsymbol{\omega}$ (via diagonal entries or eigen-decomposition of $\mathbf{S}=\tfrac{1}{2}(\mathbf{A}^T+\mathbf{A})$) and $\dot{\boldsymbol{\omega}}$ (from skew-symmetric relations), and propagates uncertainties through linear or linearized transformations. A Monte Carlo study validates the approach, showing the estimator’s performance depends on sensor geometry—greater distances from the base sensor and orthogonal layouts reduce covariance—thus supporting practical gyro-free attitude determination for spacecraft with accelerometer arrays.

Abstract

In this work a method for using accelerometers for the determination of angular velocity and acceleration is presented. Minimum sensor requirements and insights into how an array of accelerometers can be configured to maximize estimator performance are considered. The framework presented utilizes linear least squares to estimate functions that are quadratic in angular velocity. Simple methods for determining the sign of the spin axis and the linearized covariance approximation are presented and found to perform quite effectively when compared to results obtained by Monte Carlo.

Figures (4)

• Figure 1: Accelerometer $i$ is fixed in the body frame and positioned by $\boldsymbol{a}_i$. The acceleration measured by this sensor is the sum of the translational acceleration and the gyroscopic acceleration induced by the angular velocity vector $\boldsymbol{\omega}$.
• Figure 2: $\omega$ covariance and error
• Figure 3: $d\omega$ covariance and error
• Figure 4: Covariance Ellipsoid and Angular Velocity Error