Learning to estimate probabilistic attitude via matrix Fisher distribution with inertial sensor

Abstract

We propose a probabilistic framework for attitude estimation using the matrix Fisher distribution on the special orthogonal group $SO\left(3\right)$. To be specific, a deep neural network is first designed to estimate the attitude probability distributions over $SO\left(3\right)$, i.e., the unconstrained parameters of matrix Fisher distribution. The network takes common inertial measurements as input and overcomes the challenge of learning a regression model due to the topology difference between the $R^N$ and $SO\left(3\right)$. We achieve this by using negative log-likelihood loss, which provides a loss function with desirable properties, such as convexity. Subsequently, a Bayesian fusion framework is introduced for properly fusing the measurement probability distribution estimated by the network with the kinematic model of the rigid body to recover the accurate attitude, and simultaneously estimate the time-varying gyroscope bias. The overall Bayesian estimator can be divided into two key steps: uncertainty propagation based on attitude kinematics and measurement update based on the learned network outputs. Finally, extensive experiments are conducted on several datasets representing driving, flying and walking scenarios, and the results demonstrate a promising performance of the proposed method compared with the state-of-the-art ones. Moreover, we present some ablation experimental results and discuss the application of our method to vehicle dead reckoning by combining it with a constrained velocity model.