# Observer Design for Optical Flow-Based Visual-Inertial Odometry with Almost-Global Convergence

## TL;DR

This paper introduces a cascaded observer architecture combining optical flow and IMU data for monocular visual-inertial odometry, achieving almost-global stability and accurate velocity and attitude estimation.

## Contribution

It proposes a novel observer design that fuses optical flow and IMU measurements with stability guarantees, including a gradient descent method for velocity direction extraction from sparse optical flow.

## Key findings

- Observer architecture is almost globally asymptotically stable.
- Simulation results validate the effectiveness of the proposed algorithms.
- Gradient descent algorithm accurately estimates velocity direction from sparse optical flow.

## Abstract

This paper presents a novel cascaded observer architecture that combines optical flow and IMU measurements to perform continuous monocular visual-inertial odometry (VIO). The proposed solution estimates body-frame velocity and gravity direction simultaneously by fusing velocity direction information from optical flow measurements with gyro and accelerometer data. This fusion is achieved using a globally exponentially stable Riccati observer, which operates under persistently exciting translational motion conditions. The estimated gravity direction in the body frame is then employed, along with an optional magnetometer measurement, to design a complementary observer on $\mathbf{SO}(3)$ for attitude estimation. The resulting interconnected observer architecture is shown to be almost globally asymptotically stable. To extract the velocity direction from sparse optical flow data, a gradient descent algorithm is developed to solve a constrained minimization problem on the unit sphere. The effectiveness of the proposed algorithms is validated through simulation results.