# Closed-form solution to cooperative visual-inertial structure from motion

## TL;DR

This paper derives a closed-form solution for cooperative visual-inertial structure from motion with two agents, enabling scale and bias estimation without external features, and analyzes the impact of sensor biases on performance.

## Contribution

It extends the single-agent closed-form solution to cooperative scenarios, providing analytical observability, bias impact analysis, and a method for gyroscope bias calibration.

## Key findings

- The solution accurately estimates absolute scale and biases.
- Gyroscope bias significantly affects performance, accelerometer bias does not.
- The proposed bias calibration method is effective and successful in simulations.

## Abstract

This paper considers the problem of visual-inertial sensor fusion in the cooperative case and it provides new theoretical contributions, which regard its observability and its resolvability in closed form. The case of two agents is investigated. Each agent is equipped with inertial sensors (accelerometer and gyroscope) and with a monocular camera. By using the monocular camera, each agent can observe the other agent. No additional camera observations (e.g., of external point features in the environment) are considered. All the inertial sensors are assumed to be affected by a bias. First, the entire observable state is analytically derived. This state includes the absolute scale, the relative velocity between the two agents, the three Euler angles that express the rotation between the two agent frames and all the accelerometer and gyroscope biases. Second, the paper provides the extension of the closed-form solution given in [19] (which holds for a single agent) to the aforementioned cooperative case. The impact of the presence of the bias on the performance of this closed-form solution is investigated. As in the case of a single agent, this performance is significantly sensitive to the presence of a bias on the gyroscope, while, the presence of a bias on the accelerometer is negligible. Finally, a simple and effective method to obtain the gyroscope bias is proposed. Extensive simulations clearly show that the proposed method is successful. It is amazing that, it is possible to automatically retrieve the absolute scale and simultaneously calibrate the gyroscopes not only without any prior knowledge (as in [13]), but also without external point features in the environment.