Section: New Results

Sensor Fusion for state parameters identification

Participants : Agostino Martinelli, Chiara Troiani.

General theoretical results

During this year we have focused our research on two distinct domains:

• the visual-inertial structure from motion problem;

• the derivation of analytical solutions for the probability distribution of a Brownian motion that satisfies the unicycle constraint.

The research carried out on the first domain is the follow up of our previous activity. We continued to investigate the observability properties of the visual inertial structure from motion and in particular we have analyzed the case when some of the inertial sensors are missing. This analysis has never been provided before and we started this investigation at the end of last year. During this year we confirmed the validity of our preliminary analysis and we also extended them. The preliminary results were obtained by referring to the case when at least five pont features are available and showed that the observability properties of visual inertial structure from motion do not change by removing all the three gyroscopes and one accelerometer. By removing a further accelerometer, if the camera is not extrinsically calibrated, the system loses part of its observability properties. On the other hand, if the camera is extrinsically calibrated, the system maintains the same observability properties as in the standard case. These results have been published on the journal Foundations and Trends in Robotics and have also been presented at the last ICRA conference[20] .

We recently extended these results by considering the extreme case of a single point feature (i.e., not five). This analysis required to approach an open problem in control theory, called the Unknown Input Observability (UIO). In [20] we proposed a possible method to solve this UIO problem. However, we had to improve this method to deal with this extreme case (i.e., the case of one single point feature). Preliminary results on the extension of this method have been published as a research report [30] and we also plan to present them at the next American Control Conference. By applying this method to our problem, we obtained new interesting results. The new investigation allowed us to conclude that, even in the case of a single point feature, the information provided by a sensor suit composed by a monocular camera and two inertial sensors (along two independent axes and where at least one is an accelerometer) is the same as in the case of a complete inertial measurement unit (i.e., when the inertial sensors consist of three orthogonal accelerometers and three orthogonal gyroscopes). Our first objective is to validate these new results.

Regarding the second domain mentioned above, we have derived a complete analytical solution for the probability distribution of the configuration of a non-holonomic mobile robot that moves in two spatial dimensions by satisfying the unicycle kinematic constraints. The proposed solution differs from previous solutions since it is obtained by deriving the analytical expression of any-order moment of the probability distribution. To the best of our knowledge, an analytical expression for any-order moment that holds even in the case of arbitrary linear and angular speed, has never been derived before. To compute these moments, a direct integration of the Langevin equation has been carried out and each moment was expressed as a multiple integral of the deterministic motion (i.e., the known motion that would result in absence of noise).

For the special case when the ratio between the linear and angular speed is constant, the multiple integrals can be easily solved and expressed as the real or the imaginary part of suitable analytic functions. As an application of the derived analytical results, we also investigated the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed. These results have been published on the journal of statistical mechanics [13] and also as a research report [29] where we added more specific considerations about the impact of the derived results on mobile robotics.

Applications with a Micro Aerial Vehicle

We continued our previous activity about the estimation of the relative motion between two consecutive camera views in order to introduce very efficient algorithms to remove the outliers of the feature-matching process. Thanks to their inherent efficiency, the proposed algorithms are very suitable for computationally-limited robots.

In particular, during this year, we extended the previous results by removing the assumption of planar motions. In this case, to obtain useful results, we had to include one more point feature (i.e., the proposed algorithms only use two feature correspondences and gyroscopic data from IMU measurerments to compute the motion hypothesis). By exploiting this 2-point motion parametrization, we proposed two algorithms to remove wrong data associations in the feature matching process for case of a 6DoF motion. We showed that in the case of a monocular camera mounted on a quadrotor vehicle, motion priors from IMU can be used to discard wrong estimations in the framework of a 2 -point-RANSAC based approach. The proposed methods have been evaluated on both synthetic and real data and presented at the last ICRA conference [27] .