Complete analytic solution to Brownian unicycle dynamics

This paper derives a complete analytical solution for the probability distribution of the configuration of a non-holonomic vehicle that moves in two spatial dimensions by satisfying the unicycle kinematic constraints and in presence of Brownian noises. In contrast to previous solutions, the one here derived holds even in the case of arbitrary linear and angular speed. This solution 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 is carried out and each moment is 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, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed.Comment: 22 pages, 6 figures, 2 table

Closed-form solution of visual-inertial structure from motion

International audienceThis paper investigates the visual-inertial structure from motion problem. A simple closed form solution to this problem is introduced. Special attention is devoted to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The investigation is also performed in the case when the inertial data are biased, showing that, in this latter case, more images and more restrictive conditions on the trajectory are required for the problem resolvability

Visual-inertial structure from motion: observability vs minimum number of sensors

International audienceThis paper analyzes the observability properties of the visual inertial structure from motion as the number of inertial sensors is reduced. Specifically, instead of considering the standard formulation where the inertial sensors are 3 orthogonal accelerometers and 3 orthogonal gyroscopes, the sensor system here considered only consists of a monocular camera and 1 or 2 accelerometers. This analysis has never been provided before. The main result achieved in this context is that the observability properties of visual inertial structure from motion do not change by removing all the 3 gyroscopes and 1 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. This contribution clearly shows that the information provided by a monocular camera, 3 accelerometers and 3 gyroscopes is redundant. Additionally, it provides a new perspective in the framework of neuroscience to the process of vestibular and visual integration for depth perception and self motion perception. Finally, to analyze these systems with a reduced number of inertial sensors, the paper introduces a new method to derive the observability properties of a non linear system when part of its input controls is unknown. This method is a further original paper contribution in control theory

Complete analytic solution to Brownian unicycle dynamics

This paper derives 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 is carried out and each moment is 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, the paper investigates the diffusivity of the considered Brownian motion for constant and for arbitrary time-dependent linear and angular speed

Visual-inertial structure from motion: observability and resolvability

International audienceThis paper provides two novel contributions. The former regards the observability of the visual-inertial structure from motion. It is proven that, the information contained in the data provided by a monocular camera which observes a single point-feature and by an Inertial Measurement Unit (IMU) allows estimating the absolute scale, the speed in the local frame, the absolute roll and pitch angles, the biases which affect the accelerometer's and the gyroscope's measurements, the magnitude of the gravitational acceleration and the extrinsic camera-IMU calibration. The latter contribution is the derivation of a new closed form solution to determine some of the previous observable quantities by only using few camera measurements collected during a short time interval and the data provided by the IMU during the same time interval. This closed-solution allows us to investigate the intrinsic properties of the visual-inertial structure from motion and in particular to identify the conditions under which the problem has a finite number of solutions. Specifically, it is shown that the problem can have a unique solution, two distinct solutions and infinite solutions depending on the trajectory, on the number of point-features and on their layout and on the number of camera images. The proposed closed solution is finally used in conjunction with a filter based approach in order to show its benefit

General analytical condition to nonlinear identifiability and its application in viral dynamics

Identifiability describes the possibility of determining the values of the unknown parameters that characterize a dynamic system from the knowledge of its inputs and outputs. This paper finds the general analytical condition that fully characterizes this property. The condition can be applied to any system, regardless of its complexity and type of nonlinearity. In the presence of time varying parameters, it is only required that their time dependence be analytical. In addition, its implementation requires no inventiveness from the user as it simply needs to follow the steps of a systematic procedure that only requires to perform the calculation of derivatives and matrix ranks. Time varying parameters are treated as unknown inputs and their identifiability is based on the very recent analytical solution of the unknown input observability problem. Finally, when a parameter is unidentifiable, the paper also provides an analytical method to determine infinitely many values for this parameter that are indistinguishable from its true value. The condition is used to study the identifiability of two nonlinear models in the field of viral dynamics (HIV and Covid-19). In particular, regarding the former, a very popular HIV ODE model is investigated, and the condition allows us to automatically find a new fundamental result that highlights a serious error in the current state of the art.Comment: This preprint is currently under review on Transaction and Automatic Control. It is a short version of arXiv:2211.13507. It includes the definition of identifiability in the presence of time varying parameters - which is absent in arXiv:2211.1350