Agostino Martinelli

Agostino Martinelli

Observability is a fundamental structural property of any dynamic system and describes the possibility of reconstructing the state that characterizes the system from observing its inputs and outputs. Despite the huge effort made to study this property and to introduce analytical criteria able to check whether a dynamic system satisfies this property or not, there is no general analytical criterion to automatically check the state observability when the dynamics are also driven by unknown inputs. Here, we introduce the general analytical solution of this fundamental problem, often called the unknown input observability problem. This paper provides the general analytical solution of this problem, namely, it provides the systematic procedure, based on automatic computation (differentiation and matrix rank determination), that allows us to automatically check the state observability even in the presence of unknown inputs (Algorithm 6.1). A first solution of this problem was presented in the second part of the book: "Observability: A New Theory Based on the Group of Invariance" [45]. The solution presented by this paper completes the previous solution in [45]. In particular, the new solution exhaustively accounts for the systems that do not belong to the category of the systems that are "canonic with respect to their unknown inputs". The analytical derivations largely exploit several new concepts and analytical results introduced in [45]. Finally, as a simple consequence of the results here obtained, we also provide the answer to the problem of unknown input reconstruction which is intimately related to the problem of state observability. We illustrate the implementation of the new algorithm by studying the observability properties of a nonlinear system in the framework of visual-inertial sensor fusion, whose dynamics are driven by two unknown inputs and one known input.

Agostino Martinelli

This paper provides the extension of the observability rank condition and the extension of the controllability rank condition to time-varying nonlinear systems. Previous conditions to check the state observability and controllability, only account for nonlinear systems that do not explicitly depend on time, or, for time-varying systems, they only account for the linear case. In this paper, the general analytic conditions are provided. The paper shows that both these two new conditions (the extended observability rank condition and the extended controllability rank condition) reduce to the well known rank conditions for observability and controllability in the two simpler cases of time-varying linear systems and time-invariant nonlinear systems. The proposed new conditions work automatically and can deal with any system, independently of its complexity (state dimension, type of nonlinearity, etc). Simple examples illustrate both these conditions. In addition, the two new conditions are used to study the observability and the controllability properties of a lunar module. For this system, the dynamics exhibit an explicit time-dependence due to the variation of the weight and the variation of the moment of inertia. These variations are a consequence of the fuel consumption. To study the observability and the controllability properties of this system, the extended observability rank condition and the extended controllability rank condition introduced by this paper are required. The paper shows that, even under the constraint that the main rocket engine delivers constant power, the state is weakly locally controllable. Additionally, it is weakly locally observable up to the yaw angle.

Agostino Martinelli

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.

Agostino Martinelli

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.