Ruggero Frezza

Modeling and estimation of discrete-time Gaussian reciprocal processes

Discrete-time Gaussian reciprocal processes are characterized in terms of a second-order two-point boundary-value nearest-neighbor model driven by a locally correlated noise whose correlation is specified by the model dynamics. This second-order model is the analog for reciprocal processes of the standard first-order state-space models for Markov processes. The model is used to obtain a solution to the smoothing problem for reciprocal processes. The resulting smoother obeys second-order equations whose structure is similar to that of the Kalman filter for Gauss-Markov processes. It is shown that the smoothing error is itself a reciprocal process. >

Higher order linear approximations to nonlinear control systems

One traditional approach in the analysis and design of nonlinear control systems is a first order approximation by a linear system. A new approach is to use nonlinear change of coordinates and feedback to construct linear approximations that are accurate to second and higher orders. However, the algebraic calculations required to obtain these aproximations are somewhat lengthy. In this paper, the theoretical framework for finding such change of coordinates for a nonlinear system are described. A software package that symbolically solves these transformations is currently being prepared.

Gaussian reciprocal processes and self-adjoint stochastic differential equations of second order

We show that under suitable conditions the covariance of a Gaussian reciprocal process satisfies a self-adjoint linear differential equation of second order. We also give a revised definition of a linear stochastic differential equation of second order and necessary and sufficient conditions for the existence of solutions of such equations with Dirichlet boundary conditions. We close with a series of examples of the theory applied to the scalar stationary Gaussian Reciprocal processes which have been completely classified.

Motion estimation on the essential manifold

We introduce a novel perspective for viewing the “ego-motion reconstruction” problem as the estimation of the state of a dynamical system having an implicit measurement constraint and unknown inputs. Such a system happens to be “linear”, but it is defined on a space (the “Essential Manifold”) which is not a linear (vector) space.

Motorcycle trajectory reconstruction by integration of vision and MEMS accelerometers

MEMS accelerometers have the advantage with respect to traditional INS platforms of being miniaturized and economic. Cameras are, nowadays, also miniaturized and the necessity of broadcasting live video from on-board racing motorcycles solved problems such as the transmission of the video signal. The paper presents an algorithm for the accurate reconstruction of a motorcycle trajectory based on the integration of vision and MEMS accelerometers. In a previous paper it was shown that the images taken by the onboard camera on racing motorcycles were sufficient to roughly reconstruct the trajectory by model based estimation. A robust algorithm based on a cumulated Hough transform integrated in time with an appropriate dynamical model allowed for the reconstruction of the roll angle of the velocity and of an approximate trajectory of the motorcycle. Here, the algorithm is extended on one the hand to include measurements of accelerations and on the other hand to use visual landmarks to estimate biases and drifts of the dead reckoning sensors.

A virtual motorcycle driver for closed-loop simulation

The development of a motorcycle driver for virtual prototyping applications is discussed. The driver is delivered with a commercial multibody code as a tool for performing closed-loop maneuvers with virtual motorcycle models. The closed-loop controller is developed with a qualitative analysis of how a human rider controls a motorcycle. The analysis concerns handling and maneuverability, which are relevant for real and virtual vehicle performance evaluation. A motorcycle model for control design and a controller structure are developed. The model is based on a mathematical representation of common-sense rules of motorcycle riding. The virtual rider is then tested in various operating conditions to assess whether the control requirements are achieved. Criteria for evaluating driver models are briefly discussed

Achievable motorcycle trajectories

The authors show that a (simple, nonholonomic) motorcycle can exactly track a large class of smooth trajectories in the plane. Instability and nontrivial dynamic coupling make the exploration of aggressive motorcycle trajectories a rather challenging task. Previously (Hauser et al., 2004), we developed optimization techniques for constructing a suitable roll trajectory that (approximately) implements the desired plane trajectory. In that work, we found that the tracking error is usually quite small leading to the natural question: Given a smooth trajectory in the plane, does there exist a bounded roll trajectory that allows a simple motorcycle model to exactly track the plane trajectory? In this paper, we develop a technique for proving that such exact tracking is possible and apply it to a number of example cases. Our technique is based on the nonlinear system inversion work of Devasia and Paden (1998). Indeed, our algorithm is in the class that they propose. Unfortunately, we have been unable to directly use their results as the motorcycle system does not appear to satisfy the specific conditions required.

Trajectory Manifold Exploration for the PVTOL aircraft

In this paper, we study the trajectory space of the PVTOL aircraft. We show that, due to the non-minimum phase nature of the system, more aggressive trajectories may be tracked with respect to the simplified differentially flat model. Given bounded C2trajectories of the center of gravity, we show that there exists a bounded roll trajectory which implements them. We compute an approximation of such roll trajectory using a Newton method for nonlinear optimization based on a trajectory tracking projection operator.

On the Driven Inverted Pendulum

We explore the solutions of the driven inverted pendulum system l\#0308\=gsinφ-al(t)cosτ where al(·) is a bounded lateral acceleration. We show that, for lateral accelerations that are constant before some initial time, an inverted trajectory always exists and remains within a diamond shaped region in the state space. Functional analytic techniques are also developed to provide further insight into the nature of the inverted pendulum trajectories. Associated to the driven inverted pendulum is a time varying linear system. We show that this system always possesses an exponential dichotomy, allowing for the development of a successive approximation algorithm for finding the desired inverted pendulum trajectory. We show that the curve obtained from one iteration of this algorithm is a very good estimate of the required inverted trajectory. As that curve is obtained by filtering the quasi-static angle trajectory by a noncausal time varying low pass filter with weighting function with a shape similar to h(t) = exp−α0|t|, we find that the current pendulum angle is influenced by the values of the lateral acceleration within only a few seconds of the current time. These results are important as the driven inverted pendulum is a common susbsystem in systems ranging from motorcycles and bicycles to rockets and aircaft.

Some inverse eigenproblems for Jacobi and arrow matrices

We consider the problem of reconstructing Jacobi matrices and real symmetric arrow matrices from two eigenpairs. Algorithms for solving these inverse problems are presented. We show that there are reasonable conditions under which this reconstruction is always possible. Moreover, it is seen that in certain cases reconstruction can proceed with little or no cancellation. The algorithm is particularly elegant for the tridiagonal matrix associated with a bidiagonal singular value decomposition.