Gabriele Lini

Predictable Dynamics of Opinion Forming for Networks With Antagonistic Interactions

For communities of agents which are not necessarily cooperating, distributed processes of opinion forming are naturally represented by signed graphs, with positive edges representing friendly and cooperative interactions and negative edges the corresponding antagonistic counterpart. Unlike for nonnegative graphs, the outcome of a dynamical system evolving on a signed graph is not obvious and it is in general difficult to characterize, even when the dynamics are linear. In this paper, we identify a significant class of signed graphs for which the linear dynamics are however predictable and show many analogies with positive dynamical systems. These cases correspond to adjacency matrices that are eventually positive, for which the Perron-Frobenius property still holds and implies the existence of an invariant cone contained inside the positive orthant. As examples of applications, we determine cases in which it is possible to anticipate or impose unanimity of opinion in decision/voting processes even in presence of stubborn agents, and show how it is possible to extend the PageRank algorithm to include negative links.

Path Generation Using

Generation of high-quality drive paths is a significant issue for automated wheeled vehicles. To achieve this aim for a truck and trailer vehicle, the paper proposes the use of a parameterized curve primitive, the η4-spline. Using this spline, generation and shaping of smooth feasible paths is made possible as well as the transfer between arbitrary dynamic configurations of the articulated vehicle. The η4-spline is a ninth-order polynomial curve that can interpolate given Cartesian points with associated arbitrary unit tangent vector, curvature, and first and second derivatives of curvature. It depends on a set of eight (eta) parameters that can be freely chosen to modify the path shape without changing the interpolations conditions at the path endpoints. Completeness, minimality, and symmetry of the η4-spline are established. An example on a parking maneuver of the articulated vehicle is presented and the pertinent optimal path planning is also discussed.

{\mbi \eta}^4

This paper proposes a multi-optimization approach to the autonomous parking of car-like vehicles. It uses a polynomial curve primitive — the η3-spline — to build up intrinsically feasible path maneuvers over which to minimize with a weighted sum method the total length of parking paths and the moduli of the maximum path curvature and curvature derivative. The approach takes into account the mandatory constraint of obstacle avoidance and maximal steering angle and the constraint of maximal curvature derivative which is a selectable limit to ensure the desired smoothness of the parking paths. Simulation results are included for a garage parking example.

-Splines for a Truck and Trailer Vehicle

This paper proposes a method for minimum-time velocity planning with velocity, acceleration and jerk constraints and generic initial and final boundary conditions for the velocity and the acceleration. This minimum-time planning problem is relevant in the context of robotic autonomous navigation, where the iterative steering supervisor periodically replans the future mobile robot motion starting from current position, velocity and acceleration conditions. The problem is faced through discretization and its solution is based on a sequence of linear programming feasibility checks, depending on motion constraints and boundary conditions.

Multi-optimization of η 3 -splines for autonomous parking

The article presents the time-optimal trajectory planning of an automatic guided vehicle (AGV) on a given feasible path while respecting velocity, acceleration and jerk constraints. A theoretical result shows the connection for the AGV between the geometric continuity of its paths and the smoothness of its control inputs (linear velocity and steering angle of the AGV motor wheel). The solution hence proposed for the optimal planning is based on a dynamic path inversion algorithm for which first the optimal velocity profile is determined and then the optimal steering signal is derived from a geometrical construction. A set of sufficient conditions for the feasibility of the velocity planning is devised and the practical computation of the optimal velocity profile uses time-discretization and linear programming. A worked example using η3-splines illustrates the method.

Minimum-time constrained velocity planning

The article consider the Cartesian trajectory tracking of wheeled mobile robots to be performed by a hybrid control scheme with feedforward inverse control and a state feedaback that is only updated periodically and relies on a recursive convex replanning of the reference trajectory. This approach applied to the standard unicycle model is shown to maintain its efficacy also in presence of noise or unmodeled robot dynamics. Explicit, sufficient conditions are provided to ensure global boundedness of the tracking error. Experimental results are presented using Lego Mindstorm mobile robots.

Time-optimal dynamic path inversion for an automatic guided vehicle

Being able to predict the outcome of an opinion forming process is an important problem in social network theory. However, even for linear dynamics, this becomes a difficult task as soon as non-cooperative interactions are taken into account. Such interactions are naturally modeled as negative weights on the adjacency matrix of the social network. In this paper we show how the Perron-Frobenius theorem can be used for this task also beyond its standard formulation for cooperative systems. In particular we show how it is possible to associate the achievement of unanimous opinions with the existence of invariant cones properly contained in the positive orthant. These cases correspond to signed adjacency matrices having the eventual positivity property, i.e., such that in sufficiently high powers all negative entries have disappeared. More generally, we show how for social networks the achievement of a, possibily non-unanimous, opinion can be associated to the existence of an invariant cone fully contained in one of the orthants of ℝn.

Recursive convex replanning for the trajectory tracking of wheeled mobile robots

We study the modulation of the output signal of a Central Pattern Generator (CPG) through variations of the coupling parameters between the neurons. It is assumed that a given CPG possesses an exponentially stable limit cycle which originates a periodic output signal. We propose a method that allows to find the variations of the coupling parameters that change the network output in order to approximate a given reference signal. This problem is relevant in many applications. For instance, in robotics locomotion is desirable to change the output of a network in order to change the frequency or the amplitude of the legs movements. We present applications to ring networks and to the quadruped model developed by Golubitsky and coworkers.

Achieving unanimous opinions in signed social networks

The paper considers the output tracking problem for nonlinear systems whose performance output is also a flat output of the system itself. A desired output signal is sought on the actual performance output by using a feedforward inverse input that is periodically updated with discrete-time feedback of the sampled state of the system. The proposed method is based on an iterative output replanning that uses the desired output trajectory and the sampled state to replan an output trajectory whose inverse input helps in reducing the tracking error. This iterative replanning exploits the Hermite interpolating polynomials to achieve an overall arbitrarily smooth input and a tracking error that can be made arbitrarily small if the state sampling period is sufficiently small and mild assumptions are considered. Some simulation results are presented for the cases of an unicycle and a one-trailer system affected by additive noise.

Limit cycle perturbations for parametric modulation of central pattern generators

In the wide field of vehicle autonomous navigation, significant research efforts have been dedicated to the problem of optimal motion planning. The work presented in this paper faces the aspect of minimum-time velocity planning in the context of the so-called path-velocity decomposition [6] and the iterative steering navigation technique [8,5]. The robot vehicle has to travel on an assigned geometric path and the vehicle velocity on that path can be determined by assigning a minimum-time motion with arbitrary velocity/acceleration boundary conditions. Consequently, the relevant velocity planning problem is the synthesis of a velocity C 1-function that permits in minimum-time and with a bounded jerk to interpolate given velocity and acceleration at the time planning interval endpoints and to travel a given distance. The condition on the maximum jerk value permits to obtain a smooth velocity profile [3].