Ross P. Anderson

Optimal feedback guidance of a small aerial vehicle in a stochastic wind

The navigation of a small unmanned aerial vehicle is challenging due to a large influence of wind to its kinematics. When the kinematic model is reduced to two dimensions, it has the form of the Dubins kinematic vehicle model. Consequently, this paper addresses the problem of minimizing the expected time required to drive a Dubins vehicle to a prescribed target set in the presence of a stochastically varying wind. First, two analytically-derived control laws are presented. One control law does not consider the presence of the wind, whereas the other assumes that the wind is constant and known a priori. In the latter case it is assumed that the prevailing wind is equal to its mean value; no information about the variations of the wind speed and direction is available. Next, by employing numerical techniques from stochastic optimal control, feedback control strategies are computed. These anticipate the stochastic variation of the wind and drive the vehicle to its target set while minimizing the expected tim...

A jump persistent turning walker to model zebrafish locomotion

Zebrafish are gaining momentum as a laboratory animal species for the investigation of several functional and dysfunctional biological processes. Mathematical models of zebrafish behaviour are expected to considerably aid in the design of hypothesis-driven studies by enabling preliminary in silico tests that can be used to infer possible experimental outcomes without the use of zebrafish. This study is motivated by observations of sudden, drastic changes in zebrafish locomotion in the form of large deviations in turn rate. We demonstrate that such deviations can be captured through a stochastic mean reverting jump diffusion model, a process that is commonly used in financial engineering to describe large changes in the price of an asset. The jump process-based model is validated on trajectory data of adult subjects swimming in a shallow circular tank obtained from an overhead camera. Through statistical comparison of the empirical distribution of the turn rate against theoretical predictions, we demonstrate the feasibility of describing zebrafish as a jump persistent turning walker. The critical role of the jump term is assessed through comparison with a simplified mean reversion diffusion model, which does not allow for describing the heavy-tailed distributions observed in the fish turn rate.

A Stochastic Approach to Dubins Vehicle Tracking Problems

An optimal feedback control is developed for fixed-speed, fixed-altitude Unmanned Aerial Vehicle (UAV) to maintain a nominal distance from a ground target in a way that anticipates its unknown future trajectory. Stochasticity is introduced in the problem by assuming that the target motion can be modeled as Brownian motion, which accounts for possible realizations of the unknown target kinematics. Moreover, the possibility for the interruption of observations is included by assuming that the duration of observation times of the target is exponentially distributed, giving rise to two discrete states of operation. A Bellman equation based on an approximating Markov chain that is consistent with the stochastic kinematics is used to compute an optimal control policy that minimizes the expected value of a cost function based on a nominal UAV-target distance. Results indicate how the uncertainty in the target motion, the tracker capabilities, and the time since the last observation can affect the control law, and simulations illustrate that the control can further be applied to other continuous, smooth trajectories with no need for additional computation.

An approach to optimization of airport taxiway scheduling and traversal under uncertainty

Airport runways and taxiways have been identified as a key source of system-wide congestion and delay in the over-strained commercial air traffic system. To combat this growing problem, we present a novel approach for taxiway scheduling and traversal. Aircraft must traverse a taxiway, represented by a graph, from gates to their respective runways and arrive at their scheduled times while adhering to safety separation constraints. We describe a combinatorial mixed-integer linear program to determine the push-back time windows, aircraft speeds, stopping times, and in particular, traversal paths for a given graph and an imposed flight schedule as part of a single optimization problem. Safety and scheduling constraints are made robust to probabilistic deviations from the prescribed schedule and aircraft motion, and multiple objective functions are considered to examine the trade-off between taxi times and the probability of safety separation violation. Several scenarios are presented to demonstrate improvements gained from the method and possible uses for this approach.

Self-triggered sampling for second-moment stability of state-feedback controlled SDE systems

Event-triggered and self-triggered control, whereby the times for controller updates are computed from sampled data, have recently been shown to reduce the computational load or increase task periods for real-time embedded control systems. In this work, we propose a self-triggered scheme for nonlinear controlled stochastic differential equations with additive noise terms. We find that the family of trajectories generated by these processes demands a departure from the standard deterministic approach to event- and self-triggering, and, for that reason, we use the statistics of the sampled-data system to derive a self-triggering update condition that guarantees second-moment stability. We show that the length of the times between controller updates as computed from the proposed scheme is strictly positive and provide related examples.

DUBINS VEHICLE TRACKING OF A TARGET WITH UNPREDICTABLE TRAJECTORY

Motivated by a fixed-speed, fixed-altitude Unmanned Aerial Vehicle (UAV), we seek to control the turning rate of a planar Dubins vehicle that tracks an unpredictable target at a nominal standoff distance. To account for all realizations of the uncertain target kinematics, we model the target motion as a planar random walk. A Bellman equation and an approximating Markov chain that is consistent with the stochastic kinematics is used to compute an optimal control policy that minimizes the expected value of a cost function based on the nominal distance. Our results illustrate that the control can further be applied to a class of continuous, smooth trajectories with no need for further computation.Copyright

Collective Dynamics in the Vicsek and Vectorial Network Models Beyond Uniform Additive Noise

In this work, we analyze the coordination of interacting individuals in two nonlinear dynamical models that are subject to a new form of noise. Specifically, we propose extensions both to the classical Vicsek model, whereby each individual averages the orientation of its geographically proximal neighbors, and to the vectorial network model, in which the selection of neighbors is random and independent of the group geometric configuration. In the traditional forms of these models, the update rule for the individuals’ orientations is affected by additive uniform noise. Motivated by biological groups in which individuals’ turn rates exhibit sporadic and large changes, we extend the uniform additive noise model to a turn rate stochastic process. Through comprehensive numerical simulations, we demonstrate the impact of such occasional large deviations (intensity and frequency), along with the role of the neighbors’ selection process, on the coordination of the group. In addition, we establish a closed-form expression for the group polarization for the vectorial network model in the vicinity of an ordered state.

Understanding Policy Diffusion in the U.S.: An Information-Theoretical Approach to Unveil Connectivity Structures in Slowly Evolving Complex Systems

Detecting and explaining the relationships among interacting components has long been a focal point of dynamical systems research. In this paper, we extend these types of data-driven analyses to the realm of public policy, whereby individual legislative entities interact to produce changes in their legal and political environments. We focus on the U.S. public health policy landscape, whose complexity determines our capacity as a society to effectively tackle pressing health issues. It has long been thought that some U.S. states innovate and enact new policies, while others mimic successful or competing states. However, the extent to which states learn from others, and the state characteristics that lead two states to influence one another, are not fully understood. Here, we propose a model-free, information-theoretical method to measure the existence and direction of influence of one states policy or legal activity on others. Specifically, we tailor a popular notion of causality to handle the slow time-scale of policy adoption dynamics and unravel relationships among states from their recent law enactment histories. The method is validated using surrogate data generated from a new stochastic model of policy activity. Through the analysis of real data in alcohol, driving safety, and impaired driving policy, we provide evidence for the role of geography, political ideology, risk factors, and demographic and economic indicators on a states tendency to learn from others when shaping its approach to public health regulation. Our method offers a new model-free approach to uncover interactions and establish cause-and-effect in slowly-evolving complex dynamical systems.

On the Construction of Minimum-Time Tours for a Dubins Vehicle in the Presence of Uncertainties

We propose an approach to the problem of computing a minimum-time tour through a series of waypoints for a Dubins vehicle in the presence of stochasticity. In this paper, we explicitly account for kinematic nonlinearities, the stochastic drift of the vehicle, the stochastic motion of the targets, and the possibility for the vehicle to service each of the targets or waypoints, leading to a new version of the Dubins vehicle traveling salesperson problem (TSP). Based on the Hamilton–Jacobi–Bellman (HJB) equation, we first compute the minimum expected time feedback control to reach one waypoint. Next, minimum expected times associated with the feedback control are used to construct and solve a TSP. We provide numerical results illustrating our solution, analyze how the stochasticity affects the solution, and consider the possibility for on-line recomputation of the waypoint ordering in a receding-horizon manner.

The Markov-Dubins problem in the presence of a stochastic drift field

We consider the problem of navigating a small Dubins-type aerial or marine vehicle to a prescribed destination set in minimum expected time and in the presence of a stochastic drift field induced by local winds or currents. First, we present a deterministic control law that is independent of the local winds/currents and their statistics. Next, by employing numerical techniques from stochastic optimal control, we compute an optimal feedback control strategy that incorporates the stochastic variation in the wind when driving the Dubins vehicle to its destination set in minimum expected time. Our analyses and simulations offer a side-by-side comparison of the optimal deterministic and stochastic optimal feedback control laws for this problem, and they illustrate that the deterministic control can, in many cases, capture the salient features of structure of the stochastic optimal feedback control.