Cornelis C. de Visser

A sparse collocation method for solving time-dependent HJB equations using multivariate B-splines

This paper presents a sparse collocation method for solving the time-dependent Hamilton-Jacobi-Bellman (HJB) equation associated with the continuous-time optimal control problem on a fixed, finite time-horizon with integral cost functional. Through casting the problem in a recursive framework using the value-iteration procedure, the value functions of every iteration step is approximated with a time-varying multivariate simplex B-spline on a certain state domain of interest. In the collocation scheme, the time-dependent coefficients of the spline function are further approximated with ordinary univariate B-splines to yield a discretization for the value function fully in terms of piece-wise polynomials. The B-spline coefficients are determined by solving a sequence of highly sparse quadratic programming problems. The proposed algorithm is demonstrated on a pair of benchmark example problems. Simulation results indicate that the method can yield increasingly more accurate approximations of the value function by refinement of the triangulation.

Three-Dimensional Velocity Obstacle Method for Uncoordinated Avoidance Maneuvers of Unmanned Aerial Vehicles

This paper proposes a novel avoidance method called the three-dimensional velocity obstacle method. The method is designed for unmanned aerial vehicle applications, in particular to autonomously handle uncoordinated multiple encounters in an integrated airspace, by exploiting the limited space in a three-dimensional manner. The method is a three-dimensional extension of the velocity obstacle method that can reactively generate an avoidance maneuver by changing the vehicle velocity vector based on the encounter geometry. Adverse maneuvers of the obstacle are anticipated by introducing the concept of a buffer velocity set, which ensures that the ownship will diverge with sufficient space in case of sudden imminence. A three-dimensional resolution is generated by choosing the right plane for avoidance, in which the unmanned aerial vehicle conducts a pure turning maneuver. Implementation of the three-dimensional velocity obstacle method is tested in several simulations that demonstrate its capability to resol...

Wavefront reconstruction using intensity measurements for real-time adaptive optics

The crucial step in adaptive optics feedback control is the reconstruction of the wavefront from wavefront sensor measurements. One of the sensors commonly used is the (Shack-)Hartmann sensor, which makes the wavefront reconstruction problem linear, in a least-squares sense. In this paper, a new methodology is proposed to reconstruct the wavefront for real-time adaptive optics control using the complete intensity measurements provided by the sensor, instead of the classical centroid algorithm which approximates the local wavefront slopes. In addition to an outline of the new wavefront reconstruction method, its performance is illustrated via a numerical simulation study. The advantages of the new method are highlighted by comparing it, in both open- and closed-loop, with a modal reconstruction algorithm that uses local wavefront slopes.

Framework for state and unknown input estimation of linear time-varying systems

The design of unknown-input decoupled observers and filters requires the assumption of an existence condition in the literature. This paper addresses an unknown input filtering problem where the existence condition is not satisfied. Instead of designing a traditional unknown input decoupled filter, a Double-Model Adaptive Estimation approach is extended to solve the unknown input filtering problem. It is proved that the state and the unknown inputs can be estimated and decoupled using the extended Double-Model Adaptive Estimation approach without satisfying the existence condition. Numerical examples are presented in which the performance of the proposed approach is compared to methods from literature.

Safe Exploration Algorithms for Reinforcement Learning Controllers

Self-learning approaches, such as reinforcement learning, offer new possibilities for autonomous control of uncertain or time-varying systems. However, exploring an unknown environment under limited prediction capabilities is a challenge for a learning agent. If the environment is dangerous, free exploration can result in physical damage or in an otherwise unacceptable behavior. With respect to existing methods, the main contribution of this paper is the definition of a new approach that does not require global safety functions, nor specific formulations of the dynamics or of the environment, but relies on interval estimation of the dynamics of the agent during the exploration phase, assuming a limited capability of the agent to perceive the presence of incoming fatal states. Two algorithms are presented with this approach. The first is the Safety Handling Exploration with Risk Perception Algorithm (SHERPA), which provides safety by individuating temporary safety functions, called backups. SHERPA is shown in a simulated, simplified quadrotor task, for which dangerous states are avoided. The second algorithm, denominated OptiSHERPA, can safely handle more dynamically complex systems for which SHERPA is not sufficient through the use of safety metrics. An application of OptiSHERPA is simulated on an aircraft altitude control task.

Nonlinear spline wavefront reconstruction from Shack–Hartmann intensity measurements through small aberration approximations

We propose an extension of the Spline based ABerration Reconstruction (SABRE) method to Shack-Hartmann (SH) intensity measurements, through small aberration approximations of the focal spot models. The original SABRE for SH slope measurements is restricted to the use of linear spline polynomials, due to the limited amount of data, and the resolution of its reconstruction is determined by the number of lenslets. In this work, a fast algorithm is presented that directly processes the pixel information of the focal spots, allowing the employment of nonlinear polynomials for high accuracy reconstruction. In order to guarantee the validity of the small aberration approximations, the method is applied in two correction steps, with a first compensation of large, low-order aberrations through the gradient-based linear SABRE followed by compensation of the remaining high-order aberrations with the intensity-based nonlinear SABRE.

Distance-Fields-Over-Grids Method for Aircraft Envelope Determination

In this research, the concept of posing the flight envelope as a reachable set is explored further, and it is investigated whether the distance fields over grids (DFOG) method for reachability computations is a viable method for envelope determination. The DFOG method approximates the reachable set of a dynamic system by solving a number of optimal control problems over a grid in the state space. Then, the end points of the resulting optimal trajectories are used to provide an approximation of the reachable set. The DFOG method is improved by using a second-order Runge–Kutta method for the state discretization and developing a new adaptive grid-selection approach. This method is then applied to the computation of a four-dimensional reachable set of a nonlinear high-fidelity F-16 model. The results are compared to the results obtained with the level-set method, demonstrating that the DFOG method provides a less conservative approximation to the reachable set.

Filtering and Identification for Spline based Wavefront Reconstruction from Gradient Measurements in Adaptive Optics

An extension to the SABRE wavefront reconstruction method compensates for delays in Adaptive Optics feedback loops. With a constrained Kalman Filter, the wavefront aberrations are predicted using a temporal dynamic model derived through Subspace Identification.

Distributed wavefront reconstruction with SABRE for real-time large scale adaptive optics control

We present advances on Spline based ABerration REconstruction (SABRE) from (Shack-)Hartmann (SH) wavefront measurements for large-scale adaptive optics systems. SABRE locally models the wavefront with simplex B-spline basis functions on triangular partitions which are defined on the SH subaperture array. This approach allows high accuracy through the possible use of nonlinear basis functions and great adaptability to any wavefront sensor and pupil geometry. The main contribution of this paper is a distributed wavefront reconstruction method, D-SABRE, which is a 2 stage procedure based on decomposing the sensor domain into sub-domains each supporting a local SABRE model. D-SABRE greatly decreases the computational complexity of the method and removes the need for centralized reconstruction while obtaining a reconstruction accuracy for simulated E-ELT turbulences within 1% of the global methods accuracy. Further, a generalization of the methodology is proposed making direct use of SH intensity measurements which leads to an improved accuracy of the reconstruction compared to centroid algorithms using spatial gradients.

Compressive Sampling in Intensity Based Control for Adaptive Optics

Abstract The central problem in Adaptive Optics feedback control is the reconstruction of the aberrated wavefront from wavefront sensor measurements. We recently presented a novel algorithm to compute the wavefront estimate directly from (Shack-)Hartmann intensity images instead of using the classical centroid algorithm to approximate the local wavefront slopes. The novel algorithm allows a distributed linearization of the model describing the imaging process through the use of a B-spline parametrization of the wavefront. This linearization enables the estimation of the wavefront via a linear least-squares solver. A major bottleneck of this new algorithm is the computational complexity that stems from the large number of pixels with each pixel giving rise to one row of the overdetermined set of equations. In this paper, a compression method is proposed to speed up this new reconstruction method by only using a small percentage of the given intensities to make it applicable for real-time Adaptive Optics. Numerical simulations for open- and closed-loop show that reducing the data on the one hand dramatically reduces the number of measurements, but on the other hand does not cause any significant loss in accuracy or robustness of the reconstructed wavefront estimate.