Coen C. de Visser

Coen Visser, Alexander Heinlein, Bianca Giovanardi

Physics-Informed Neural Networks (PINNs) have emerged as a tool for approximating the solution of Partial Differential Equations (PDEs) in both forward and inverse problems. PINNs minimize a loss function which includes the PDE residual determined for a set of collocation points. Previous work has shown that the number and distribution of these collocation points have a significant influence on the accuracy of the PINN solution. Therefore, the effective placement of these collocation points is an active area of research. Specifically, available adaptive collocation point sampling methods have been reported to scale poorly in terms of computational cost when applied to high-dimensional problems. In this work, we address this issue and present the Point Adaptive Collocation Method for Artificial Neural Networks (PACMANN). PACMANN incrementally moves collocation points toward regions of higher residuals using gradient-based optimization algorithms guided by the gradient of the PINN loss function, that is, the squared PDE residual. We apply PACMANN for forward and inverse problems, and demonstrate that this method matches the performance of state-of-the-art methods in terms of the accuracy/efficiency tradeoff for the low-dimensional problems, while outperforming available approaches for high-dimensional problems. Key features of the method include its low computational cost and simplicity of integration into existing physics-informed neural network pipelines. The code is available at https://github.com/CoenVisser/PACMANN.

Jasper van Beers, Coen de Visser

Ensuring the reliability and validity of data-driven quadrotor model predictions is essential for their accepted and practical use. This is especially true for grey- and black-box models wherein the mapping of inputs to predictions is not transparent and subsequent reliability notoriously difficult to ascertain. Nonetheless, such techniques are frequently and successfully used to identify quadrotor models. Prediction intervals (PIs) may be employed to provide insight into the consistency and accuracy of model predictions. This paper estimates such PIs for polynomial and Artificial Neural Network (ANN) quadrotor aerodynamic models. Two existing ANN PI estimation techniques - the bootstrap method and the quality driven method - are validated numerically for quadrotor aerodynamic models using an existing high-fidelity quadrotor simulation. Quadrotor aerodynamic models are then identified on real quadrotor flight data to demonstrate their utility and explore their sensitivity to model interpolation and extrapolation. It is found that the ANN-based PIs widen considerably when extrapolating and remain constant, or shrink, when interpolating. While this behaviour also occurs for the polynomial PIs, it is of lower magnitude. The estimated PIs establish probabilistic bounds within which the quadrotor model outputs will likely lie, subject to modelling and measurement uncertainties that are reflected through the PI widths.

Prashant Solanki, Nikolaus Vertovec, Yannik Schnitzer et al.

Recent approaches to leveraging deep learning for computing reachable sets of continuous-time dynamical systems have gained popularity over traditional level-set methods, as they overcome the curse of dimensionality. However, as with level-set methods, considerable care needs to be taken in limiting approximation errors, particularly since no guarantees are provided during training on the accuracy of the learned reachable set. To address this limitation, we introduce an epsilon-approximate Hamilton-Jacobi Partial Differential Equation (HJ-PDE), which establishes a relationship between training loss and accuracy of the true reachable set. To formally certify this approximation, we leverage Satisfiability Modulo Theories (SMT) solvers to bound the residual error of the HJ-based loss function across the domain of interest. Leveraging Counter Example Guided Inductive Synthesis (CEGIS), we close the loop around learning and verification, by fine-tuning the neural network on counterexamples found by the SMT solver, thus improving the accuracy of the learned reachable set. To the best of our knowledge, Certified Approximate Reachability (CARe) is the first approach to provide soundness guarantees on learned reachable sets of continuous dynamical systems.

Bram Strack van Schijndel, Sihao Sun, Coen de Visser

This paper presents a novel method for fast and robust detection of actuator failures on quadrotors. The proposed algorithm has very little model dependency. A Kalman filter estimator estimates a stochastic effectiveness factor for every actuator, using only onboard RPM, gyro and accelerometer measurements. Then, a hypothesis test identifies the failed actuator. This algorithm is validated online in real-time, also as part of an active fault tolerant control system. Loss of actuator effectiveness is induced by ejecting the propellers from the motors. The robustness of this algorithm is further investigated offline over a range of parameter settings by replaying real flight data containing 26 propeller ejections. The detection delays are found to be in the 30 to 130 ms range, without missed detections or false alarms occurring.

A. Narasimhan, C. C. de Visser, C. de Wagter et al.

This paper aims to develop a Fault Tolerant Control (FTC) architecture, for the case of a damaged actuator for a multirotor UAV that can be applied across multirotor platforms based on their Attainable Virtual Control Set (AVCS). The research is aimed to study the AVCS and identify the parameters that limit the controllability of multirotor UAV post an actuator failure. Based on the study of controllability, the requirements for a FTC is laid out. The implemented control solution will be tested on a quadrotor, Intel Shooting Star UAV platform in indoor and outdoor flights using only the onboard sensors. The attitude control solution is implemented with reduced attitude control, and the control allocation is performed with pseudo-inverse based model inversion with sequential desaturation to ensure tilt priority. The model is identified with an offline Ordinary Least Squares routine and subsequently updated with the Recursive Least Squares method. An offline calibration routine is implemented to correct IMU offset distance from the centre of rotation to correct for accelerometer bias caused by the high-speed spin after failure in a quadrotor.

Sihao Sun, Matthias Baert, Bram Adriaan Strack van Schijndel et al.

This study has developed a fault-tolerant controller that is able to recover a quadrotor from arbitrary initial orientations and angular velocities, despite the complete failure of a rotor. This cascaded control method includes a position/altitude controller, an almost-global convergence attitude controller, and a control allocation method based on quadratic programming. As a major novelty, a constraint of undesirable angular velocity is derived and fused into the control allocator, which significantly improves the recovery performance. For validation, we have conducted a set of Monte-Carlo simulation to test the reliability of the proposed method of recovering the quadrotor from arbitrary initial attitude/rate conditions. In addition, real-life flight tests have been performed. The results demonstrate that the post-failure quadrotor can recover after being casually tossed into the air.

Willem Eerland, Coen de Visser, Erik-Jan van Kampen

We define a SDP framework based on the RLSTD algorithm and multivariate simplex B-splines. We introduce a local forget factor capable of preserving the continuity of the simplex splines. This local forget factor is integrated with the RLSTD algorithm, resulting in a modified RLSTD algorithm that is capable of tracking time-varying systems. We present the results of two numerical experiments, one validating SDP and comparing it with NDP and another to show the advantages of the modified RLSTD algorithm over the original. While SDP requires more computations per time-step, the experiment shows that for the same amount of function approximator parameters, there is an increase in performance in terms of stability and learning rate compared to NDP. The second experiment shows that SDP in combination with the modified RLSTD algorithm allows for faster recovery compared to the original RLSTD algorithm when system parameters are altered, paving the way for an adaptive high-performance non-linear control method.

Kirk Y. W. Scheper, Sjoerd Tijmons, Coen C. de Visser et al.

Evolutionary Robotics allows robots with limited sensors and processing to tackle complex tasks by means of sensory-motor coordination. In this paper we show the first application of the Behaviour Tree framework to a real robotic platform using the Evolutionary Robotics methodology. This framework is used to improve the intelligibility of the emergent robotic behaviour as compared to the traditional Neural Network formulation. As a result, the behaviour is easier to comprehend and manually adapt when crossing the reality gap from simulation to reality. This functionality is shown by performing real-world flight tests with the 20-gram DelFly Explorer flapping wing Micro Air Vehicle equipped with a 4-gram onboard stereo vision system. The experiments show that the DelFly can fully autonomously search for and fly through a window with only its onboard sensors and processing. The success rate of the optimised behaviour in simulation is 88% and the corresponding real-world performance is 54% after user adaptation. Although this leaves room for improvement, it is higher than the 46% success rate from a tuned user-defined controller.