Global Nonlinear Model Identification with Multivariate Splines

At present, model based control systems play an essential role in many aspects of modern society. Application areas of model based control systems range from food processing to medical imaging, and from process control in oil refineries to the flight control systems of modern aircraft. Central to a model based control system is a mathematical model of the physical system or process that is being controlled. The field of science concerned with the identification of models of physical systems is called system identification. In this thesis, a new methodology is proposed for the identification of models of nonlinear systems using multivariate simplex splines. This new methodology has the potential to increase the performance of any model based control system by improving the quality of system models. Multivariate simplex splines consist of polynomial basis functions, called B-form polynomials, which are defined on geometric structures called simplices. Every simplex supports a single B-form polynomial which itself consists of a linear combination of Bernstein basis polynomials. Each individual Bernstein basis polynomial is scaled by a single coefficient called a B-coefficient. The B-coefficients have a special property in the sense that they have a unique spatial location inside their supporting simplex. This spatial structure, also known as the B-net, provides a number of unique capabilities that add to the desirability of the simplex splines as a tool for data approximation. For example, the B-net simplifies local model modification by directly relating specific model regions to subsets of B-coefficients involved in shaping the model in those regions. This particular capability has the potential to play an important role in future adaptive model based control systems. In such a control system, an on-board simplex spline model can be locally adapted in real time to reflect changes in system dynamics. The approximation power of the multivariate simplex splines can be increased by joining any number of simplices together into a geometric structure called a triangulation. Triangulations come in many shapes and sizes, ranging from configurations consisting of just two simplices to configurations containing millions of simplices. Triangulations can be optimized by locally increasing or decreasing the density of simplices to reflect local system complexity. The new methodology was applied in the identification of a complete set of aerodynamic models for the Cessna Citation II laboratory using flight data obtained during seven test flights. In total, 247 flight test maneuvers were flown which together provided a significant coverage of the flight envelope of the Citation II. The complete identification dataset consisted of millions of measurements on more than sixty flight parameters. More than 2000 prototype spline based aerodynamic models were identified using a newly developed, highly optimized software implementation of the simplex spline identification algorithm. Using the developed methods for simplex spline model validation it was proved that the models are both accurate and of guaranteed numerical stability inside the spline domain. The identification and validation results of the simplex spline models were compared with those of ordinary polynomial models identified using standard identification methods. These results showed that the multivariate simplex spline based aerodynamic models were of significantly higher quality than the aerodynamic models based on ordinary polynomials.Control & OperationsAerospace Engineerin

Horizontal and Vertical Wind Measurements from GOCE Angular Accelerations

Because of the highly accurate accelerometers, the GOCE mission has proven to be a unique source of thermosphere neutral density and cross-wind data. In the current methods, in which only the horizontal linear accelerations are used, the vertical winds cannot be obtained. In the algorithm proposed in this paper, angular accelerations derived from the individual gradiometer accelerations are used to obtain the vertical wind speeds as well. To do so, the measured angular rate and acceleration are combined to find a measurement of the torque acting on the spacecraft. This measurement is then corrected for modeled control torque applied by the magnetic torquers, aerodynamic torque, gravity gradient torque, solar radiation pressure torque, the torque caused by the misalignment of the thrust with respect to the center of gravity, and magnetic torque caused by the operation of several different subsystems of the spacecraft bus. Since the proper documentation of the magnetic properties of the payload were not available, a least squares estimate is made of one hard- and one soft-magnetic dipole pertaining to the payload, on an aerodynamically quiet day. The model for aerodynamic torque uses moment coefficients from Monte-Carlo Test Particle software ANGARA. Finally the neutral density, horizontal cross-wind, and vertical wind are obtained from an iterative process, in which the residual forces and torques are minimized. It is found that, like horizontal wind, the vertical wind responds strongly to geomagnetic storms. This response is observed over the whole latitude range, and shows seasonal variations.Astrodynamics & Space MissionsControl & Simulatio

Attitude Estimation of a Quadcopter with one fully damaged rotor using on-board MARG Sensors

Quadcopters are becoming increasingly popular across diverse sectors. Since rotor damages occur frequently, it is essential to improve the attitude estimation and thus ultimately the ability to control a damaged quadcopter. This research is based on a state-of-the-art method that makes it possible to control the quadcopter despite the total failure of a single rotor, where the attitude and position of the quadcopter are provided by an external system. In the present research, a novel attitude estimator called Adaptive Fuzzy Complementary Kalman Filter (AFCKF) has been developed and validated that works independently of any external systems. It is able to estimate the attitude of a quadcopter with one fully damaged rotor while only relying on the on-board MARG (Magnetometer, Accelerometer, Rate Gyroscope) sensors. The AFCKF provides significantly better attitude estimates for flights with a damaged rotor than mainstream filters, estimating the roll and pitch of the quadcopter with an RMS error of less than 1.7 degrees and a variance of less than 2 degrees. The proposed filter also provides accurate yaw estimates despite the fast spinning motion of the damaged quadcopter, and thus outperforms existing methods at the cost of only a small increase in computation.Control & Simulatio

GOCE Aerodynamic Torque Modeling

In recent studies thermospheric densities and cross-winds have been derived from linear acceleration measurements of the gradiometer on board the GOCE satellite. Our current work is aimed at analyzing also the angular accelerations, in order to improve the thermosphere density and wind data by allowing for the estimation of more unknown parameters. On this poster an overview is provided of the modeling efforts involved in isolating the aerodynamic torque. The intermediate result is a comparison of modeled and measured torques. Each box contains a plot of the torque from a specific source, compared to the measured torque, on October 16th, 2013. A short description of the model for each torque is also provided.Astrodynamics & Space MissionsControl & Simulatio

System Identification using the Multivariate Simplotope B-Spline

In recent research efforts the multivariate simplex spline has shown great promise in system identification applications. It has high approximation power, while its linearity in the parameters allows for computationally efficient estimation of the coefficients. In this paper the multivariate simplotope spline is derived from this spline, and compared to its simplex counterpart in a system identification setting. Contrary to the simplex spline, the simplotope spline allows the user to incorporate expert knowledge of the system in his models. Whereas in the first spline all variables are included in a complete polynomial, in the latter the user can split the variables in decoupled subsets. By fitting models to specifically designed test functions it is shown that this can indeed improve the approximation performance in terms of both the error metrics and the number of B-coefficients required. This comes at the price of a higher total degree, and therefore an increased sensitivity to Runge's phenomenon in case of poor data distribution. Finally an attempt is made to apply the proposed methods to a set of flight data of the DelFly II, a flapping wing micro aerial vehicle. It is found that the used data set is not suitable for global system identification, as the data in concentrated in low-dimensional clusters in the five-dimensional state space. Therefore it is advised that a more suitable data set is obtained to validate the simplotope spline in a system identification setting.Control & SimulationAstrodynamics & Space Mission

Torque model verication for the GOCE satellite

Astrodynamics & Space MissionsControl & Simulatio

Characterization of Thermospheric Vertical Wind Activity at 225- to 295-km Altitude Using GOCE Data and Validation Against Explorer Missions

Recently, the horizontal and vertical cross wind at 225- to 295-km altitude were derived from linear acceleration measurements of the Gravity field and steady-state Ocean Circulation Explorer satellite. The vertical component of these wind data is compared to wind data derived from the mass spectrometers of the Atmosphere Explorer C and E and Dynamics Explorer 2 satellites. From a statistical analysis of the 120-s moving-window standard deviation of the vertical wind (σ(Vz)), no consistent discrepancy is found between the accelerometer-derived and the mass spectrometer-derived data. The validated Gravity field and steady-state Ocean Circulation Explorer data are then used to investigate the influence of several parameters and indices on the vertical wind activity. To this end, the probability distribution of σ(Vz) is plotted after distributing the data over bins of the parameter under investigation. The vertical wind is found to respond strongly to geomagnetic activity at high latitudes, although the response settles around a maximum standard deviation of 50 m/s at an Auroral Electrojet index of 800. The dependence on magnetic local time changes with magnetic latitude, peaking around 4:30 over the polar cap and around 01:30 and 13:30 in the auroral oval. Seasonal effects only become visible at low to middle latitudes, revealing a peak wind variability in both local summer and winter. The vertical wind is not affected by the solar activity level.Astrodynamics & Space MissionsControl & Simulatio

Horizontal and vertical thermospheric cross-wind from GOCE linear and angular accelerations

Thermospheric wind measurements obtained from linear non-gravitational accelerations of the Gravity field and steady-state Ocean Circulation Explorer (GOCE) satellite show discrepancies when compared to ground-based measurements. In this paper the cross-wind is derived from both the linear and the angular accelerations using a newly developed iterative algorithm. The two resulting data sets are compared to test the validity of wind derived from angular accelerations and quantify the uncertainty in accelerometer-derived wind data. In general the difference is found to be less than 50 m/s vertically after high-pass filtering, and 100 m/s horizontally. A sensitivity analysis reveals that continuous thrusting is a major source of uncertainty in the torque-derived wind, as are the magnetic properties of the satellite. The energy accommodation coefficient is identified as a particularly promising parameter for improving the consistency of thermospheric cross-wind data sets in the future. The algorithm may be applied to obtain density and cross-wind from other satellite missions that lack accelerometer data, provided the attitude and orbit are known with sufficient accuracy.Astrodynamics & Space MissionsControl & Simulatio

Blade Element Theory Model for UAV Blade Damage Simulation

From fault-tolerant control to failure detection, blade damage simulation is integral for developing and testing failure-resilient modern unmanned aerial vehicles. Existing approaches assume partial loss of rotor effectiveness or reduce the problem to centrifugal forces resulting from the shift in the propeller centre of gravity. In this study, a white-box blade damage model based on Blade Element Theory is proposed, integrating both mass and aerodynamic effects of blade damage. The model serves as plug-in to the nominal system model, enables the simulation of any degree of blade damage and does not require costly experimental data from failure cases. A complementary methodology for the identification of the airfoil lift and drag coefficients is also presented. Both contributions were demonstrated with the Bebop 2 drone platform and validated with static test stand wrench measurements obtained at 3 levels of blade damage (0%, 10%, 25%) in a dedicated wind tunnel experimental campaign with velocities up to 12 m/s. Results indicate high accuracy in simulating a healthy propeller. In the presence of blade damage, the model exhibits a relative error between 5% and 24% at high propeller rotational speeds and between 15% and 75% at low propeller rotational speeds.Control & Simulatio

Unreal Success: Vision-Based UAV Fault Detection and Diagnosis Framework

Online fault detection and diagnosis (FDD) enables Unmanned Aerial Vehicles (UAVs) to take informed decisions upon actuator failure during flight, adapting their control strategy or deploying emergency systems. Despite the camera being a ubiquitous sensor on-board of most commercial UAVs, it has not been used within FDD systems before, mainly due to the nonexistence of UAV multi-sensor datasets that include actuator failure scenarios. This paper presents a knowledge-based FDD framework based on a lightweight LSTM network and a single layer neural network classifier that fuses camera and Inertial Measurement Unit (IMU) information. Camera data are pre-processed by first computing its optical flow with RAFT-S, a state-of-the-art deep learning model, and then extracting features with the backbone of MobileNetV3-S. Short-Time Fourier Transform is applied on the IMU data for obtaining their time-frequency information. For training and assessing the proposed framework, UUFOSim was developed: an Unreal Engine-based simulator built on AirSim that allows the collection of high-fidelity photo-realistic camera and sensor information, and the injection of actuator failures during flight. Data were collected in simulation for the Bebop 2 UAV with 16 failure cases. Results demonstrate the added value of the camera and the complementary nature of both sensors with failure detection and diagnosis accuracies of 99.98% and 98.86%, respectively.Control & Simulatio