E. De Weerdt

Delft University of Technology, Delft, South Holland, Netherlands

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Publications (19)7.36 Total impact

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    ABSTRACT: The computation of the reachable set of states of a given dynamic system is an important step to verify its safety during operation. There are different methods of computing reachable sets, namely interval integration, capture basin, methods involving the minimum time to reach function, and level set methods. This work deals with interval integration to compute subpavings to over or under approximate reachable sets of low dimensional systems. The main advantage of this method is that, compared to guaranteed integration, it allows to control the amount of over-estimation at the cost of increased computational effort. An algorithm to over and under estimate sets through subpavings, which potentially reduces the computational load when the test function or the contractor is computationally heavy, is implemented and tested. This algorithm is used to compute inner and outer approximations of reachable sets. The test function and the contractors used in this work to obtain the subpavings involve guaranteed integration, provided either by the Euler method or by another guaranteed integration method. The methods developed were applied to compute inner and outer approximations of reachable sets for the double integrator example. From the results it was observed that using contractors instead of test functions yields much tighter results. It was also confirmed that for a given minimum box size there is an optimum time step such that with a greater or smaller time step worse results are obtained. © 2014 Springer Basel.
    Full-text · Article · Aug 2014 · Mathematics in Computer Science
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    ABSTRACT: The appearance variation cue captures the variation in texture in a single image. Its use for obstacle avoidance is based on the assumption that there is less such variation when the camera is close to an obstacle. For videos of approaching frontal obstacles, it is demonstrated that combining the cue with optic flow leads to better performance than using either cue alone. In addition, the cue is successfully used to control the 16-g flapping-wing micro air vehicle DelFly II.
    Full-text · Article · Apr 2012 · IEEE Transactions on Robotics
  • E. De Weerdt · E. Van Kampen · Q.P. Chu · J.A. Mulder
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    ABSTRACT: When using interval analysis, the bounds of an inclusion function are often non-tight due to dependency effects. The benefit of Taylor Models (TMs) or Verified Taylor Series (VTSs) is the use of higher order derivatives terms, significantly reducing the dependency effect. In this paper, it is assumed that the required information to derive these inclusion functions is obtained using automatic differentiation. The drawback of TMs and VTSs is that not all available information is used, resulting in non-optimal inclusion functions. In this paper the Polynomial Inclusion Function (PIF) is presented, which is guaranteed to form equal or shaper enclosures than any (combination of) Taylor Model(s) defined using the same set of information. The PIF is derived for the one dimensional case. Extension to n-dimensional functions is performed via application of the PIF to every dimension independently. The performance of the PIF is compared to that of Verified Taylor Series for multiple (non-linear) functions and is shown to yield to superior inclusions. Moreover, unlike with TMs or VTSs, increasing the order of the PIF will always sharpen its bounds.
    No preview · Article · Jan 2012 · Reliable Computing
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    ABSTRACT: A visual cue is introduced that exploits the visual appearance of a single image to estimate the proximity to an obstacle. In particular, the appearance variation cue captures the variation in texture and / or color in the image, and is based on the assumption that there is less such variation when the camera is close to an obstacle. Random sampling is applied in order to evaluate the appearance variation fast enough for use in robotics. It is demonstrated that the randomly sampled appearance variation cue can be complementary to optic flow for obstacle detection; combining the two visual cues leads to better obstacle detection performance. Random sampling leads to sufficient computational efficiency for the cue's utilization in autonomous flight: a speed-up of a factor ~100 is attained, which allows the successful control of the 16-gram flapping wing MAV DelFly II.
    Full-text · Conference Paper · Jan 2011
  • E. de Weerdt · E. van Kampen · Q.P. Chu · J.A. Mulder
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    ABSTRACT: Trajectory optimization has been a large field of research for many years. The drawback is that for non-convex, constrained problems practically all available solvers cannot guarantee that the globally optimal trajectory is found. Interval analysis based solvers however can provide this guarantee. Interval analysis has been applied to trajectory optimization before, but the previously presented methods suffered from major drawbacks which limited their application to small scale problems. In this paper a new interval based method is introduced which incorporates state parameterization to prevent explicit integration. The performance of the proposed method is demonstrated by applying it to a spacecraft formation flying optimization problem. The results are compared with a gradient based solver and it is shown that the interval method is guaranteed to find the global optimal solution. Finally the first steps for another new trajectory optimization method based on interval analysis and direct collocation are presented.
    No preview · Conference Paper · Jan 2011
  • E. de Weerdt · E. van Oort · E. van Kampen · Q.P. Chu
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    ABSTRACT: Level set methods are used to determine the reachable set for a given time domain. In the aerospace industry level set methods are used to determine for example flight envelopes. A new level set method is presented which uses interval analysis to give guaranteed bounds on the solution. Unlike the current grid-based methods the new method does not use a grid and has a lower computation complexity. The guaranteed bounds are a product of using interval analysis. Both an inner and outer bound can be derived. The time step and accuracy are easily controlled by the user and can be automatically adapted during processing. Initial tests show that the new method provides accurate solutions and does so using lower computational loads. The new method is tested on a one-dimensional and a two-dimensional test bed, showing that correct outer bounds of the reachable sets can be computed. © 2011 by Delft University of Technology. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.
    No preview · Conference Paper · Jan 2011
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    ABSTRACT: Estimating multimodal pilot model parameters from experimental data requires solving a global nonlinear optimization problem with many local minimums. With traditional parameter estimation techniques, the solution depends on the initial parameter estimate and a local optimum can be found instead of the global optimum. In this paper, the parameter optimization is performed by using the theory of interval analysis, which describes the properties of intervals of numbers instead of crisp numbers. Interval analysis has been shown to bean excellent tool for global nonlinear optimization and it can guarantee that the global minimum of the cost function is found. The interval optimization method is applied to data from an experiment investigating the role of optic flow and the influence of physical motion cues during control of self-motion. A comparison between gradient-based and interval optimization shows that the interval method can find the global minimum of the cost function, resulting in the optimal set of model parameters, whereas gradient-based methods often converge to a local minimum.
    No preview · Article · Jan 2010 · Journal of Guidance Control and Dynamics
  • E. Van Kampen · E. De Weerdt · O.P. Chu · J.A. Mulder
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    ABSTRACT: This paper provides improvements on the interval integer ambiguity resolution algorithm BOUNDS, which can give theoretical guarantees on finding the correct integers. The BOUNDS algorithm is validated by applying it to real GPS data and comparing it to the LAMBDA method. The first improvement is a transformation of the search space, which reduces dependency. Secondly the process of using multiple frequencies is explained and demonstrated. The third improvement, called contractions, quickly reduces the search space and increases solution accuracy. Finally, matrix computations are introduced to decrease the computational load. Previous research into BOUNDS indicated that defining correct noise bands is an important step of the algorithm. This is investigated in more detail, discussing several noise iteration schemes and noise models based on the signal-to-noise ratio. Application of the improved BOUNDS algorithm to the attitude determination problem for a static baseline shows considerable reduction in computation time and improvement in accuracy.
    No preview · Article · Sep 2009
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    ABSTRACT: A new trajectory optimization algorithm for the Terminal Area Energy Management phase is presented based on interval analysis. Through a branch and bound strategy, interval analysis is able to yield guaranteed and rigorous bounds on the global minimum, i.e., on the best possible trajectory. It does so by using intervals instead of crisp numbers and interval arithmetic instead of crisp arithmetic. Even the numerical roundoff errors introduced by computers are considered and do not affect the rigor of the solution. The steering commands of the vehicle are optimized in order to regulate the kinematic and potential energies of the vehicle while aligning it with the runway. Normalized total energy is used as independent variable. Additionally, an interval algorithm for determining the initial gate of the Terminal Area Energy Management phase is presented, which is mathematically guaranteed to enclose the true solution and that facilitates the execution of sensitivity analyses.
    No preview · Conference Paper · Aug 2009
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    Conference Paper: Fuzzy Simplex Splines
    E De Weerdt · C C De Visser · Q P Chu · J A Mulder
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    ABSTRACT: Multivariate simplex splines are capable of approximating non-linear input-output mappings with great accuracy, as can neural networks. A new method is proposed which com-bines the best qualities of simplex splines and polynomial neural networks while circumventing their drawbacks. The new method generates mappings which are continuously differentiable, use a stable polynomial basis, and are not restricted to contiguous configurations of simplices. The method is shown to perform equally well or better than both polynomial neural networks and standard simplex splines for a highly non-linear identification problem. An iterative polynomial order optimization scheme has been applied to all three methods to increase their performance. Finally the setup of the new method creates the opportunity for simplex vertex optimization using standard (gradient) methods.
    Full-text · Conference Paper · Jul 2009
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    E. de Weerdt · Q.P. Chu · J.A. Mulder
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    ABSTRACT: The problem of output optimization within a specified input space of neural networks (NNs) with fixed weights is discussed in this paper. The problem is (highly) nonlinear when nonlinear activation functions are used. This global optimization problem is encountered in the reinforcement learning (RL) community. Interval analysis is applied to guarantee that all solutions are found to any degree of accuracy with guaranteed bounds. The major drawbacks of interval analysis, i.e., dependency effect and high-computational load, are both present for the problem of NN output optimization. Taylor models (TMs) are introduced to reduce these drawbacks. They have excellent convergence properties for small intervals. However, the dependency effect still remains and is even made worse when evaluating large input domains. As an alternative to TMs, a different form of polynomial inclusion functions, called the polynomial set (PS) method, is introduced. This new method has the property that the bounds on the network output are tighter or at least equal to those obtained through standard interval arithmetic (IA). Experiments show that the PS method outperforms the other methods for the NN output optimization problem.
    Full-text · Article · May 2009 · IEEE Transactions on Neural Networks
  • E. De Weerdt · Q.P. Chu · J.A. Mulder
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    ABSTRACT: Spacecraft formation flying is the new trend in space missions. Increased flexibility, lower cost, enhanced capabilities and fail-safe concepts are the main drivers of spacecraft formations. One important aspect of operating formations of spacecraft is trajectory planning. In this paper interval analysis is applied to find the set of global optimal trajectories such that the consumed fuel is minimal and/or equalized over the spacecraft in the formation. Moreover, interval analysis is applied to find the minimal time in which the rotation can be performed. Interval analysis implements interval arithmetic to guarantee that the global optimum is found and can easily incorporate inequality and equality constraints. As an initial validation test, the case of sequential constrained spacecraft formation rotations in free space is chosen. The proposed method is shown to solve the problem rigorously even in the case of acceleration constraints and fuel constraints. Moreover, application of the proposed method is not limited to the test case. Future work will consider unconstrained rotations and planet orbiting spacecraft formation. Copyright © 2009 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
    No preview · Article · Jan 2009
  • E. Van Kampen · E. De Weerdt · Q.P. Chu · J.A. Mulder
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    ABSTRACT: GPS carrier phase measurements can be used to determine the attitude of an aircraft fitted with multiple GPS antennae. For this to succeed, an optimization needs to be performed to find the integer ambiguities corresponding to the number of complete carrier wave cycles between the antennae. In this paper an interval based algorithm for solving these ambiguities is proposed. The theory of interval analysis is used to encapsulate the phase measurements with an interval noise band, such that the true phase is inside this band. This algorithm has been successful for static baselines with low noise, but the goal of this paper is to evaluate the performance of this algorithm for highly dynamic baselines with higher noise levels. For this purpose the algorithm is applied to GPS phase measurements recorded during a test flight with a Cessna Citation II jet. Results show that multiple epochs of data are required to fix the ambiguities. Once the ambiguities are resolved, the attitude of the aircraft can be determined with high accuracy.
    No preview · Conference Paper · Jan 2009
  • E. De Weerdt · E.J. Van Kampen · Q.P. Chu
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    ABSTRACT: Abstract of WO 2009095134 (A1) The invention relates to a transmitter-receiver system comprising at least three transmitters and at least a first receiver and a second receiver, wherein the receivers are connected to a computing device that is arranged to analyse signals that said receivers receive from said transmitters and to calculate length and attitude information of an imaginary baseline connecting said receivers depending on at least carrier phase information of said signals using interval analysis
    No preview · Patent · Jan 2009
  • E. Van Kampen · E. De Weerdt · Q. P. CHU · J.A. Mulder
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    ABSTRACT: A new algorithm is proposed for resolving the integer ambiguities encountered in GNSS carrier phase observations. The algorithm is based on Interval Analysis, which deals with computations on intervals of numbers instead of crisp numbers. The advantage of optimization algorithms using interval analysis is that they will detect all solutions within a search space, as compared to the classical algorithms that only find a local solution closest to some initial point. In this paper we show how interval analysis can be used to resolve integer ambiguities by taking a geometric approach, resulting in a nonlinear optimization problem. Using interval analysis it can be guaranteed that the found integers are the correct ones.
    No preview · Conference Paper · Dec 2008
  • E. De Weerdt · E. Van Kampen · D. Van Gemert · Q.P. Chu · J.A. Mulder
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    ABSTRACT: Through the years researchers have developed many different forms of spacecraft attitude controllers ranging from simple linear controllers to highly nonlinear ones. For Nonlinear Dynamic Inversion controllers, the tracking performance depends on the model on the plant dynamics. In this paper we explore the response of a controlled satellite with liquid sloshing and apply neural networks to create an adaptive NDI controller. Feedforward neural networks are used to model any unknown system dynamics. The fuel motion is modeled using a mechanical model often used in the field of liquid sloshing. The equations of motion of the combined satellite/fuel system are derived and a simulation is constructed. Results in the form of tracking performance for both the standard and the adaptive NDI controller will be shown using a model of SloshSat, an experimental liquid sloshing satellite of ESA and the Netherlands Agency for Aerospace Programs. The results will demonstrate that an adaptive controller is indeed needed and that the proposed NDI controller with neural network is capable of excellent reference tracking in case of fuel sloshing.
    No preview · Conference Paper · Jan 2008
  • E. De Weerdt · Q.P. Chu · J.A. Mulder
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    ABSTRACT: The paper is about Reconfigurable Flight Control (RFC) using Reinforcement Learning techniques. It presents the results of a feasibility study in which a continuous state and action space Advantage-learning controller is designed for the F-16 fighter. The controller consists of an Advantage-learning algorithm, a neural network for storage of the stateaction value function, and a neural network output optimization algorithm based on interval analysis. A neural network is chosen because of its capability of storing the (highly) nonlinear input-output mapping of the value function. During flight, the controller computes the optimal action based on the information stored in the state-action value function. This computation is, when using neural networks, a global non-linear optimization problem. We propose to use Interval Analysis to determine all solutions with probability one. The global optimization algorithm is developed for both radial basis function neural networks and feedforward neural networks. Results of the optimization algorithm in terms of network size versus computation time are presented as well as the results of the Advantage-learning controller applied to the F-16 fighter. The longitudinal aircraft model is taken to reduce the complexity of the control problem although the RFC system is also applicable to the complete aircraft. Results show that the optimization algorithm can have a 'high' computational load for larger neural networks but it is guaranteed to find all optimal actions. The results also prove that the combination of interval analysis, neural networks, and Advantage-learning produces a working RFC system.
    No preview · Article · Jan 2007
  • E. de Weerdt · Q.P. Chu · J.A. Mulder
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    ABSTRACT: In the field of reconfigurable flight control (RFC) one can identify two streams in types of failures. One stream deals with actuator failures and the other deals with the more general structural failures. Many researchers have successfully implemented RFC systems that can cope with actuator failures. However, the techniques used in these RFC systems are often not applicable to the second type of failures. Many RFC systems use some type of failure detection and isolation/identification (FDI) system which, in case of structural failures, is hard to construct properly since not all possible failures are known. For structural failures, the variety of possible failures is infinite and subsequent aircraft characteristics are very hard to predict. In this paper a RFC system based on non-linear dynamic inversion (NDI) in combination with on-line aerodynamic model identification is proposed. This type of RFC system circumvents the need for a FDI system by using neural networks to keep the internal model of the NDI controller up to date. The assumption is made that the structural failures are purely aerodynamically, e.g. no large mass variations are encountered. With this assumption the basic kinematic relations for an aircraft are valid even after a failure. This means that only identification of the total aerodynamic force and moment coefficients input-output mappings is necessary. On-line estimation of the states and total aerodynamic force and moment coefficients (using an extended Kalman filter (EKF)) provides the input-output data for the on-line identification of the aerodynamic model. The main advantage of using neural networks is that the internal structure of the aerodynamic model does not have to be defined. Keeping the input span of the neural networks general will ensure that all possible dependencies between input and output can be learned, resulting in an accurate aerodynamic model. A drawback of neural networks is that they suffer from the 'recency'-effect. Neural networks tend to 'forget' knowledge which has been learned in the past if that knowledge is no longer in the recent input-output (IO) pairs presented to the network. To prevent the recency effect the aerodynamic model is defined by a nine dimensional hyperbox structure. Each hyperbox contains neural networks which learn a small local IO-mapping of the aerodynamic model. Experimental results show that an accuracy of > 99% in the approximation of the aerodynamic model can be obtained throughout the flight envelope with off-line learning using the Levenberg-Marquardt algorithm. The aircraft under consideration is the F-16 and wind tunnel test data has been used for the off-line training. NDI control using the off-line learned aerodynamic model is shown to be stable and good performance is possible. The NDI does, however, react inefficiently to (small) modelling errors and sensor noise. A more robust version of the used NDI control system would improve performance. Further research has to be performed to investigate the on-line performance of the proposed RFC system. Copyright © 2005 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved.
    No preview · Conference Paper · Aug 2005
  • E. de Weerdt · Q. P. Chu · J. A. Mulder
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    ABSTRACT: In the field of reconfigurable flight control (RFC) one can distinguish two types of failures. One type deals with actuator failures and the other with the more general case of structural failures. Many researchers have successfull y implemented RFC systems that can cope with actuator failures ((3), (4), (7), etc.). However, the techniques use d in these RFC systems are often not applicable to the second type of failures. Many RFC systems use some form of failure detection and isolation/identification (FDI) appr oach based on a predetermined finite set of failure cases. In case of structural failures, however, the number of possibl e failure cases is infinite and the corresponding flight dynami c characteristics are very hard to predict. In this paper a RFC system based on non-linear dy- namic inversion (NDI) (5) (1) in combination with on-line aerodynamic model identification is proposed. This type of RFC system circumvents the need for predetermined failure cases by using neural networks to keep the internal model of the NDI controller up to date. The assumption is made that the aerodynamic effects of structural failures ar e much more important compared to the changes of mass, mass distribution and the rotational inertias. This means that only identification of the total aerodynamic force and moment coefficients input-output mappings is necessary. On-line estimation of the states and total aerodynamic force and moment coefficients (using an extended Kalman filter (EKF)) provides the input-output data for the on-line identification of the aerodynamic model. The main advan- tage of using neural networks is that the internal structure of the aerodynamic model does not have to be defined. Keeping the input span of the neural networks general will ensure that all possible dependencies between input and output can be learned, resulting in an accurate aerodynamic model. A drawback of neural networks is that they suffer from the 'recency'-effect. Neural networks tend to 'forget ' knowledge which has been learned in the past if that knowledge is no longer in the recent input-output (IO) pairs presented to the network (2). The larger the input/output dimensions and range, the larger this effect. To reduce the recency effect the aerodynamic model is defined by a nine dimensional hyperbox structure. Each hyperbox contains neural networks with smaller input/output dimensions and range, which learn a small local IO-mapping of the aerodynamic model. Experiments have been performed using the non-linear model of the F-16 aircraft taken from (6). Experimental re- sults show that an accuracy of > 99% in the approximation of the aerodynamic model can be obtained throughout the flight envelope with off-line learning using the Levenberg- Marquardt algorithm. Wind tunnel test data has been used for the off-line training and subsequent validation. NDI control using the off-line learned aerodynamic model is shown to be stable and good performance is possible. The NDI is, however, sensitive to modeling errors and sensor noise. A more robust version of the used NDI control system is needed to improve performance. Further research is needed to investigate the on-line performance of the proposed RFC system.
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