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Proceedings ArticleDOI

Methods for trajectory generation in a magnetic-levitation system under constraints

23 Jun 2010-pp 945-950
TL;DR: In this paper, the authors discuss two methods to generate trajectories for a magnetic-levitation (Maglev) system in the presence of constraints and compare each method's performance, based on the notion of differential flatness and spline parametrisation of every signal.
Abstract: In this work, we discuss two methods to generate trajectories for a magnetic-levitation (Maglev) system in the presence of constraints and compare each method's performance. The methods are based on the notion of differential flatness and spline parametrisation of every signal. The first method uses the nonlinear model of the plant, which turns out to belong to the class of flat systems. The second method uses a linearised version of the plant model around an operating point. In every case, a continuous-time description is used. Experimental results on a real Maglev system, reported here, show that, in most scenarios, the nonlinear and linearised model produce almost similar, indistinguishable trajectories.
Citations
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Journal ArticleDOI
TL;DR: In this paper, a predictive control strategy for UAVs in the presence of bounded disturbances is proposed to prove the feasibility of such a real-time optimization-based control design and demonstrate its tracking capabilities for the nonlinear dynamics with respect to a reference trajectory.

63 citations


Cites methods from "Methods for trajectory generation i..."

  • ...To overcome these issues, B-spline functions (De Doná et al., 2009; Suryawan et al., 2010) have been used....

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Journal ArticleDOI
TL;DR: This paper addresses a predictive control strategy for a particular class of multi-agent formations with a time-varying topology to guarantee tracking capabilities with respect to a reference trajectory which is pre-specified for an agent designed as the leader.
Abstract: This paper addresses a predictive control strategy for a particular class of multi-agent formations with a time-varying topology. The goal is to guarantee tracking capabilities with respect to a reference trajectory which is pre-specified for an agent designed as the leader. Then, the remaining agents, designed as followers, track the position and orientation of the leader. In real-time, a predictive control strategy enhanced with the potential field methodology is used in order to derive a feedback control action based only on local information within the group of agents. The main concern is that the interconnections between the agents are time-varying, affecting the neighborhood around each agent. The proposed method exhibits effective performance validated through some illustrative examples.

34 citations


Cites methods from "Methods for trajectory generation i..."

  • ...Using the flatness theory (Fliess et al., 1995), (Van Nieuwstadt and Murray, 1998), (Suryawan et al., 2010), the system is parameterized in terms of a finite set of variables zl(t) and a finite number of their derivatives: xl(t) = ξ(zl(t), żl(t), · · · , zl,(q)(t)), (19) ul(t) = η(zl(t), żl(t), ·…...

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Proceedings ArticleDOI
01 Jul 2017
TL;DR: This paper extends some previous work on trajectory generation for UAV using differential flatness in combination with B-splines parametrization to generate feasible flat trajectories for nonlinear UAV dynamics while ensuring continuous constraint validation.
Abstract: This paper extends some previous work on trajectory generation for UAV (Unmanned Aerial Vehicles) using differential flatness in combination with B-splines parametrization. The originality of this work resides in the geometrical interpretations of the B-splines properties and their use in generating feasible flat trajectories for nonlinear UAV dynamics while ensuring continuous constraint validation. Of particular interest (and difficulty) are constraints involving system inputs since often the mapping between the input and the flat output space is strongly nonlinear. The tools used and the results obtained are exemplified over a particular UAV dynamical system and can be generalized to any nonlinear system admitting a flat description.

26 citations


Cites background from "Methods for trajectory generation i..."

  • ...ro pursued in this paper, is the generation of a timeparametrized trajectory using differential flatness [12], [13] which takes explicitly into account the nonlinear system dynamics....

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Journal ArticleDOI
TL;DR: An output feedback control scheme is proposed for reference position trajectory tracking tasks on the flexible mechanical system and the differential flatness structural property of the system is employed for the synthesis of the controller and the signal estimation approach.
Abstract: This paper presents an application of a nonlinear magnetic levitation system to the problem of efficient active control of mass–spring–damper mechanical systems. An output feedback control scheme is proposed for reference position trajectory tracking tasks on the flexible mechanical system. The electromagnetically actuated system is shown to be a differentially flat nonlinear system. An extended state estimation approach is also proposed to obtain estimates of velocity, acceleration and disturbance signals. The differential flatness structural property of the system is then employed for the synthesis of the controller and the signal estimation approach presented in this work. Some experimental and simulation results are included to show the efficient performance of the control approach and the effective estimation of the unknown signals.

25 citations

Proceedings ArticleDOI
16 Jun 2015
TL;DR: This paper addresses some alternatives to classical trajectory generation for an autonomous vehicle which needs to pass through a priori given way-points by using differential flatness for trajectory generation and B-splines for the flat output parametrization.
Abstract: This paper addresses some alternatives to classical trajectory generation for an autonomous vehicle which needs to pass through a priori given way-points. Using differential flatness for trajectory generation and B-splines for the flat output parametrization, the current study concentrates on constraint relaxations and on obstacle avoidance conditions. The results are validated through simulations over standard UAV dynamics.

24 citations

References
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Book
01 Jan 1978
TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.
Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.

10,258 citations


"Methods for trajectory generation i..." refers background in this paper

  • ...The following are some properties of B-Splines [ 14 ], [15]....

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Book
25 Aug 1995
TL;DR: This chapter discusses the construction of B-spline Curves and Surfaces using Bezier Curves, as well as five Fundamental Geometric Algorithms, and their application to Curve Interpolation.
Abstract: One Curve and Surface Basics.- 1.1 Implicit and Parametric Forms.- 1.2 Power Basis Form of a Curve.- 1.3 Bezier Curves.- 1.4 Rational Bezier Curves.- 1.5 Tensor Product Surfaces.- Exercises.- Two B-Spline Basis Functions.- 2.1 Introduction.- 2.2 Definition and Properties of B-spline Basis Functions.- 2.3 Derivatives of B-spline Basis Functions.- 2.4 Further Properties of the Basis Functions.- 2.5 Computational Algorithms.- Exercises.- Three B-spline Curves and Surfaces.- 3.1 Introduction.- 3.2 The Definition and Properties of B-spline Curves.- 3.3 The Derivatives of a B-spline Curve.- 3.4 Definition and Properties of B-spline Surfaces.- 3.5 Derivatives of a B-spline Surface.- Exercises.- Four Rational B-spline Curves and Surfaces.- 4.1 Introduction.- 4.2 Definition and Properties of NURBS Curves.- 4.3 Derivatives of a NURBS Curve.- 4.4 Definition and Properties of NURBS Surfaces.- 4.5 Derivatives of a NURBS Surface.- Exercises.- Five Fundamental Geometric Algorithms.- 5.1 Introduction.- 5.2 Knot Insertion.- 5.3 Knot Refinement.- 5.4 Knot Removal.- 5.5 Degree Elevation.- 5.6 Degree Reduction.- Exercises.- Six Advanced Geometric Algorithms.- 6.1 Point Inversion and Projection for Curves and Surfaces.- 6.2 Surface Tangent Vector Inversion.- 6.3 Transformations and Projections of Curves and Surfaces.- 6.4 Reparameterization of NURBS Curves and Surfaces.- 6.5 Curve and Surface Reversal.- 6.6 Conversion Between B-spline and Piecewise Power Basis Forms.- Exercises.- Seven Conics and Circles.- 7.1 Introduction.- 7.2 Various Forms for Representing Conics.- 7.3 The Quadratic Rational Bezier Arc.- 7.4 Infinite Control Points.- 7.5 Construction of Circles.- 7.6 Construction of Conies.- 7.7 Conic Type Classification and Form Conversion.- 7.8 Higher Order Circles.- Exercises.- Eight Construction of Common Surfaces.- 8.1 Introduction.- 8.2 Bilinear Surfaces.- 8.3 The General Cylinder.- 8.4 The Ruled Surface.- 8.5 The Surface of Revolution.- 8.6 Nonuniform Scaling of Surfaces.- 8.7 A Three-sided Spherical Surface.- Nine Curve and Surface Fitting.- 9.1 Introduction.- 9.2 Global Interpolation.- 9.2.1 Global Curve Interpolation to Point Data.- 9.2.2 Global Curve Interpolation with End Derivatives Specified.- 9.2.3 Cubic Spline Curve Interpolation.- 9.2.4 Global Curve Interpolation with First Derivatives Specified.- 9.2.5 Global Surface Interpolation.- 9.3 Local Interpolation.- 9.3.1 Local Curve Interpolation Preliminaries.- 9.3.2 Local Parabolic Curve Interpolation.- 9.3.3 Local Rational Quadratic Curve Interpolation.- 9.3.4 Local Cubic Curve Interpolation.- 9.3.5 Local Bicubic Surface Interpolation.- 9.4 Global Approximation.- 9.4.1 Least Squares Curve Approximation.- 9.4.2 Weighted and Constrained Least Squares Curve Fitting.- 9.4.3 Least Squares Surface Approximation.- 9.4.4 Approximation to Within a Specified Accuracy.- 9.5 Local Approximation.- 9.5.1 Local Rational Quadratic Curve Approximation.- 9.5.2 Local Nonrational Cubic Curve Approximation.- Exercises.- Ten Advanced Surface Construction Techniques.- 10.1 Introduction.- 10.2 Swung Surfaces.- 10.3 Skinned Surfaces.- 10.4 Swept Surfaces.- 10.5 Interpolation of a Bidirectional Curve Network.- 10.6 Coons Surfaces.- Eleven Shape Modification Tools.- 11.1 Introduction.- 11.2 Control Point Repositioning.- 11.3 Weight Modification.- 11.3.1 Modification of One Curve Weight.- 11.3.2 Modification of Two Neighboring Curve Weights.- 11.3.3 Modification of One Surface Weight.- 11.4 Shape Operators.- 11.4.1 Warping.- 11.4.2 Flattening.- 11.4.3 Bending.- 11.5 Constraint-based Curve and Surface Shaping.- 11.5.1 Constraint-based Curve Modification.- 11.5.2 Constraint-based Surface Modification.- Twelve Standards and Data Exchange.- 12.1 Introduction.- 12.2 Knot Vectors.- 12.3 Nurbs Within the Standards.- 12.3.1 IGES.- 12.3.2 STEP.- 12.3.3 PHIGS.- 12.4 Data Exchange to and from a NURBS System.- Thirteen B-spline Programming Concepts.- 13.1 Introduction.- 13.2 Data Types and Portability.- 13.3 Data Structures.- 13.4 Memory Allocation.- 13.5 Error Control.- 13.6 Utility Routines.- 13.7 Arithmetic Routines.- 13.8 Example Programs.- 13.9 Additional Structures.- 13.10 System Structure.- References.

4,552 citations


"Methods for trajectory generation i..." refers background or methods in this paper

  • ...The following are some properties of B-Splines [14], [ 15 ]....

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  • ...Given specified way-points in time, and using standard spline interpolation techniques (see, e.g., [ 15 ]), we can find a suitable reference trajectory y ref for the flat output of a...

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Journal ArticleDOI
TL;DR: In this paper, the authors introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous feedback, which subsumes the physical properties of a linearizing output and provides another nonlinear extension of Kalman's controllability.
Abstract: We introduce flat systems, which are equivalent to linear ones via a special type of feedback called endogenous. Their physical properties are subsumed by a linearizing output and they might be regarded as providing another nonlinear extension of Kalman's controllability. The distance to flatness is measured by a non-negative integer, the defect. We utilize differential algebra where flatness- and defect are best defined without distinguishing between input, state, output and other variables. Many realistic classes of examples are flat. We treat two popular ones: the crane and the car with n trailers, the motion planning of which is obtained via elementary properties of plane curves. The three non-flat examples, the simple, double and variable length pendulums, are borrowed from non-linear physics. A high frequency control strategy is proposed such that the averaged systems become flat.

3,025 citations


"Methods for trajectory generation i..." refers background in this paper

  • ...In the case of linear systems, a system is flat if and only if it is controllable [1], [13]....

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  • ...REVIEW OF FLATNESS Differential flatness [13] is a property of some controlled (linear or nonlinear) dynamical systems, often encountered in applications, which allows for a complete parameterisation of all system variables (inputs and states) in terms of a finite number of variables, called flat outputs, and a finite number of their time derivatives....

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Book
26 May 2004
TL;DR: This chapter discusses linear time-Invariant SISO Systems, MIMO Systems, and Flatness and Optimal Trajectories.
Abstract: Illustrating the power, simplicity, and generality of the concept of flatness, this reference explains how to identify, utilize, and apply flatness in system planning and design. The book includes a large assortment of exercises and models that range from elementary to complex classes of systems. Leading students and professionals through a vast array of designs, simulations, and analytical studies on the traditional uses of flatness, Differentially Flat Systems contains an extensive amount of examples that showcase the value of flatness in system design, demonstrate how flatness can be assessed in the context of perturbed systems and apply static and dynamic feedback controller design techniques.

717 citations


"Methods for trajectory generation i..." refers background in this paper

  • ...For a class of dynamical systems, one very useful tool is the notion of flatness, where every system’s signal can be characterised as a function of a set of variables (called flat outputs) and their derivatives [ 1 ]....

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  • ...In the case of linear systems, a system is flat if and only if it is controllable [ 1 ], [13]....

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Journal ArticleDOI
TL;DR: A nonlinear model for magnetic levitation systems which is validated with experimental measurements and a real-time implementation of this model based on differential geometry is developed.
Abstract: In this paper, the authors propose a nonlinear model for magnetic levitation systems which is validated with experimental measurements. Using this model, a nonlinear control law based on differential geometry is firstly synthesized. Then, its real-time implementation is developed. In order to highlight the performance of the proposed control law, experimental results are given.

285 citations