Modelling and system identification of active magnetic bearing systems

Active magnetic bearing (AMB) systems have recently attracted much attention in the rotating machinery industry due to their advantages over traditional bearings such as fluid film and rolling element bearings. The AMB control system must provide robust performance over a wide range of machine operating conditions and over the machine lifetime in order to make this technology commercially viable. An accurate plant model for AMB systems is essential for the aggressive design of control systems. In this paper, we propose two approaches to obtain accurate AMB plant models for the purpose of control design: physical modelling and system identification. The former derives a model based upon the underlying physical principles. The latter uses input – output data without explicitly resorting to physical principles. For each problem, a brief summary of the theoretical derivation and assumptions is given. Experimental results based on data collected from an AMB test facility at the United Technologies Research Center provide a vehicle for a comparison of the two approaches.     

On the conditions for the coincidence of two cubic Bézier curves

In a recent article, Wang et al. [2] derive a necessary and sufficient condition for the coincidence of two cubic Bézier curves with non-collinear control points. The condition reads that their control points must be either coincident or in reverse order. We point out that this uniqueness of the control points for polynomial cubics is a straightforward consequence of a previous and more general result of Barry and Patterson, namely the uniqueness of the control points for rational Bézier curves. Moreover, this uniqueness applies to properly parameterized polynomial curves of arbitrary degree.     

A network approach to the modelling of active magnetic bearings

The modelling of active magnetic bearings based on a network approach is considered. Unlike in the standard modelling approach, where a linearization of the current-force relation for the centred shaft position is used, network models permit to include the position dependence of the bearing force in the force model. This becomes necessary when model based controllers are used to stabilize a magnetically supported shaft in tracking applications.

The approach is based on the well known application of network models to magnetic circuits. Further simplifying assumptions are discussed which allow one to obtain a network with a limited number of lumped parameters describing the magnetic behaviour of a magnetic bearing. The modelling of a combined radial and axial bearing serves as an example for the application of the proposed approach. Furthermore, the fitting of the network based model to measured characteristic force curves is discussed. In this context, a method for including saturation effects in the model is sketched.     

Approximate Bézier curves by cubic LN curves

In order to derive the offset curves by using cubic Bézier curves with a linear field of normal vectors (the so-called LN Bézier curves) more efficiently, three methods for approximating degree n Bézier curves by cubic LN Bézier curves are considered, which includes two traditional methods and one new method based on Hausdorff distance. The approximation based on shifting control points is equivalent to solving a quadratic equation, and the approximation based on L₂ norm is equivalent to solving a quartic equation. In addition, the sufficient and necessary condition of optimal approximation based on Hausdorff distance is presented, accordingly the algorithm for approximating the degree n Bézier curves based on Hausdorff distance is derived. Numerical examples show that the error of approximation based on Hausdorff distance is much smaller than that of approximation based on shifting control points and L₂ norm, furthermore, the algorithm based on Hausdorff distance is much simple and convenient.     

Approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces

An attractive method for approximating rational triangular Bézier surfaces by polynomial triangular Bézier surfaces is introduced. The main result is that the arbitrary given order derived vectors of a polynomial triangular surface converge uniformly to those of the approximated rational triangular Bézier surface as the elevated degree tends to infinity. The polynomial triangular surface is constructed as follows. Firstly, we elevate the degree of the approximated rational triangular Bézier surface, then a polynomial triangular Bézier surface is produced, which has the same order and new control points of the degree-elevated rational surface. The approximation method has theoretical significance and application value: it solves two shortcomings-fussy expression and uninsured convergence of the approximation-of Hybrid algorithms for rational polynomial curves and surfaces approximation.     

A planar cubic Bézier spiral

A planar cubic Bézier curve segment that is a spiral, i.e., its curvature varies monotonically with arc-length, is discussed. Since this curve segment does not have cusps, loops, and inflection points (except for a single inflection point at its beginning), it is suitable for applications such as highway design, in which the clothoid has been traditionally used. Since it is polynomial, it can be conveniently incorporated in CAD systems that are based on B-splines, Bézier curves, or NURBS (nonuniform rational B-splines) and is thus suitable for general curve design applications in which fair curves are important.     

Improved bounds on the magnitude of the derivative of rational Bézier curves

In this paper we derive some new derivative bounds of rational Bézier curves according to some existing identities and inequalities. The comparison of the new bounds with some existing ones is also presented.     

Correspondence and (2,1)-Bézier Surfaces

Tom Sederberg's method of moving curves (surfaces) is a new and effective tool for implicitizing curves (surfaces). From our point of view, the curve (surface) can be defined by using moving curves (surfaces) which in algebraic geometry are called correspondences. It turns out that from this definition we can easily derive both parametric and implicit representations of the curve (surface). In this paper, we investigate the geometry of the bi-degree (2,1)-Bézier surface and study the relationship between singularities and correspondences. We also characterize all the possible singular curves in terms of the control points of the surface.     

Conditions for coincidence of two cubic Bézier curves

This paper presents a necessary and sufficient condition for judging whether two cubic Bézier curves are coincident: two cubic Bézier curves whose control points are not collinear are coincident if and only if their corresponding control points are coincident or one curve is the reversal of the other curve. However, this is not true for degree higher than 3. This paper provides a set of counterexamples of degree 4.     

Toric Surface Patches

We define a toric surface patch associated with a convex polygon, which has vertices with integer coordinates. This rational surface patch naturally generalizes classical Bézier surfaces. Several features of toric patches are considered: affine invariance, convex hull property, boundary curves, implicit degree and singular points. The method of subdivision into tensor product surfaces is introduced. Fundamentals of a multidimensional variant of this theory are also developed.     

Dual bases for Wang-Bézier basis and their applications

This paper presents the dual bases for Wang-Bézier curves with a position parameter L, which include Bézier curve, Wang-Ball curve and some intermediate curves. The Marsden identity and the transformation formulas from Bézier curve to Wang-Bézier curve are also given. These results are useful for the application of Wang-Bézier curve and their popularization in Computer Aided Geometric Design.     

Nonnegative surface fitting with Powell-Sabin splines

Algorithms are presented for fitting a nonnegative Powell-Sabin spline to a set of scattered data. Existing necessary and sufficient nonnegativity conditions for a quadratic polynomial on a triangle are used to compose a set of necessary and sufficient nonnegativity constraints for the PS-spline. The PS-spline is expressed as a linear combination of locally supported basis functions, of which the Bernstein-Bézier representation is considered to improve the efficiency. Numerical examples illustrate the profit of nonnegative surface fitting with Powell-Sabin splines.     

Smooth polynomial approximation of spiral arcs

Constructing fair curve segments using parametric polynomials is difficult due to the oscillatory nature of polynomials. Even NURBS curves can exhibit unsatisfactory curvature profiles. Curve segments with monotonic curvature profiles, for example spiral arcs, exist but are intrinsically non-polynomial in nature and thus difficult to integrate into existing CAD systems. A method of constructing an approximation to a generalised Cornu spiral (GCS) arc using non-rational quintic Bézier curves matching end points, end slopes and end curvatures is presented. By defining an objective function based on the relative error between the curvature profiles of the GCS and its Bézier approximation, a curve segment is constructed that has a monotonic curvature profile within a specified tolerance.     

Computing the minimum distance between two Bézier curves

A sweeping sphere clipping method is presented for computing the minimum distance between two Bézier curves. The sweeping sphere is constructed by rolling a sphere with its center point along a curve. The initial radius of the sweeping sphere can be set as the minimum distance between an end point and the other curve. The nearest point on a curve must be contained in the sweeping sphere along the other curve, and all of the parts outside the sweeping sphere can be eliminated. A simple sufficient condition when the nearest point is one of the two end points of a curve is provided, which turns the curve/curve case into a point/curve case and leads to higher efficiency. Examples are shown to illustrate efficiency and robustness of the new method.     

An extension of the Bézier model

In this paper, we first construct a new kind of basis functions by a recursive approach. Based on these basis functions, we define the Bézier-like curve and rectangular Bézier-like surface. Then we extend the new basis functions to the triangular domain, and define the Bernstein-Bézier-like surface over the triangular domain. The new curve and surfaces have most properties of the corresponding classical Bézier curve and surfaces. Moreover, the shape parameter can adjust the shape of the new curve and surfaces without changing the control points. Along with the increase of the shape parameter, the new curve and surfaces approach the control polygon or control net. In addition, the evaluation algorithm for the new curve and triangular surface are provided.     

Rate of convergence in simultaneous approximation for Szász-Mirakyan-Durrmeyer operators

In the present paper, we study the rate of convergence in simultaneous approximation for the Szász-Mirakyan-Durrmeyer operators by using the decomposition technique of functions of bounded variation.     

Necessary and sufficient conditions for rational quartic representation of conic sections

Conic section is one of the geometric elements most commonly used for shape expression and mechanical accessory cartography. A rational quadratic Bézier curve is just a conic section. It cannot represent an elliptic segment whose center angle is not less than π$\pi$. However, conics represented in rational quartic format when compared to rational quadratic format, enjoy better properties such as being able to represent conics up to 2π$2\pi$ (but not including 2π$2\pi$) without resorting to negative weights and possessing better parameterization. Therefore, it is actually worth studying the necessary and sufficient conditions for the rational quartic Bézier representation of conics. This paper attributes the rational quartic conic sections to two special kinds, that is, degree-reducible and improperly parameterized; on this basis, the necessary and sufficient conditions for the rational quartic Bézier representation of conics are derived. They are divided into two parts: Bézier control points and weights. These conditions can be used to judge whether a rational quartic Bézier curve is a conic section; or for a given conic section, present positions of the control points and values of the weights of the conic section in form of a rational quartic Bézier curve. Many examples are given to show the use of our results.     

Simultaneous approximation for Bézier variant of Szász-Mirakyan-Durrmeyer operators

We study the rate of convergence in simultaneous approximation for the Bézier variant of Szász-Mirakyan-Durrmeyer operators by using the decomposition technique of functions of bounded variation.     

PDE triangular Bézier surfaces: Harmonic, biharmonic and isotropic surfaces

We approach surface design by solving second-order and fourth-order Partial Differential Equations (PDEs). We present many methods for designing triangular Bézier PDE surfaces given different sets of prescribed control points and including the special cases of harmonic and biharmonic surfaces. Moreover, we introduce and study a second-order and a fourth-order symmetric operator to overcome the anisotropy drawback of the harmonic and biharmonic operators over triangular Bézier surfaces.