Rida T. Farouki’s research while affiliated with University of California, Davis and other places

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Publications (87)


Real-time needle guidance for venipuncture based on optical coherence tomography
  • Article

November 2021

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6 Reads

Computer Methods in Biomechanics and Biomedical Engineering Imaging & Visualization

Rida T. Farouki

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Rachel Ward Rohlen

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David B. Smith

An algorithm for real–time venipuncture needle guidance is described, using an optical coherence tomography (OCT) probe that emits light pulses at fixed angular intervals along a cone, giving accurate distance measurements to points on the blood vessel. Using this data, a method is developed to visually display the blood vessel for needle guidance. A least–squares fit to a general quadric surface, specified by a symmetric matrix, is performed. For a cylindrical blood vessel, this provides an estimate for its orientation, from which its location and radius can be determined. The algorithm is compatible, in efficiency and robustness, with real–time implementation.


Construction of rational curves with rational arc lengths by direct integration

September 2019

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9 Reads

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7 Citations

Computer Aided Geometric Design

A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r′(ξ)=(u2(ξ)−v2(ξ),2u(ξ)v(ξ))/w2(ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r′(ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates.


A general framework for solving inverse dynamics problems in multi-axis motion control

May 2019

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24 Reads

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5 Citations

ISA Transactions

An inverse dynamics compensation (IDC) scheme for the execution of curvilinear paths by multi-axis motion controllers is proposed. For a path specified by a parametric curve r(ξ), the IDC scheme computes a real-time path correction Δr(ξ) that (theoretically) eliminates path deviations incurred by the inertia and damping of the machine axes. To exploit the linear time-invariant nature of the dynamic equations, the correction term is computed as a function of elapsed time t, and the corresponding curve parameter values ξ are only determined as the final step of the IDC scheme, through a real-time interpolator algorithm. It is shown that, in general, the correction term for P, PI, and PID controllers consists of derivative, natural, and integral terms (the integrand of the latter involving only the path r(ξ), and not its derivatives). The use of lead segments to minimize transient effects associated with the initial conditions is also discussed, and the performance of the method is illustrated by simulation results. The IDC scheme is expressed in terms of a linear differential operator formalism to provide a clear, general, and systematic development, amenable to further adaptations and extensions.


Existence of Pythagorean-hodograph quintic interpolants to spatial G1 Hermite data with prescribed arc lengths

February 2019

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11 Reads

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23 Citations

Journal of Symbolic Computation

A unique feature of polynomial Pythagorean–hodograph (PH) curves is the ability to interpolate G ¹ Hermite data (end points and tangents) with a specified total arc length. Since their construction involves the solution of a set of non–linear equations with coefficients dependent on the specified data, the existence of such interpolants in all instances is non–obvious. A comprehensive analysis of the existence of solutions in the case of spatial PH quintics with end derivatives of equal magnitude is presented, establishing that a two–parameter family of interpolants exists for any prescribed end points, end tangents, and total arc length. The two free parameters may be exploited to optimize a suitable shape measure of the interpolants, such as the elastic bending energy.


Reduced difference polynomials and self-intersection computations

May 2018

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26 Reads

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4 Citations

Applied Mathematics and Computation

A reduced difference polynomial f(u,v)=(p(u)−p(v))/(u−v) may be associated with any given univariate polynomial p(t), t ∈ [0, 1] such that the locus f(u,v)=0 identifies the pairs of distinct values u and v satisfying p(u)=p(v). The Bernstein coefficients of f(u, v) on [0, 1]² can be determined from those of p(t) through a simple algorithm, and can be restricted to any subdomain of [0, 1]² by standard subdivision methods. By constructing the reduced difference polynomials f(u, v) and g(u, v) associated with the coordinate components of a polynomial curve r(t)=(x(t),y(t)), a quadtree decomposition of [0, 1]² guided by the variation-diminishing property yields a numerically stable scheme for isolating real solutions of the system f(u,v)=g(u,v)=0, which identify self-intersections of the curve r(t). Through the Kantorovich theorem for guaranteed convergence of Newton–Raphson iterations to a unique solution, the self-intersections can be efficiently computed to machine precision. By generalizing the reduced difference polynomial to encompass products of univariate polynomials, the method can be readily extended to compute the self-intersections of rational curves, and of the rational offsets to Pythagorean–hodograph curves.


Efficient high-speed cornering motions based on continuously-variable feedrates. II. Implementation and performance analysis
  • Article
  • Publisher preview available

January 2017

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81 Reads

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17 Citations

The International Journal of Advanced Manufacturing Technology

A novel high-speed cornering strategy for piecewise-linear motions is proposed, based on the G 2 continuous Pythagorean–hodograph (PH) rounding segments and continuously-variable feedrates described in Part I of this two-part paper. This strategy employs an acceleration-limited approach to feedrate scheduling, including smooth deceleration and acceleration profiles along linear segments entering and leaving the rounded segments. A G-code part program parsing software package has been developed, that automatically identifies toolpath corners and inserts appropriately-sized PH quintic corner rounding segments, with associated feedrate suppression ratios. The method has been tested on a three-axis CNC mill with an open-architecture controller incorporating real-time interpolators that realize smoothly varying feedrates along the PH corner curves. Tests on a representative selection of toolpaths yield substantial savings (up to 40 %) in execution times, compared to the traditional “full stop” strategy for unmodified sharp corners.

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Efficient high-speed cornering motions based on continuously-variable feedrates. I. Real-time interpolator algorithms

December 2016

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55 Reads

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26 Citations

The International Journal of Advanced Manufacturing Technology

The problem of high-speed traversal of sharp toolpath corners, within a prescribed geometrical tolerance 𝜖, is addressed. Each sharp corner is replaced by a quintic Pythagorean–hodograph (PH) curve that meets the incoming/outgoing path segments with G 2 continuity, and deviates from the exact corner by no more than the prescribed tolerance 𝜖. The deviation and extremum curvature admit closed-form expressions in terms of the corner angle 𝜃 and side-length L, allowing precise control over these quantities. The PH curves also permit a smooth modulation of feedrate around the corner by analytic reduction of the interpolation integral. To demonstrate this, real-time interpolator algorithms are developed for three model feedrate functions. Specifying the feedrate as a quintic polynomial in the curve parameter accommodates precise acceleration continuity, but has no obvious geometrical interpretation. An inverse linear dependence on curvature offers a purely geometrical specification, but incurs slight initial and final tangential acceleration discontinuities. As an alternative, a hybrid form that incorporates the main advantages of these two approaches is proposed. In each case, the ratio f=Vmin/V0f=V_{\min }/V_{0} of the minimum and nominal feedrates is a free parameter, and the improved cornering time is analyzed. This paper develops the basic cornering algorithms—their implementation and performance analysis are described in detail in a companion paper.


Rational rotation-minimizing frames—Recent advances and open problems

January 2016

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77 Reads

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28 Citations

Applied Mathematics and Computation

Recent developments in the basic theory, algorithms, and applications for curves with rational rotation-minimizing frames (RRMF curves) are reviewed, and placed in the context of the current state-of-the-art by highlighting the many significant open problems that remain. The simplest non-trivial RRMF curves are the quintics, characterized by a scalar condition on the angular velocity of the Euler-Rodrigues frame (ERF). Two different classes of RRMF quintics can be identified. The first class of curves may be characterized by quadratic constraints on the quaternion coefficients of the generating polynomials; by the root structure of those polynomials; or by a certain polynomial divisibility condition. The second class has a strictly algebraic characterization, less well-suited to geometrical construction algorithms. The degree 7 RRMF curves offer more shape freedoms than the quintics, but only one of the four possible classes of these curves has been satisfactorily described. Generalizations of the adapted rotation-minimizing frames, for which the angular velocity has no component along the tangent, to directed and osculating frames (with analogous properties relative to the polar and binormal vectors) are also discussed. Finally, a selection of applications for rotation-minimizing frames are briefly reviewed - including construction of swept surfaces, rigid-body motion planning, 5-axis CNC machining, and camera orientation control.


Algorithm 952

October 2015

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29 Reads

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9 Citations

ACM Transactions on Mathematical Software

The implementation of a library of basic functions for the construction and analysis of planar quintic Pythagorean-hodograph (PH) curves is presented using the complex representation. The special algebraic structure of PH curves permits exact algorithms for the computation of key properties, such as arc length, elastic bending energy, and offset (parallel) curves. Single planar PH quintic segments are constructed as interpolants to first-order Hermite data (end points and derivatives), and this construction is then extended to open or closed C² PH quintic spline curves interpolating a sequence of points in the plane. The nonlinear nature of PH curves incurs a multiplicity of formal solutions to such interpolation problems, and a key aspect of the algorithms is to efficiently single out the unique “good” interpolant among them.


A real-time surface interpolator methodology for precision CNC machining of swept surfaces

July 2015

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30 Reads

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24 Citations

The International Journal of Advanced Manufacturing Technology

A real-time surface interpolator is developed to machine a family of swept surfaces directly from their high-level procedural definitions. All the computations required for machining are performed in real time based on the exact surface geometry, including tool path planning, tool path interpolation, tool offsetting, and tool path step-over to achieve a prescribed scallop height. A G-code command (G05) is introduced to concisely communicate the precise surface geometry and all necessary process parameters to the controller. The swept surface interpolator offers profound accuracy and efficiency advantages over the traditional approach of generating voluminous piecewise–linear/circular tool path approximations as a preprocessing step. For example, in one instance, a 36,000-line piecewise-linear (G01) approximate part program file is replaced by a 3-line exact swept surface (G05) part program file. The methodology is verified by machining a variety of swept surface forms in aluminum and wax, using a 3-axis milling machine with the surface interpolator incorporated into an open-architecture CNC controller.


Citations (70)


... In the present paper we extend the previous ideas to a residuum based approach, c.f. [20]. Basically, rational PH curves are obtained by integrating expressions of type λ(t)F(t) where λ(t) is a rational function and F(t) = A(t)iA * (t) ∈ H[t] is the typical vector valued polynomial that is also used for the construction of polynomial PH curves, [1]. ...

Reference:

Optimal interpolation with spatial rational Pythagorean hodograph curves
Construction of rational curves with rational arc lengths by direct integration
  • Citing Article
  • September 2019

Computer Aided Geometric Design

... However, this optimization may not suffice for applications requiring high-speed execution of complex curvilinear paths, in which the smoothing influence of inertia improvement in tracking accuracy is possible, even when the P gain is successively reduced toward the open-loop limit. A further extension of this approach, to accommodate PI and PID controllers, was subsequently developed in [27]: in these cases, the correction term includes a single irreducible integral, which can nevertheless be evaluated in real time by means of an efficient adaptive quadrature scheme. ...

A general framework for solving inverse dynamics problems in multi-axis motion control
  • Citing Article
  • May 2019

ISA Transactions

... However, computing the initial septic PH curve, which represent the initial posture of the soft continuum manipulator is not obvious but a quintic PH curve is. For that, the shapes are first modeled with PH quintic curves using the method described [12] [31]. Then With degree elevation process, the septic PH curves are derived from the quintic; ones without any loss of pertinence [28]. ...

Existence of Pythagorean-hodograph quintic interpolants to spatial G1 Hermite data with prescribed arc lengths
  • Citing Article
  • February 2019

Journal of Symbolic Computation

... Also, it implies that b(t) has zero reach (shortest distance to the cut locus) at S, so no smooth (trimmed) tube or offset can be constructed using b(t) as the generator [10]. To compute the potential self-intersection of an integral cubic, we could resort to numerical algorithms for Bézier curves of an arbitrary degree, such as those by Lasser [11] or Farouki [12], based on subdivision. However, for the specific cubic case, simple closed-form formulas are well known. ...

Reduced difference polynomials and self-intersection computations
  • Citing Article
  • May 2018

Applied Mathematics and Computation

... With the real-time implementation of this system by defining the tool path with the help of PH curves, this method can accurately determine the tool wear status. Thus, it can provide both fault detection and real-time adjustment and control by adding it to embedded systems [17,18] (for more applications and recent studies on PH curves, see [19][20][21][22][23][24][25][26][27][28][29][30]). ...

Efficient high-speed cornering motions based on continuously-variable feedrates. II. Implementation and performance analysis

The International Journal of Advanced Manufacturing Technology

... Corner smoothing algorithms improve the geometric continuity of intersections by replacing them with splines [3][4][5], or by using filters to generate smooth velocity, acceleration, and jerk curves [6,7]. ...

Efficient high-speed cornering motions based on continuously-variable feedrates. I. Real-time interpolator algorithms

The International Journal of Advanced Manufacturing Technology

... Furthermore, Petrone (1999) applied Bernstein polynomials to density estimation under a Bayesian nonparametric framework. Regarding convergence of Bernstein approximation form, Farouki (1999) investigated its convergent inversion approximations. All such published pieces of intensive work on Bernstein polynomials regarding their error and convergence offer strong indication about the important need for use of Bernstein polynomials in CAGD. ...

Convergent inversion approximations for polynomials in Bernstein form
  • Citing Article
  • February 2000

Computer Aided Geometric Design

... The term 'rotation minimising frame' is used in Isogonal Moulding Surfaces (Mesnil et al. 2015), where it is key to holding the movement of the generatrix within this 'rotation minimising frame'. A mathematical approach to defining the 'rotation minimising frame' has received special attention from a computational and mathematical point of view (Farouki 2016;Wang 2008) in the field of Computer Aided Design (CAD). ...

Rational rotation-minimizing frames—Recent advances and open problems
  • Citing Article
  • January 2016

Applied Mathematics and Computation

... Producing cutter sweeping envelopes in five-axis milling using sphere families with two parameters is discussed to increase accuracy in machining operations of canal surfaces [197]. In order to increase accuracy in swept surface machining operations, a real-time surface interpolator algorithm is proposed [198]. The sweep plane method for global collision detection involving workpiece geometry modification in five-axis NC machining operations is examined to detect and prevent the collision between workpiece and cutting tool in machining operations [199]. ...

A real-time surface interpolator methodology for precision CNC machining of swept surfaces
  • Citing Article
  • July 2015

The International Journal of Advanced Manufacturing Technology

... One approach is to increase the real-time arc length estimation accuracy 18 or provide a fitted arc length and parameter relation. 21 In another approach, an arc length parametrized tool path for CNC machining is also proposed, such as that put forward by Farouki,22,23 who presents Pythagorean-hodograph (PH) spline paths whose arc length is just a piecewise polynomial. Chen and Khan 24 propose an approach to generate arc length parameterized B-spline tool paths with high accuracy to address the disparity between the arc displacement and the spline parameter. ...

Short communication: Arc lengths of rational Pythagorean–hodograph curves
  • Citing Article
  • March 2015

Computer Aided Geometric Design