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

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.

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.

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.

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.

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.

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.

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 $f=V_{\min }/V_{0}$ 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.

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.

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 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.