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

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

Solution of a quadratic quaternion equation with mixed coefficients

A comprehensive analysis of the morphology of the solution space for a special type of quadratic quaternion equation is presented. This equation, which arises in a surface construction problem, incorporates linear terms in a quaternion variable and its conjugate with right and left quaternion coefficients, while the quadratic term has a quaternion coefficient placed between the variable and its conjugate. It is proved that, for generic coefficients, the equation has two, one, or no solutions, but in certain special instances the solution set may comprise a circle or a 3-sphere in the quaternion space $\mathbb{H}$. The analysis yields solutions for each case, and intuitive interpretations of them in terms of the four-dimensional geometry of the quaternion space $\mathbb{H}$.Comment: 19 pages, to appear in the Journal of Symbolic Computatio

Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves

AbstractA “double” Pythagorean-hodograph (DPH) curve r(t) is characterized by the property that |r′(t)| and |r′(t)×r″(t)| are both polynomials in the curve parameter t. Such curves possess rational Frenet frames and curvature/torsion functions, and encompass all helical polynomial curves as special cases. As noted by Beltran and Monterde, the Hopf map representation of spatial PH curves appears better suited to the analysis of DPH curves than the quaternion form. A categorization of all DPH curve types up to degree 7 is developed using the Hopf map form, together with algorithms for their construction, and a selection of computed examples of (both helical and non-helical) DPH curves is included, to highlight their attractive features. For helical curves, a separate constructive approach proposed by Monterde, based upon the inverse stereographic projection of rational line/circle descriptions in the complex plane, is used to classify all types up to degree 7. Criteria to distinguish between the helical and non-helical DPH curves, in the context of the general construction procedures, are also discussed

Minkowski products of unit quaternion sets

The Minkowski product of unit quaternion sets is introduced and analyzed, motivated by the desire to characterize the overall variation of compounded spatial rotations that result from individual rotations subject to known uncertainties in their rotation axes and angles. For a special type of unit quaternion set, the spherical caps of the 3-sphere $S^3$ in $\mathbb{R}^4$, closure under the Minkowski product is achieved. Products of sets characterized by fixing either the rotation axis or rotation angle, and allowing the other to vary over a given domain, are also analyzed. Two methods for visualizing unit quaternion sets and their Minkowski products in $\mathbb{R}^3$ are also discussed, based on stereographic projection and the Lie algebra formulation. Finally, some general principles for identifying Minkowski product boundary points are discussed in the case of full-dimension set operands.Comment: 29 pages, 1 figur

Evaluating the boundary and covering degree of planar Minkowski sums and other geometrical convolutions

AbstractAlgorithms are developed, based on topological principles, to evaluate the boundary and “internal structure” of the Minkowski sum of two planar curves. A graph isotopic to the envelope curve is constructed by computing its characteristic points. The edges of this graph are in one-to-one correspondence with a set of monotone envelope segments. A simple formula allows a degree to be assigned to each face defined by the graph, indicating the number of times its points are covered by the Minkowski sum. The boundary can then be identified with the set of edges that separate faces of zero and non-zero degree, and the boundary segments corresponding to these edges can be approximated to any desired geometrical accuracy. For applications that require only the Minkowski sum boundary, the algorithm minimizes geometrical computations on the “internal” envelope edges, that do not contribute to the final boundary. In other applications, this internal structure is of interest, and the algorithm provides comprehensive information on the covering degree for different regions within the Minkowski sum. Extensions of the algorithm to the computation of Minkowski sums in R3, and other forms of geometrical convolution, are briefly discussed

Construction of C 2 Pythagorean-hodograph interpolating splines by the homotopy method

The complex representation of polynomial Pythagorean-hodograph (PH) curves allows the problem of constructing a C 2 PH quintic “spline” that interpolates a given sequence of points p 0 , p 1 ,..., p N and end-derivatives d 0 and d N to be reduced to solving a “tridiagonal” system of N quadratic equations in N complex unknowns. The system can also be easily modified to incorporate PH-spline end conditions that bypass the need to specify end-derivatives. Homotopy methods have been employed to compute all solutions of this system, and hence to construct a total of 2 N +1 distinct interpolants for each of several different data sets. We observe empirically that all but one of these interpolants exhibits undesirable “looping” behavior (which may be quantified in terms of the elastic bending energy , i.e., the integral of the square of the curvature with respect to arc length). The remaining “good” interpolant, however, is invariably a fairer curve-having a smaller energy and a more even curvature distribution over its extent-than the corresponding “ordinary” C 2 cubic spline. Moreover, the PH spline has the advantage that its offsets are rational curves and its arc length is a polynomial function of the curve parameter.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/41719/1/10444_2005_Article_BF02124754.pd