Interpolation and Extrapolation of Optimally-Fitted Kinematic Error Model for Five-Axis Machine Tools

Machine tool geometric errors are frequently corrected by populating compensation tables that contain position-dependent offsets to each commanded axis position. While each offset can be determined by directly measuring the individual geometric error at that location, it is often more efficient to compute the compensation using a volumetric error model derived from measurements across the entire workspace. However, interpolation and extrapolation of measurements, once explicit in direct measurement methods, become implicit and obfuscated in the curve fitting process of volumetric error methods. The drive to maximize model accuracy while minimizing measurement sets can lead to significant model errors in workspace regions at or beyond the range of the metrology equipment. In this paper, a novel method of constructing machine tool volumetric error models is presented in which the characteristics of the interpolation and extrapolation errors are constrained. Using a typical five-axis machine tool compensation methodology, a constraint bounding the tool tip modeled error slope is added to the error model identification process. By including this constraint over the entire space, the geometric errors over the interpolation space are still well-identified. Also, the model performance over the extrapolation space is consistent with the behavior of the geometric error model over the interpolation space. The methodology is applied to an industrial five-axis machine tool. In the experimental implementation, for measurements outside of the measured region, an unconstrained model increases the mean residual by 40% while the constrained model reduces the mean residual by 40%.

Constrained null controllability for distributed systems and applications to hyperbolic-like equations

We consider linear control systems of the form y′(t) = Ay(t) + Bu(t) on a Hilbert space Y . We suppose that the control operator B is bounded from the control space U to a larger extrapolation space containing Y . The aim is to study the null controllability in the case where the control u is constrained to lie in a bounded subset Γ ⊂ U. We obtain local constrained controllability properties. When (etA)t∈ℝ is a group of isometries, we establish necessary conditions and sufficient ones for global constrained controllability. Moreover, when the constraint set Γ contains the origin in its interior, the local constrained property turns out to be equivalent to a dual observability inequality of L1 type with respect to the time variable. In this setting, the study is focused on hyperbolic-like systems which can be reduced to a second order evolution equation. Furthermore, we treat the problem of determining a steering control for general constraint set Γ in nonsmooth convex analysis context. In the case where Γ contains the origin in its interior, a steering control can be obtained by minimizing a convenient smooth convex functional. Applications to the wave equation and Euler-Bernoulli beams are presented.