The volumetric accuracy of machine tools is represented by a map of position and orientation error vectors of the tool over the volume concerned. Numerical compensation for volumetric error is possible in many latest commercial CNCs for machine tools. This paper reviews indirect measurement schemes for machine tool kinematics, in which the tool center position is measured as the superposition of error motions of linear or rotary axes. Each error motion can be separately identified by best-fitting a set of measured tool center positions to the kinematic model of machine tools. Indirect measurement schemes for the kinematics of three orthogonal linear axes, as well as the fiveaxis kinematics with two rotary axes, will be reviewed.

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Abstract:Kinematic errors due to geometric inaccuracies in five-axis machining centers cause deviations in tool positions and orientation from commanded values, which consequently affect geometric accuracy of the machined surface. As is well known in the machine tool industry, machining of a cone frustum as specified in NAS979 standard is a widely accepted final perforance test for five-axis machining centers. A critical issue with this machining test is, however, that the influence of the machine's error sources on the geometric accuracy of the machined cone frustum is not fully understood by machine tool builders and thus it is difficult to find causes of machining errors. To address this issue, this paper presents a simulator of machining geometric errors in fiveaxis machining by considering the effect of kinematic errors on the three-dimensional interference of the tool and the workpiece. Kinematic errors of a five-axis machining center with tilting rotary table type are first identified by a DBB method. Using an error model of the machining center with identified kinematic errors and considering location and geometry of the workpiece, machining geometric error with respect to the nominal geometry of the workpiece is predicted and evaluated.In an aim to improve geometric accuracy of the machined surface, an error compensation for tool position and orientation is also presented. Finally, as an example, the machining of a cone frustum by using a straight end mill, as described in the standard NAS979, is considered in case studies to experimentally verify the prediction and the compensation of machining geometric errors in fiveaxis machining.

The R-test is an instrument to measure three-dimensional displacement of a precision sphere attached to a spindle relative to a work table by using three displacement sensors. Its application to error calibration for five-axis machine tools has been studied in both academia and industry. For the simplicity in calculating the sphere center displacement, all conventional R-test devices use contact-type displacement sensors with a flat-ended probe. Conventional contact-type R-test may be potentially subject to the influence of the friction or the dynamics of supporting spring in displacement sensors particularly in dynamic measurement. This paper proposes a non-contact R-test with laser displacement sensors. First, a new algorithm is proposed to calculate the three-dimensional displacement of sphere center by using non-contact displacement sensors. The compensation of measurement error of a laser displacement sensor due to the curvature of target sphere is incorporated. Then, the measurement uncertainty of four laser displacement sensors with different measuring principles is experimentally investigated in measuring the geometry of a sphere in order to select the laser displacement sensor most suitable for the application to a non-contact R-test. A prototype non-contact R-test device is developed for the verification of the proposed algorithm for non-contact R-test. Experimental case studies of error calibration of 1) static *Manuscript Click here to view linked References 2 and 2) dynamic error motions of rotary axes in a five-axis machine tool with the developed non-contact R-test prototype are presented. Its measurement performance is compared to the conventional contact-type R-test device.

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This paper presents the rank minimization approach to solve general bilinear matrix inequality (BMI) problems.Due to the NP-hardness of BMI problems, no proposed algorithm that globally solves general BMI problems is a polynomial-time algorithm. We present a local search algorithm based on the semidefinite programming (SDP) relaxation approach to indefinite quadratic programming, which is analogous to the well-known relaxation method for a certain class of combinatorial problems. Instead of applying the branch and bound (BB) method for global search, a linearization-based local search algorithm is employed to reduce the relaxation gap. Furthermore, a random search approach is introduced along with the deterministic approach. Four numerical. experiments are presented to show the search performance of the proposed approach.