Effects of number of digits in large-scale multilateration

Highlights

• Sequential large scale multilateration using a TI was studied.

• A simulation protocol is proposed in order to effects of number of digits in large-scale multilateration.

• The use of multi-precision libraries is recommended to master the uncertainty propagations.

Abstract

Since many years ago, multilateration has been used in precision engineering notably in machine tool and coordinate measuring machine calibration. This technique needs, first, the use of laser trackers or tracking interferometers, and second, the use of nonlinear optimization algorithms to determine point coordinates. Research works have shown the influence of the experimental configuration on measure precision in multilateration. However, the impact of floating-point precision in computations on large-scale multilateration precision has not been addressed. In this work, the effects of numerical errors (rounding and cancellation effects) due to floating-point precision (number of digits) were studied. In order to evaluate these effects in large-scale multilateration, a multilateration measurement system was simulated. This protocol is illustrated with a case study where large distances (≤20 m) between pairs of target points were simulated. The results show that the use of multi-precision libraries is recommended to control the propagation of uncertainties during the multilateration computation.

Introduction

In the last 40 years, large-scale measurement or dimensional metrology has been an active research field in the world. Several keynote papers have been published highlighting the scientific advances in this research field [[1], [2], [3]]. These works have been supported by the increase of large mechanical systems (aircraft wings, wind turbine, rotor blades, mechanical structures such as nuclear reactors …). In order to produce and control the constitutive parts of these mechanical systems, large machine tools and coordinate measuring machines are required. In turn, these mechanisms also need to be calibrated and compensated [4]. In consequence, a new measuring technique appeared: multilateration. In the context of digital enterprise, measurement systems using this technique are being used increasingly in precision engineering. Maropoulos et al. [5] defined this research field as a priority in the context of measurement-assisted assembly.

The multilateration technique is based on the computation of the coordinates of a given point using either four measurement devices simultaneously or a single device sequentially. This computation is feasible if the position coordinates of the measurement devices and the distances to the target point are known. The coordinates of the target point are calculated as the intersection of four spheres. Each sphere is centered at the measurement devices position and its radius is defined by the measured distance. The intersection of the first two spheres generates a circle. Two points can be derived by intersecting this circle and the third sphere. The last sphere allows to obtain the coordinates of the target point.

Two types of instruments are chiefly used in the multilateration technique. The first type is the laser tracker, which measures simultaneously a distance by interferometry or absolute distance-meter and two angles by encoders. The second type is the Tracking Interferometer (TI, such as laser tracer) which measures by interferometry the distance between its standard sphere and the Spherically Mounted Retroreflector (SMR) without taking into account the laser dead zone.

The multilateration technique has been used in numerous research areas such as electrical and electronic engineering, telecommunications and aerospace engineering … Around 200 research papers related to this topic can be found in the Web of Science database. In the field of machine compensation and calibration, the multilateration principle was introduced by Schwenke et al. [6] in order to calibrate a machine structure using a TI. Muralikrishnan et al. [7] wrote a survey of the literature about the use of TIs in large-scale dimensional metrology. That paper is focused on error modeling, measurement uncertainty, performance evaluation and standardization. Norrdine [8] proposed an algebraic approach to solve nonlinear problems in multilateration.

Norrdine derives the spatial coordinates of the unknown points as a function of the distances to other known points using a system of quadratic equations. In the case of sequential multilateration using a TI, the “other known points” (positions of the laser tracer) and the laser dead zone are in fact not known. However, it helps to explain the quadratic equations of a global multilateration problem. Gao et al. [9] summarized the multi-axis coordinate measurement methods such as triangulation and multilateration. To determine the TI position coordinates, Chen et al. [10] proposed a calibration procedure based on additional measurements comparatively to the classic multilateration procedure. Camboulives et al. [11] presented a calibration procedure of a 3D working space based on multilateration using only one TI. The reference datum system used in this procedure was built from the successive locations of a single TI independently of the machine reference frame.

A considerable amount of research has been done on the multilateration using laser trackers. The impact of variations in system configuration (laser tracker positions) on the volumetric measurement error was studied by Zhang et al. [12]. In their article, the authors propose an optimization of the system configuration and a self-calibration planning to reduce the propagation of volumetric measurement errors. This reduction was obtained by increasing the number of laser tracker stations. In the same way, a new procedure to calibrate an articulated arm coordinate measuring was presented in that paper. Wang et al. [13] used a genetic algorithm to optimize the laser tracker positions of multilateration measurements. Santolaria et al. [14] proposed a self-calibration algorithm of four laser trackers. The same research team presented a work about the different techniques and factors that affect the measurement accuracy of laser trackers used in machine tool volumetric verification [15] and proposed different calibration strategies based on network measurements [16]. Recent papers propose the calibration of coordinate measuring machines [17] and machine-tools [[18], [19], [20]] using this measurement principle.

In short, most of the research works about multilateration precision are centered around the influence of the experimental configuration (the positions of the measurement device and of the target points as well as the uncertainties of the temperature, pressure and humidity sensors) on measure precision [21,22]. However, the effects of the number of digits used during computations in large-scale multilateration precision have not been addressed [23].

In response to this shortcoming, the aim of this paper is to bring to the fore the impact of the number of digits used in computation on large-scale sequential multilateration using a TI. In order to do this, two numerical experiments simulating multilateration-based measurements using a TI were performed. These experiments were conducted using the multilateration model and the measurement configuration described in Section 2. The first experiment, which is detailed in Section 3, aimed to determine the impact of numerical errors in the course of the nonlinear least-squares algorithm used in multilateration. This was evaluated by solving the floating-point calculations of the optimization problem using different numbers of digits (10–20). The obtained results were compared with the nominal solution of the problem. The second experiment, detailed in Section 4, was aimed to evaluate the combined effects of the measuring uncertainties together with numerical errors.