International Journal of Aerospace Engineering

Volume 2018 (2018), Article ID 3582508, 10 pages

https://doi.org/10.1155/2018/3582508

## Experimental Verification of a Simple Method for Accurate Center of Gravity Determination of Small Satellite Platforms

Department of Industrial Engineering, University of Bologna, Via Fontanelle 40, 47121 Forlì, Italy

Correspondence should be addressed to Dario Modenini; ti.obinu@ininedom.oirad

Received 8 September 2017; Revised 8 February 2018; Accepted 27 February 2018; Published 19 April 2018

Academic Editor: Franco Bernelli-Zazzera

Copyright © 2018 Dario Modenini et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We propose a simple and relatively inexpensive method for determining the center of gravity (CoG) of a small spacecraft. This method, which can be ascribed to the class of suspension techniques, is based on dual-axis inclinometer readings. By performing two consecutive suspensions from two different points, the CoG is determined, ideally, as the intersection between two lines which are uniquely defined by the respective rotations. We performed an experimental campaign to verify the method and assess its accuracy. Thanks to a quantitative error budget, we obtained an error distribution with simulations, which we verified through experimental tests. The retrieved experimental error distribution agrees well with the results predicted through simulations, which in turn lead to a CoG error norm smaller than 2 mm with 95% confidence level.

#### 1. Introduction

The growing interest for the development of light, small, highly capable spacecraft (S/C) platforms for a wide range of missions demands for a boost in performance from the standards established by the multitude of low-cost micro/nanosatellites. Often developed as part of university educational programs, they have been dominating this segment in the last two decades. In this respect, it is known that accurate attitude and orbit control systems rely on the precise knowledge of the spacecraft CoG. However, the development of such a class of S/C is highly cost-driven, whereas methods for measuring the CoG commonly employed for larger platforms [1], being highly accurate, require rather complex and expensive equipment. Thus, cost-effective and easy-to-implement alternatives shall be pursued.

Typically, the methods for measuring CoG of an S/C fall into two broad categories, that is, static methods and dynamic methods [2]. Static methods are often based on the pivoting axis system: the payload under test (PUT) is mounted on an instrument featuring a pivoting axis. In principle, the offset of the CoG from the pivoting axis can be retrieved by measuring the force acting on a point at a certain distance from the axis itself, once the total mass of the payload is known. Complete CoG localization is then obtained by repeating the measurement after rotating the PUT. The most accurate instruments exploiting the static balancing principle consist of rotary platforms featuring a closed loop self-balancing controller, to hold the platform to its neutral position [3]. The torque required for rebalancing is the measured output from which the CoG location can be retrieved, leading to submillimeter accuracies. Another common static measurement method is the one of multipoint weighting, achieved by placing the PUT over a multipoint weight platform equipped with 3 (or 4) high accuracy force transducers. The forces measured by the transducers, whose locations are known, allow to compute the in-plane coordinates of the CoG. This concept is employed at NOVA test facility (Utah University), to measure the mass properties of nanosatellites, with a reported accuracy of 1 mm in localizing the CoG [4].

Dynamic methods are based on the principle of dynamic balancing: the PUT is placed on a spin balance which estimates the CoG location by measuring the centrifugal forces. High sensitivity, however, is achieved at high rotational speeds, which makes such method of limited applicability for space vehicles CoG measurements [1].

Despite various measurement instruments based on all methods listed above are commercially available, these are quite expensive: even when aiming at a relatively low total weight capacity and moderate accuracy, the cost reaches several thousands of Euros. The concept of suspending a body for measuring its CoG, which is pursued in this work, is certainly not new, rather one of the oldest. Suspension was employed for example in NASA X-38 project [5]. In that case, the CoG localization was obtained combining weight distribution (as for a multipoint weight method) with inclination measurements. Recent examples involving the suspension concept are the trifilar torsional pendulum [6], and the photogrammetry technique [7], applied by NASA engineers to locate Orion capsule CoG. The trifilar pendulum is a quite simple mechanism, allowing the joint determination of the CoG and the inertia matrix. The reported accuracy in locating the CoG is 1.5 mm, but this was obtained after a careful calibration of the mechanism and the use of a tricoordinate measuring machine to determine the distance between some predefined points [6]. In [7], the authors suspended a full-scale Orion crew module from an asymmetric bifilar lifting strap and retrieved the CoG position from triangulation of the plumb lines. These, in turn, were determined from a set of images, gathered by a multicamera system, and processed through a set of custom-designed data reduction functions. Authors’ indications suggest for an accuracy in the order of few millimeters.

In this paper, we aim at the experimental verification of the method devised by the authors in [8], which relies upon two consecutive monofilar suspensions of the object under test to determine its CoG, using as measured quantities the angle output from a dual-axis inclinometer. To this end, we first generalize the method relaxing some of the constraints outlined in the original formulation. The experimental verification approach is that of applying the method to determine the barycenter of a known mass distribution, that is, a proof mass. To enforce experiment repeatability and smooth systematic errors, we perform measurements from several couple of suspension points. The error of the method is then quantified as the distance between the computed barycenter of the proof mass and the true one.

The main contribution of this work is twofold: (1) to investigate an extremely low-cost method for determining the CoG, with minimum hardware and calibration requirements, with an accuracy suitable for many practical applications, and (2) to provide a comprehensive error analysis which is validated through experiments. To this end, the paper is organized as follows: first, the double suspension method is outlined (Section 2). Then, an error budget is presented, first qualitatively to justify the experimental setup design (Section 3) and later quantitatively by introducing the test facility and the assumed statistical distributions of errors (Section 4). The verification method is then presented in Section 5, which combines Monte Carlo error analysis and experiments. Once the theory is set, results are presented in Section 6, and finally, our conclusions are drawn in Section 7.

#### 2. The Double Suspension Method

In recalling and generalizing the method presented in [8], we first define the inclinometer frame of reference. Consider the inclinometer in Figure 1, with top face up; is perpendicular to the top face, with outward positive, is directed in the direction of the cable connection, and completes the right-handed frame.