Moment of Inertia Measurement of a CubeSat Through Object Motion Tracking of Trifilar Pendulum Oscillations
Authors:   University College Dublin, Dublin, Ireland, Stijnen Bas , Thomson Joseph , Somers Eoghan , McKeown David 
Published: January 2024

Every satellite project follows roughly the same steps, starting with requirements definition and preliminary design through to the final design, manufacturing, assembly, test and verification, launch, and operation. At every step, a milestone review is required with the status of the design verification against requirements. This however is commonly non-trivial when considering the Attitude Determination and Control Systems (ADCS) of the Spacecraft. This difficulty is compounded by the small budgets commonly available for CubeSat projects. The replication of the space environment and hardware testing spacecraft dynamics becomes too costly and time consuming that it is replaced solely by simulations and Computer Aided Design (CAD) models. It is however possible to measure properties such as Moment of Inertia (MoI) and Centre of Mass (CoM) using relatively low-cost test setups as will be outlined in this paper.
Relying on CAD models for CoM and MoI calculation is often lacking when complex parts, such as harnessing, are frequently modelled incorrectly, simplified, or omitted. This limits the usefulness of the estimations these models can give. Accurate measurements are required by launch authorities or to understand the spacecraft dynamics for control purposes. In this paper the focus will be on the Moment of Inertia measurement carried out for a 2U CubeSat, EIRSAT-1, designed and built as part of the ESA Fly Your Satellite mission.
By presenting the design of the test equipment, it is hoped that this relatively simple test device for CubeSats will make it easier and faster for small teams (start-ups and University teams) to test and validate MoI requirements of their own satellites and reduce the repetition of the work designing bespoke systems.
To measure MoI a trifilar pendulum is used [1], the pendulum is made up out of an aluminium extrusion structure from which three cables hang down and suspend a platform on which a test object can be placed. By giving the test platform a small displacement such that it oscillates around its centre point, small angle approximations can be assumed [2] leaving a motion in only three degrees of freedom; displacement in X, displacement in Y and rotation around Z. with X and Y two axes perpendicular to each other in the plane of the platform and Z pointing vertically upwards out of the platform. The displacement in X and Y direction is called sway and needs to be accounted for when measuring the oscillation around the Z axis. From the period of the oscillations the properties of the pendulum and the mass of the test object the MoI can be calculated.
In the suggested design the pendulum’s motion is captured by a camera looking up at the bottom of the platform and positioned such that the centre of the platform is in view. On this surface of the platform a number of coloured tracking markers are placed. During a test run the recorded video captures the motion of the coloured markers in all three degrees of freedom. After the test, the video can be analysed in open-source video software Blender and used to first determine the sway. This is done by tracking the centre of the platform. When looking for pure rotation the centre of the platform should be stationary. Hence the displacement centre of the platform per frame will provide the user with the displacement in X and Y. Then the same video can be used to track other markers that are not the centre. Markers that are further away from the centre give a more accurate measurement. In order to find the pure rotational motion of these markers the sway is simply subtracted from the motions of these outer markers. The result is a x-position and a y-position per frame is found and can be converted to polar coordinates using the inverse tangent function, the result is a plot of the sinusoidal motion of the pendulum
The benefit of this non-contact method is that the pendulum has complete freedom to oscillate without being restrained by either a centring device or other contact measurement sensor. A number of tests have been performed to characterise the pendulum and the error varies with mass and inertia of the test object. When the pendulum platform mass and MoI are relatively large compared to the test object it should be taken into account in the final calculation where the MoI of the test object is the measured MoI minus the MoI of the platform.
This method allows the user to accurately within 10% of the actual value determine the MoI around one axis, to measure all the MoI’s and their products the test can be repeated in at least 6 in different configurations aligning different test object axes with the vertical axis of the pendulum.
[1] Previati G., Gobbi M. and Mastinu G., Measurement of the inertia tensor - a overview, 73rd Annual Conference of the Society of Allied Weight Engineers, Inc., SAWE 2014, Society of Allied Weight Engineers, pp. 1-23 (2014).
[2] Du Bois, J.L., Lieven, N.A.J., Adhikari, S., Error Analysis in Trifilar Inertia Measurements, Exp Mech 49, 533–540, 2009