Mike Baker, director at Sherborne Sensors, details the science behind the influence of location on the accuracy of inertial sensors, and offers a more simplistic explanation of how gravity can affect sensor calibration accuracy
The calibration accuracy of many sensors is fundamentally dependent upon the force of gravity at the site of operation. Because of the principles on which they work, the sensitivity of accelerometers, inclinometers, force transducers and load cells is fundamentally proportional to the force of gravity where they are being used; their absolute sensitivity may well differ when in situ from that of their place of manufacture. The acceleration due to gravity varies across the Earth’s surface due to a number of circumstances and, in the extreme, may well translate to a variation of up to 0.5% depending on where in the world it is measured.
For example, electronic weigh scales that use load cells as weight sensors effectively measure the force of gravity acting upon a mass. If on-site gravity compensation is not taken into consideration, the scales will have an error proportional to the difference between the acceleration due to gravity between the installation and original calibration sites.
Sir Isaac Newton’s Law of Universal Gravitation states that: “Every point mass attracts every other point mass by a force pointing along the line intersecting both points. The force is proportional to the product of the two masses and inversely proportional to the square of the distance between them.”
Mathematically, the force due to gravity is expressed by the formula: F= Gm1m2/r2. (Where F is the force between masses, G is the Gravitational Constant, m1 and m2 are the first and second masses and r is the distance between the centres of the masses).
Although Einstein’s Theory of General Relativity has since superseded this law, it continues to be applied, unless there is a need for extreme precision or when dealing with gravitation for massive and dense objects.
The Gravitational Constant G is actually very difficult to measure but, in 2010, CODATA, (Committee on Data for Science and Technology) recommended the value of G = 6.67384 x 10-11 m3 kg-1 s-2, with an uncertainty of 1 part in 8,300. Thus, with knowledge of the mass of the Earth and its radius, the force due to gravity can be ascertained.
It should be noted that the Law of Universal Gravitation defines a mass as a point mass; the Earth is neither of uniform shape nor even mass distribution and cannot be treated as such unless to a first approximation. Gravity therefore varies in proportion to latitude, (Figure 1). Also, the height above sea level of land masses varies, so acceleration is proportional to altitude too, (Figure 2).
Thirdly, the Earth is spinning on its axis and consequently the force of gravity at the equator is reduced by the centripetal force, the effect diminishing to zero at the poles.
What it all means
Typically a weighing instrument is calibrated using test masses. But as has been shown, the forces produced by test masses will vary according to location. Most accurate weighing systems will have a means of adjustment built-in to allow for multi-location deployment throughout the world.
Inertial inclinometers and accelerometers also use gravity as their fundamental calibration reference, but calibration will only be truly valid when the sensors are used at the original calibration site.
When making accurate measurements and unless the sensors can be calibrated locally, it is essential to consider the latitude and the altitude at which the sensors will be installed and adjust the sensitivity of the sensors accordingly if the manufacturer’s calibration data is to be the sole source of reference.