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What Do Inertial Sensors Measure?

Paul G Savage Strapdown Associates, Inc.

September 18, 2005

Inertial navigation systems contain gyros to measure changes in angular orientation and accelerometers to measure changes in velocity (acceleration). To properly define angular orientation and velocity change one must define what the change is relative to. For gyros and accelerometers, what are their outputs referenced to? Consider an observer on the earth who has no concept of earth rotation and perceives the heavens as rotating around the earth. He defines "total angular rate" as angular rotation relative to the earth. He believes that gyros measure angular rate but not total angular rate. By experiment he has found previously that when a gyro is stationary on the earth (i.e., when the gyro's total angular rate is zero), the gyro output equals the negative of the angular rate of the heavens. Defining this as "heavenly rate", he concludes that gyros measure total angular rate minus heavenly rate. To find total angular rate, he adds heavenly rate to the gyro output. The observer's analysis yields useful results, but also introduces some interesting questions. We now know that on other planets, heavenly rate is different than on planet earth. Will the gyro output equal total angular rate (relative to the other planet) minus earth's heavenly rate or minus the planet's heavenly rate (assuming that the gyro was manufactured on the earth)? How does the gyro know the heavenly rate value to subtract from its output? Today we know that gyros actually measure angular rate relative to non-rotating inertial space (e.g., the heavens). Angular rate relative to the earth (or relative to any rotating body) is then calculated as the gyro output minus the inertial rotation rate of the earth (or body). The concept of "total angular rate" is not used today when describing angular motion relative to the earth. But what is non-rotating inertial space that the gyro output is referenced to? Is it really the heavens? How does a gyro know about the heavens so that it can provide its output relative to them? Reference 1 explores these questions in more detail on a scientific basis. Interestingly, today's definition for acceleration (rate of change of velocity) in the inertial navigation community has not yet reached the level of sophistication used to describe rotation. Engineers commonly use the term "total acceleration" to describe acceleration relative to the earth. Accelerometers are used to measure acceleration. From experiment we have learned that the output from a stationary accelerometer on the earth will equal the negative of earth's gravity at the acceleration location. Thus, a common conception is that accelerometers measure total acceleration minus gravity. To find total acceleration one must add gravity to the accelerometer output. We know of course, that the gravity component to add is the local gravity value at the accelerometer location whether on the earth or elsewhere. But how does the accelerometer know the value of gravity to subtract from its output? What does the accelerometer really measure? Reference 2 explores this question in more depth.

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Regarding gravity and accelerometers, a rather bizarre misunderstanding by the general public (fostered by public relations people in a well known US government agency) is that the space within a vehicle orbiting the earth has no gravity (i.e., the so-called "micro-gravity" environment). In reality, space surrounding the earth and within vehicles in earth orbit do have gravity; for low orbit vehicles such as the Space Shuttle, the gravity magnitude is very close to the gravity value on earth's surface (i.e., 32 feet per second per second). In fact, it is the gravity within an earth orbiting spacecraft that produces the centripetal acceleration required to maintain the satellite orbit around the earth. How could such a misconception have originated? Possibly from the term "micro-g" used by engineers to characterize the environment in space. Micro-g means one millionth of a "g". A g is a unit of acceleration generated by an applied force. For example, if a rocket's motor is ignited it will produce a force that accelerates the rocket (i.e., the rocket's velocity will change at a rate equal to the acceleration generated by the applied rocket motor force). One g of applied force acceleration corresponds to a 32 feet per second change in velocity each second due to applied force. Ten gs would correspond to 320 feet per second per second acceleration. The value for one g unit has been chosen by the scientific community to equal the value of gravity on the earth's surface, i.e., 32 feet per second per second. One g of force acceleration output would also be measured by a stationary accelerometer on the earth's surface with its input axis up in response to one g of force pushing up on the accelerometer (through its mount) to balance gravitational acceleration downward that exists on the earth's surface. A micro-g is a very small acceleration due to force, and characterizes the acceleration of a space vehicle with very little externally applied force. Given the term micro-g, one can imagine a non-engineer inquiring on the meaning of g in the term micro-g. Without the above more complicated explanation, the simple answer might have been "gravity". Hence, the term "micro-gravity" to describe the environment in space and the erroneous belief that gravity is micro small in space. References 1. "What Do Gyros Measure?", Paul G. Savage, Strapdown Associates, Inc.,

January 4, 2006, Available on the Internet at www.strapdownassociates.com 2. "What Do Accelerometers Measure?", Paul G. Savage, Strapdown Associates, Inc.,

May 8, 2005, Available on the Internet at www.strapdownassociates.com