Strapdown Inertial Navigation: What an Accelerometer Actually Measures

An accelerometer is often described casually as a device that “measures acceleration.” That description is incomplete and, in inertial navigation, can be seriously misleading. An ideal accelerometer does not directly measure the kinematic acceleration a of its case. It measures the non-gravitational force per unit mass required to make an internal proof mass follow the motion of the instrument.

That quantity is called specific force. In an inertial frame,

(1)

where g is gravitational acceleration. Equivalently,

(2)

This deceptively simple relationship is the translational heart of inertial navigation. The accelerometers supply f in the body frame. The navigation computer rotates that measurement into a navigation frame, restores gravity, applies the rotating-frame corrections derived in INS02, and integrates to obtain velocity and position [1, 2, 3].

The physics is easiest to understand by asking what happens to a small proof mass inside the sensor. A supported accelerometer sitting motionless on a table reports a nonzero specific force even though its coordinate acceleration is zero. The same accelerometer in ideal ballistic free fall reports zero even though it is accelerating toward the Earth at nearly g. These are not paradoxes; they are exactly what Newton’s second law predicts once gravitational and non-gravitational forces are separated.

1 Learning objectives

After completing this entry, the reader should be able to:

1. explain physically why an accelerometer does not directly measure coordinate acceleration;
2. derive the specific-force equation from Newton’s second law applied to an accelerometer proof mass;
3. distinguish gravitational force from the non-gravitational support force measured by an accelerometer;
4. explain why a stationary accelerometer reports approximately 1g;
5. explain why an ideal freely falling accelerometer reports zero;
6. calculate accelerometer outputs for elevators, vehicles, rockets, and other accelerating systems;
7. distinguish specific force from gravity and from apparent centrifugal terms;
8. express specific force in body and navigation coordinates;
9. show why a stationary level IMU in NED coordinates has f_D ≈−g;
10. explain how accelerometers provide roll and pitch information during static alignment;
11. connect the accelerometer measurement to the inertial-frame velocity equation;
12. identify which sensor imperfections are deliberately postponed to INS18.

2 The proof-mass model

Consider an accelerometer rigidly attached to a vehicle. Inside the accelerometer is a small proof mass of mass m. In a MEMS accelerometer the proof mass is usually suspended by compliant structures and its displacement is sensed capacitively. Closed-loop instruments may apply electrostatic forces to keep the proof mass near a null position. The detailed implementation varies, but the essential mechanics is the same [4, 1].

Let the proof mass have inertial acceleration a. Suppose the forces on it are separated into two classes:

1. gravitational force mg;
2. all non-gravitational forces, whose vector sum is F_ng.

Newton’s second law gives

(3)

Rearrange:

(4)

Divide by the proof-mass mass:

(5)

The left-hand side is the non-gravitational force per unit mass. Define

(6)

Therefore

(7)

This is the ideal accelerometer measurement equation in an inertial frame.

3 Why specific force has units of acceleration

Specific force is a force divided by mass, so its SI units are

(8)

That is why accelerometer outputs are naturally reported in acceleration units even though the physical quantity is force per unit mass.

A common engineering unit is g₀, the standard acceleration of gravity,

(9)

Thus an ideal stationary accelerometer near the Earth’s surface often has an output magnitude close to 1g₀.

4 Case 1: an accelerometer resting on a table

Choose a local Cartesian frame with +z upward. Near the Earth’s surface,

(10)

The accelerometer is stationary, so

(11)

Specific force is therefore

f	= a − g	(12)
	= 0 − (−gez)	(13)
	= gez .	(14)

The accelerometer reports an upward specific force of approximately 1g.

Why? The proof mass would fall if unconstrained. The sensor structure must exert an upward non-gravitational force on the proof mass to keep it motionless relative to the case. The accelerometer senses that support interaction.

This is the same physics as a bathroom scale. A scale does not directly measure gravitational force; it measures the contact force required to support the body.

5 Case 2: ideal free fall

Now release the accelerometer in vacuum and neglect all forces except gravity. Its inertial acceleration is

(15)

Hence

(16)

The proof mass and sensor housing accelerate together under gravity. No support force is required to keep the proof mass centered relative to the case, so the ideal accelerometer output is zero.

This is the Newtonian expression of the physical idea behind weightlessness in orbit. An orbiting spacecraft can have substantial coordinate acceleration toward the Earth while its occupants and accelerometers experience nearly zero specific force.

6 Case 3: an accelerating elevator

Return to the local +z upward convention. Let the elevator have vertical acceleration

(17)

Then

f	= a − g	(18)
	= (az + g)ez.	(19)

Thus

(20)

For g = 9.81 m∕s²:

The readings are also exactly what a passenger would describe as feeling heavier or lighter.

7 The accelerometer senses non-gravitational interactions

The definition

(21)

provides the most useful physical interpretation. Examples of non-gravitational forces include:

• Normal force from a table or vehicle structure;
• thrust transmitted through a rocket or aircraft structure;
• aerodynamic forces;
• spring forces;
• contact forces from a tire, rail, or mechanical guide;
• electrostatic forces used to rebalance a precision accelerometer.

Gravity is intentionally excluded from F_ng because the accelerometer cannot distinguish the gravitational acceleration of the proof mass from the gravitational acceleration of the housing. Both fall together.

Thus a useful conceptual statement is

(22)

8 A rocket example

Suppose a rocket is instantaneously vertical and accelerating upward at

(23)

Near the surface of the Earth,

(24)

Then

(25)

The ideal accelerometer therefore reports approximately

(26)

The rocket’s coordinate acceleration is only 5.0 m∕s² upward, but its accelerometer senses a larger specific force because thrust must both overcome gravity and produce the upward acceleration.

9 Specific force is a vector

The scalar elevator examples are useful, but a strapdown INS operates with three-axis vectors. In an inertial frame,

(27)

If the accelerometer triad is aligned with the body frame b, the IMU reports the body-resolved components

(28)

The navigation computer often needs the same physical specific-force vector expressed in the navigation frame n:

(29)

INS01 showed that this is a passive coordinate transformation: the physical vector is unchanged; only its components are re-expressed.

10 The basic inertial-frame navigation equation

Begin with the ideal inertial-frame specific-force equation

(30)

Since

(31)

we obtain

(32)

But the accelerometers provide body components f^b. Transform them into the inertial frame:

(33)

Therefore

(34)

This is the cleanest form of strapdown translational mechanization. It says:

1. measure specific force in the body frame;
2. use attitude to rotate it into the desired frame;
3. restore gravity;
4. integrate acceleration to obtain velocity.

Rotating Earth-fixed and local-level coordinates add the terms derived from the transport theorem in INS02. Those terms do not change what the accelerometer measures; they change the differential equation used to interpret that measurement in a rotating coordinate frame.

11 Stationary IMU in local NED coordinates

The North-East-Down navigation convention deserves special attention because its signs are initially counterintuitive.

Let

(35)

because the D axis points downward.

For a stationary IMU whose local coordinate acceleration is approximately zero,

(36)

Therefore

fn	= an − gn	(37)
	≈−gn	(38)
	= .	(39)

Thus a level stationary accelerometer in NED coordinates has approximately a negative Down specific-force component. Physically, the support force points upward.

This sign is an excellent implementation sanity check.

12 Static leveling intuition

A stationary accelerometer triad measures approximately

(40)

Therefore the measured specific-force direction identifies local vertical. This is why accelerometers can determine roll and pitch during coarse static alignment.

However, gravity alone provides no information about rotation around the vertical axis. If the body is rotated in yaw while remaining level, the measured gravity direction in body coordinates is unchanged. Yaw must come from another reference, such as Earth rate, a magnetic field, GNSS velocity, or an external heading source.

That observability issue will be treated formally in INS09.

13 Accelerometers and apparent weight

Suppose a person of mass m stands on a scale in an elevator. The scale measures the normal force N. With +z upward,

(41)

Hence

(42)

But this is exactly the vertical specific force:

(43)

Thus an accelerometer and a scale are closely related physical instruments: both respond to non-gravitational support force rather than to gravity directly.

This is why “apparent weight” is a useful intuition for specific force.

14 What about centrifugal acceleration?

INS02 showed that rotating coordinate frames introduce terms such as

(44)

It is important not to confuse these coordinate-kinematic terms with the accelerometer measurement itself.

The proof mass responds to physical non-gravitational forces. The navigation equations may combine gravitation and the Earth’s centrifugal contribution into a conventional local gravity model. Different navigation texts therefore distinguish carefully among:

• gravitational acceleration or gravitation;
• centrifugal acceleration associated with Earth rotation;
• normal or effective gravity used by an Earth-fixed navigation model.

INS05 will derive these distinctions in detail. For the present article, the important principle is that the definition of g must be consistent with the frame and mechanization equation in which it is used [2, 3].

15 What the accelerometer cannot tell you by itself

An accelerometer measurement does not uniquely determine the vehicle’s coordinate acceleration unless gravity and attitude are already known.

From

(45)

three ingredients are required:

1. the measured body-frame specific force f^b;
2. the attitude transformation C_bⁿ;
3. the gravity model gⁿ.

A tilt error therefore corrupts translational navigation even when the accelerometer itself is perfect. If the estimated attitude tilts gravity by a small angle δ𝜃, the navigation computer can interpret part of the large vertical support force as horizontal acceleration.

For a small tilt,

(46)

A one-degree attitude error gives approximately

(47)

That false horizontal acceleration is about 17 milli-g even with a perfect accelerometer. This coupling is one reason attitude accuracy is so important in inertial navigation.

16 Ideal measurement versus a real IMU

The ideal relation is

(48)

A real three-axis accelerometer is better represented schematically by

(49)

where

• S represents scale-factor error;
• M represents axis nonorthogonality and cross-axis sensitivity;
• b_a is accelerometer bias;
• n_a is measurement noise.

Temperature dependence, vibration rectification, saturation, quantization, and other effects may also matter. These errors are deferred to INS18 so that the present article can isolate the underlying physics.

17 The strapdown interpretation chain

The accelerometer is only the first step in the translational mechanization.

The ideal chain is

(50)

The important distinction is between the measured quantity and the reconstructed kinematic quantity:

(51)

Instead,

(52)

18 Common misconceptions

18.1 “A stationary accelerometer should read zero”

A stationary accelerometer is not in free fall. Its support structure continuously applies a non-gravitational force to its proof mass. Near the Earth’s surface, its ideal output magnitude is approximately g.

18.2 “An accelerometer measures gravity”

An accelerometer does not directly measure gravitational acceleration. In ideal free fall its output is zero even though gravitational acceleration is nonzero. Gravity is inferred through a model and through the behavior of supported objects.

18.3 “Zero accelerometer output means zero acceleration”

No. Zero specific force means the accelerometer is following a locally ballistic trajectory. It may have substantial coordinate acceleration.

18.4 “The navigation computer can integrate accelerometer data directly”

Not generally. The body-frame specific-force vector must first be transformed using attitude, and gravity and rotating-frame terms must be handled consistently.

18.5 “The accelerometer gives yaw when the vehicle is stationary”

A static accelerometer provides the local vertical direction, which constrains roll and pitch. Gravity is invariant under yaw rotations about the vertical, so yaw is not observable from accelerometers alone.

19 Connection to the next lessons

INS03 has established the translational sensor equation

(53)

The next main lesson, INS04, asks the analogous rotational question:

(54)

INS05 will then return to the other half of the present equation and derive gravity models for a rotating, oblate Earth. Once attitude, gyro measurements, specific force, and gravity are all established, the series can assemble the complete strapdown mechanization without treating any term as a black box.

20 Summary

For an ideal proof mass of mass m,

(55)

Therefore the accelerometer measures the non-gravitational force per unit mass,

(56)

A supported stationary accelerometer reads approximately 1g, while an ideal freely falling accelerometer reads zero. The body-frame measurement must be rotated using the attitude solution before it can contribute to navigation:

(57)

In an inertial frame,

(58)

For a stationary local-level system using NED coordinates,

(59)

Specific force is therefore the quantity that connects accelerometer physics to the translational equations of strapdown inertial navigation.

References