Strapdown Inertial Navigation: Gravity, Gravitation, and the Rotating Earth

Gravity enters inertial navigation in a deceptively simple-looking place. Once accelerometer specific force has been transformed into the navigation frame, the mechanization adds a gravity vector before integrating velocity. In local north-east-down coordinates one often sees

(1)

The symbol gⁿ can make gravity look like a single external correction that is simply inserted into the equations. Physically, however, several ideas are hiding inside it.

Earth’s mass distribution produces gravitation. Earth also rotates. A navigation equation written in a frame fixed to that rotating Earth contains a centrifugal contribution. The vector conventionally called local gravity combines these effects, with smaller corrections added when greater geophysical fidelity is required [1, 2, 3].

This entry develops that structure from Newtonian gravitation and the rotating-frame derivative derived in INS02. The goal is not merely to quote a gravity formula. The goal is to understand why the form of the strapdown velocity equation depends on exactly what quantity the gravity model represents.

1 Learning objectives

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

1. distinguish gravitation from effective gravity;
2. derive the point-mass gravitational acceleration from a scalar potential;
3. derive the centripetal acceleration of a point fixed to the rotating Earth;
4. identify the outward centrifugal term that appears when Newton’s law is rewritten in an Earth-fixed rotating frame;
5. define effective gravity as the combination of gravitation and centrifugal acceleration;
6. explain why an accelerometer at rest on Earth measures approximately the negative of effective gravity;
7. derive the latitude dependence of the centrifugal contribution on a spherical Earth;
8. explain why an oblate rotating ellipsoid is a better reference model than a nonrotating sphere;
9. use the WGS-84 Somigliana normal-gravity formula;
10. estimate the first-order change of gravity with altitude;
11. derive the ECEF strapdown velocity equation first with an explicit centrifugal term and then with effective gravity absorbed into g^e;
12. interpret the corresponding local NED velocity equation;
13. identify gravity-model errors that can produce velocity and position drift.

2 Terminology: gravitation is not exactly the same as gravity

The words gravity and gravitation are often used interchangeably in introductory mechanics. In navigation and geodesy it is useful to distinguish them.

In this entry we use the following convention.

(2)

and

(3)

For the simplest rotating-Earth model,

(4)

The second term is outward from Earth’s rotation axis because

(5)

is inward toward that axis. Therefore its negative is the centrifugal acceleration of the rotating description.

Some texts use slightly different terminology. For example, the word “gravity” may already imply a normal-gravity model on a rotating ellipsoid. The equations are unambiguous as long as the quantity being modeled is stated explicitly.

3 Newtonian gravitation from a potential

Start with the simplest Earth model: a spherically symmetric body of mass M. Outside such a body, Newton’s shell theorem allows The Gravitational Field to be treated as if all mass were concentrated at Earth’s center.

Let

(6)

where G is the Newtonian gravitational constant and M is Earth’s mass. The gravitational potential per unit mass is

(7)

The gravitational acceleration is the negative gradient of the potential:

(8)

Because the potential depends only on the radial distance r,

(9)

Now

(10)

Therefore

(11)

The acceleration is directed toward Earth’s center and has magnitude

(12)

This point-mass result is the first gravity model in the navigation hierarchy. It is simple, physically transparent, and useful for deriving error trends. It is not accurate enough to represent the real rotating, oblate Earth at high precision.

4 The gravity gradient in the spherical model

The same equation immediately explains why gravity decreases with height. Let

(13)

Differentiate with respect to r:

(14)

For a small altitude increment h relative to an Earth radius R,

(15)

Hence

(16)

Taking R ≈ 6.37 × 10⁶ m and g ≈ 9.81 m∕s² gives a vertical gradient near the surface of about

(17)

Thus a 1000 m increase in height reduces gravity by roughly

(18)

in this first-order spherical estimate.

5 A point fixed to Earth is not fixed in inertial space

Now include Earth’s rotation. Let the Earth-fixed frame e rotate relative to inertial frame i with angular velocity

(19)

Consider a point whose Earth-fixed coordinates are constant. It is stationary in ECEF coordinates, but it moves in a circle around Earth’s rotation axis when viewed from inertial space.

From the transport theorem developed in INS02, with Earth rotation treated as constant,

(20)

Differentiating again in inertial space gives

(21)

This vector points inward toward the rotation axis. It is the centripetal acceleration required for the point to remain fixed on the rotating Earth.

At the equator, its magnitude is approximately

(22)

Using

(23)

and

(24)

we obtain

(25)

This is small compared with 9.8 m∕s², but it is far too large to ignore in inertial navigation.

6 Surface force balance and what the accelerometer senses

The distinction between gravitation and effective gravity can be seen directly from Newton’s second law for an IMU resting on Earth.

Suppose the non-gravitational support force exerted by the vehicle structure on the IMU is F_support. In inertial space,

(26)

For an Earth-fixed point,

(27)

Therefore

(28)

Define effective gravity by

(29)

Then

(30)

Since an ideal accelerometer measures non-gravitational force per unit mass,

(31)

This result generalizes the INS03 statement that a supported accelerometer reads approximately 1g. On a rotating Earth, the supported accelerometer is reacting to effective gravity, not to gravitation alone.

7 The centrifugal acceleration in the rotating Earth frame

INS02 derived the general relation between inertial acceleration and acceleration observed in a rotating frame. For the Earth-fixed frame, assuming constant Earth rotation,

(32)

Newton’s law in inertial space is

(33)

Substitute the first equation into the second and solve for the Earth-fixed acceleration:

(34)

The last term is the outward centrifugal acceleration in the Earth-fixed equation:

(35)

This is the same physical geometry discussed in INS02. The sign changes because we have rearranged the inertial acceleration relation to solve for acceleration measured in the rotating frame.

The effective-gravity definition

(36)

therefore gives

(37)

This compact form is the key reason Normal gravity models are convenient in Earth-fixed navigation equations: the centrifugal contribution has already been absorbed into g.

8 Centrifugal acceleration versus latitude on a spherical Earth

The latitude dependence can be derived geometrically. Consider a spherical Earth of radius R. At latitude ϕ, the perpendicular distance from the rotation axis is

(38)

The magnitude of the centrifugal acceleration is

(39)

The vector points horizontally away from the rotation axis, not directly away from Earth’s center except at the equator.

Resolve this vector into local north and down components on a spherical Earth. The result is

(40)

The north component is negative in the northern hemisphere, so the centrifugal contribution points slightly southward toward the equator. The down component is negative because centrifugal acceleration acts upward relative to the local radial direction.

If the point-mass gravitation vector is purely down,

(41)

then the effective gravity in the spherical model is

(42)

Two useful limiting cases follow immediately.

At the equator, ϕ = 0:

(43)

At the pole, ϕ = 90^∘:

(44)

Thus rotation reduces apparent gravity most strongly at the equator and not at all at the rotation axis.

9 Why a spherical Earth is still not enough

Earth is not spherical. Its rotation and long-term hydrostatic structure produce an oblate shape, with an equatorial radius larger than the polar radius. This affects navigation gravity in two coupled ways.

First, the distance from Earth’s center depends on latitude. Since gravitational attraction scales roughly as 1∕r², the shorter polar radius tends to increase gravitation near the poles.

Second, the centrifugal acceleration itself depends on distance from the rotation axis and vanishes at the poles.

The observed increase in normal gravity from equator to pole therefore contains both rotational and geometric effects. The equatorial centrifugal acceleration of approximately

(45)

is substantial, but it is not the whole reason normal gravity differs between equator and pole.

A convenient navigation reference model is therefore a rotating oblate ellipsoid whose gravity field is chosen consistently with its shape and angular velocity. WGS-84 provides such a reference system [5, 2].

10 WGS-84 reference ellipsoid

Several WGS-84 parameters are central to inertial navigation and geodesy. The semi-major axis is

(46)

and the flattening is

(47)

The first eccentricity squared is

(48)

The conventional Earth rotation rate is

(49)

WGS-84 also specifies the geocentric gravitational constant

(50)

These parameters are not independent decoration. The ellipsoid geometry, rotation, and reference gravity field are designed to work together.

11 Normal gravity on the reference ellipsoid

For many strapdown applications, the gravity vector is approximated by normal gravity, the gravity field of the chosen rotating reference ellipsoid. At the ellipsoid surface, Somigliana’s formula can be written

(51)

where ϕ is geodetic latitude, γ_e is normal gravity at the equator, and k is the Somigliana constant [4, 5].

For WGS-84, useful values are

(52)

(53)

At selected latitudes this gives approximately

The pole-to-equator difference is about

(54)

That is more than five thousandths of g, which is enormous on the scale of inertial-navigation errors.

12 Normal gravity above the ellipsoid

A navigation system rarely remains exactly on the reference ellipsoid. Gravity must therefore be adjusted for height.

The simplest physical approximation comes from the spherical result already derived:

(55)

More accurate normal-gravity formulas include latitude-dependent height terms and ellipsoidal geometry. Groves and Jekeli provide forms suitable for navigation implementations [2, 3].

For conceptual work near Earth’s surface, it is useful to remember the approximate free-air gradient

(56)

A 1 km altitude error therefore corresponds to a gravity-magnitude error of order

(57)

if height dependence is ignored entirely.

13 Normal gravity is not the complete real gravity field

The WGS-84 normal field is deliberately smooth. The actual gravity field differs because Earth’s mass distribution is irregular.

Important deviations include:

• large-scale spherical-harmonic structure of the terrestrial gravity field;
• regional gravity anomalies caused by variations in crust and mantle density;
• mountains, ocean trenches, and local terrain;
• deflection of the vertical, where the true gravity direction differs slightly from the reference-ellipsoid normal;
• tidal effects from the Moon and Sun;
• temporal redistribution of atmosphere, oceans, groundwater, and ice.

The required gravity fidelity depends on the navigation problem. For short-duration or lower-grade inertial systems, a normal-gravity model may be more than adequate. High-accuracy inertial navigation, inertial surveying, submarine navigation, and gravimetry can require substantially more detailed models.

The important architectural idea is that the mechanization needs a gravity vector appropriate to the chosen navigation frame and fidelity level. The source of that vector can evolve without changing the fundamental strapdown equations.

14 Gravity in local NED coordinates

For a local north-east-down frame,

(58)

In the simplest normal-gravity mechanization, the local vertical is chosen so that the reference normal gravity points along the down axis. Then

(59)

This sign is easy to mishandle. NED uses positive down, so gravity has a positive third component.

A level stationary accelerometer, by contrast, measures the support specific force

(60)

The two vectors cancel in the translational mechanization of a stationary vehicle.

If a higher-fidelity gravity model predicts horizontal gravity components, then g_N and g_E need not be exactly zero. Those components can represent deflection of the vertical relative to the reference ellipsoid.

15 Deriving the ECEF strapdown velocity equation

We can now connect the gravity physics directly to the navigation equations.

Let

(61)

be the velocity of body b relative to Earth e, resolved in Earth-fixed coordinates. The accelerometers provide

(62)

Transforming specific force into ECEF gives

(63)

Start with Newton’s law in an inertial frame:

(64)

Using the rotating-frame acceleration relation from INS02 and resolving the result in ECEF gives

(65)

This form displays all the physics explicitly.

(66)

Now define effective Earth-fixed gravity as

(67)

The velocity equation becomes

(68)

Nothing has disappeared physically. The centrifugal term has been absorbed into the definition of g^e.

16 Deriving the local NED velocity equation

The navigation frame n is not fixed to ECEF. As the vehicle moves over Earth’s curved surface, the local NED frame rotates relative to ECEF with transport rate

(69)

The derivative of velocity components in NED therefore differs from the ECEF derivative by another transport term. Applying the transport theorem gives the local-level velocity equation

(70)

Each term now has a clear origin.

(71)

is the non-gravitational specific force measured by the accelerometers and rotated into NED.

(72)

is effective gravity, usually supplied by a normal-gravity or higher-fidelity model.

(73)

is the Coriolis contribution associated with Earth rotation.

(74)

appears because the local NED frame itself rotates as the vehicle moves over Earth.

The centrifugal acceleration due to Earth rotation is not written separately because it is already part of gⁿ.

17 A stationary NED sanity check

Consider a level IMU fixed to Earth. Its Earth-relative velocity is zero:

(75)

Its navigation acceleration must also be zero:

(76)

The velocity equation therefore reduces to

(77)

Hence

(78)

For a level body aligned with NED,

(79)

so

(80)

This is an extremely valuable software test. A static mechanization should not accelerate merely because the accelerometers report approximately −g along the body down axis. The transformed specific force and gravity model should cancel.

18 What happens if centrifugal acceleration is counted twice?

A common conceptual implementation error is to use a gravity model that already includes centrifugal acceleration and then also subtract the explicit centrifugal term in the ECEF velocity equation.

Suppose g^e is effective gravity:

(81)

The correct ECEF equation is

(82)

If an implementation additionally subtracts

(83)

then the centrifugal effect has been applied twice.

At the equator the erroneous acceleration magnitude is of order

(84)

which would produce a velocity error of roughly

(85)

after only one minute if it were allowed to integrate uncompensated.

The cure is not a sign tweak. The cure is to state clearly whether the gravity function returns gravitation alone or effective gravity.

19 Gravity-model error and navigation drift

Suppose the gravity model has a small constant error

(86)

If all other quantities are ideal, the resulting velocity error initially satisfies

(87)

Therefore

(88)

Integrating once more,

(89)

For example, a constant vertical gravity error of

(90)

would produce, in this simple uncoupled estimate, a velocity error after 60 s of

(91)

and a position error of

(92)

The full INS error dynamics are more complicated because attitude, velocity, position, Earth curvature, and gravity are coupled. INS19 will derive those coupled dynamics. The simple estimate already shows why gravity-model errors integrate into navigation errors.

20 A useful implementation hierarchy

For an educational or engineering strapdown implementation, the gravity model can be developed in stages.

1. Point mass. Use −μr∕r³. This is ideal for verifying frame transformations and signs.
2. Rotating spherical Earth. Add the centrifugal contribution explicitly or absorb it into an effective-gravity function.
3. WGS-84 normal gravity. Use geodetic latitude and height with a rotating reference ellipsoid. This is a practical baseline for many navigation systems.
4. Higher-fidelity gravity. Add spherical harmonics, gravity anomalies, deflection of the vertical, terrain, or time-varying corrections when the mission requires them.

At every stage, the interface should make the physical meaning of the returned vector explicit.

A function named simply

gravity(position)

can be ambiguous. Internally, it is often safer to distinguish concepts such as

gravitation_ecef(position)
effective_gravity_ecef(position)
normal_gravity_ned(latitude, height)

so the mechanization cannot accidentally count centrifugal acceleration twice.

21 Relation to the attitude solution

Gravity also plays a second role in inertial navigation. It is not only a translational acceleration model. During static or quasi-static alignment, the accelerometers provide the direction opposite local gravity:

(93)

This makes gravity a reference vector for estimating roll and pitch. However, gravity alone cannot determine yaw because rotating the body about the gravity vector leaves the measured vector unchanged.

INS09 will combine gravity with Earth rotation to derive coarse alignment and gyrocompassing. The present lesson provides the physical gravity vector needed for that derivation.

22 Where the series goes next

INS06 will develop the Earth-fixed and local navigation frames in greater detail. It will derive Earth rate resolved in NED and begin the geometry behind transport rate.

The subsequent sequence will then use the pieces established so far:

• INS07 will derive attitude kinematics from gyro measurements;
• INS08 will develop quaternion propagation;
• INS09 will combine gravity and Earth rate for initial alignment;
• INS10 will return to translational dynamics in an inertial frame;
• INS11 will derive the complete Earth-fixed rotating-frame navigation equations;
• INS12 will assemble an ECEF strapdown mechanization;
• INS13 and INS14 will derive local-level transport rate and ellipsoidal position kinematics;
• INS15 will assemble the complete NED strapdown mechanization.

23 Summary

Newtonian gravitation for a spherical Earth begins with

(94)

Because Earth rotates, a point fixed to Earth has inertial centripetal acceleration

(95)

When Newton’s law is written in an Earth-fixed frame, the corresponding outward centrifugal acceleration is

(96)

Effective gravity is therefore

(97)

On the WGS-84 reference ellipsoid, normal gravity varies with geodetic latitude. Somigliana’s formula gives

(98)

The ECEF velocity equation can be written with gravitation and centrifugal acceleration shown separately,

(99)

or more compactly using effective gravity,

(100)

In local NED coordinates the corresponding navigation equation is

(101)

The essential implementation question is therefore not merely “what is g?” It is “what physical effects are already contained in the gravity vector supplied to this particular navigation equation?”

References