Strapdown Inertial Navigation: Earth Rotation and Navigation Frames

A strapdown inertial navigation system does not evolve one vector in one coordinate system. It continually moves information among several frames that rotate with respect to one another. The gyroscopes measure body angular rate relative to inertial space. Position is usually reported relative to the rotating Earth. Velocity is often represented in a local north-east-down frame. Accelerometer specific force begins in body axes and must be rotated into the chosen navigation frame before it can be integrated.

The purpose of this entry is to make those frames and their relative angular rates explicit. The key chain is

(1)

where i denotes an Earth-centered inertial frame, e an Earth-centered Earth-fixed frame, n a local north-east-down navigation frame, and b the vehicle body frame. The corresponding relative angular rates are

(2)

The first is Earth rotation. The second is the transport rate, caused by motion over Earth’s curved surface. The third is the vehicle’s rotation relative to the local navigation frame. Their combination determines the rate that must be removed from the inertial gyro measurement before the body attitude relative to NED can be propagated [1, 2, 3].

1 Learning objectives

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

1. distinguish ECI, ECEF, local NED, and body frames physically and mathematically;
2. interpret the notation ω_ab^c as the angular velocity of frame b relative to frame a, resolved in frame c;
3. write the ECI-to-ECEF coordinate transformation and its time derivative;
4. derive the ECEF-to-NED direction cosine matrix from the local north, east, and down basis vectors;
5. derive Earth rate resolved in NED;
6. derive the WGS-84 meridian and prime-vertical radii of curvature used in local position kinematics;
7. derive the geodetic rates dϕ∕dt, dλ∕dt, and dh∕dt from NED velocity;
8. derive the NED transport-rate vector ω_enⁿ;
9. form the total navigation-frame inertial rate ω_inⁿ;
10. convert inertial gyro measurements ω_ib^b into body rate relative to NED;
11. derive the strapdown attitude equation from frame-rate composition;
12. show how the ECEF velocity equation transforms into the NED velocity equation;
13. identify coordinate singularities and frame-selection issues near the poles;
14. construct numerical checks that detect frame-order and sign errors in navigation software.

2 Frame notation

Throughout this series, a superscript indicates the frame in which vector components are resolved. A direction cosine matrix

(3)

maps components from frame a into frame b:

(4)

Angular velocity uses two frame subscripts and one resolution superscript. In

(5)

the first subscript is the reference frame, the second subscript is the rotating frame, and the superscript gives the frame in which the vector components are written. Thus

(6)

means the angular velocity of the Earth-fixed frame e relative to inertial frame i, resolved in NED coordinates.

This convention is worth enforcing rigorously. A physical angular-velocity vector is independent of coordinates, but its numerical components are not. For example,

(7)

The vector is the same Earth rotation in both equations. Only the coordinate representation has changed.

3 The Earth-centered inertial frame

An Earth-centered inertial frame has its origin at Earth’s center of mass, but its axes do not rotate with the solid Earth. For navigation derivations it is convenient to choose the zⁱ axis approximately along Earth’s spin axis and to choose the xⁱ and yⁱ axes fixed relative to inertial space.

No Earth-centered realization is perfectly inertial over arbitrary time scales. Earth’s spin axis precesses and nutates, and the planet moves around the Sun. For the strapdown derivations in this series, ECI is the reference frame in which Newton’s second law takes its simplest form. Higher-fidelity astronomical realizations can be introduced when required.

In an inertial frame the ideal translational equation from INS03 and INS05 has the clean form

(8)

There is no Coriolis term and no centrifugal term because those are consequences of expressing the dynamics in rotating coordinates. This simplicity makes ECI valuable conceptually even when the operational navigation solution is maintained in ECEF or NED.

4 The Earth-centered Earth-fixed frame

The Earth-centered Earth-fixed frame shares the same origin as ECI, but its axes rotate with Earth. We use

(9)

for the angular velocity of ECEF relative to ECI. To first order for the present discussion its magnitude is the WGS-84 nominal Earth rotation rate

(10)

The magnitude corresponds to about

(11)

That rate is small compared with typical vehicle body rates, but it is not small compared with precision inertial sensor biases. It is therefore a first-order navigation quantity.

4.1 A simple ECI-to-ECEF transformation

Let 𝜃 be the Earth rotation angle measured about the common z axis. With the passive coordinate convention of INS01, one convenient idealized transformation is

(12)

Therefore

(13)

Differentiating the matrix while using d𝜃∕dt = Ω_E gives

(14)

where

(15)

The negative sign is a consequence of the passive transformation. The ECEF axes rotate positively relative to inertial space, so the components of an inertially fixed vector appear to rotate oppositely when written in ECEF coordinates.

5 Geodetic position on the WGS-84 ellipsoid

Local navigation frames are attached to a position on a reference ellipsoid. Let

(16)

For an ellipsoid with semi-major axis a and eccentricity squared e², define the prime-vertical radius of curvature

(17)

and the meridian radius of curvature

(18)

The geodetic coordinates map to ECEF position through

(19)

These equations distinguish geodetic latitude from the geocentric angle discussed in INS05E2. The local vertical used by a normal-gravity navigation frame is aligned with the ellipsoid Normal, not generally with the radius vector from Earth’s center.

6 The local north-east-down frame

The NED frame is attached to the vehicle’s local geodetic position. Its axes are

(20)

North is tangent to the reference ellipsoid in the direction of increasing geodetic latitude. East is tangent in the direction of increasing longitude. Down points opposite the ellipsoid outward normal.

6.1 NED basis vectors expressed in ECEF

The outward ellipsoid-normal unit vector is

(21)

Therefore the down unit vector is

(22)

A unit vector tangent to increasing longitude is

(23)

The north vector completing the right-handed NED triad is

(24)

For NED axes the handedness relation is

(25)

6.2 ECEF-to-NED direction cosine matrix

From INS01, the rows of C_eⁿ are the NED basis vectors expressed in ECEF coordinates. Hence

(26)

As always for a proper DCM,

(27)

The body-to-NED transformation can therefore be assembled through intermediate frames. For example,

(28)

or, if attitude is first known relative to ECI,

(29)

This is a practical expression of the composition rule derived in INS01.

7 Earth rate resolved in NED

Earth rotation is simplest in ECEF coordinates:

(30)

To obtain the same physical vector in NED components,

(31)

Using the third column of C_eⁿ gives

(32)

Several immediate checks are useful.

At the equator,

(33)

Earth rate points north in local coordinates.

At the North Pole,

(34)

Earth rate points upward, which is negative down.

At 45^∘ latitude,

(35)

These are the same Earth-rate components used in the gyroscope examples of INS04E1.

8 Position kinematics from NED velocity

Let local velocity relative to Earth be

(36)

A displacement northward by ds_N changes geodetic latitude according to the local meridian radius of curvature:

(37)

Dividing by time gives

(38)

An eastward displacement follows a parallel of latitude whose local radius is (R_N + h) cos ϕ. Therefore

(39)

so

(40)

Finally, because the third NED axis points down while ellipsoidal height is positive upward,

(41)

Together,

(42)

This is the local-level position mechanization introduced in INS00, now derived from ellipsoid geometry.

9 Why the local navigation frame rotates

Suppose a vehicle moves north while maintaining zero roll, pitch, and heading relative to local NED. The vehicle is following Earth’s curved surface. The local down direction at its new location is not parallel to the local down direction at its old location. Therefore the NED frame has rotated relative to ECEF even though the vehicle did not deliberately turn relative to local level.

The same effect occurs during eastward motion. In addition to following a curved parallel, the directions called north and east change with longitude. This rotation of the local navigation frame caused by translational motion is the transport rate.

10 Derivation of the NED transport rate

Let

(43)

be the angular velocity of the local NED frame relative to ECEF, resolved in NED coordinates.

There are two geometric contributions.

10.1 Change of latitude

Increasing geodetic latitude rotates the local NED frame about its east axis. With the NED sign convention, the corresponding angular velocity is

(44)

The negative sign can be checked at the equator. Moving north tips the local down vector toward north, which corresponds to a negative rotation about east under the right-hand rule.

10.2 Change of longitude

Increasing longitude rotates the local meridian around Earth’s spin axis. The angular velocity associated with this change is

(45)

The Earth spin axis resolved in NED has the same geometry as Earth rate, except without the factor Ω_E:

(46)

Therefore the longitude contribution is

(47)

Adding latitude and longitude contributions gives

(48)

Now substitute the position-rate equations:

(49)

(50)

The result is the standard NED transport rate

(51)

11 Total navigation-frame angular rate relative to inertial space

Angular velocities add when they are expressed in the same coordinates. Since ECEF rotates relative to ECI and NED rotates relative to ECEF,

(52)

Substituting the Earth-rate and transport-rate expressions gives

(53)

This vector is one of the most important reference-rate quantities in a local-level strapdown navigator.

11.1 Numerical scale example

Consider

(54)

Using WGS-84,

(55)

Earth rate is

(56)

while transport rate is

(57)

Therefore

(58)

The transport-rate magnitude in this example is about 9.69^∘∕h, which is not negligible compared with Earth’s 15.04^∘∕h rotation rate. Fast motion over Earth can therefore make transport rate a major part of the attitude reference-rate correction.

12 From inertial gyro rate to body rate relative to NED

An ideal strapdown gyro measures

(59)

the body frame relative to inertial space, resolved in body coordinates.

But attitude C_bⁿ describes the body relative to the navigation frame. The desired relative angular velocity is therefore

(60)

The angular-rate chain is

(61)

All terms must be represented in the same frame before subtraction. Resolving in body coordinates,

(62)

Hence

(63)

Using the previous result,

(64)

This equation explains why a navigation-grade gyro correction depends on both current position and current velocity. Latitude determines Earth rate in NED, while velocity and ellipsoid curvature determine transport rate.

13 Derivation of the strapdown attitude equation

INS04 introduced the result

(65)

We can now interpret every term.

Start with the composition

(66)

Differentiate:

(67)

For the inertial-to-navigation passive transformation,

(68)

For body axes rotating relative to inertial space,

(69)

Substitution gives

(70)

Since C_iⁿC _bⁱ = C _bⁿ,

(71)

Finally,

(72)

The first term advances attitude using the inertial angular rate measured by the gyros. The second term removes the rotation of the navigation frame itself.

14 ECEF velocity dynamics

INS05 showed that when effective gravity already contains centrifugal acceleration, the ECEF velocity equation is

(73)

This equation contains Earth rate but not transport rate. That is because ECEF is a global Earth-fixed frame. It rotates with Earth, but it does not rotate further as the vehicle moves across the surface.

Transport rate appears only after the velocity coordinates are expressed in the moving local NED frame.

15 Deriving the NED velocity equation from ECEF

Start from

(74)

Differentiate:

(75)

Because NED rotates relative to ECEF at ω_enⁿ,

(76)

Thus

(77)

Substitute the ECEF equation:

(78)

Use

(79)

and the cross-product transformation identity from INS01,

(80)

Therefore

(81)

The transport-rate term has now appeared explicitly because the coordinate frame used for velocity moves over the ellipsoid with the vehicle.

16 The full local-level mechanization loop

The frame structure developed in this entry closes several feedback loops in the navigation computation.

Position provides ϕ, λ, and h. These determine the local ECEF-to-NED transformation, Earth-rate components, curvature radii, and gravity model. Velocity provides v_N and v_E, which determine transport rate. Earth rate and transport rate correct the gyro measurements and enter the velocity equation. Updated attitude rotates accelerometer specific force into NED. Updated velocity then advances position.

A local-level strapdown navigator therefore has the coupled structure

(82)

Every quantity in these equations has now been connected either to sensor physics, rotating-frame kinematics, Earth geometry, or gravity modeling.

17 Body-frame conventions

The body frame b is fixed to the IMU or vehicle. In this series the default convention is forward-right-down:

(83)

This convention is common in aerospace inertial navigation, but not universal. Robotics software often uses forward-left-up or other conventions. A DCM that is algebraically correct for one body-axis convention can be physically wrong for another.

A robust implementation should therefore document three items independently:

1. the physical direction of each sensor axis;
2. the coordinate convention expected by the navigation mechanization;
3. the exact DCM or permutation used to map sensor coordinates into the mechanization body frame.

Axis convention errors often mimic attitude-sign or gravity-sign errors and can be difficult to isolate after integration begins.

18 Local-level singularity near the poles

The longitude-rate equation contains

(84)

As |ϕ|→ 90^∘, longitude becomes poorly conditioned because all meridians converge at the pole. The transport-rate down component also contains

(85)

This is not a physical singularity in vehicle motion. It is a coordinate singularity of the conventional local-level parameterization.

High-latitude systems can avoid the problem by using ECEF mechanization, wander-azimuth frames, grid navigation frames, or other local coordinate constructions designed to remain well conditioned near the poles [1, 2].

19 Implementation checks

Frame mathematics is unusually vulnerable to sign, transpose, and ordering mistakes. Several tests should be built into a strapdown implementation.

19.1 DCM orthogonality

Every proper frame transform should satisfy

(86)

19.2 Round-trip transformation

For an arbitrary vector,

(87)

should reproduce the original vector to numerical precision.

19.3 Earth-rate limiting cases

At the equator,

(88)

At the North Pole,

(89)

A code result with the opposite down sign usually indicates either an ENU/NED mismatch or an incorrect ECEF-to-local DCM.

19.4 Zero-velocity transport rate

If

(90)

then

(91)

A stationary Earth-fixed vehicle still sees Earth rate, but it has no transport rate.

19.5 Pure north motion

If v_E = 0,

(92)

Only the east-axis transport component remains.

19.6 Pure east motion

If v_N = 0,

(93)

This makes a useful test of both the north and down transport components.

20 What the IMU knows and what the navigator must know

A useful conceptual distinction is that the IMU alone does not know latitude, longitude, ellipsoid curvature, or transport rate. The gyroscopes provide ω_ib^b. The accelerometers provide f _ib^b. The navigation computer supplies the Earth and geometry model needed to interpret those inertial measurements in an Earth-relative frame.

For a local-level mechanization the computer must maintain or compute

(94)

This separation is fundamental. The sensors measure inertial quantities. The Earth model supplies the rotating reference frame in which navigation outputs are desired.

21 Connection to subsequent strapdown lessons

The next articles use the frame structure developed here repeatedly.

INS07 will use

(95)

to develop attitude propagation in greater detail.

INS08 will reformulate the same rotational kinematics using quaternions and finite incremental rotations.

INS09 will use gravity and Earth rate as reference vectors for initial alignment.

INS12 will construct the complete ECEF mechanization, where no local transport-rate correction appears explicitly.

INS13 and INS14 will return to local-level transport rate and ellipsoidal position kinematics in greater numerical detail.

INS15 will assemble the full NED attitude, velocity, and position mechanization into one continuous system.

22 Summary

The essential frame relationships are

(96)

Earth rate in NED is

(97)

Transport rate is

(98)

The local geodetic position rates are

(99)

The gyro rate needed for attitude relative to NED is

(100)

The corresponding attitude equation is

(101)

Finally, the NED velocity equation is

(102)

The purpose of the frame hierarchy is now clear. Earth rotation, motion over the ellipsoid, body motion, gravity, and IMU measurements all refer to different physical relationships. Strapdown navigation works only when those relationships are transformed into common coordinates with the correct frame ordering and angular-rate signs.

References