Strapdown Inertial Navigation: Roadmap, Frames, and the Mechanization Problem

A strapdown inertial navigation system estimates attitude, velocity, and position by integrating measurements from gyroscopes and accelerometers that are rigidly attached to a moving vehicle. No external reference is required for the short-term propagation itself. This is both the great strength and the central difficulty of inertial navigation: every navigation quantity must be reconstructed from local measurements of rotation and specific force while the vehicle, the navigation frame, and the Earth are all moving relative to one another.

The purpose of this opening article is to establish the physical picture, notation, coordinate-frame conventions, and final equations that the rest of the series will derive in detail. Nothing in the complete strapdown equations should eventually appear as an unexplained “correction term.” Earth rotation, transport rate, gravity, Coriolis acceleration, and the attitude transformation all arise from ordinary mechanics and rotating-frame kinematics.

The central processing chain is

(1)

The treatment follows the physical and mathematical structure used in standard inertial-navigation references, especially Titterton and Weston [1], Groves [2], Jekeli [3], and the modern integration-algorithm treatment of Savage [4, 5].

1 Learning objectives

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

1. explain what a strapdown inertial navigation system estimates and why integration is required;
2. distinguish a physical vector from the coordinates used to represent that vector;
3. identify the inertial, Earth fixed, local navigation, and body frames used throughout the series;
4. interpret the notation C_bⁿ, v _ebⁿ, and ω _ib^b;
5. explain physically what an ideal gyroscope and an ideal accelerometer measure;
6. explain why an accelerometer at rest on a table measures approximately one g even though its translational acceleration is zero;
7. state the complete attitude, velocity, and position mechanization equations that the later lessons will derive;
8. identify where Earth rotation, navigation frame transport, gravity, and Coriolis terms enter the mechanization;
9. explain why attitude error immediately contaminates translational navigation;
10. describe the sequence of topics needed to progress from raw IMU data to a working strapdown inertial navigator.

2 What inertial navigation is trying to estimate

For a vehicle moving near the Earth, the navigation problem is usually separated into three state groups:

1. attitude: orientation of the vehicle body frame relative to a chosen navigation frame;
2. velocity: translational velocity of the vehicle relative to the Earth, resolved in the navigation frame;
3. position: location of the vehicle, often represented by latitude, longitude, and height.

In a local North-East-Down navigation frame, one common state description is

(2)

where C_bⁿ describes attitude, v _ebⁿ is Earth relative velocity resolved in the navigation frame, and (ϕ,λ,h) are geodetic latitude, longitude, and ellipsoidal height.

The navigation computer does not measure these quantities directly. Its primary inertial inputs are angular motion from the gyroscopes and specific force from the accelerometers.

3 Why the system is called strapdown

Early inertial navigation systems often mounted accelerometers on a mechanically stabilized platform. Gyroscopes drove gimbals so that the sensor platform remained aligned with a useful reference frame while the vehicle rotated around it. In that architecture the mechanical platform performed much of the coordinate transformation physically.

A strapdown system removes the stabilized platform. The inertial measurement unit is rigidly attached, or “strapped down,” to the vehicle. As a result, the sensors rotate whenever the vehicle rotates. The coordinate transformation that once occurred mechanically must now be performed mathematically in the navigation computer [1, 2].

This changes the problem fundamentally:

(3)

High rate digital computation and compact solid state sensors make the strapdown architecture practical, but they also expose the navigator directly to finite rotation effects such as coning and sculling. Those topics appear later in the series.

4 A vector is not the same thing as its coordinates

This distinction is essential throughout inertial navigation.

A physical vector, such as velocity v, exists independently of the coordinate frame used to describe it. Its numerical components change when the basis changes. We therefore write a superscript to indicate the frame in which the components are resolved:

(4)

for body frame components and

(5)

for navigation frame components.

The two component sets describe the same geometric vector and are related by a direction cosine matrix:

(6)

Throughout this series, C_bⁿ is a passive coordinate transformation: it maps components resolved in frame b into components resolved in frame n.

Because the transformation represents an orthonormal rotation,

(7)

The detailed construction of direction cosine matrices and their relation to basis vectors is the subject of INS01.

5 The four principal coordinate frames

A strapdown navigator near Earth commonly uses four frames.

5.1 Inertial frame i

The inertial frame is an ideal nonaccelerating, nonrotating frame suitable for Newton’s laws. In practical terrestrial navigation an Earth centered inertial approximation is commonly used over the relevant time scales.

The point of introducing i is conceptual as well as computational: gyroscopes fundamentally sense body rotation relative to inertial space, and Newton’s second law has its cleanest form in an inertial frame.

5.2 Earth frame e

The Earth centered, Earth fixed frame rotates with the Earth. Its origin is near the Earth’s center of mass and its axes remain fixed relative to the terrestrial surface.

The Earth frame is therefore not inertial. Its angular velocity relative to the inertial frame is the Earth rotation vector

(8)

The conventional Earth rotation magnitude used in navigation is approximately

(9)

which is about 15.04 degrees per hour.

5.3 Navigation frame n

The local navigation frame is attached to the vehicle’s current location. This series uses the common North-East-Down convention:

(10)

Even if the Earth were not spinning, this frame would rotate as the vehicle moves over the curved Earth. That motion generated angular rate is the transport rate.

Thus the navigation frame rotates relative to inertial space for two distinct reasons:

(11)

The first term is Earth rotation; the second is navigation frame transport due to motion over the Earth.

5.4 Body frame b

The body frame is fixed to the vehicle and normally aligned with convenient vehicle axes. A common aerospace convention is forward-right-down.

The IMU measurements are naturally resolved in this frame. Consequently, the navigator must continuously estimate how b is oriented relative to n.

6 Angular rate notation

Angular velocity notation is a common source of confusion, so this series uses an explicit three index convention:

(12)

It means:

angular velocity of frame b relative to frame a, resolved into coordinates of frame c.

Examples are

(13)

for body rotation relative to inertial space, resolved in body coordinates, and

(14)

for Earth rotation relative to inertial space, resolved in the local navigation frame.

This explicit notation makes equations longer, but it prevents physically different angular rates from being mistaken for one another.

7 What the gyroscopes measure

An ideal three axis gyroscope measures the angular velocity of the body frame relative to inertial space, resolved in body coordinates:

(15)

A real gyro reports something closer to

(16)

The exact sensor error model is deferred to INS18.

The important physical point for now is that a gyro attached to an object sitting motionless on Earth’s surface does not ideally read zero. The body is rotating through inertial space with the Earth. In a sufficiently accurate gyro, Earth rotation is observable and can even be used for heading alignment.

8 What the accelerometers measure

An accelerometer does not directly measure gravitational acceleration, and it does not directly measure the coordinate acceleration d²r∕dt².

An ideal accelerometer measures specific force: the nongravitational force per unit mass acting on the proof mass. In an inertial frame,

(17)

where a is inertial acceleration and g is gravitational acceleration.

The body mounted accelerometers provide body resolved specific force,

(18)

A real accelerometer measurement can be represented schematically as

(19)

8.1 The table top thought experiment

Suppose an accelerometer rests on a table. Its translational acceleration relative to the table is approximately zero, yet the accelerometer reads approximately one g upward along the local vertical axis.

Why?

Gravity pulls the accelerometer proof mass downward, while the housing and support structure exert an upward contact force that prevents free fall. The accelerometer senses the nongravitational support force per unit mass.

By contrast, in ideal ballistic free fall,

(20)

so

(21)

The proof mass becomes locally weightless and the accelerometer ideally reads zero.

This distinction between gravity and specific force is one of the central physical ideas in inertial navigation and is developed carefully in INS03.

9 Why attitude must be solved before translational acceleration

Accelerometers report f^b, but the velocity state is normally maintained in another frame such as NED or ECEF. Therefore the specific force vector must be transformed:

(22)

This simple equation explains why attitude errors are so dangerous. If C_bⁿ is wrong, the navigator resolves the measured force into the wrong directions.

A small tilt error δ𝜃 causes a component of the approximately vertical gravity supporting specific force to leak into a horizontal channel. For small angles,

(23)

Thus even a tiny attitude error can produce a persistent false horizontal acceleration. INS19 will turn this observation into a complete linear error model and derive Schuler dynamics.

10 The simplest navigation equation: inertial frame first

Before Earth rotation and local level geometry are introduced, the translational mechanics are remarkably simple.

Newton’s second law per unit mass is

(24)

Since the accelerometers measure specific force in body coordinates,

(25)

Therefore

(26)

This is the conceptual core of strapdown translational navigation:

(27)

The complexity of terrestrial navigation arises because practical coordinates such as ECEF and NED are themselves rotating and because gravity is not perfectly uniform.

11 The rotating frame problem

If a vector is differentiated in a rotating frame, its time derivative differs from the derivative seen in an inertial frame. The fundamental transport theorem is

(28)

Applying this relation repeatedly to position and velocity produces the familiar noninertial acceleration terms:

(29)

(30)

and, when the rotation rate itself changes,

(31)

These become the Coriolis, centrifugal, and Euler acceleration terms. INS02 derives them from rotating basis vectors rather than introducing them by name.

12 The three equations the series is building toward

The rest of the course derives and implements three coupled differential equations: attitude, velocity, and position.

12.1 Attitude equation

With the passive transformation convention used here,

(32)

The skew-symmetric matrix [a]_× is defined so that

(33)

The first term propagates body rotation measured by the gyroscopes. The second accounts for rotation of the navigation frame itself.

Since

(34)

the attitude solution must ultimately compensate both Earth rotation and transport rate.

INS07 derives the DCM equation, while INS08 derives the equivalent quaternion propagation equation.

12.2 Velocity equation

For an Earth relative velocity maintained in NED coordinates, the standard local level mechanization has the form

(35)

Every term has a physical origin:

• C_bⁿf _ib^b: measured specific force rotated from body coordinates into navigation coordinates;
• gⁿ: gravity restored because accelerometers do not measure gravity directly;
• 2ω_ieⁿ × v _ebⁿ: Coriolis acceleration associated with Earth rotation;
• ω_enⁿ × v _ebⁿ: correction associated with rotation of the local navigation frame as it moves over Earth.

Depending on the exact gravity definition, centrifugal effects are conventionally absorbed into the gravity model. INS05 and INS11 make that bookkeeping explicit.

12.3 Position equation

For geodetic latitude ϕ, longitude λ, and ellipsoidal height h,

(36)

(37)

and, with NED coordinates,

(38)

Here R_M and R_N are the meridian and prime vertical radii of curvature of the reference ellipsoid.

These relations are not arbitrary conversion formulas. They follow from Differential Geometry of motion over the ellipsoid and are derived in INS14.

13 The mechanization is a coupled feedback process

The attitude, velocity, and position equations cannot be solved independently.

Position determines latitude and height, which influence gravity and the resolved Earth rotation rate. Velocity and position determine transport rate. Earth rate and transport rate enter attitude propagation. Attitude determines how measured specific force is projected into the navigation frame. That projected force changes velocity, and velocity changes position.

This coupling is why consistent frame notation and timing are essential in implementation.

14 Continuous equations versus actual IMU data

The differential equations above are mathematically continuous. A real digital IMU normally delivers sampled data, often as increments over a finite time interval:

(39)

from the gyroscopes and

(40)

from the accelerometers.

Therefore a working navigator must approximate the continuous dynamics over each finite sample interval. Straightforward first order integration is not always sufficient because finite rotations do not commute and because translational increments occur while the body is rotating.

This leads to the advanced strapdown corrections:

• coning corrections for accumulated rotation;
• sculling corrections for velocity increments acquired under rotation;
• high resolution position corrections sometimes called scrolling in the strapdown literature.

Savage’s two part integration algorithm treatment is a standard source for these subjects [4, 5]. They are deferred to INS16 and INS17 so the continuous physics is fully understood first.

15 What a pure INS can and cannot know

A strapdown INS is a dead reckoning system. Once initialized, it can propagate motion without receiving an external signal. This gives it excellent short term continuity and makes it resistant to radio frequency outages.

However, integration also accumulates sensor error. Biases in gyroscopes and accelerometers do not average away automatically. Instead they drive growing errors in attitude, velocity, and position.

A simple accelerometer bias example illustrates this. If a constant horizontal acceleration error b_a is integrated once,

(41)

Integrating again gives

(42)

Gyro bias can be even more damaging because it first creates attitude error and then misprojects gravity into the horizontal channels.

Consequently, practical navigation systems commonly aid the INS with GNSS, barometers, magnetometers, odometers, cameras, radar, lidar, or other external measurements. Those aiding systems do not replace the strapdown mechanization; they estimate and correct its errors.

This course deliberately postpones Kalman filtering until the deterministic mechanization and error physics are understood.

16 Three thought experiments to carry through the series

16.1 Stationary IMU on the Earth

A motionless IMU on a laboratory table is not dynamically trivial.

Its accelerometers measure the support specific force associated with gravity, while sufficiently sensitive gyroscopes measure Earth rotation. A correct mechanization should nevertheless maintain nearly zero Earth relative velocity and fixed position after alignment.

This is one of the most useful numerical acceptance tests for a strapdown navigator.

16.2 IMU in free fall

In ideal free fall,

(43)

The accelerometers therefore read zero even though the vehicle accelerates gravitationally. The navigation computer must recover gravitational acceleration from its gravity model.

This thought experiment is the clearest test of the difference between acceleration and specific force.

16.3 Vehicle undergoing pure rotation

Suppose the vehicle rotates in place without translating. The gyroscopes measure angular motion, the attitude changes, but the true velocity and position remain fixed.

A correct mechanization must continuously rotate the measured support specific force vector while still producing zero translational motion after gravity and rotating frame terms are handled consistently.

This example exposes attitude sign, frame order, and gravity projection mistakes very quickly.

17 Series roadmap

The Strapdown Inertial Navigation series is organized so that each term in the final mechanization equations is derived from simpler physics before it is implemented.

18 A notation reference for the rest of the series

19 Implementation discipline

Several habits will be maintained throughout the series because they prevent common strapdown navigation errors.

1. Every vector quantity carries an explicit resolving frame until the context is completely unambiguous.
2. Every angular velocity identifies both frames and the resolving frame.
3. The passive DCM convention is not changed between articles.
4. NED uses positive Down, so ḣ = −v_D.
5. Gravity model definitions state explicitly whether centrifugal effects are included.
6. Continuous time derivations are separated from sampled data algorithms.
7. Analytic or synthetic truth cases are used before complicated flight data.
8. Every mechanization implementation is tested first with stationary, pure rotation, and simple constant acceleration cases.

These conventions are not stylistic decoration. Most implementation failures in inertial navigation are ultimately frame, sign, timing, or model consistency errors.

20 Summary

A strapdown INS reconstructs navigation states from body fixed measurements. Its ideal measurements are

(44)

from the gyroscopes and

(45)

from the accelerometers.

The central physical relationship for translation is

(46)

so specific force must be transformed into the navigation frame and combined with an appropriate gravity model before velocity can be propagated.

The local level attitude equation is

(47)

with

(48)

The corresponding NED velocity equation is

(49)

Position is propagated through ellipsoidal geometry:

(50)

The remaining articles derive each of these relationships from first principles, then convert them into a discrete strapdown algorithm that can be validated numerically.

References