Strapdown Inertial Navigation: Attitude Kinematics from Gyroscope Measurements

A strapdown inertial navigation system has no mechanically stabilized platform. The inertial sensors rotate with the vehicle, so the navigation computer must continuously reconstruct the orientation of the body frame from gyroscope measurements. This reconstruction is the attitude mechanization.

The physical input is the inertial body angular rate measured by the gyro triad,

(1)

For navigation relative to the local NED frame, this is not yet the rate that should drive the attitude solution. INS06 showed that the navigation frame itself rotates relative to inertial space with

(2)

Therefore the body rate relative to NED is

(3)

This one physical angular-rate vector can drive several mathematical attitude representations. Four are especially important in inertial navigation:

1. the direction cosine matrix (DCM);
2. three Euler Angles;
3. a unit quaternion;
4. a finite rotation vector.

They do not describe four different attitudes. They are four coordinate descriptions of the same element of the three-dimensional rotation group SO(3) [1, 2, 3, 4].

1 Learning objectives

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

1. explain why a gyroscope measures angular rate rather than attitude;
2. convert the gyro measurement ω_ib^b into the body rate relative to NED;
3. derive the continuous DCM attitude equation;
4. derive the 3-2-1 Euler-angle rate equations from the body angular-rate vector;
5. identify the Euler-angle singularity at pitch 𝜃 = ±90^∘;
6. define a scalar-first Hamilton quaternion that represents C_bⁿ;
7. derive quaternion kinematics from the same body-relative angular rate;
8. construct an exact finite quaternion increment from an angular increment;
9. relate the matrix exponential, Rodrigues formula, and a finite rotation vector;
10. state the Bortz rotation-vector differential equation and its small-angle form;
11. compare the numerical advantages and disadvantages of DCM, Euler-angle, quaternion, and rotation-vector propagation;
12. formulate practical invariants and unit tests for an attitude mechanization.

2 The attitude problem is a kinematic problem

Let C_bⁿ transform body-resolved vector components into navigation-frame components,

(4)

At one instant, C_bⁿ answers the geometric question: how are the body axes oriented relative to the navigation axes? The gyroscope answers a different question: how rapidly is that orientation changing?

The attitude mechanization must therefore integrate angular velocity on the geometry of rotations.

For scalar translation, integrating velocity is straightforward because ordinary vector addition commutes. Three-dimensional rotations are different. If R_x and R_y are finite rotations about two different axes, then in general

(5)

This noncommutativity is why three gyro channels cannot generally be integrated as three independent scalar angles.

3 From gyro rate to navigation-relative body rate

The ideal gyro triad supplies

(6)

The local navigation frame has inertial angular rate

(7)

Resolved in body coordinates,

(8)

Angular velocities add according to their frame relationships,

(9)

Thus

(10)

A stationary vehicle fixed to Earth is an important limiting case. If the body remains aligned with NED, then

(11)

and therefore

(12)

The gyro can measure Earth rate while the Earth-relative attitude remains constant.

4 Direction cosine matrix kinematics

The DCM is the most direct representation of the coordinate transformation itself. Its columns are the body basis vectors resolved in navigation coordinates.

Let

(13)

Each body basis vector rotates relative to the navigation frame according to

(14)

Resolve the angular rate in body coordinates. Using the skew-symmetric cross-product matrix introduced in INS01,

(15)

collecting the three differentiated columns gives

(16)

This is the relative-rate form of the DCM attitude equation.

4.1 Recovering the full strapdown equation

Insert

(17)

Then

= Cbn[ω ibb] ×− Cbn[C nbω inn] ×.

(18)

The cross-product transformation identity is

(19)

Therefore

(20)

Using INS06,

(21)

The first term rotates the body relative to inertial space. The second term removes the rotation of the navigation frame itself.

5 Finite DCM propagation

Suppose the corrected body-relative angular rate is approximately constant over a short interval Δt. Define the body-resolved angular increment

(22)

The DCM differential equation then has the finite solution

(23)

The matrix exponential is a true finite rotation. Let

(24)

Rodrigues’ formula gives

(25)

For small α,

(26)

The second-order term is important because I + [Δ𝜃]_× is not exactly orthogonal.

5.1 A two-sided update using raw gyro rate

If the inertial body increment and navigation-frame increment are handled separately, a useful short-interval form is

(27)

This form makes the physics visible. The body is advanced by the gyro-measured inertial rotation on the right, while the navigation axes are advanced on the left. Higher-order algorithms refine how the two increments are formed within the sample interval [4].

6 Euler angles: a minimal but singular description

Euler angles use only three numbers. For navigation, a common convention is the 3-2-1 yaw-pitch-roll sequence

(28)

The attitude matrix may be written

(29)

using the passive DCM convention of INS01.

Let the body rate relative to NED, resolved in body coordinates, be

(30)

The three Euler-angle rates do not equal p, q, and r. Each elemental Euler rotation is taken about a different intermediate axis.

6.1 Deriving the Euler-rate mapping

The roll contribution is already about the final body x axis,

(31)

The pitch rotation, after the roll rotation is accounted for, resolves in body axes as

(32)

The yaw contribution resolves as

(33)

Adding the three contributions gives

(34)

Inverting the matrix gives the familiar 3-2-1 Euler kinematics,

(35)

(36)

and

(37)

6.2 The Euler-angle singularity

As

(38)

both tan 𝜃 and sec 𝜃 become unbounded. The physical attitude and angular velocity remain perfectly finite. It is the coordinate description that becomes singular.

This is often called gimbal lock. It is a representation singularity, not a failure of rigid-body mechanics.

Euler angles remain valuable for display, interpretation, and moderate-angle applications. They are usually a poor choice for the internal attitude state of a general strapdown navigator.

7 Quaternion attitude representation

A unit quaternion represents attitude with four numbers and one norm constraint. In this article a scalar-first Hamilton quaternion is written

(39)

The quaternion represents the same body-to-navigation transformation as C_bⁿ.

One DCM consistent with this convention is

(40)

The signs in quaternion formulas depend on whether a text uses active or passive rotations, scalar-first or scalar-last storage, and Hamilton or alternate multiplication conventions. A navigation implementation should state its convention explicitly and test it against a known one-axis rotation.

8 Quaternion differential equation

Form a pure-vector quaternion from the body-relative angular rate,

(41)

For the convention above, quaternion kinematics are

(42)

Let

(43)

Then

(44)

The 4 × 4 rate matrix is skew-symmetric. Therefore the exact continuous dynamics preserve unit norm.

Indeed,

(qT q)	= 2qT	(45)
	= qT Ω(ω)q	(46)
	= 0.	(47)

Numerical integration can still cause small norm drift, so practical implementations often renormalize after each update.

8.1 Using the raw gyro and navigation rates separately

The relative-rate equation can also be written directly in terms of the gyro measurement and navigation-frame inertial rate,

(48)

This is the quaternion analogue of the two-term DCM equation.

9 Finite quaternion propagation

Suppose a corrected angular increment over one IMU interval is

(49)

Define

(50)

The exact unit quaternion for that finite rotation is

(51)

The attitude update is

(52)

For very small α,

(53)

10 Rotation vectors and the exponential map

A finite rotation may also be represented by a three-component rotation vector

(54)

where u is the rotation axis and α is the rotation angle.

The corresponding DCM increment is

(55)

Because

(56)

its exponential series can be regrouped into Rodrigues’ formula,

(57)

Thus the rotation vector, DCM exponential, and incremental quaternion are three descriptions of the same finite rotation.

11 Why the rotation vector is not simply the integral of angular rate

If the angular velocity keeps a constant direction, then the finite rotation vector is simply

(58)

For general three-dimensional motion, the direction of the angular rate changes, finite rotations do not commute, and the relationship is more complicated.

For the convention

(59)

Bortz’s rotation-vector equation may be written [5, 4]

(60)

where

(61)

For small rotation vectors,

(62)

so

(63)

The cross-product terms are the beginning of the finite-rotation corrections that later lead to coning algorithms.

12 One physical maneuver in four representations

Consider a body already aligned with NED. After reference-frame correction, suppose the body rotates about its down axis at

(64)

for

(65)

Then

(66)

12.1 Euler-angle view

At zero roll and pitch,

(67)

so

(68)

12.2 Quaternion view

The incremental quaternion is

(69)

12.3 Rotation-vector view

The increment is simply

(70)

12.4 DCM view

Rodrigues’ formula gives

(71)

All four descriptions represent the same physical one-degree yaw rotation.

13 Why forward Euler integration damages a DCM

A tempting discrete approximation is

(72)

Let

(73)

Even if C_k^T C _k = I,

Ck+1T C k+1	≈ (I + S)T (I + S)	(74)
	= (I − S)(I + S)	(75)
	= I − S2.	(76)

The error is second order in the angular increment but accumulates over time. A finite exponential update stays orthogonal by construction.

This provides a general numerical lesson:

(77)

14 Quaternion normalization and finite increments

A first-order quaternion integration step has the form

(78)

The new vector q_k+1^∗ will generally not have exactly unit norm in finite-precision arithmetic. A common correction is

(79)

Using an exact incremental quaternion is usually preferable when a finite angular increment is already available from the IMU.

15 Gyro angular increments rather than continuous rates

A real digital IMU commonly reports either an angular rate sampled over a finite interval or an integrated angular increment

(80)

This makes finite rotation updates natural. However, a simple componentwise integral does not capture every effect of a changing rotation axis within the interval. The noncommutative corrections required for high-dynamic motion are called coning corrections. They will be developed later from the same rotation-vector structure introduced here.

16 Representation comparison

The phrase “no singularity” for a quaternion means there is no attitude singularity analogous to Euler gimbal lock. Quaternions have the double-cover property

(81)

representing the same physical attitude.

A finite rotation vector is also not globally unique. Axis-angle representations repeat after full turns, and the principal logarithm becomes ambiguous for rotations near 180^∘. In strapdown algorithms, rotation vectors are therefore especially useful as local finite increments rather than as an unrestricted global attitude coordinate.

17 A stationary-Earth consistency test

One of the strongest attitude-mechanization unit tests uses a body fixed to Earth.

Suppose

(82)

Then

(83)

The ideal gyro measures

(84)

The reference-rate correction yields

ωnbb	= ω ibb − C nbω ien	(85)
	= 0.	(86)

Every representation must therefore remain constant:

(87)

If a stationary Earth-fixed simulated IMU causes attitude to rotate at approximately Earth rate, the navigation-frame correction is missing or has the wrong sign.

18 Gyro bias and attitude drift

Let the measured gyro rate contain a small constant bias,

(88)

Over a short interval in which the frame geometry can be treated as fixed, the corresponding small attitude error grows approximately as

(89)

This is the rotational analogue of integrating an accelerometer bias into velocity error. The consequences are even more severe because attitude error rotates gravity into the horizontal acceleration channels. INS19 will derive the resulting coupled navigation error dynamics.

19 Implementation checks

A robust attitude implementation should contain invariants that can be checked automatically.

19.1 DCM checks

Verify

(90)

A determinant near one by itself is not enough. Orthogonality must also be checked.

19.2 Quaternion checks

Verify

(91)

and confirm

(92)

when quaternion and DCM propagators are run in parallel.

19.3 One-axis exact solution

For constant rate r about the down axis,

(93)

has an exact closed-form solution. DCM, quaternion, Euler-angle, and rotation-vector implementations should all reproduce it.

19.4 Stationary Earth-fixed solution

After Earth-rate correction, a body fixed to the local NED frame should have zero relative angular rate and constant attitude.

19.5 Round-trip coordinate transformations

For nonsingular attitudes, convert

(94)

and

(95)

The reconstructed DCM should agree with the original to numerical precision.

20 Connection to the rest of the strapdown mechanization

Attitude is not an isolated output. The accelerometers measure specific force in body coordinates,

(96)

The attitude solution supplies the transformation

(97)

The transformed specific force then enters the NED velocity equation,

(98)

An attitude error is therefore immediately an acceleration error. This is why attitude propagation is the first major numerical operation performed on each new gyro sample.

21 Connection to subsequent INS lessons

INS08 will deepen the quaternion and finite-increment treatment, with special attention to discrete propagation and numerical implementation.

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

INS15 will place the attitude propagator inside the complete NED attitude, velocity, and position mechanization.

INS16 and INS17 will move from ideal continuous rates to sampled IMU increments, including finite-rotation, coning, and sculling corrections.

INS18 and INS19 will introduce sensor errors and derive the resulting attitude and navigation error dynamics.

22 Summary

The gyro measures

(99)

while local-level attitude propagation requires

(100)

The DCM equation is

(101)

For 3-2-1 Euler angles,

(102)

For the scalar-first Hamilton quaternion convention used here,

(103)

A finite rotation vector produces the DCM increment

(104)

and the equivalent quaternion increment

(105)

The representation changes, but the underlying physics does not. The navigation computer is integrating one angular-velocity vector on the geometry of three-dimensional rotations.

References