A Primer on the Differential Calculus of 3D Orientations

II. VECTORS AND COORDINATE SYSTEMS NOTATION

• Orientations are defined by the coordinate mappings they induce between coordinate systems, with relative orientation represented by ΦBA ∈ SO(3).Coordinate tuples transform through the associated orientation mapping and rotation-matrix representation.Source: §II · p. 1, §III · p. 2

III. THEORY

• 3D orientations form the Lie group SO(3), which lacks ordinary addition and subtraction needed by filtering and optimization.The exponential map connects SO(3) to its tangent space, isomorphic to R3.Source: §III · p. 2
• Choosing a basis extends the tangent-space map to R3 and defines the exponential exp: R3 → SO(3).On Bπ(0), the exponential is bijective onto all non-180° orientations, with inverse logarithm log: SO(3)∗ → Bπ(0).Source: §III · p. 2
• The boxplus and boxminus operators place actual orientations on SO(3) and their differences on R3.This separation enables smooth embedding of rotational quantities in filtering and optimization frameworks.Source: §III · p. 2
• Orientation differentials are defined by replacing ordinary addition and subtraction with boxplus and boxminus.These constructions extend differential calculus to mappings involving orientations and Euclidean quantities.Source: §III · p. 2

IV. IMPLEMENTATION-INDEPENDENT IDENTITIES

• Concatenating the exponential with the coordinate mapping yields Rodrigues’ formula and identifies ϕ with the relative rotation vector.The rotation vector is associated with the relative orientation of two coordinate systems.Source: §IV · pp. 2–3
• The exponential commutes with coordinate mappings through the adjoint-related identity exp(Φ(ϕ)) = Φ ◦ exp(ϕ) ◦ Φ−1.This expresses how exponential-map rotations transform under a change of coordinates.Source: §IV · p. 3
• The derivative of the exponential map is represented by the Jacobian Γ(ϕ) ∈ R3×3.This Jacobian supplies a reusable differential for orientation-related derivatives.Source: §IV · p. 3

V. QUATERNION IMPLEMENTATION

• Unit quaternions implement orientations as Φ = (q0, ˇq) under the Hamilton convention, subject to q0^2 + ∥ˇq∥^2 = 1.The quaternion has a real part q0 and a three-dimensional imaginary part ˇq.Source: §V · p. 3

A. Coordinates Mapping and Rotation Matrix

• For relative orientation ΦBA, quaternion operations map vector coordinates between arbitrary coordinate systems.The resulting mapping provides the basis for the associated rotation matrix.Source: A. Coordinates Mapping and Rotation… · p. 3
• The quaternion coordinate-mapping expression directly yields the associated rotation matrix.The implementation therefore supplies both coordinate transformation and matrix representations of the same orientation.Source: A. Coordinates Mapping and Rotation… · p. 3

B. Concatenation

• Quaternion concatenation is represented explicitly through scalar and vector components, including the vector cross-product term.Source: B. Concatenation · p. 3

C. Exponential and Logarithm

• For ϕ ∈ R3, the exponential map provides a unit-quaternion representation, while its logarithm recovers the corresponding minimal orientation differential.Source: C. Exponential and Logarithm · p. 3
• Angles below 10^-4 rad require a small-angle approximation to avoid numerical instabilities in the exponential-map implementation.Source: C. Exponential and Logarithm · p. 3

D. Consistency Tests

• Consistency tests compare the implemented operators with Rodriguez’ formula and should include varied samples, especially very small angles.Source: D. Consistency Tests · p. 3

VI. SIMPLE MODELING EXAMPLE

• The modeling example estimates robot position, velocity, and orientation by predicting with an IMU and updating with a pose sensor.Source: §VI · p. 3

A. Continuous Time Description

• The continuous-time IMU model includes additive biases, white-noise processes, gravity, and bias- and noise-corrected acceleration and angular velocity.Source: A. Continuous Time Description · pp. 3–4

• Euler-Forward discretization advances the formulation over a time increment ∆t, while allowing other discretization schemes.Source: B. Euler-Forward Discretization · p. 4
• The discretized IMU measurements are bias- and noise-corrected.Source: B. Euler-Forward Discretization · p. 4
• The discrete Gaussian noise ˆni is distributed as N(0, Ri/∆t) when Ri is the continuous white-noise density.Source: B. Euler-Forward Discretization · p. 4

C. Differentiation

• The process-model Jacobians are derived with F relative to the state and G relative to process noise using orientation identities and the chain rule.Source: C. Differentiation · p. 4

D. Measurement

• GPS position and orientation measurements define the measurement model with discrete Gaussian measurement noise.Source: D. Measurement · p. 4
• Measurement Jacobians H and J are derived using orientation identities and the zero expectation of orientation measurement noise.Source: D. Measurement · p. 4

E. Hints for the EKF Implementation

• The EKF uses the ⊟ operator for innovation residuals and the ⊞ operator for state updates instead of ordinary subtraction and addition.Source: E. Hints for the EKF Implementation · p. 4

VII. CONCLUSION

• The document presents these orientation identities and differentials as tools for computing Jacobians of more complex models through the chain rule.Source: §VII · p. 4

1) Time Derivative of Orientation:

• Angular velocity is defined for a moving coordinate system relative to an inertial observer, with a negative sign matching active rotations measured by typical IMUs.Source: 1) Time Derivative of Orientation · p. 5
• A stated limit enables derivation of the time derivative of an orientation using previously established composition and differential identities.Source: 1) Time Derivative of Orientation · p. 5

2) Derivative of Inverse:

• The derivative of an inverse orientation is derived from orientation composition and inverse-related identities.Source: 2) Derivative of Inverse · p. 5

3) Derivative of Coordinate Map:

• The coordinate-map derivative is formulated from the paper’s orientation identities, providing the basis for subsequent orientation differentials.Source: 3) Derivative of Coordinate Map · p. 5

4) Concatenation - Left:

• The concatenation derivative is derived separately with respect to the left orientation.Source: 4) Concatenation - Left · p. 5

5) Concatenation - Right:

• The concatenation derivative is likewise derived with respect to the right orientation, using an additional identity.Source: 5) Concatenation - Right · p. 5

6) Exponential Derivative:

• The exponential derivative is obtained by differentiating exp(xϕ) through the chain rule and an alternative limit-based route, then comparing both derivatives at x = 1.Source: 6) Exponential Derivative · p. 6
• The two exponential-derivative routes are combined with eq. (74) to obtain a matrix equation and, after substituting C(ϕ), eq. (33).Source: 6) Exponential Derivative · p. 6

A. Rodriguez’ Formula

• The appendix states properties of C(ϕ) = C(exp(ϕ)) for arbitrary t, ϕ, and v.Source: A. Rodriguez’ Formula · p. 6
• For fixed ϕ, the curve Cϕ(t) := C(tϕ) is introduced, and a coordinate change extends the differential identity’s range.Source: A. Rodriguez’ Formula · p. 6
• The resulting matrix differential equation has a matrix-exponential solution valid for arbitrary ϕ.Source: A. Rodriguez’ Formula · p. 6
• The derived solution is shown equivalent to eq. (21) using series expansions.Source: A. Rodriguez’ Formula · p. 6

• The proof begins by identifying the target adjoint-related identity to establish.Source: B. Concatenation and Exponential –… · p. 6
• Uniqueness of the exponential reduces the proof to verifying that the conjugated expression satisfies the defining properties, beginning with eq. (12).Source: B. Concatenation and Exponential –… · p. 6
• The derivative requirement in eq. (13) is separately verified for the conjugated expression.Source: B. Concatenation and Exponential –… · p. 6
• Because the conjugated map satisfies both uniquely defining properties, it is equivalent to exp(Φ(ϕ)).Source: B. Concatenation and Exponential –… · p. 6