Notation and Nomenclature

\(x \quad\quad \text{Real scalar value}\)

\(\mathbf{x} \quad\quad \text{Real column vector}\)

\(x_i \quad\quad i\text{'th scalar component of column vector } \mathbf{x}\)

\(\mathbf{M} \quad\quad \text{Real matrix}\)

\(\mathbf{I} \quad\quad \text{Identity matrix}\)

\(\mathbf{I}_i \quad\quad i\text{-th column vector of identity matrix}\)

\(\mathbf{F}_{\alpha} \quad \ \text{ Right-handed coordinate frame } \alpha\)

\(\mathbf{t}^\gamma_{\beta \alpha} \quad \text{Vector with reference frame } \beta \text{ object frame } \alpha \text{ resolved in frame } \gamma \text{ axes}\)

\(\boldsymbol{\psi}_{\beta \alpha} \quad \text{Euler rotation from frame } \beta \text{ to frame } \alpha\)

\(\mathbf{t}_{\beta \alpha} \quad \text{Vector from frame } \beta \text{'s origin to frame } \alpha \text{'s origin}\)

\(\mathbf{R} \quad\quad SO(3) \text{ rotation matrix}\)

\(\mathbf{R}^{\alpha}_\beta \quad\quad \text{Rotation from } F_\beta \text{ to } F_\alpha\)

\(\mathbf{T} \quad\quad SE(4) \text{ transformation matrix}\)

\(\mathbf{T}^{\alpha}_\beta \quad\quad \text{Transformation from } F_\beta \text{ to } F_\alpha\)

\(\left[ \ \cdot \ \right]_\times \quad \text{Skew matrix operator}\)

\(\left[ \ \cdot \ \right]^{-}_{\times} \quad \text{Inverse skew matrix operator}\)

\(\hat{x} \quad\quad \ \ \text{Estimated quantity of } x\)

\(\tilde{x} \quad\quad \ \ \text{Measured quantity of } x\)

\(|| \mathbf{x} ||^2_{\mathbf{A}} \quad \text{2-norm of } \mathbf{x} \ \text{weighted by the covariance matrix } \mathbf{A}. \ \text{Equivalent to } \mathbf{x}^T \mathbf{A}^{-1} \mathbf{x}.\)