Table 2.

Description of notations and symbols used in the formulation.

Notation	Description
$\left\{inf\right\}$	Inertial navigational frame.
$\left\{sbf\right\}$	Sensor-body frame.
$s\in R$	Scaler part of quaternions.
$v\in (x,y,z)$	Vector part of quaternions, where $v\in R^3$.
q	Unit quaternion.
$q^{\ast }$	Conjugate quaternion.
⨂	Multiplication operation
$v^{inf}$	Vector in the inertial navigational frame.
$v^{sbf}$	Vector in the sensor body frame.
$i,j,$ and k	A quaternion basis elements.
$\alpha ,\beta ,\gamma$ and $\delta$	Quaternion real numbers.
$q_{inf}^{sbf}$	The unit-vector quaternion encoding rotation from the inertial navigational frame to the body frame of the sensor.
$\alpha$	The amount of rotation that should be performed about the vector part.
${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$	Elements ${\sigma }_1$, ${\sigma }_2$, and ${\sigma }_3$ thought of as a vector about which rotation should be performed.
$\varphi$	The angle of rotation.
$ϵ$	Unit vector representing the axis of rotation.
$Q\left(q\right)$	Rotation matrix.
Q	Four-dimensional vector space over the real numbers $R^4$.
$NED$	North–east down
$\psi$	Rotation around yaw.
$\xi$	Rotation around pitch.
$\phi$	Rotation around roll.
$atan2$	Computes the principal value of the argument function applied to the complex number in the quaternion.
$\delta \varphi$	Prior gyros bias errors. The error between estimated gyroscope bias and true gyroscope bias.
$\delta \psi$	Euler angles errors.
x	State vector of the proposed filter.
$e_q$	Error quaternions.
e	Attitude error.
${\dot{x}}_k=A_k{x}_k+E{u}_k$	The state equation for the attitude estimation system.
$u_k$	The noise vector, which refers to the noise related to the rotation error angle.
${\zeta }_{\delta \Psi }\left(t\right)$	Noise error, true bias random walk.
${\zeta }_{\Delta \Phi }\left(t\right)$	Noise error, estimated bias random walk.
${\widehat{w}}_{b/n^{\prime }}^b$	The estimated rotation rate.
${\widetilde{acc}}^{sbf}$	Output of accelerometer.
${\widetilde{mag}}^{sbf}$	Output of magnetometer.
y	Measurement of the combination of the accelerometer and magnetometer.
${\eta }_{{\mathrm{acc}}^{sbf}}\mathrm{and}{\eta }_{{\mathrm{mag}}^{sbf}}$	The measurement independent zero-mean Gaussian white-noise.
$m^n$ and $g^n$	True magnetic and gravity vector.
${\eta }_{ac{c}^{sbf}}^2\mathrm{and}{\eta }_{ma{g}^{sbf}}^2$	The variance of measurement noise.
$M_R$	Covariance matrix.
$h\left(q\right)$	Represents the nonlinear equations that convert the magnetometer reference vector $r_{mag}$ ∈ $R^3$ and accelerometer reference vector $r_{acc}$ ∈ $R^3$ from INF to the SBF.
$\rho$	Sigma points.
$\lambda$	Represents the scaling parameter that shows the sigma points spread around the column vectors of the covariance matrix.
$K_{x_j}^{\mathrm{prior}}$	The prior estimates of covariance.
$\rho$′	Posterior sigma points.
$W_i^m$ and $W_i^j$	Used to calculate the mean and covariance of the posterior sigma points.
$\mu$	Determines the spread of the $\rho$ around ${\widehat{x}}_k^{\mathrm{prior}}$ and $\beta$ accentuate the weighting on the zeroth $\rho$.
$\left(\tilde{y}\right)$	Residual error.
$D(.)$	Distribution constructed by the kernel density estimate.
wO,t (.)	Weight assigned to each activity performed in the various dedicated zones $Z_i$.