Notation & glossary

Every equation on this site uses the symbols below. Conventions follow OpenVINS, which shares this lab’s lineage — if you arrive from there, nothing needs re-learning.

Frames

Symbol	Frame	Notes
$\{G\}$	global	gravity-aligned; yaw and position are unobservable
$\{I\}$	base IMU	the reference clock and the trajectory’s body frame
$\{I_{s_j}\}$	auxiliary IMU $j$	rigidly attached to $\{I\}$
$\{C_i\}$	camera $i$	optical frame; $z$ forward
$\{B\}$	calibration board	fixed in $\{G\}$

Transforms

A rotation is written ${}^{A}_{B}\mathbf{R}$ and takes a vector expressed in $\{B\}$ into $\{A\}$. A translation ${}^{A}\mathbf{p}_{B}$ is the origin of $\{B\}$ expressed in $\{A\}$. Together:

$${}^{A}\mathbf{p} = {}^{A}_{B}\mathbf{R}\, {}^{B}\mathbf{p} + {}^{A}\mathbf{p}_{B}$$

The result file uses a different spelling

Calibration output writes T_C0toI for the transform taking points from $\{C_0\}$ into $\{I\}$ — the same object as ${}^{I}_{C_0}\mathbf{T}$. The XtoY naming reads left-to-right in the direction of the mapping. See Result formats.

Calibration state

Symbol	Quantity	Dimension	Per
$\mathbf{f}_i = [f_u\ f_v\ c_u\ c_v]$	camera intrinsics	4	camera
$\mathbf{d}_i$	distortion coefficients	4+	camera
${}^{I}_{C_i}\mathbf{T}$	camera extrinsics	6	camera
$t_{d,i}$	camera–IMU time offset	1	camera
$t_{r,i}$	rolling-shutter readout	1	RS camera
$\mathbf{D}_w,\ \mathbf{D}_a$	gyro / accel scale + misalignment	6 each	IMU
$\mathbf{T}_g$	g-sensitivity	9	IMU
${}^{I}_{I_{s_j}}\mathbf{T}$	auxiliary IMU extrinsics	6	aux IMU
$t_{d,j}$	IMU–IMU time offset	1	aux IMU
$\mathbf{b}_g,\ \mathbf{b}_a$	gyro / accel biases	3 each	IMU

The full state $\mathbf{x}$ stacks the trajectory with all of the above; the cost that acts on it is in Calibration graph.

Measurements

Symbol	Measurement
$\boldsymbol{\omega}_m$	measured angular velocity
$\mathbf{a}_m$	measured specific force
$\mathbf{z}_{i,k}$	pixel observation of a board corner
$\mathbf{r}$	residual (subscripted by factor type)
$\boldsymbol{\Sigma}$	measurement covariance

Time

Symbol	Meaning
$t_I$	timestamp on the base-IMU clock — the reference
$t_C$	timestamp as reported by a camera driver
$t_d$	offset such that $t_I = t_C + t_d$
$t_r$	whole-frame readout time of a rolling-shutter camera

The sign convention matters: td is added to the sensor’s own stamp to reach base-IMU time. The result file states it explicitly as (t_Ib = t_C + td) next to each value.

Conventions worth stating once

• Quaternions are JPL-convention, as in OpenVINS.
• Error states are defined multiplicatively for rotations, $\mathbf{R} \approx \hat{\mathbf{R}}\,\exp(\delta\boldsymbol{\theta}^\wedge)$, and additively for everything else.
• Board points are exact — the target is treated as rigid and known, so board geometry errors appear as a scale bias on translations rather than as noise.