Glossary · glass-lio course

Bias: A slowly drifting offset added to a sensor's reading, in the body frame: \(\mathbf{b}^{g}\) for the gyroscope, \(\mathbf{b}^{a}\) for the accelerometer. It is modelled as a random walk and estimated alongside the pose. A known stationary interval can reveal gyro bias; separating accelerometer bias from gravity and attitude requires additional information and suitable excitation.
Body frame: The frame attached to the IMU, written \(B\). The gyroscope and accelerometer measure in it, and biases live in it. \(\mathbf{R}_{WB}\) rotates body-frame vectors into the world frame.
Coasting: Keeping the IMU prediction as the pose when a scan's registration is rejected, and not inserting that scan into the map. A slightly stale pose recovers on the next scan; a wrong pose in the map does not.
Degeneracy: Geometry that leaves some direction unconstrained, such as sliding along a corridor. It shows up as a small eigenvalue of \(\mathbf{H}_t = \sum_i w_i\,\mathbf{n}_i\mathbf{n}_i^{\top}\); glass-lio tests the smallest eigenvalue divided by the trace against a gate in the loose path. This tests pure translation with rotation fixed; it does not certify full pose observability.
Deskew: Undoing the motion that happened while a LiDAR sweep was being captured, by moving every point into the frame of one instant using the gyro-integrated rotation.
Exp and Log maps: The maps between a rotation vector in \(\mathbb{R}^3\) and a rotation matrix in SO(3). \(\mathrm{Exp}\) turns a small rotation such as \(\boldsymbol{\omega}\,\Delta t\) into a rotation; \(\mathrm{Log}\) turns a rotation back into a vector, which is how rotation residuals are measured.
Factor: One term in a least-squares cost that ties variables to a measurement, weighted by that measurement's information: a point-to-plane residual, an IMU preintegration residual, a bias random walk.
Filter: An estimator that keeps only the current state and its covariance, and marginalizes the past at every step. FAST-LIO2's iterated EKF is one.
Fixed-lag window: An estimator that keeps the last \(N\) states, re-linearises all of them at every solve, and marginalizes the oldest into a prior when it drops out.
Information matrix: The inverse of a covariance. In the normal equations each sensor contributes \(\mathbf{J}^{\top}\boldsymbol{\Sigma}^{-1}\mathbf{J}\), so the more information a sensor carries, the more it decides the answer.
Jacobian: The matrix of first derivatives of a residual with respect to the state, taken on the manifold for rotations. It is what linearises each factor for Gauss-Newton.
Loose coupling: Using the IMU only to predict the pose, then letting LiDAR registration solve alone. See tight coupling.
Marginalization: Integrating out variables. For a nonsingular linear Gaussian problem, the Schur complement of the information matrix gives an exact Gaussian prior on the survivors. In a nonlinear problem this applies to a local linearization, whose approximation is retained in the prior.
Measurement model: The equation for what a sensor reports in terms of the true state, bias and noise. For the IMU: \({}_{B}\tilde{\boldsymbol{\omega}} = {}_{B}\boldsymbol{\omega} + \mathbf{b}^{g} + \boldsymbol{\eta}^{g}\) and \({}_{B}\tilde{\mathbf{a}} = \mathbf{R}_{WB}^{\top}({}_{W}\mathbf{a} - {}_{W}\mathbf{g}) + \mathbf{b}^{a} + \boldsymbol{\eta}^{a}\). Lesson 1 covers both.
Normal equations: The linear system \(\mathbf{H}\,\delta\mathbf{x} = \mathbf{b}\), with \(\mathbf{H} = \sum \mathbf{J}^{\top}\boldsymbol{\Sigma}^{-1}\mathbf{J}\) and \(\mathbf{b}=-\sum\mathbf{J}^{\top}\boldsymbol{\Sigma}^{-1}\mathbf{r}\), solved at each Gauss-Newton step. Stacking LiDAR and IMU terms into one \(\mathbf{H}\) is the sensor fusion.
Point-to-plane residual: The signed distance from a transformed scan point to a plane in the map, \(\mathbf{n}^{\top}(\mathbf{R}\mathbf{p} + \mathbf{t} - \mathbf{q})\). One per laser return.
Preintegration: Integrating the IMU samples between two keyframes once, into relative rotation, velocity and position deltas that do not depend on the starting state, so they need not be re-integrated when that state changes. Bias changes are handled by a first-order correction.
Retraction: Applying a small update vector to a state on a manifold, for example \(\mathbf{R} \leftarrow \mathbf{R}\,\mathrm{Exp}(\delta\boldsymbol{\phi})\), so a rotation stays a rotation.
Specific force: What an accelerometer actually measures: acceleration minus gravity, \(\mathbf{a} - \mathbf{g}\), expressed in the body frame. At rest it reads \(-\mathbf{R}^{\top}\mathbf{g}\), about +9.81 m/s² upward: the table pushing up.
Tight coupling: Putting LiDAR and IMU residuals into the same normal equations and solving for the pose, velocity, biases and gravity together, so the IMU takes over wherever the geometry is degenerate.
World frame: The fixed frame the map and trajectory live in, written \(W\), with gravity \({}_{W}\mathbf{g} \approx [0,\,0,\,-9.81]^{\top}\).