## 2 Prerequisites  
### IMU State Prediction and Pre-Integration  
Usually, the frequency of IMU is much higher than lidar. IMU can obtain the acceleration and angular velocity at each time step, which can be used to predict the next state of IMU through integration.  
In general, the IMU states at time \(t\) can be modeled as:  
\[\mathbf{X_{t}^{W}}=[\mathbf{p_{r}},\mathbf{v_{t}},\mathbf{R_{t}},\mathbf{b_{t}}] \tag{1}\]  
where \(\mathbf{X_{t}^{W}}\) represents the state in the world frame \(W\) at time \(t\); \(\mathbf{p_{r}}\), \(\mathbf{v_{t}}\), and \(\mathbf{R_{t}}\) represent the position, velocity, and rotation matrix, respectively, at time \(t\); and \(\mathbf{\hat{u}_{t}}\) and \(\mathbf{\hat{\omega}_{t}}\) are the acceleration and angular velocity of the raw IMU measurements (\(\mathbf{\hat{u}_{t}}\) and \(\mathbf{\hat{\omega}_{t}}\) are affected by a slowly varying bias, \(\mathbf{b_{t}}\)).  
The next state, \(\mathbf{X_{t+\Delta t}^{W}}\), is predicted through the integration of IMU, where \(\Delta t\) is the interval between two consecutive IMU measurements. The state prediction value of the IMU is then used to infer the motion of the vehicle. The velocity, position, and rotation of the vehicle at time \(t+\Delta t\) can be computed by Equation (2) [27]:  
\[\begin{split}\mathbf{v}_{t+\Delta t}&=\mathbf{v}_{t}+g\Delta t +\mathbf{R}_{t}(\hat{\mathbf{a}}_{t}-\mathbf{b}_{t}^{\alpha}-\mathbf{m}_{t}^{\alpha})\Delta t\\ \mathbf{p}_{t+\Delta t}&=\mathbf{p}_{t}+\mathbf{v}_{t}\Delta t+ \frac{1}{2}g\Delta t^{2}+\frac{1}{2}\mathbf{R}_{t}(\hat{\mathbf{a}}_{t}-\mathbf{b}_{t}^{ \alpha}-\mathbf{n}_{t}^{\alpha})\Delta t^{2}\\ \mathbf{q}_{t+\Delta t}&=\mathbf{q}_{t}\otimes\left[ \begin{array}{c}\frac{1}{2}\Delta t(\hat{\mathbf{\omega}}_{\Delta t}-\mathbf{b}_{ \Delta t}^{\alpha}-\mathbf{n}_{\Delta t}^{\omega})\\ 1\end{array}\right]\end{split} \tag{2}\]  
where \(\mathbf{g}\) is gravitational acceleration; \(\mathbf{b}_{t}^{\alpha}\) is the varying bias of the acceleration \(\hat{\mathbf{a}}_{t}\); and \(\mathbf{n}_{t}^{\alpha}\) is the white noise of the acceleration \(\hat{\mathbf{a}}_{t}\). The quaternion \(\mathbf{q}_{t}\) under Hamilton notation (which corresponds to \(\mathbf{R}_{t}\)) is used; \(\otimes\) is used for the multiplication of two quaternions; \(\mathbf{b}_{t}^{\alpha}\) is the varying bias of the angular velocity \(\hat{\mathbf{\omega}}_{t}\); and \(\mathbf{m}_{t}^{\alpha}\) is the white noise of the angular velocity \(\hat{\mathbf{\omega}}_{t}\).  
The motion state of IMU between the \(i\)-th and the \(j\)-th time steps can be represented by the IMU pre-integrated measurement value \(\Delta\mathbf{p}_{ij}\), \(\Delta\mathbf{v}_{ij}\), \(\Delta\mathbf{q}_{ij}\), which can be computed by:  
\[\begin{split}\Delta\mathbf{v}_{ij}&=\mathbf{R}_{t}^{T}(\bm {v}_{j}-\mathbf{v}_{i}-g\Delta t_{ij})\\ \Delta\mathbf{p}_{ij}&=\mathbf{R}_{t}^{T}\left(\mathbf{p}_{j}- \mathbf{p}_{i}-\mathbf{v}_{i}\Delta t_{ij}-\frac{1}{2}g\Delta t_{ij}^{2}\right)\\ \Delta\mathbf{q}_{ij}&=\mathbf{q}_{i}^{-1}\otimes\mathbf{q}_{j} =\prod\limits_{k=t}^{j-1}\left[\begin{array}{c}\frac{1}{2}\Delta t_{ij}(\hat {\mathbf{\omega}}_{k}-\mathbf{b}_{k}^{\alpha}-\mathbf{n}_{k}^{\omega})\\ 1\end{array}\right]\end{split} \tag{3}\]  
where \(\Delta\mathbf{p}_{ij}\), \(\Delta\mathbf{v}_{ij}\), \(\Delta\mathbf{q}_{ij}\) is the position, velocity, and rotation matrix of the IMU, respectively, which has the covariance \(\mathbf{C}_{\mathbf{l}_{j}}^{\mathbf{l}_{j}}\) in the error-state model. Due to space limitations, we invite readers to refer to the descriptions in [26; 34] to understand the detailed derivation of IMU pre-integration.