IMU as an Input vs. a Measurement of the State in Inertial-Aided State Estimation

TL;DR

This paper methodically compares treating an IMU as an input to a motion model against treating it as a measurement of the state in a continuous-time state estimation framework and shows that continuous-time techniques and the treatment of the IMU as a measurement of the state are promising areas of further research.

Abstract

Treating IMU measurements as inputs to a motion model and then preintegrating these measurements has almost become a de-facto standard in many robotics applications. However, this approach has a few shortcomings. First, it conflates the IMU measurement noise with the underlying process noise. Second, it is unclear how the state will be propagated in the case of IMU measurement dropout. Third, it does not lend itself well to dealing with multiple high-rate sensors such as a lidar and an IMU or multiple asynchronous IMUs. In this paper, we compare treating an IMU as an input to a motion model against treating it as a measurement of the state in a continuous-time state estimation framework. We methodically compare the performance of these two approaches on a 1D simulation and show that they perform identically, assuming that each method's hyperparameters have been tuned on a training set. We also provide results for our continuous-time lidar-inertial odometry in simulation and on the Newer College Dataset. In simulation, our approach exceeds the performance of an imu-as-input baseline during highly aggressive motion. On the Newer College Dataset, we demonstrate state of the art results. These results show that continuous-time techniques and the treatment of the IMU as a measurement of the state are promising areas of further research. Code for our lidar-inertial odometry can be found at: https://github.com/utiasASRL/steam_icp

Figures (11)

• Figure 1: In this factor graph, we consider a case where we would like to marginalize several states out of the full Bayesian posterior. The triangles represent states and the black dots represent factors. This factor graph could potentially correspond to doing continuous-time state estimation with binary motion prior factors, unary measurement factors, and a unary prior factor on the initial state $\mathbf{x}_0$
• Figure 2: This figure depicts the results of our preintegration, which can incorporate heterogeneous factors. The resulting joint Gaussian factor in \ref{['eq:joint_gaussian_factor']} can be thought of as two unary factors, one each for $\mathbf{x}_k$ and $\mathbf{x}_{k+1}$, and an additional binary factor between $\mathbf{x}_k$ and $\mathbf{x}_{k+1}$.
• Figure 3: The estimated trajectories of IMU-as-input and IMU-as-measurement are plotted alongside the ground-truth trajectory, which is sampled from white-noise-on-jerk motion prior with $Q_c = 1.0$. Both methods were pretrained on a hold-out validation set of simulated trajectories
• Figure 4: This figure depicts 1000 simulated trajectories sampled from a white-noise-on-jerk (WNOJ) prior where $\check{\mathbf{x}}_{0} = [0.0~0.0~1.0]^T$, $\check{\mathbf{P}}_{0} = \text{diag}(0.001, 0.001, 0.001)$, $Q_c = 1.0$
• Figure 5: This figure compares the performance of the baseline IMU-as-input approach vs. our proposed IMU-as-measurement approach that leverages a GP motion prior. The ground-truth trajectories are sampled from a white-noise-on-jerk (WNOJ) prior as shown in Figure \ref{['fig:traj_ad_0']}. The IMU input covariance $Q_k$ for the IMU-as-input method was trained on a validation set, with a value of $0.00338$. The parameters of our proposed method was also trained on the same validation set, with resulting values of $\sigma^2 = 1.0069$, $\alpha = 0.0$. $R_{\text{pos}}$ for both methods was chosen to match the simulated noise added to the position measurements. Similarly, $R_{\text{acc}}$ for the IMU-as-measurement approach was set to match the simulated noise added to the acceleration measurements. In the bottom row, the chi-squared bounds have a different size because the dimension of the state in IMU-as-measurement is greater (it includes acceleration), and so the dimension of the chi-squared distribution increases, resulting in a tighter chi-squared bound