Organizations: School of Surveying and Geo-Informatics, Tongji University, Shanghai 200092, China · School of Automation and Intelligent Sensing, Shanghai Jiao Tong University, Shanghai 200240, China · School of Robotics, Wuhan University, Wuhan 430072, China
Geometric filters have recently been introduced to improve the accuracy and consistency of inertial-based integrated navigation systems. Error states were defined through specific group operations, introducing state correlations in error definition, which were lacked in the additive error used by a conventional indirect Kalman filter. The desirable consistent filtering models can be obtained based on specific geometric errors. For zero-velocity measurements expressed in the reference frame, this paper derives left-error process and measurement models from invariant filtering, two-frame-group filtering, and equivariant filtering. Importantly, a new group operation is introduced for the left tangent-group equivariant error. The analysis shows that the two-frame-group invariant extended Kalman filter (TFG-IEKF) and the tangent-group equivariant filter (TG-EqF) do not offer a significant consistency advantage over the invariant extended Kalman filter (IEKF). Experiments with an INS/ZUPT measurement system show that, under small initial attitude errors, the conventional indirect extended Kalman filter (EKF) achieves loop-closure position errors below 0.1% of the traveled distance, while the three geometric filters achieve comparable positioning accuracy.
Figures & tables
Figure 1: Data-acquisition system and total-station point measurements.
Specification
Gyroscope
Accelerometer
Noise random walk
0.015deg/h
0.017(m/s)/h
Bias instability
0.5deg/h
10μg
Measurement range
500deg/s
8g
Table 1: Specifications of the MEMS IMU.
Figure 2: Indoor experiment and positioning result.
Segment
1
2
3
4
5
6
7
8
9
10
11
12
13
Reference
2.145
3.117
3.071
3.115
3.088
3.586
3.581
3.116
3.124
3.074
3.067
2.040
10.872
Measured
2.154
3.138
3.085
3.136
3.104
3.603
3.580
3.124
3.137
3.075
3.091
2.031
10.884
Absolute error
0.009
0.021
0.014
0.021
0.016
0.017
0.001
0.008
0.013
0.001
0.024
0.009
0.009
Table 2: Errors in distances between adjacent indoor control points (m).
Figure 3: Positioning results for the six outdoor experiments.
Data
EKF
IEKF
TFG-IEKF
TG-EqF
m
%D
m
%D
m
%D
m
%D
1
0.047
0.023
0.071
0.035
0.072
0.035
0.070
0.034
2
0.190
0.065
0.205
0.070
0.215
0.073
0.241
0.082
3
0.173
0.060
0.097
0.034
0.096
0.033
0.097
0.034
4
0.077
0.031
0.083
0.033
0.078
0.031
0.088
0.035
5
0.052
0.027
0.031
0.016
0.033
0.018
0.031
0.017
Table 3: Loop-closure position errors for the outdoor experiments.
Data
Length (m)
Speed (m/s)
Airborne (s)
Stationary (s)
ZUPTs
1
205.50
1.27
10.52
4.00
16
2
293.72
1.26
8.46
2.36
28
3
290.23
1.22
8.95
2.34
27
4
254.31
1.22
8.68
2.44
24
5
188.89
0.90
5.97
2.88
33
6
183.52
1.02
6.16
2.86
29
Table 4: Statistics of the outdoor experimental datasets.
Error (m)
EKF
IEKF
TFG-IEKF
TG-EqF
Horizontal position
5.88
5.71
4.83
5.60
Height
6.13
0.48
0.27
0.63
Table 5: Loop-closure results for Data 1 with an initial roll error of 60\SIUnitSymbolDegree .
Figure 4: Trajectories of the four filters with an initial roll error of 60\SIUnitSymbolDegree .
Global navigation systems require state estimation algorithms that handle Earth's curvature, Earth's rotation, and gravitational variations. These factors can typically be neglected in local navigation algorithms for robots, drones, etc. In classical error-state Kalman Filtering (ESKF) the error state dynamics are trajectory-dependent. Invariant ESKFs utilize Lie Group symmetries to represent the error, which can render error propagation trajectory-independent for group-affine systems. Choosing between a standard filter (where position and velocity errors are defined additively in the navigation frame), a left-invariant filter (where errors are represented in the body frame) and a right-invariant filter (where errors are represented in the navigation/world frame) depends on system dynamics and sensor configuration. This note presents the mathematical formulas for four classical and invariant ESKFs for globally applicable aided inertial navigation systems. It is intended to serve as a systematic reference for comparison and implementation.
Antonia Hager, Torleiv H. Bryne
Airbus Central Research and Technology, Taufkirchen, Germany · Department of Engineering Cybernetics, Norwegian University of Science and Technology, Trondheim, Norway
Autonomous Underwater Vehicles (AUVs) navigating without GPS typically fuse inertial measurements with acoustic Doppler Velocity Log (DVL) velocities and pressure-derived depth. Posing the navigation state on a Lie group improves accuracy and consistency. However, state-of-the-art filters based on the Invariant Extended Kalman Filter (IEKF) append the Inertial Measurement Unit (IMU) biases as a Euclidean extension, which breaks the group-affine structure required for exact log-linear error dynamics, causing the reported covariance to degrade alongside the estimate. We apply the Tangent-Group (TG) symmetry, which carries the biases within the geometry of the state space, to derive an Equivariant Filter (EqF) for this system, leaving zero linearization error in the navigation states and second-order error only in the biases. We develop an equivariant output model for the DVL, whose update incurs only third-order linearization error, together with a direct pressure output. Monte Carlo simulations benchmark the TG-EqF against a Two-Frame-Group IEKF and a Multiplicative EKF. The TG-EqF reduces error by 18--25% against both alternatives in each of attitude, velocity, and position. The main benefit is in the covariance it estimates: its Average Normalized Estimation Error Squared (ANEES) stays closer to its nominal value of one than that of the others. Offline analysis on AUV field data corroborates the findings of the simulations, demonstrating reduced position drift.
Arihant Lunawat, Pieter van Goor, Frank Dellaert +1
The University of Sydney Sydney, Australia · Georgia Tech Atlanta, USA
Closed-loop (or feedback) error-state Kalman filters with their relatives and offspring are the state-of-the-art in modern aided inertial navigation research. Estimated inertial navigation system (INS) errors are continually fed back to the INS to correct the nominal system state before subsequent predictions. Conversely, in safety-critical aeronautical applications, open-loop (or feedforward) systems are an undisputed standard, where the inertial mechanization is strictly decoupled to allow for operational independence and fault isolation of computing units. We assess the performance impacts of this architectural choice beyond qualitative system-safety justifications using a standard inertial mechanization in geodetic coordinates and direct position aiding. Simulations using a variety of inertial sensor error characteristics, ranging from consumer to navigation grade systems, showcase the trade-off between smooth information fusion for high-end IMUs using an open-loop filter and the inherent long-term stability of the closed-loop architecture.
Antonia Hager, Torleiv H. Bryne
Airbus Central Research and Technology, Taufkirchen, Germany