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## EKF SLAM Inconsistency: Previous Literature

EKF inconsistency is defined as the fact that the filter returns a covariance matrix that is too optimistic [48], leading to inaccurate estimates. EKF inconsistency in the con- text of SLAM has been the object of many papers, see e.g. [46,47,49–56]. Theoretical analysis [49,53,56] reveals inconsistency is caused by the inability of EKF to reflect the nobservable degrees of freedom of SLAM raised in the above section. Indeed, the filter tends to erroneously acquire information along the directions spanned by these unobservable degrees of freedom. The Observability Constrained (OC)-EKF [52,56] constitutes one of the most advanced solutions to remedy this problem and has been fruitfully adapted, e.g. for VIO, cooperative localization, and unscented Kalman filter [5,45,46]. The idea is to pick a linearization point that is such that the unobservable subspace “seen » by the filter is of appropriate dimension.

**Specific Application to SLAM**

The specific application to SLAM was released on Arxiv in 2015 [10] as part of Axel Barrau’s PhD and encountered immediate successes reported in [43,58–62], although the work was never published elsewhere. More precisely, [9] notice back the SLAM problem bears a nontrivial Lie group structure. [10] formalize the group introduced in [9] and call it SEl+1(3), and proved that for odometry based SLAM, using the right invariant error of SEl+1(3) and devising an EKF based on this error, i.e., a Right-Invariant EKF (RIEKF), the linearized system possesses the desirable properties of the linear case, since it automatically correctly captures unobservable directions for SLAM. Thus, virtually all properties of the linear Kalman filter regarding unobservability may be directly transposed: the information about unobservable directions is non-increasing (see Proposition 2), the dimension of the unobservable subspace has appropriate dimension (this relates to the result of OC- EKF [5,45,56]), the filter’s output is invariant to linear unobservable transformations, even if they are stochastic and thus change the EKF’s covariance matrix along unobservable directions [58]. The right-invariant error for the proposed Lie group structure was also recently shown to lead to deterministic observers having exponential conver- gence properties in [63]. Along the same lines, using the right-invariant error of the group SE2(3), [42] propose alternative consistent IEKF for visual inertial SLAM and VIO applications [59–62,62]. In particular, [62] demonstrate that an alternative Invariant MSCKF based on the right-invariant error of SE2(3) naturally enforces the state vector to remain in General Theory 33 the unobservable subspace, a consistency property which is preserved when consider- ing point and line features [62], or when a network of magnetometers is available [61].

### Chapter’s Organization

Section 3 presents the general theory. Section 4 applies the theory to the general prob- lem of navigation in the absence of absolute measurements, as typically occurs in GPS- denied environments. Section 5 is dedicated to SLAM and compares the proposed EKF to conventional EKF, OC-EKF [56], robocentric mapping filter [64] and iSAM [65,66]. Preliminary ideas and results can be found in the preprint [10] posted on Arxiv in 2015. Although the present chapter is a major rewrite, notably including a novel gen- eral theory encompassing the particular application to SLAM of [10], [10] is a technical report serving as preliminary material for the present chapter. Matlab codes used for the chapter are available at https://github.com/CAOR-MINES-ParisTech/esde.

#### Literature Review of Kalman Filtering on Manifolds

Filtering on manifolds is historically motivated by aerospace applications where one seeks to estimate (besides other quantities) the orientation of a body in space. Much work has been devoted to making the EKF work with orientations, namely quaternions or rotation matrices. The idea is to make the EKF estimate an error instead of the state directly, leading to error state EKFs [72–75] and their UKF counterparts [22,41,76]. The set of orientations of a body in space is the Lie group SO(3) and efforts devoted to estimation on SO(3) have paved the way to EKF on Lie groups, see [18,34,42,43,77,78] and unscented Kalman filtering on Lie groups, see [19,22,60,79–82]. ie groups play a prominent role in robotics [83] and have drawn increasing atten- tion for computer vision and robotics applications [42,84,85]. In the context of state es- timation and localization, viewing poses as elements of the Lie group SE(3) has proved relevant [7,86–91]. The use of the novel Lie group SE2(3) introduced in [42] has led to drastic improvement of Kalman filters for robot state estimation [42,43,59,91–95]. Similarly, using group SEk(n) introduced for SLAM in [9,10] makes EKF consistent or convergent [10,58,62,63,96,97]. Specifically for visual inertial odometry purpose, in [80], the authors devise an UKF that takes advantage of the Lie group structure of the robot’s (quadrotor) pose SE(3), and uses a probability distribution directly defined on the group (the distributions in [12]) to generate the sigma points, which is akin to the general unscented Kalman filtering on manifolds of [98]. Finally, there has been attempts to devise UKFs respecting natural symmetries of the systems’ dynamics, namely the invariant UKF, see [35,99].

**Table of contents :**

Acknowledgements/Remerciements

List of Publications

**1 Introduction **

1 Contributions of the Thesis

2 Organization of the Manuscript

**I Unscented Kalman Filtering on Manifold & Lie Groups **

**2 Introduction to Part I **

1 The Problem of Bayesian Filtering and the Kalman Filter

2 Nonlinear Filtering: the Extended Kalman Filter (EKF)

3 Nonlinear Filtering: the Unscented Kalman Filter (UKF)

4 Introduction to Filtering on Manifolds

5 Matrix Lie Groups

6 Nonlinear & Invariant Observers on Lie Groups

7 Invariant Extended Kalman Filter

**3 Exploiting Symmetries to Design EKFs with Consistency Properties **

1 Introduction

2 Contributions

3 General Theory

4 Application to Multi-Sensor Fusion for Navigation

5 Simulation Results

6 Conclusion

**4 A New Approach to Unscented Kalman Filtering on Manifolds **

1 Introduction

2 Unscented Kalman Filtering on Parallelizable Manifolds

3 Application to UKF on Lie Groups

4 UKF-M Implementation

5 Extension to General Manifolds

6 Concluding Remarks

5 Invariant Kalman Filtering for Visual Inertial SLAM 59

1 Introduction

2 Visual Inertial SLAM Problem Modeling

3 Proposed Algorithms

4 Contents

4 Simulation Results

5 Experimental Results

6 Conclusion

**6 Unscented Kalman Filter on Lie Groups for Visual Inertial Odometry **

1 Introduction

2 Problem Modeling

3 Unscented Based Inferred Jacobian for UKF-LG Update

4 Proposed Filters

5 Experimental Results

6 Conclusion

**II Measurement Noise Estimation for Kalman Filter Tuning **

**7 Introduction to Part II **

1 Illustrative Example

2 Content of Part II

**8 AI-IMU Dead-Reckoning **

1 Introduction

2 Relation to Previous Literature

3 IMU and Problem Modelling

4 Kalman Filtering with Pseudo-Measurements

5 Proposed AI-IMU Dead-Reckoning

6 Experimental Results

7 Conclusion

**9 A New Approach to 3D ICP Covariance Estimation **

1 Introduction

2 Proposed Approach

3 Practical Covariance Computation

4 Experimental Results

5 Complementary Experimental Results

6 Conclusion

**III Using Deep Learning to Extract Information from an IMU **

**10 Introduction to Part III **

1 Relations between Neural Networks and the Kalman Filter

2 Designed “Deep” Kalman Filter with Neural Networks

3 Adding Information to Kalman Filter with Neural Networks

**11 RINS-W: Robust Inertial Navigation System on Wheels **

1 Introduction

2 Inertial Navigation System & Sensor Model

3 Specific Motion Profiles for Wheeled Systems

4 Proposed RINS-W Algorithm

5 Results on Car Dataset

6 Conclusion

**12 Denoising IMU Gyroscopes with Deep Learning for Attitude Estimation **

1 Introduction

2 Kinematic & Low-Cost IMU Models

3 Learning Method for Denoising the IMU

4 Experiments

5 Discussion

6 Conclusion

**13 Additional Results for Inertial Navigation of Cars**

1 Introduction

2 Experimental Results

3 Further Experimental Results

4 Conclusion

**14 Conclusion of the Thesis **

**Bibliography**