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**Networked Inertial Navigation with Constraints Generated by Neural Networks**

*Yingjie Hu, Yohannes Ketema, Dept. of Aerospace Engineering & Mechanics, University of Minnesota, Twin Cities; Robert McGovern III, Jacob Hanson, James Jean IV, Alec Jonason, Dept. of Neurosurgery, University of Minnesota, Twin Cities; Demoz Gebre-Egziabher, Dept. of Aerospace Engineering & Mechanics, University of Minnesota, Twin Cities*

**Location:** Beacon B

Peer Reviewed |

Peer Reviewed

This paper deals with the subject of networked inertial navigation. The term networked inertial navigation refers to the PNT technique whereby the measurements from a network of individual inertial measurement units (IMU) are fused to generate a navigation solution while mitigating the drift error inherent in the process of integrating accelerations and angular rates. Kinematic constraints between the IMUs in the network are exploited to help reduce the overall drift error of the PNT solution. Perhaps one of the best-known examples of this is the zero-velocity update (ZUPT) used in the personal navigation problem [1]. This paper deals with one key challenge encountered in networked inertial navigation: unknown or poorly known constraints and discusses two different sensor fusion topologies of applying constraints.

In most prior work associated with networked inertial navigation, the kinematic constraints used are assumed to be known a priori. They are also assumed to be known perfectly. However, this limits the utility of networked inertial navigation because some constraints are difficult to derive precisely and be known a priori. In addition, for unknown or poorly known stochastic constraints, the common and conservative way of dealing with them is to formulate them as inequality constraints (such as upper bound constraints) to account for the poor knowledge of the constraint. Projection approach is a common way of solving inequality constraints, which essentially transforms the inequality constraint to an equality constraint using KKT(Karush-Kuhn-Tucker) conditions, and the constraint update is obtained by projecting the state onto the constraint surface. However, the projection approach is limited in the sense that it will stop searching once it reaches the constraint surface whereas the true state may be inside the constraint surface.

In this work, we propose two approaches to deal with the unknown or poorly known stochastic constraints. First, deep learning techniques can be used to gain valuable information from data. Thus, kinematic constraints can be learnt from motion data using deep learning approaches and the learned constraints can be used as measurement updates within an estimator (extended Kalman Filter or Particle Filter). In addition, the learnt constraints come with error statistics from the testing datasets. Thus, the deep-learning approach fills the void of the unknown or poorly known constraints and enables us to learn those constraints with some error uncertainly from data. Second, when those poorly unknown constraints are formulated as inequality constraints (such as upper bound constraints), instead of projecting the state onto the constraint surface, we argue a more systematic way of incorporating inequality constraints in particle filtering framework based on the pdf truncation method [2]. In pdf-truncation, the constraints are used to reshape the covariance matrix (or the probability density function) of the navigation states derived during the time update using the inertial navigation equations. Both the deep-learning-based extended Kalman filtering and pdf-truncation-based particle filtering approaches are presented and discussed either in experiment or simulation.

One important factor that affects the quality of the solution obtained from a networked inertial navigation system is the overall network topology used for the fusion of the IMU measurements. The information fusion can be done in a centralized or de-centralized fashion. In the centralized topology, the constraints determined by deep learning are transformed to pseudo-measurements of relative difference between the navigation states generated using the measurement from each IMU. In the decentralized topology, the constraints are used to synthesize the pseudo-measurements of the absolute navigation states. Since deep learning is used to generate rather simple constraints, error statistics associated with the constraint is available. This is used as the measurement noise statistics for the pseudo-measurement.

Simulation Studies: We use a simple simulation study to demonstrate and highlight the approach of pdf-truncation in particle filtering framework discussed above. The simulation consists of an IMU moving in a 2-D plane and two inequality constraints are imposed on its global X and Y direction in the 2-D plane. The pdf-truncation approach is implemented in a particle filtering framework. Simulation results show that the constrained position is well-bounded inside the constraint surface. Next a follow-up simulation will be implemented to compare this particle-filtering-based pdf truncation approach with the projection approach.

Experimental Validation: The performance of the networked inertial navigation with deep-learning constraints is validated on a two-IMU network for the personal navigation problem. The network consists of one IMU mounted on the foot and another on the shin. The foot IMU can make use of zero-velocity-update as its measurement updates. There are no external measurements for the IMU on the shin. A multi-layer perceptron neural network is trained over an approximately 1 hour length of human walking data. This learning approach takes as input the inertial signals from the foot and shin and is able to predict the relative distance constraints between the foot and shin during the walking session, thus it is more informative than the inequality constraint which only gives an upper bound on the relative distance between foot and shin. To benchmark our algorithm, an optical camera system is used to produce ground truth data for the experiments. The algorithm was implemented in both centralized and de-centralized sensor fusion architectures. The test results have two main components: 1. In the centralized sensor fusion architecture, the foot position estimate is not improved and even get slightly worse than without the constraints. The shin position estimate is much more improved than without the constraints. 2. In the decentralized fusion architecture, the foot position accuracy remains unchanged because it is only used to synthesize the pseudo measurements. The shin position estimate is much more improved than without the constraints. And we evaluate the advantages and disadvantages of the two sensor fusion topologies.

Conclusion: This paper described two approaches of dealing with unknown or poorly known constraints in a networked inertial navigation system using (1) a deep-learning-based constraint learning method and (2) a pdf-truncation-based particle filtering method. Simulation and experimental results show that the two approaches are able to effectively mitigate the error drifts inherent in the networked inertial system.

[1] Skog, Isaac, et al. "Zero-velocity detection—An algorithm evaluation." IEEE transactions on biomedical engineering 57.11 (2010): 2657-2666.

[2] Simon, Dan, and Donald L. Simon. "Constrained Kalman filtering via density function truncation for turbofan engine health estimation." International Journal of Systems Science 41.2 (2010): 159-171.

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