• Systems and control
• Navigation and localization
• Radar tracking algorithms
• Optimization on manifolds
• Machine learning
• Industrial products
Industrial applications :
• Crane control with the world leader MANITOWOC (POTAIN). See this picture
where I am experimenting the real-time controller on a 30 meters high crane. It has also been successfully tested and implemented on various cranes.
• Guidance and navigation with Safran Electronics & Defense, the No. 1 company in Europe for inertial navigation systems (INS). Our algorithm is embedded in the new drone Patroller (see picture below, courtesy of Safran Electronics & Defense).
Below is displayed the yaw estimation error evolution over time when using our Invariant EKF for various initial errors from 5 up to 90 degrees:
Albeit similar for small initial errors, the conventional EKF has much more trouble converging when initial errors are large, see below.
• Sensor fusion for feet motion estimation with the startup FeetMe
• Radar tracking with THALES.
• High pressure polyethylene reactor of TOTAL PETROCHEMICALS. Carling- France.
Invariant Extended Kalman Filtering
The Kalman filter is an optimal tool to estimate some physical quantities, that are governed by linear differential equations. For nonlinear systems, the engineers resort to the extended Kalman filter (EKF), that works well in practice
but does not possess any optimality properties and whose efficiency in practice may prove aleatory for challenging problems. Since its early implementation in the Apollo program in the sixties, though, it has been the state of the art for fusion of inertial measurement units (IMU) with other sensors, especially in the aerospace industry.
The invariant extended Kalman filter (IEKF), is a variant that improves the EKF for navigation, localization, and some related problems.
It has been successfully implemented by Safran Electronics and Defense (formerly Sagem) for a high performance inertial navigation application where it has led to substantial improvements over the existing industrial methods. It has been the object of various patents by this company. One patent was awarded a Sagem Innovation Award, as the most innovative patent filed by the company in 2015.
For more information about IEKF theory, please start with our Annual Review paper
Properties of the Invariant Kalman filter
The main point about the theory is that for a large class of systems related to navigation and robotics, the IEKF possesses properties akin to linear systems and to the linear Kalman filter
• A large class of nonlinear systems, named "group affine", possesses properties that remind of linear systems. Notably, the evolution between two trajectories is actually governed by a linear equation. This intriguing property is called "log-linearity" of the error as it is the log of the error that evolves via linear differential equations.
• For group affine systems, the IEKF possesses convergence guarantees.
• If the covariance matrix of the error is singular, then the estimate is physically restrained in a subspace of the state space. Only the IEKF - as opposed to conventional EKF - preserves this property under a set of conditions. For an example see (this article
). This is important as nearly singular covariance matrices are often encountered in high performance navigation, and eigenvalues of the covariance matrix may drop at any time after a combination of observations.
• The conventional EKF has difficulties in incorporating relative observations between state variables, as they lead to inconsistency of the estimates. This has caused its decline for Simultaneous Localization and Mapping (SLAM) in robotics. When using the Invariant EKF, relative observations are seamlessly handled. See this pdf
• For linear systems, it is possible to build a map through a finite dimensional integration F once and for all, such that for all initial condition x the vector Fx is the solution to the system at arbitrary predefined time t. This is called preintegration property. We have shown that all group affine systems possess the preintegration property, although being highly nonlinear.
• One of the biggest success of the theory is to have introduced the group SE_2(3) that makes the inertial navigation (IMU) equations a group affine system.
• We have also introduced the group SE_k(d), which makes the SLAM problem a group affine system.
See the PhD thesis of Axel Barrau for more details (pdf
The invariant EKF approach differs from the literature revolving around "EKF on Lie groups" in that we are not pursuing how to accomodate the state representation but rather how to recover properties akin to the linear case. In particular, the state error we use depends on what we observe, not only on the state. Note that the IEKF inherently handles Lie groups state spaces, and may be considered an EKF on Lie groups when applied to an arbitrary system.
Optimization on manifolds
The geometry in algorithms can also stem from probabilistic considerations (information geometry), or from the intrinsic geometry of the problem. Indeed, in optimization and machine learning, some constraints on the search space (orthogonality constraints, low rank etc.) make the search space a submanifold. This has led to the realm of optimization on manifolds. The search space can generally be endowed with several geometries, but here again, the relevant geometries are often encoded in the invariances of the problem.
I introduced stochastic gradient and obtained the first convergence results when the search space is a Riemannian manifold in this paper
In partnership with Manitowoc (Potain Cranes), one of the world leading crane manifacturers, I have developped a controller for real-time assistance of crane operators in the field of construction. Owing to the complexity of construction sites, that notably involve men at work, full automation is deemed undesirable and operators must be kept in the loop. The goal of the controller is then to track the velocity reference from the operator, while guaranteeing the absence of sway, that is, undesirable residual oscillations. Our method only uses built-in crane sensors. As a result the state is unobservable and the method is largely open-loop. It builds on the celebrated flatness theory, coupled with a hierarchical control architecture based on a time-scale separation
where modern variable-speed drives ensure fast closed-loop control of motor speeds, whereas our method allows to generate feedforward feasible anti-sway trajectories for the slower mechanical part. We experimented the approach on a mockup as well as on various cranes, including a 40 meters high tower crane. Tests conducted on crane operators with various levels of experience and one crane instructor confirm the benefits of the controller.
See this paper