Henrique Marra Menegaz

A Systematization of the Unscented Kalman Filter Theory

In this paper, we propose a systematization of the (discrete-time) Unscented Kalman Filter (UKF) theory. We gather all available UKF variants in the literature, present corrections to theoretical inconsistencies, and provide a tool for the construction of new UKFs in a consistent way. This systematization is done, mainly, by revisiting the concepts of Sigma-Representation, Unscented Transformation (UT), Scaled Unscented Transformation (SUT), UKF, and Square-Root Unscented Kalman Filter (SRUKF). Inconsistencies are related to 1) matching the order of the transformed covariance and cross-covariance matrices of both the UT and the SUT; 2) multiple UKF definitions; 3) issue with some reduced sets of sigma points described in the literature; 4) the conservativeness of the SUT; 5) the scaling effect of the SUT on both its transformed covariance and cross-covariance matrices; and 6) possibly ill-conditioned results in SRUKFs. With the proposed systematization, the symmetric sets of sigma points in the literature are formally justified, and we are able to provide new consistent variations for UKFs, such as the Scaled SRUKFs and the UKFs composed by the minimum number of sigma points. Furthermore, our proposed SRUKF has improved computational properties when compared to state-of-the-art methods.

A new smallest sigma set for the Unscented Transform and its applications on SLAM

In this work we propose a new set of sigma points for the Unscented Transform that uses the minimum number of points. We than compare this new set with the symmetric set, the reduced set, and the spherical set. Simulations comparing this sets are done to verify the properties of this set and to verify their transforms. Lastly, we simulate each of these sets in a recursive filter for SLAM. The results show that our set is a better choice for a non symmetric prior distribution and still a good alternative for symmetric prior distributions.

Unscented Kalman Filters for Estimating the Position of an Automotive Electronic Throttle Valve

This paper presents an application of unscented Kalman filters (UKFs) to an automotive electronic throttle device. The motivation of this study is on estimating the position of the throttle device when measurements of the position are inaccessible, e.g., due to failures in the sensor of position. In this case, an external wattmeter is connected in the circuitry to measure the power consumed by the throttle, and this information feeds UKFs to produce the estimation for the position. Experimental data support the findings of this paper. Almost all of the brand-new vehicles based on spark-ignition combustion engines have an electronic throttle valve to control the power produced by the engine. The electronic throttle has a unique sensor for measuring the position of the throttle valve, and this feature can represent a serious problem when the sensor of position fails. As an attempt to prevent the effects of a failure from such a sensor, we present an algorithm (UKF) combined with the use of an additional sensor, i.e., a wattmeter. The wattmeter is detached from the throttles structure but is arranged to measure the electric power consumed by the throttle. Measurements of the power consumption then feed the UKF. This filter then produces an estimation of the position of the throttle valve. Experimental data illustrate the practical benefits of our approach.

Interacting multiple model unscented filter for tracking a ballistic missile during its boost phase

Boost-phase detection of a ballistic missile is an attractive option for defense systems because boosting rockets are easy to detect and, in this phase, countermeasures are less effective. Nevertheless, it raises technical criticalities due to the short burn-out times and to the unknown features and behavior of the missile. The boosting trajectory of a ballistic missile, in fact, can be divided in at least three phases, each one characterized by a different steering strategy: the vertical arc, the pitch maneuver, and the gravity turn. Furthermore, the overall trajectory is very sensitive to variations in the parameters of the model and in the kick angle chosen during the pitch maneuver. Therefore, the tracking of a boosting missile presents two major difficulties: the identification of the missile parameters (ballistic coefficient, mass rate, thrust over weight ratio, etc.), and the determination of the steering strategy. The multiple models filter approach has been adopted in this paper to cope with these issues. In particular, an improved version of the Interacting Multiple Model Unscented Filter (IM-MUF) for tracking a missile has been introduced. Essentially, a modified form of the filter Markov Transition Matrix (MTM), ad hoc designed for the problem of tracking a ballistic missile during its boost phase, is proposed. The probabilities of the state transitioning from one dynamic model to another are given by the MTM elements. In the traditional approach, these probabilities are supposed constant and known. In this work these conservative hypotheses have been relaxed and these probabilities have been considered to be functions of the estimated state vector, thanks to specific features of the missile trajectory. A numerical, three-dimensional Monte Carlo simulation has been implemented in order to reconstruct a state vector composed by the position, velocity and missile parameters vectors. The modified IMMUF runs several models, each one accounting for one of the phases of flight with a different value of thrust direction, and combines estimations in order to provide the final estimation. Noisy measurements are obtained from a radar. The results show that the proposed IMMUF with the variable MTM outperform the traditional form of the IMMUF, by giving more accurate estimation.

An EKF-based approach for estimating leg stiffness during walking

The spring-like behavior is an inherent condition for human walking and running. Since leg stiffness kleg is a parameter that cannot be directly measured, many techniques has been proposed in order to estimate it, most of them using force data. This paper intends to address this problem using an Extended Kalman Filter (EKF) based on the Spring-Loaded Inverted Pendulum (SLIP) model. The formulation of the filter only uses as measurement information the Center of Mass (CoM) position and velocity, no a priori information about the stiffness value is known. From simulation results, it is shown that the EKF-based approach can generate a reliable stiffness estimation for walking.

Scaled Minimum Unscented Multiple Hypotheses Mixing Filter

This work brings two new contributions. First, it introduces the Scaled Minimum Unscented Multiple Hypotheses Mixing Filter, a novel filter for hybrid dynamical systems that 1) uses a new minimum set of sigma points along with the scaled unscented transform in a hybrid framework; 2) can estimate the Markovian Transition Probability Matrix in real-time; 3) features a pruning step that reduces the filters computational effort and prevents its estimates from being degraded by very unlikely hypotheses; and 4) has a mixing step with merging depth greater than one. Second, we present a result revealing the conservativeness of one of the scaled unscented transform forms.