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In this paper, we establish a connection between Reid{'}s HOMHT and the modern Random Finite Set (RFS)/ Finite Set Statistics (FISST) based methods for Multi-Target Tracking. We start with an RFS description of the Multi-Target probability density function (MT-pdf), and derive the prediction, and update equations of the MT-tracking problem in the RFS framework from a belief space perspective. We show that the RFS pdf has a hypothesis dependent structure that is similar to the HOMHT hypotheses structure. In particular, we examine the different hypotheses, and derive the hypotheses update equations under the FISST recursions, and clearly show its relationship to the classical HOMHT hypotheses and hypothesis weight update formula, thereby establishing a connection between the methods.

Consider a space object in an orbit about the earth. An uncertain initial state can be represented as a point cloud which can be propagated to later times by the laws of Newtonian motion. If the state of the object is represented in Cartesian earth centered inertial (Cartesian-ECI) coordinates, then even if initial uncertainty is Gaussian in this coordinate system, the distribution quickly becomes non-Gaussian as the propagation time increases. Similar problems arise in other standard fixed coordinate systems in astrodynamics, e.g. Keplerian and to some extent equinoctial. To address these problems, a local "Adapted STructural (AST)'' coordinate system has been developed in which uncertainty is represented in terms of deviations from a "central state". Given a sequence of angles-only measurements, the iterated nonlinear extended (IEKF) and unscented (IUKF) Kalman filters are often the most appropriate variants to use. In particular, they can be much more accurate than the more commonly used non-iterated versions, the extended (EKF) and unscented (UKF) Kalman filters, especially under high eccentricity. In addition, iterated Kalman filters can often be well-approximated by two new closed form filters, the observation-centered extended (OCEKF) and unscented (OCUKF) Kalman filters., Comment: 37 pages, 8 figures, 2 tables

Various methods have been proposed for the nonlinear filtering problem, including the extended Kalman filter (EKF), iterated extended Kalman filter (IEKF), unscented Kalman filter (UKF) and iterated unscented Kalman filter (IUKF). In this paper two new nonlinear Kalman filters are proposed and investigated, namely the observation-centered extended Kalman filter (OCEKF) and observation-centered unscented Kalman filter (OCUKF). Although the UKF and EKF are common default choices for nonlinear filtering, there are situations where they are bad choices. Examples are given where the EKF and UKF perform very poorly, and the IEKF and OCEKF perform well. In addition the IUKF and OCUKF are generally similar to the IEKF and OCEKF, and also perform well, though care is needed in the choice of tuning parameters when the observation error is small. The reasons for this behaviour are explored in detail., Comment: 17 pages, 2 figures, 3 tables

In multi-object tracking one may encounter situations were at any time step the number of possible hypotheses is too large to generate exhaustively. These situations generally occur when there are multiple ambiguous measurement returns that can be associated to many objects. This paper contains a newly developed approach that keeps the aforementioned situations computationally tractable. Utilizing a hypothesis level derivation of the Finite Set Statistics (FISST) Bayesian recursions for multi-object tracking we are able to propose a randomized method called randomized FISST (R-FISST). Like our previous methods, this approach utilizes Markov Chain Monte Carlo (MCMC) methods to sample highly probable hypotheses, however, the newly developed (R-FISST) can account for hypotheses containing multiple births and death within the MCMC sampling. This alleviates the burden of having to exhaustively enumerate all birth and death hypotheses and makes the method more equipped to handle spawn scenarios. We test our method on Space Situational Awareness (SSA) scenarios with spawn events., Comment: arXiv admin note: substantial text overlap with arXiv:1603.04096

In this paper, we present a randomized version of the finite set statistics (FISST) Bayesian recursions for multi-object tracking problems. We propose a hypothesis level derivation of the FISST equations that shows that the multi-object tracking problem may be considered as a finite state space Bayesian filtering problem, albeit with a growing state space. We further show that the FISST and Multi-Hypothesis Tracking (MHT) methods for multi-target tracking are essentially the same. We propose a randomized scheme, termed randomized FISST (R-FISST), where we sample the highly likely hypotheses using Markov Chain Monte Carlo (MCMC) methods which allows us to keep the problem computationally tractable. We apply the R-FISST technique to a fifty-object birth and death Space Situational Awareness (SSA) tracking and detection problem. We also compare the R-FISST technique to the Hypothesis Oriented Multiple Hypothesis Tracking (HOMHT) method using an SSA example., Comment: 27 pages, 15 figures

In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange--d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver., Comment: 22 pages, 3 figures

In this paper we use an affine connection formulation to study an optimal control problem for a class of nonholonomic, under-actuated mechanical systems. In particular, we aim at minimizing the norm-squared of the control input to move the system from an initial to a terminal state. We consider systems evolving on general manifolds. The class of nonholonomic systems we study in this paper includes, in particular, wheeled-type vehicles, which are important for many robotic locomotion systems. The two special aspects of this optimal control problem are the nonholonomic constraints and under-actuation. Nonholonomic constraints restrict the evolution of the system to a distribution on the manifold. The nonholonomic connection is used to express the constrained equations of motion. Furthermore, it is used to take variations of the cost functional. Many robotic systems are under-actuated since control inputs are usually applied through the robot's internal configuration space only. While we do not consider symmetries with respect to group actions in this paper, the fact that the system is under-actuated is taken into account in our problem formulation. This allows one to compute reaction forces due to any inputs applied in directions orthogonal to the constraint distribution. We illustrate our ideas by considering a simple example on a three-dimensional manifold., Comment: 8 pages, 1 figure

In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange--d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver., Comment: IEEE Conference on Decision and Control, 2006 6 pages, 2 figures. v3: improved exposition, and shortened paper to 6 pages v2: revised to include numerical examples on SO(3)