Lyapunov stable penalty methods for imposing holonomic constraints in multibody system dynamics

This paper presents stability and convergence results on a novel approach for imposing holonomic constraints for a class of multibody system dynamics. As opposed to some recent techniques that employ a penalty functional to approximate the Lagrange multipliers, the method herein defines a penalized dynamical system using penalty-augmented kinetic and potential energies, as well as a penalty dependent constraint violation dissipation function. In as much as the governing equations are not typically cocreive, the usual convergence criteria for linear variational boundary value problems are not directly applicable. Still numerical simulations by various researchers suggest that the method is convergent and stable. Despite the fact that the governing equations are nonlinear, the theoretical convergence of the formulation is guaranteed if the multibody system is natural and conservative. Likewise, stability and asymptotic stability results for the penalty formulation are derived from well-known stability results available from classical mechanics. Unfortunately, the convergence theorem is not directly applicable to dissipative multibody systems, such as those encountered in control applications. However, it is shown that the approximate solutions of a typical dissipative system converge to a nearby collection of trajectories that can be characterized precisely using a Lyapunov/Invariance Principle analysis. In short, the approach has many advantages as an alternative to other computational techniques:

Minimum distance and optimal transformations on SO(N)

The group of special (or proper) orthogonal matrices, SO(N), is used throughout engineering mechanics in the analysis and representation of mechanical systems. In this paper, a solution is presented for the optimal transformation between two elements of SO(N). The transformation is assumed to occur during a specified finite time, and a cost function that penalizes the transformation rates is utilized. The optimal transformation is found as a constant-rate rotation in each of the principal planes relating the two elements. Although the kinematics of SO(N) are nonlinear and governed by Poisson’s equation, the solution is found to be a linear function of the generalized principal angles. This is made possible by the extension of principal-rotation kinematics from three-dimensional rotations to the general SO(N) group. This extension relates the N-dimensional angular velocity to the derivatives of the principal angles. The cost of the optimal transformation, the square root of the sum of the principal angles squared, also provides a useful measure for the angular distance between two elements of SO(N).     

Low-Thrust Transfers to Southern $$L_2$$ L 2 Near-Rectilinear Halo Orbits Facilitated by Invariant Manifolds

Journal of Optimization Theory and Applications - In this paper, we investigate the manifolds of three Near-Rectilinear Halo Orbits (NRHOs) and optimal low-thrust transfer trajectories using a...     

A perturbation method for estimation of dynamic systems

A novel framework called the Perturbed Jth Moment Extended Kalman Filter (PJMEKF), based on a classical perturbation technique is proposed for estimating the states of a nonlinear dynamical system from sensor measurements. This method falls under a class of architectures under investigation primarily to study the interplay of major issues in nonlinear estimation such as nonlinearity, measurement sparsity, and initial condition uncertainty in an environment with low levels of process noise. Taylor series expansion of the departure motion dynamics about the best estimate is used to derive a series representation of the unforced motion. It is found that such series representation evolves as a set of differential equations that force each other in a cascade manner, adding up to give the unforced motion (in a so-called “triangular” structure). This formal perturbation solution for the departure motion dynamics is used in deriving the differential equations governing the time evolution of the high order statistical moments of the estimation error. These tensor differential equations are found to possess a similar high order triangular structure in addition to being symmetric (in N tensorial dimensions and we appropriately term the evolution equations as Tensor Lyapunov Equations of statistical moment perturbations). Elegance of the tensor differential equations thus derived is accompanied by the computational advantages due to symmetry in all tensorial dimensions. A vector matrix representation of tensors is proposed with which the representation and solution of the tensor differential equations can be carried out effectively. Approximations are introduced to incorporate low levels of process noise forcing function in the propagation phase of the moment equations. The statistics thus propagated are used in a filtering framework to estimate the state vector of a nonlinear system from noisy measurements, within the traditional Kalman update paradigm. The Kalman gain thus determined is utilized in updating all high order moments in preparation for the subsequent propagation phase leading to improved estimation accuracy. The filter developed is applied to an orbit estimation problem and comparisons are presented with classical extended Kalman filter.     

Perturbation methods based upon varying action integrals

Direct perturbation solution procedures to non-linear systems are developed from variational statements derived from the principle of invariance of the action integral under infinitesimal transformations. Solution procedures that are the variational equivalent of the classical perturbation methods of strained parameters, the KBM method of averaging and the method of multiple time scales are presented.     

The J-invariant and Tits algebras for groups of inner type E 6

A connection between the indices of the Tits algebras of a split linear algebraic group G and the degree one parameters of its motivic J-invariant was introduced by Quéguiner-Mathieu, Semenov and Zainoulline through use of the second Chern class map in the Riemann-Roch theorem without denominators. In this paper we extend their result to higher Chern class maps and provide applications to groups of inner type E ₆.     

Twisted gamma filtration and algebras with orthogonal involution

For the Grothendieck group of a split simple linear algebraic group, the twisted γ-filtration provides a useful tool for constructing torsion elements in -rings of twisted flag varieties. In this paper, we construct a non-trivial torsion element in the γ-ring of a complete flag variety twisted by means of a PGO-torsor. This generalizes the construction in the HSpin case previously obtained by Zainoulline.