A simple suboptimal least-squares algorithm for attitude determination with multiple sensors

Three-axis attitude determination is equivalent to finding a coordinate transformation matrix which transforms a set of reference vectors fixed in inertial space to a set of measurement vectors fixed in the spacecraft. The attitude determination problem can be expressed as a constrained optimization problem. The constraint is that a coordinate transformation matrix must be proper, real, and orthogonal. A transformation matrix can be thought of as optimal in the least-squares sense if it maps the measurement vectors to the reference vectors with minimal 2-norm errors and meets the above constraint. This constrained optimization problem is known as Wahba's problem. Several algorithms which solve Wahba's problem exactly have been developed and used. These algorithms, while steadily improving, are all rather complicated. Furthermore, they involve such numerically unstable or sensitive operations as matrix determinant, matrix adjoint, and Newton-Raphson iterations. This paper describes an algorithm which minimizes Wahba's loss function, but without the constraint. When the constraint is ignored, the problem can be solved by a straightforward, numerically stable least-squares algorithm such as QR decomposition. Even though the algorithm does not explicitly take the constraint into account, it still yields a nearly orthogonal matrix for most practical cases; orthogonality only becomes corrupted when the sensor measurements are very noisy, on the same order of magnitude as the attitude rotations. The algorithm can be simplified if the attitude rotations are small enough so that the approximation sin(theta) approximately equals theta holds. We then compare the computational requirements for several well-known algorithms. For the general large-angle case, the QR least-squares algorithm is competitive with all other know algorithms and faster than most. If attitude rotations are small, the least-squares algorithm can be modified to run faster, and this modified algorithm is faster than all but a similarly specialized version of the QUEST algorithm. We also introduce a novel measurement averaging technique which reduces the n-measurement case to the two measurement case for our particular application, a star tracker and earth sensor mounted on an earth-pointed geosynchronous communications satellite. Using this technique, many n-measurement problems reduce to less than or equal to 3 measurements; this reduces the amount of required calculation without significant degradation in accuracy. Finally, we present the results of some tests which compare the least-squares algorithm with the QUEST and FOAM algorithms in the two-measurement case. For our example case, all three algorithms performed with similar accuracy

Finite element formulation for modelling nonlinear viscoelastic elastomers

Nonlinear viscoelastic response of reinforced elastomers is modeled using a three-dimensional mixed finite element method with a nonlocal pressure field. A general second-order unconditionally stable exponential integrator based on a diagonal Padé approximation is developed and the Bergström–Boyce nonlinear viscoelastic law is employed as a prototype model. An implicit finite element scheme with consistent linearization is used and the novel integrator is successfully implemented. Finally, several viscoelastic examples, including a study of the unit cell for a solid propellant, are solved to demonstrate the computational algorithm and relevant underlying physics

Enabling scalable stochastic gradient-based inference for Gaussian processes by employing the Unbiased LInear System SolvEr (ULISSE)

In applications of Gaussian processes where quantification of uncertainty is of primary interest, it is necessary to accurately characterize the posterior distribution over covariance parameters. This paper proposes an adaptation of the Stochastic Gradient Langevin Dynamics algorithm to draw samples from the posterior distribution over covariance parameters with negligible bias and without the need to compute the marginal likelihood. In Gaussian process regression, this has the enormous advantage that stochastic gradients can be computed by solving linear systems only. A novel unbiased linear systems solver based on parallelizable covariance matrix-vector products is developed to accelerate the unbiased estimation of gradients. The results demonstrate the possibility to enable scalable and exact (in a Monte Carlo sense) quantification of uncertainty in Gaussian processes without imposing any special structure on the covariance or reducing the number of input vectors.Comment: 10 pages - paper accepted at ICML 201

Renyi Entropies of Interacting Fermions from Determinantal Quantum Monte Carlo Simulations

Entanglement measures such as the entanglement entropy have become an indispensable tool to identify the fundamental character of ground states of interacting quantum many-body systems. For systems of interacting spin or bosonic degrees of freedom much recent progress has been made not only in the analytical description of their respective entanglement entropies but also in their numerical classification. Systems of interacting fermionic degrees of freedom however have proved to be more difficult to control, in particular with regard to the numerical understanding of their entanglement properties. Here we report a generalization of the replica technique for the calculation of Renyi entropies to the framework of determinantal Quantum Monte Carlo simulations -- the numerical method of choice for unbiased, large-scale simulations of interacting fermionic systems. We demonstrate the strength of this approach over a recent alternative proposal based on a decomposition in free fermion Green's functions by studying the entanglement entropy of one-dimensional Hubbard systems both at zero and finite temperatures.Comment: 11 pages, 10 figure