A Jacobian-Based Algorithm for Planning Attitude Maneuvers Using Forward and Reverse Rotations

Algorithms for planning quasistatic attitude maneuvers based on the Jacobian of the forward kinematic mapping of fully-reversed (FR) sequences of rotations are proposed in this paper. An FR sequence of rotations is a series of finite rotations that consists of initial rotations about the axes of a body-fixed coordinate frame and subsequent rotations that undo these initial rotations. Unlike the Jacobian of conventional systems such as a robot manipulator, the Jacobian of the system manipulated through FR rotations is a null matrix at the identity, which leads to a total breakdown of the traditional Jacobian formulation. Therefore, the Jacobian algorithm is reformulated and implemented so as to synthesize an FR sequence for a desired rotational displacement. The Jacobian-based algorithm presented in this paper identifies particular six-rotation FR sequences that synthesize desired orientations. We developed the single-step and the multiple-step Jacobian methods to accomplish a given task using six-rotation FR sequences. The single-step Jacobian method identifies a specific FR sequence for a given desired orientation and the multiple-step Jacobian algorithm synthesizes physically feasible FR rotations on an optimal path. A comparison with existing algorithms verifies the fast convergence ability of the Jacobian-based algorithm. Unlike closed-form solutions to the inverse kinematics problem, the Jacobian-based algorithm determines the most efficient FR sequence that yields a desired rotational displacement through a simple and inexpensive numerical calculation. The procedure presented here is useful for those motion planning problems wherein the Jacobian is singular or null.

An accurate numerical integration scheme for finite rotations using rotation vector parametrization

A numerical integration procedure for rotational motion using a rotation vector parametrization is explored from an engineering perspective by using rudimentary vector analysis. The incremental rotation vector, angular velocity and acceleration correspond to different tangent spaces of the rotation manifold at different times and have a non-vectorial character. We rewrite the equation of motion in terms of vectors lying in the same tangent space, facilitating vector space operations consistent with the underlying geometric structure. While any integration algorithm (that works within a vector space setting) may be used, we presently employ a family of explicit Runge-Kutta algorithms to solve this equation. While this work is primarily motivated out of a need for highly accurate numerical solutions of dissipative rotational systems of engineering interest, we also compare the numerical performance of the present scheme with some of the invariant preserving schemes, namely ALGO-C1, STW, LIEMIDEA] and SUBCYC-M. Numerical results show better local accuracy via the present approach vis-a-vis the preserving algorithms. It is also noted that the preserving algorithms do not simultaneously preserve all constants of motion. We incorporate adaptive time-stepping within the present scheme and this in turn enables still higher accuracy and a `near preservation' of constants of motion over significantly longer intervals. (C) 2010 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

Carbon-13 relaxation study of chain segmental motion and side-chain internal rotations of poly(isobutyl methacrylate) in solution

The dynamics of poly(isobutyl methacrylate) in toluene solution has been examined by C-13 spin-lattice relaxation time and NOE measurements as a function of temperature. The experiments were performed at 50.3 and 100.6 MHz. The backbone carbon relaxation data have been analyzed using the Dejean-Laupretre-Monnerie (DLM) model, which describes the dynamical processes in the backbone in terms of conformational transitions and bond librations. The relaxation data of the side chain nuclei have been analyzed by assuming different motional models, namely, unrestricted rotational diffusion, three site jumps, and restricted rotational diffusion. The different models have been compared for their ability to reproduce the experimental spin-lattice relaxation times and also to predict the behavior of NOE as a function of temperature. Conformational energy calculations have been carried out on a model compound by using the semiempirical quantum chemical method, AM1, and the results confirm the validity of the motional models used to describe the side-chain motion.

Efficient QR Decomposition Using Low Complexity Column-wise Givens Rotation (CGR)

QR decomposition (QRD) is a widely used Numerical Linear Algebra (NLA) kernel with applications ranging from SONAR beamforming to wireless MIMO receivers. In this paper, we propose a novel Givens Rotation (GR) based QRD (GR QRD) where we reduce the computational complexity of GR and exploit higher degree of parallelism. This low complexity Column-wise GR (CGR) can annihilate multiple elements of a column of a matrix simultaneously. The algorithm is first realized on a Two-Dimensional (2 D) systolic array and then implemented on REDEFINE which is a Coarse Grained run-time Reconfigurable Architecture (CGRA). We benchmark the proposed implementation against state-of-the-art implementations to report better throughput, convergence and scalability.

A frame-invariant scheme for the geometrically exact beam using rotation vector parametrization

While frame-invariant solutions for arbitrarily large rotational deformations have been reported through the orthogonal matrix parametrization, derivation of such solutions purely through a rotation vector parametrization, which uses only three parameters and provides a parsimonious storage of rotations, is novel and constitutes the subject of this paper. In particular, we employ interpolations of relative rotations and a new rotation vector update for a strain-objective finite element formulation in the material framework. We show that the update provides either the desired rotation vector or its complement. This rules out an additive interpolation of total rotation vectors at the nodes. Hence, interpolations of relative rotation vectors are used. Through numerical examples, we show that combining the proposed update with interpolations of relative rotations yields frame-invariant and path-independent numerical solutions. Advantages of the present approach vis-a-vis the updated Lagrangian formulation are also analyzed.

A Global Constraint On Relative Rotation To Avoid Lumped Compliant Mechanisms In Topology Optimization

Some of the well known formulations for topology optimization of compliant mechanisms could lead to lumped compliant mechanisms. In lumped compliance, most of the elastic deformation in a mechanism occurs at few points, while rest of the mechanism remains more or less rigid. Such points are referred to as point-flexures. It has been noted in literature that high relative rotation is associated with point-flexures. In literature we also find a formulation of local constraint on relative rotations to avoid lumped compliance. However it is well known that a global constraint is easier to handle than a local constraint, by a numerical optimization algorithm. The current work presents a way of putting global constraint on relative rotations. This constraint is also simpler to implement since it uses linearized rotation at the center of finite-elements, to compute relative rotations. I show the results obtained by using this constraint oil the following benchmark problems - displacement inverter and gripper.

Stability Of Large Damped Flexible Spacecraft With Stored Angular Momentum

Vibrational stability of large flexible structurally damped spacecraft carrying internal angular momentum and undergoing large rigid body rotations is analysed modeling the systems as elastic continua. Initially, analytical solutions to the motion of rigid gyrostats under torque-free conditions are developed. The solutions to the gyrostats modeled as axisymmetric and triaxial spacecraft carrying three and two constant speed momentum wheels, respectively, with spin axes aligned with body principal axes are shown to be complicated. These represent extensions of solutions for simpler cases existing in the literature. Using these solutions and modal analysis, the vibrational equations are reduced to linear ordinary differential equations. Equations with periodically varying coefficients are analysed applying Floquet theory. Study of a few typical beam- and plate-like spacecraft configurations indicate that the introduction of a single reaction wheel into an axisymmetric satellite does not alter the stability criterion. However, introduction of constant speed rotors deteriorates vibrational stability. Effects of structural damping and vehicle inertia ratio are also studied.

EPR observation of phase transitions in calcium cadmium acetate hexahydrate: Using Mn2+ and Cu2+ ions as probe

Results of temperature dependence of EPR spectra of Mn2+ and Cu2+ ions doped calcium cadmium acetate hexahydrate (CaCd(CH3COO)4•6H2O) have been reported. The investigation has been carried out in the temperature range between room temperature ( 300 K) and liquid nitrogen temperature. A I-order phase transition at 146 ± 0.5 K has been confirmed. In addition a new II-order phase transition at 128 ± 1 K has been detected for the first time. There is evidence of large amplitude hindered rotations of CH3 groups which become frozen at 128 K. The incorporation of Cu2+ and Mn2+ probes at Ca2+ and Cd2+ sites respectively provide evidence that the phase transitions are caused by the molecular rearrangements of the common coordinating acetate groups between Ca2+ and Cd2+ sites. In contradiction to the previous reports of a change of symmetry from tetragonal to orthorhombic below 140 K, the symmetry of the host is concluded to remain tetragonal in all the three observed phases between room temperature and liquid nitrogen temperature.

ESR Studies of Phase Transitions in Double Propionates: Dicalcium Barium Propionate Ca2Ba(C2H5COO)6

Single crystal [(111) and (100) planes], and powder ESR of Mn2+ (substituting for Ca2+) in Ca2Ba(C2H5COO)6 in the temperature range 220°C to -160°C shows (i) a doubling of both the physically and chemically inequivalent sites, and a change in the magnitude (150 G at -6°C to 170 G at -8°C) as well as the orientation of the D tensor across the -6°C transition and (ii) an inflection point in the D vs T plot across the -75°C transition. The oxygen octahedra around the Ca2+ sites are inferred to undergo alternate rotations, showing the participation of the carboxyl oxygens in the -6°C transition. A relation of the type D=D0(1+αT+βT2) seems to fit the D variation satisfactorily.

NMR relaxation study of disorder in the mixed system BP x GPI(1-x) and the low temperature transition from classical to quantum rotation

1H NMR spin-lattice relaxation time (T1) studies have been carried out in the temperature range 100 K to 4 K, at two Larmor frequencies 11.4 and 23.3 MHz, in the mixed system of betaine phosphate and glycine phosphite (BPxGPI(1-x)), to study the effects of disorder on the proton group dynamics. Analysis of T1 data indicates the presence of a number of inequivalent methyl groups and a gradual transition from classical reorientations to quantum tunneling rotations. At lower temperatures, microstructural disorder in the local environments of the methyl groups, result in a distribution in the activation energy (Ea) and the torsional energy gap (E01). For certain values of x, the magnetisation recovery shows biexponential behaviour at lower temperatures.

Macromechanical behaviour of composite laminates

An improved higher order transverse shear deformation theory is employed to arrive at modified constitutive relations which can be used in the flexural, buckling and vibration analysis of laminated plates and shells. The strain energy for such systems is then expressed in terms of the displacements and the rotations for ready reference and use. Numerical values of vibration frequencies are obtained using this formulation employing Ritz's method of analysis. The results are compared with those available in the literature to validate the analysis presented.

Molecular Dynamics in [(CH8)aN]z Cd14 A Proton Magnetic Relaxation Study

second moment measurements are carried out on [(CH,),N], CdI, in the temperature range 77 to 400 K. The results are interpreted based on a molecular dynamical model of randomly reorienting methyl groups and isotropically tumbling tetramethyl ammonium group. The relaxation data show contributions from spin-rotation interaction at high temperatures and presence of inequivalent methyl groups. The correlation times and associated activation energies, connected with this model, are calculated from the data. The structure in the absorption line and in the free-induction decay signal at 77 K indicates the possibility of tunnelling motion of the methyl groups. Im Temperaturbereich 77 bis 400 K werden an [(CH,),N],CdI, Protonen-Spin-Gitter-Relaxationsexperimente (bei Larmorfrequenzen von 10,20 und 30 MHz) und Messungen des zweiten Moments durchgefiihrt. Die Ergebnisse werden an Hand eines molekularen dynamischen Modells sich statistisch umorientierender Methylgruppen und isotrop taumelnder Tetramethyl-Ammoniumgruppen interpretiert. Die Relaxationswerte zeigen Beitriige von Spin-Rotations-Wechselwirkung bei hohen Temperaturen und die Anwesenheit von inaquivalenten Methylgruppen. Die Korrelationszeiten und verknupften Aktivierungsenergien, die mit diesem Model1 verbunden sind, werden am den Werten berechnet. Die Struktur in der Absorptionslinie und im Abklingsignal der freien Induktion bei 77 K zeigt die Moglichkeit einer Tunnelbewegung der Methylgruppen.

ESR studies of phase transitions in double propionates

Detailed ESR investigations of Mn2+ substituting for Ca2+ in Ca2Sr(C2H5COO)6 (DSP), Ca2Pb(C2H5COO)6 (DLP) and Ca2Ba(C2H5COO)6 (DBP), in single crystals and powders, over the temperature range from 200°C to -180°C have been carried out to study the successive phase transitions in these compounds. (DSP: [Tetragonal] ← 8.5°C → [tetragonal, ferroelectric] [tetragonal] ← -169°C → [monoclinic, ferroelectric]; DLP : [tetragonal] ← 60°C → [tetragonal, ferroelectric] ← -71.5°C → [monoclinic, ferroelectric]; [Cubic] ← -6°C → [orthorhombic] ← -75°C → [?]). Spectra have been analysed in terms of axial spin Hamiltonians and the temperature dependences of the parameters studied. In DSP and DLP across the I ↔ II transition, new physically and chemically inequivalent sites appear indicating the disappearance of the diad axes on which the propionate groups are located, bringing out the connection between the motional states of the propionate groups and the occurence of ferroelectricity. The II ↔ III transition also causes chemically inequivalent sites to develop, indicating that the transitions may not be isomorphous as believed previously. In DBP, the -6°C transition leads to (i) a doubling of both physically and chemically inequivalent sites (ii) a small (150 G at -6°C to 170 G at -8°C), but abrupt change in the magnitude of the zero-field splitting tensor D, and (iii) displacements of the orientations of the D tensors. Results are interpreted in terms of alternate rotations of the oxygen octahedra, showing participation of the carboxyl oxygens in the transition. No drastic changes in the parameters occur across the -75°C transition consistent with its second order nature. Similarities and dissimilarities of the ESR spectra of the three compounds in relation to the phase transitions, are discussed.

Is 3'-nucleotide rigid?

Conformational analysis of nucleic acids and polynucleotides is far more complex than that of proteins and polypeptides, due to five single bond rotations in addition to the sugar puckerings in the monomer. Sundaralingam1 proposed the concept of the 'rigid' nucleotides from analysis of crystal structure data, with the flexibility allowed only about the phosphodiester bonds. However, the crystal structure of deoxyguanosine-5'−phosphate2,3 indicates at gt conformation about the C-4'−C-5' bond against gg in a conformationally rigid nudeotide1. Jack et al. 4 considered the flexibility of nucleotides in tRNA about the C-4'−C-5' bond, thereby introducing the concept of 'non-rigid' ribonucleotides. Conformational flexibility of the f uranose ring in DNA and RNA and their energetics using classical and quantum chemical methods have been reported5−8. We have examined the flexibility of 3'-nucleotides. alpha, the most important of the conformational parameters defining the 3'-end of a nucleotide unit9, has a value in the range 195°−270° in all the 3'-nucleotides, dinucleoside monophosphates and higher oligomers which have been surveyed. A survey of the proposed structures of polyribonudeotides10,11 also shows the values of a to be greater than 200°. However, the structures proposed for B-DNA by Arnott and Hukins12,13 and D-DNA by Arnott et al. 14 have values of alpha of 155° and 141° respectively, much lower than the lowest observed value. The structure for B-DNA has two strong, short contacts (C-2'...OP-1 = 2.64 Å and HC-2"...OP-1 = 1.79 Å) which lead to an energetically unfavourable conformation. Hence, it is of interest to investigate whether, by allowing flexibility to the sugar moiety in the nucleotide unit, it is possible to make the structure energetically favourable. Here, conformational energy calculations were carried out to determine the range of alpha which would give rise to energetically favoured conformations with different sugar puckerings. Our analysis has shown that the theoretically obtained range is nearly the same as the preferred range in crystals, indicating the flexibility of the 3'-nucleotides.