Orientation Of Rigid Body Research Articles

Symmetrical Parameterization of Rigid Body Transformations for Biomolecular Structures.

Assessing preferred relative rigid body position and orientation is important in the description of biomolecular structures (such as proteins) and their interactions. In this article, we extend and apply the "symmetrical parameterization," which we recently introduced in the kinematics community, to address problems in structural biology. We also review parameterization methods that are widely used in structural biology to describe relative rigid body motions (in particular, orientations) as a basis for comparison. The new symmetrical parameterization is useful in describing the relative biomolecular rigid body motions, where the parameters are symmetrical in the sense that the subunits of a complex biomolecular structure are described in the same way for the corresponding motion and its inverse. The properties of this new parameterization, singularity analysis, and inverse kinematics are also investigated in more detail. Finally, parameterization is applied to real biomolecular structures and a potential application to structure modeling of symmetric macromolecules to show the efficacy of the symmetrical parameterization in the field of computational structural biology.

Adaptive Nonlinear Boundary Control of a Hybrid Euler–Bernoulli Beam with Coupled Rigid and Flexible Dynamics

This paper establishes adaptive nonlinear boundary control of an Euler–Bernoulli beam with rigid and flexible hybrid motion in the plane. In the previous work, a nonlinear boundary control was designed based on partial differential equation model that stabilizes the beam vibration while regulating the rigid body position and orientation. Here, the control law in the previous work is redesigned to establish an adaptive boundary controller that is able to compensate for system parametric uncertainties. Lyapunov analysis shows that the proposed adaptive control achieves asymptotic stability for the hybrid beam system. Simulation results are used to illustrate the efficacy of the proposed controller.

Real-time body orientation estimation based on two-layer stochastic filter architecture

The article presents real time rigid body orientation estimation using inertial and magnetic sensors. Based on the review of orientation estimation literature we suggest, as possible alternative to standard approaches, novel two-layer stochastic estimation filter architecture based on Kalman and particle filters combined into two layers. Two-layer architecture was chosen because it enables greater applicability via upgrade of already implemented Kalman or particle filters. Four two-layer filter architectures were designed, each one enabling different layer interaction. Estimation of human head orientation was chosen as a case example. Simulated data and batch head orientation measurement data were used to test the proposed architectures in terms of accuracy and computational efficiency and to select the best one in terms of aforementioned performance parameters. Selected architecture was then implemented in real time for human-computer interaction and was tested on several practical applications. Obtained results are presented and discussed and future research directions suggested.

High-accuracy modeling of antibody structures by a search for minimum-energy recombination of backbone fragments.

Current methods for antibody structure prediction rely on sequence homology to known structures. Although this strategy often yields accurate predictions, models can be stereo-chemically strained. Here, we present a fully automated algorithm, called AbPredict, that disregards sequence homology, and instead uses a Monte Carlo search for low-energy conformations built from backbone segments and rigid-body orientations that appear in antibody molecular structures. We find cases where AbPredict selects accurate loop templates with sequence identity as low as 10%, whereas the template of highest sequence identity diverges substantially from the query's conformation. Accordingly, in several cases reported in the recent Antibody Modeling Assessment benchmark, AbPredict models were more accurate than those from any participant, and the models' stereo-chemical quality was consistently high. Furthermore, in two blind cases provided to us by crystallographers prior to structure determination, the method achieved <1.5 Ångstrom overall backbone accuracy. Accurate modeling of unstrained antibody structures will enable design and engineering of improved binders for biomedical research directly from sequence. Proteins 2016; 85:30-38. © 2016 Wiley Periodicals, Inc.

Improved quaternion-based integration scheme for rigid body motion

Rotation quaternions are frequently used for describing the orientation of non-spherical rigid bodies. Their compact representation by four numbers and disappearance of numerical problems, such as gimbal lock, are reasons for using them. We describe an improvement of a predictor–corrector direct multiplication (PCDM) method for numerically integrating the rigid body equations of motion with rotation quaternions. The method only uses quaternions to describe the orientation, so no rotation matrices are needed in the implementation. A predictor–corrector approach is used to update the quaternions each time step, such that no renormalization is needed at the end of the time step. The PCDM method suggested by Zhao and Van Wachem is improved such that forces and torques are calculated at the correct time using position and orientation information at that same time. This is achieved by using a leapfrog approach in the improved version, in which the linear and angular velocities and rotation quaternions are defined at half time steps, while whole time step information of these quantities is calculated as part of the improved integration scheme. The improved PCDM scheme is compared with the original implementation for rotational kinetic energy conservation, accuracy of object orientation and angular velocity, and rate of convergence for different time steps. With the modifications that we propose, the improved method has a true second-order rate of convergence, without the need for explicit renormalization of the quaternions. Furthermore, the method is applicable to problems with position and velocity dependent torques, while still only a single force/torque evaluation is needed per time step. For objects experiencing torque, the improved PCDM method performs better than the original method, now showing a true second-order rate of convergence, and much smaller errors in the prediction of object orientation and angular velocity while still requiring only a single torque evaluation per time step.

Distributed event-triggered cooperative attitude control of multiple groups of rigid bodies on manifold SO(3)

In this note, the distributed event-triggered cooperative attitude control of multiple groups of rigid bodies evolved on nonlinear manifold SO(3) is investigated. The distributed event-triggered mechanism is introduced into the controller design to reduce the communication pressure. It is verified that, the orientations of rigid bodies in each group converge to a common value with zero angular velocities, and these common values in all groups achieve an anti-consensus configuration under the proposed intra- and inter-group cooperative control law. Simulation results show the effectiveness of the proposed method.

Inverse kinematics of a heavy duty manipulator with 6-DOF based on dual quaternion

An iterative method is introduced successfully to solve the inverse kinematics of a 6-DOF manipulator of a tunnel drilling rig based on dual quaternion, which is difficult to get the solution by Denavit-Hartenberg (D-H) based methods. By the intuitive expression of dual quaternion to the orientation of rigid body, the coordinate frames assigned to each joint are established all in the same orientation, which does not need to use the D-H procedure. The compact and simple form of kinematic equations, consisting of position equations and orientation equations, is also the consequence of dual quaternion calculations. The iterative process is basically of two steps which are related to solving the position equations and orientation equations correspondingly. First, assume an initial value of the iterative variable; then, the position equations can be solved because of the reduced number of unknown variables in the position equations and the orientation equations can be solved by applying the solution from the position equations, which obtains an updated value for the iterative variable; finally, repeat the procedure by using the updated iterative variable to the position equations till the prescribed accuracy is obtained. The method proposed has a clear geometric meaning, and the algorithm is simple and direct. Simulation for 100 poses of the end frame shows that the average running time of inverse kinematics calculation for each demanded pose of end-effector is 7.2 ms on an ordinary laptop, which is good enough for practical use. The iteration counts 2–4 cycles generally, which is a quick convergence. The method proposed here has been successfully used in the project of automating a hydraulic rig.

A constraint violation suppressing formulation for spatial multibody dynamics with singular mass matrix

The constraint violation problem of spatial multibody systems is analyzed in this paper. The mass matrix is singular in the equations of motion when the Euler parameters with the normalization constraints are used to describe the orientation of the spatial rigid body. The constrained and weighted least-squares-based geometrical projection method is implemented to suppress the constraint violation during numerical integration, and the explicit correction formulation can be obtained by the block matrix inversion scheme. The mass matrix weighted correction formulation gives the physically consistent energy norm, but it needs the mass matrix to be positive definite. To extend the physically consistent correction formulation for solving spatial multibody systems’ constraint violation problems with a singular mass matrix, a Modified Mass-Orthogonal Projection Method (MMOPM) and a Generalized Physical Orthogonal Projection Method (GPOPM) are proposed. MMOPM modifies the mass matrix directly by adding a penalty factor matrix which appears in the mass-orthogonal projection method and leads to a positive definite weight matrix that satisfies the block matrix inversion scheme condition. GPOPM is a generalization of the physical orthogonal projection method where the constrained least-squares method is weighted by the positive semi-definite mass matrix and the correction formulation is given by using the generalized block matrix inversion scheme. Numerical results show the feasibility and accuracy of the presented MMOPM and GPOPM. The constraints in position and velocity can reach machine precision during numerical integration. The elimination of violation of position constrains may require few iterations, while the violation of velocity constraints is removed in one step, and GPOPM is more accurate in velocity correction than MMOPM.

Pose Error Analysis Model Based on Binocular Vision for Rigid-Body

In order to satisfy the orientationmeasuring requirements of rigid-body such as work piece, cutting tool inindustry and medical instrumentsfor invasive surgery, this paper presents a binocularvision detection technique based on spatial position information of markers toextract rigid-body pose information and analyzes the pose accuracy of rigid-bodyusing the principalcomponent analysis(PCA) and the least square method(LSM) when spatial position error of markers exist. The simulation experiment demonstrates themaximum angle error of orientation is about 0.59 degree when the position error ofmarkers satisfy the Gaussian distribution with the meanis zero and thestandard deviation is 0~3mm. The experimental results verify this method canrobustly solve the orientation of rigid body using the position information ofmarkers with position errors, and it provides a theoretical and experimentalbasis for orientation measurement of rigid body. http://dx.doi.org/10.11591/telkomnika.v12i12.6136

Rigid Body Orientation Analysis Model Based on Stereo Vision

In order to satisfy the orientation measuring requirements of rigid-body such as work piece, cutting tool in industry, the paper presents a binocular vision detection technique based on spatial position information of markers to extract rigid-body pose information and analyzes the pose accuracy of rigid-body using the principal component analysis (PCA) when spatial position error of markers exist. The simulation experiment demonstrates the maximum angle error of orientation is about 0.59 degree when the position error of markers satisfy the Gaussian distribution with the mean is zero and the standard deviation is 0~3mm. The experimental results verify this method can robustly solve the orientation of rigid body using the position information of markers with position errors, and it provides a theoretical and experimental basis for orientation measurement of rigid body.

High-resolution structure of the Shigella type-III secretion needle by solid-state NMR and cryo-electron microscopy.

We introduce a general hybrid approach for determining the structures of supramolecular assemblies. Cryo-electron microscopy (cryo-EM) data define the overall envelope of the assembly and rigid-body orientation of the subunits while solid-state NMR (ssNMR) chemical shifts and distance constraints define the local secondary structure, protein fold and inter-subunit interactions. Finally, Rosetta structure calculations provide a general framework to integrate the different sources of structural information. Combining a 7.7-Å cryo-EM density map and 996 ssNMR distance constraints, the structure of the Type-III Secretion System (T3SS) needle of Shigella flexneri is determined to a precision of 0.4 Å. The calculated structures are cross-validated using an independent dataset of 691 ssNMR constraints and STEM measurements. The hybrid model resolves the conformation of the non-conserved N-terminus, that occupies a protrusion in the cryo-EM density, and reveals conserved pore residues forming a continuous pattern of electrostatic interactions, thereby suggesting a mechanism for effector protein translocation.

Dynamic modeling and nonlinear boundary control of hybrid Euler–Bernoulli beam system with a tip mass

This paper presents dynamic modeling and Lyapunov-based boundary control of a hybrid Euler–Bernoulli beam. The beam is hybrid in the sense that it holds both rigid and elastic motions. The beam is equipped with actuators hub at one end and it carries the payload as the tip mass at the free end. The distributed parameter dynamic equations (i.e. partial differential equations governing the hybrid beam motion) are derived using Hamilton’s principle. The dynamic model consists of four distributed parameter dynamic equations, representing the beam vibration and rigid motion in the plane, coupled to the discrete dynamic boundary equations. This paper uses the system equations to achieve model-based control laws that asymptotically stabilize the beam vibration while driving the rigid body position and orientation to the desired set point. The control system applies three forces/torque at the hub to regulate the rigid body position/orientation and a transverse force at the free end to suppress the beam vibration. In this regard, the control system only applies actuation and makes measurement at the beam boundary, thus excluding the need for distributed actuators or sensors. Furthermore, the proposed method directly uses the system equations for the control design without model truncation to rule out spillover instabilities. The closed-loop system stability is shown through Lyapunov-based analysis. Numerical simulations demonstrate the effectiveness of the proposed method.

GPCRTooKit: A Computational Platform for Structural Comparison of GPCR Crystal Structures and Homology Model Refinement

G-protein coupled receptors (GPCRs) constitute an important family of transmembrane receptors that regulate major life processes including sensory perception, cell proliferation and hormonal regulation. In lieu of the recent surge in GPCR crystal structures, structure based drug design methods are becoming more viable for GPCRs. However there is a huge disparity in the number of GPCR structures available and the number of GPCRs being studied in research. Thus homology models play an important role in a wide variety of GPCR research ranging from drug design to studying the functional mechanisms in these receptors. GPCR homology models do not capture the critical structural differences between the template and the GPCR being investigated.We have developed GPCRToolKit that lays the computational framework for building GPCR based modeling tools. We have developed a robust and accurate homology model refinement method called LITiCon2.0, based on optimizing the helical translation, rotational orientation, tilt and gyration angles of the seven helices. The structural differences between two GPCRs stem mostly from the rigid body degrees of freedom. The algorithm is highly parallel to enhance the computational speed. We have tested and validated this method by refining the homology models of several class A GPCRs with known crystal structures, one of them being the chemokine CXCR4 using β2-adrenergic receptor as template. We have also analyzed the statistical distribution of translations, tilts and rotations of TM helices in all available GPCR crystal structures. We found that TM3 and TM7 showed relatively less variation in rigid body orientation compared to the other helices. Overall, TM4 and TM5 showed the highest diversity in spatial orientation among the TM domains. These results along with the description and validation of the LITiCon2.0 method will be presented.

On the choice of coordinates for computational multibody dynamics

AbstractThe choice of coordinates for the description of multibody dynamics has a strong impact on the form of the equations of motion. In the talk two alternative formulations are compared: (i) joint coordinates along with Euler angles for the orientation of rigid bodies, and (ii) redundant coordinates where the orientation of rigid bodies is described in terms of direction cosines. In the case of multibody systems with tree structure the use of generalized coordinates yields equations of motion in the form of ordinary differential equations. In contrast to that, the choice of redundant coordinates yields differential‐algebraic equations. The two alternative formulations are compared and their influence on the numerical time integration is highlighted. (© 2011 Wiley‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim)

Construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body

We consider the problem of construction of optimal laws of variation in the angular momentum vector of a dynamically symmetric rigid body so as to ensure the transition of the rigid body from an arbitrary initial angular position to the required final angular position. For the functionals to be minimized, we use combined performance functionals, one of which characterizes the expenditure of time and of the squared modulus of the angular momentum vector in a given proportion, while the other characterizes the expenditure of time and momentum of the modulus of the angular momentum vector necessary to change the rigid body orientation. The control (the vector of the rigid body angular momentum) is assumed to be bounded in the modulus. The problem is solved by using Pontryagin’s maximum principle and the quaternion differential equation [1, 2] relating the vector of the dynamically symmetric rigid body angular momentum to the quaternion of orientation of the coordinate system rotating with respect to the rigid body about its dynamical symmetry axis at an angular velocity proportional to the angular momentum vector projection on the axis. The use of such a model of rotational motion leads to the problem of optimal control with the moving right end of the trajectory and significantly simplifies the analytic study of the problem of construction of optimal laws of variation in the angular momentum vector, because this model explicitly exploits the body angular momentum quaternion (control) instead of the rigid body absolute angular velocity quaternion. We construct general analytic solutions of the differential equations for the boundary-value problems which form systems of nine nonlinear differential equations. It is shown that the process of solving the differential boundary-value problems is reduced to solving two scalar algebraic transcendental equations. We obtain, as explicit functions of time, dependencies for the orientation quaternion, the absolute angular velocity vector, and the vector of the rigid body angular momentum, which describe the program optimal controlled motion of the rigid body. We construct the corresponding laws of variation in the program controlling moments for the rigid body (spacecraft) and geometrically interpret the controlled angular motion of the rigid body. This paper generalizes and develops the results obtained in [3].

SymmRef: a flexible refinement method for symmetric multimers.

Symmetric protein complexes are abundant in the living cell. Predicting their atomic structure can shed light on the mechanism of many important biological processes. Symmetric docking methods aim to predict the structure of these complexes given the unbound structure of a single monomer, or its model. Symmetry constraints reduce the search-space of these methods and make the prediction easier compared to asymmetric protein-protein docking. However, the challenge of modeling the conformational changes that the monomer might undergo is a major obstacle. In this article, we present SymmRef, a novel method for refinement and reranking of symmetric docking solutions. The method models backbone and side-chain movements and optimizes the rigid-body orientations of the monomers. The backbone movements are modeled by normal modes minimization and the conformations of the side-chains are modeled by selecting optimal rotamers. Since solved structures of symmetric multimers show asymmetric side-chain conformations, we do not use symmetry constraints in the side-chain optimization procedure. The refined models are re-ranked according to an energy score. We tested the method on a benchmark of unbound docking challenges. The results show that the method significantly improves the accuracy and the ranking of symmetric rigid docking solutions. SymmRef is available for download at http:// bioinfo3d.cs.tau.ac.il/SymmRef/download.html.

Rosetta FlexPepDock ab-initio: simultaneous folding, docking and refinement of peptides onto their receptors.

Flexible peptides that fold upon binding to another protein molecule mediate a large number of regulatory interactions in the living cell and may provide highly specific recognition modules. We present Rosetta FlexPepDock ab-initio, a protocol for simultaneous docking and de-novo folding of peptides, starting from an approximate specification of the peptide binding site. Using the Rosetta fragments library and a coarse-grained structural representation of the peptide and the receptor, FlexPepDock ab-initio samples efficiently and simultaneously the space of possible peptide backbone conformations and rigid-body orientations over the receptor surface of a given binding site. The subsequent all-atom refinement of the coarse-grained models includes full side-chain modeling of both the receptor and the peptide, resulting in high-resolution models in which key side-chain interactions are recapitulated. The protocol was applied to a benchmark in which peptides were modeled over receptors in either their bound backbone conformations or in their free, unbound form. Near-native peptide conformations were identified in 18/26 of the bound cases and 7/14 of the unbound cases. The protocol performs well on peptides from various classes of secondary structures, including coiled peptides with unusual turns and kinks. The results presented here significantly extend the scope of state-of-the-art methods for high-resolution peptide modeling, which can now be applied to a wide variety of peptide-protein interactions where no prior information about the peptide backbone conformation is available, enabling detailed structure-based studies and manipulation of those interactions.

A Nonlinear Filtering Approach for the Attitude and Dynamic Body Acceleration Estimation Based on Inertial and Magnetic Sensors: Bio-Logging Application

This paper addresses the problem of rigid body orientation and Dynamic Body Acceleration (DBA) estimation. This work is applied in bio-logging, an interdisciplinary research area at the intersection of animal behavior and bioengineering. The proposed approach combines a quaternion-based nonlinear filter with the Levenberg Marquardt Algorithm (LMA). The algorithm has a complementary structure design that exploits measurements from a three-axis accelerometer, a three-axis magnetometer, and a three-axis gyroscope. Attitude information is necessary to calculate the animal's DBA in order to evaluate its energy expenditure. Some numerical simulations illustrate the nonlinear filter performance. Some quantitative assessments prove this efficiency such as the time constant of the filter ( ) and the rms magnitude of the quaternion error ( ). Moreover, the effectiveness of the algorithm is experimentally demonstrated. In the experiments a domestic animal is equipped with an Inertial Measurement Unit (MTi-G), which provides a truth attitude for comparison with the complementary nonlinear filter. The rms difference between the filter and MTi-G outputs in the free movement experiments is within 0.392 rms on roll, 0.577 rms on pitch, and 2.521 rms on yaw.

On the Convex Parameterization of Constrained Spacecraft Reorientation

Guiding the rotational motion of a spacecraft subject to constraints on its permissible orientation often leads to nonconvex optimal control problems. In this paper, we consider convex parameterizations of sets associated with the constrained rigid body orientations. We then elaborate on ramifications of such a parameterization in the development of steering laws for autonomous spacecraft reorientation that are based on convex optimization algorithms. Such problems appear in almost every space science mission equipped with heat- or light-sensitive instruments, e.g., cryogenically cooled infrared telescopes, star trackers, and low-energy ion composition analyzers. An example, demonstrating the viability of the proposed algorithm for constrained multiple spacecraft reorientations concludes the work presented here.

Real-Time Body Orientation Estimation Based on Two-Layer Stochastic Filter Architecture

The article presents real time rigid body orientation estimation using inertial and magnetic sensors. Based on the review of orientation estimation literature we suggest, as possible alternative to standard approaches, novel two-layer stochastic estimation filter architecture based on Kalman and particle filters combined into two layers. Two-layer architecture was chosen because it enables greater applicability via upgrade of already implemented Kalman or particle filters. Four two-layer filter architectures were designed, each one enabling different layer interaction. Estimation of human head orientation was chosen as a case example. Simulated data and batch head orientation measurement data were used to test the proposed architectures in terms of accuracy and computational efficiency and to select the best one in terms of aforementioned performance parameters. Selected architecture was then implemented in real time for human-computer interaction and was tested on several practical applications. Obtained results are presented and discussed and future research directions suggested.