A New Approach to Generating Ruled Surfaces with Rigid Body Motion

Dual numbers are introduced by Clifford in the mid-nineteenth century and they are systematically applied to kinematics by Study (1903) and Kotelnikov (1895). Any point on the dual unit sphere (DUS) corresponds to a straight line in QUOTE (real three space), and vice versa. By this way, there is a one to one correspondence between the dual curves on DUS and the one parameter rigid body motions in QUOTE . Using Cayley Mapping (McCarthy, 1990) we get a relation between the dual rotation matrices and the dual skew symmetric matrices. In this paper, this relation is given by the exponential mapping which can be called the dual exponential mapping. Keywords: Kinematics, study mapping, cayley mapping, dual exponential mapping.

<i>Robots and Screw Theory: Applications of Kinematics and Statics to Robotics</i>

Virtually all docking methods include some local continuous minimization of an energy/scoring function in order to remove steric clashes and obtain more reliable energy values. In this paper, we describe an efficient rigid-body optimization algorithm that, compared to the most widely used algorithms, converges approximately an order of magnitude faster to conformations with equal or slightly lower energy. The space of rigid body transformations is a nonlinear manifold, namely, a space which locally resembles a Euclidean space. We use a canonical parametrization of the manifold, called the exponential parametrization, to map the Euclidean tangent space of the manifold onto the manifold itself. Thus, we locally transform the rigid body optimization to an optimization over a Euclidean space where basic optimization algorithms are applicable. Compared to commonly used methods, this formulation substantially reduces the dimension of the search space. As a result, it requires far fewer costly function and gradient evaluations and leads to a more efficient algorithm. We have selected the LBFGS quasi-Newton method for local optimization since it uses only gradient information to obtain second order information about the energy function and avoids the far more costly direct Hessian evaluations. Two applications, one in protein-protein docking, and the other in protein-small molecular interactions, as part of macromolecular docking protocols are presented. The code is available to the community under open source license, and with minimal effort can be incorporated into any molecular modeling package.

In this study, we introduced the vectorial moments as a new curves as [Formula: see text]-dual curve, where [Formula: see text], constructed by the Frenet vectors of a regular curve in Euclidean 3-space and we gave the Frenet apparatus of [Formula: see text]-dual curves and also we applied to helices and curve pairs of constant breadth.

In this paper, we first give the basic information about octonions and present the Euclidean rotation matrix formed by an octonion in seven‐dimensional Euclidean space. Next, we define and introduce the ‐module and dual vectors using dual numbers. Then, we provide the transformation that maps the points on the unit dual sphere one‐to‐one with the directed lines in . We also define a subset of the unit dual sphere, demonstrating that each element of this subset corresponds to two intersecting perpendicular directed lines in seven‐dimensional Euclidean space. Following that, we introduce dual octonions with their basic algebraic properties and examine rigid body (screw) motions in seven‐dimensional Euclidean space using dual octonions. Finally, we define an operator and express that this operator transforms two perpendicular intersecting directed lines in seven‐dimensional Euclidean space into two perpendicular intersecting directed lines.

We report on the validation of our combined respiratory and rigid body motion compensation strategy through acquisitions of the Data Spectrum anthropomorphic phantom, and investigation of clinical efficacy and robustness in 25 cardiac perfusion patient studies, employing a visual tracking system (VTS). The heart and liver was filled with a 2:1 concentration of Tc-99m and two sets of SPECT data were acquired. Each set of SPECT data consisted of a rest-perfusion baseline frame-mode emission acquisition, a Beacon (Philips, Cleveland, OH) transmission acquisition, and a list-mode emission acquisition. Respiratory motion was simulated during the list-mode acquisitions using the Quasar (Modus Medical Devices Inc. ON, Canada). Rigid-body motion was introduced in one of the two list-mode acquisitions by rotating the phantom around the x-axis and translating the phantom in x, y, and z. Patient volunteers with written consent were similarly acquired and asked to execute some predefined body motion during the list-mode acquisition. Motion tracking was performed using 6 near infrared Vicon cameras in combination with 7 retro-reflective markers, 5 placed on the chests of both patient volunteers and phantom, 2 placed on the abdomen of patient volunteers, and 2 placed on the vertical motion stage of the Quasar to simulate abdominal phantom motion. Processing steps included, down sampling VTS positional data to 10 Hz (100 ms) and synchronized with 100 ms SPECT frames, separating rigid body and respiratory motion and estimating 6 DOF rigid body motion, amplitude bin 100 ms frames into respiratory projection sets, reconstruct with rigid body motion compensation respiratory projection sets, estimated respiratory motion employing intensity based estimation, combine rigid body and respiratory motion, and reconstruct with combined compensation. We showed for both phantom and patient acquisitions that combined respiratory and rigid body motion compensation improve the visual appearance of slices.

The Procrustes regression model provides a statistical framework to assess the errors in image registration (in arbitrary dimensions) from “landmark” data. The same mathematics can be used to determine the errors in calculated motions of rigid bodies in Euclidean space. Perhaps the scientifically most compelling example of rigid body motion is tectonic plates. Tectonic plates, to a first approximation, move as rigid bodies on the surface of the Earth. The estimation of the past configuration of the tectonic plates, and the errors in these reconstructions, is integral to the understanding of the past history of the Earth. Because tectonic plates are restricted to the surface of the Earth, the Procrustes regression model does not apply and the relevant model is called spherical regression. The Procrustes and spherical regression models are mathematically simple and, because of this simplicity, beautiful theorems about the properties of their estimates can be proven. The previous chapter (Chang T, Image registration, rigid bodies, and unknown coordinate systems. In: Pham H (ed) Springer handbook of engineering statistics, Springer-Verlag, London, pp. 571–590, 2006) discusses many of these results. Using the spherical regression model, interesting insights into the properties of tectonic plate reconstructions are discussed in (Chang, J Geophys Res 92(B7):6319–6329, 1987; Chang, Int Stat Rev 61:299–316, 1993). We will not replicate these results here. Rather the focus of this chapter is the underlying mathematics used to establish the results. These problems are intrinsically geometric and the proper use of geometry is important to their understanding. We will explain these points. This chapter is motivated by a request from a geophysics friend to explain, in as elementary fashion as feasible, the mathematics behind his work. This chapter will not replicate the formal proofs that appear elsewhere (see, for example, Chang, Ann Stat 14(3):907–924, 1986, Rivest, Ann Stat 17(1):307–317, 1989, or Chang and Ko, Ann Stat 23(5):1823–1847, 1995). This chapter is aimed at the scientist who wants to understand on a heuristic level why the results are true, without reading mathematically complete proofs! The type of data that is actually used to estimate plate reconstructions (experimentally determined “marine magnetic anomaly lineation” locations) is not of the form that is modeled in the spherical regression model. We discuss in Sect. 53.6 the type of data that is actually collected and how to analyze it using the mathematical principles we will discuss here. For the nongeophysicist, this section can be used as an example of the use of the mathematical principles of this chapter in a different and complex data setting. Furthermore, although the rigid plate hypothesis is a simplification, one must first understand what types of reconstruction errors are consistent with rigid plates before deciding if the errors one is observing are in fact evidence of nonrigidity. Section 53.6 discusses some work of this type. Essentially, this chapter is a case study in the use of tangent space approximations and some elementary ideas from differential geometry. Other authors have used similar geometrical approaches. For example, Rivest (J Biomech 38:1604–1611, 2005) and Oualkacha and Rivest (Biometrika 99:585–598, 2012) developed statistical methods for human motion studies derived from sensors placed on the human body as it moves. Chang and Rivest (Ann Stat 29(3):784–814, 2001) extended to Stiefel manifolds the work outlined here and in (Chang T, Image registration, rigid bodies, and unknown coordinate systems. In: Pham H (ed) Springer handbook of engineering statistics, Springer-Verlag, London, pp. 571–590, 2006) and used the results to reanalyze a data set on vector cardiograms. Patrangenaru has numerous papers (e.g., Mardia and Patrangenaru (Ann Stat 33(4):1666–1699, 2005)) developing statistical methods for comparing images when projective, rather than rigid, transformations are allowed. Indeed, the author believes that the engineering disciplines have multiple problems of a geometric nature and hopes that the approaches used here can be helpful in studying them. This chapter has been written so that it can be read independently of the preceding chapter (Chang T, Image registration, rigid bodies, and unknown coordinate systems. In: Pham H (ed) Springer handbook of engineering statistics, Springer-Verlag, London, pp. 571–590, 2006).

Previous chapter Next chapter Other Titles in Applied Mathematics Nonholonomic Motion of Rigid Mechanical Systems from a DAE Viewpoint5. Constrained Rigid Bodiespp.45 - 59Chapter DOI:https://doi.org/10.1137/1.9780898719536.ch5PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt In this chapter, we derive the equations of motion of a constrained rigid body, or system of rigid bodies, in direct continuation of what was done in Chapter 4 in the unconstrained case. Our goal is to obtain a primary DAE formulation, namely, the system (5.49) below or (5.18) if only one body is involved. In addition, several other versions of these systems are given for cases involving certain special features. Specifically, while the main system (5.49) is a DAE on the configuration space, and hence a DAE on a manifold, the system (5.51) is a simpler DAE in Euclidian space in the case when extensions of the constraint function and of the force and moment fields are readily available. Alternately, geometric constraints lead to a special form of (5.49) given by the system (5.56). Although we set up the equations of motion, we do not address here the existence of solutions for the corresponding systems. This will be done later in Chapters 6 and 7, where, among other things, existence theorems for initial value problems are proved for DAEs in Euclidian space (Chapter 6) and manifolds (Chapter 7). 5.1 Kinematic Constraints We will retain here the same notation developed in Chapter 4 for the unconstrained motion of a rigid body ℬ. But, for the sake of brevity, it is useful to introduce the simplification Fm (t,u,p, u ˙ , p ˙ ) ≔2H (p)⊤ N (t,u,p, u ˙ , p ˙ ) +8H (p)⊤ H (p) H ( p ˙ )⊤ JH ( p ˙ ) p, 5.1 where the subscript "m" is a reminder that Fm represents a moment force. With this the equations of unconstrained motion (4.22) assume the more concise form Mü= Fr (t,u,p, u ˙ , p ˙ ), 4H (p) ⊤ JH (p) p ¨ = Fm (t,u,p, u ˙ , p ˙ ). 5.2 The unconstrained state space is S0 = R1 ×T ( R3 × S3 ) , with the understanding that (t, (u,p), ( u ˙ , p ˙ ) ) and (t, (u,−p), ( u ˙ ,− p ˙ ) ) in R1 ×T ( R3 × S3 ) correspond to the same state. Previous chapter Next chapter RelatedDetails Published:2000ISBN:978-0-89871-446-3eISBN:978-0-89871-953-6 https://doi.org/10.1137/1.9780898719536Book Series Name:Other Titles in Applied MathematicsBook Code:OT68Book Pages:vii + 136Key words:dynamics, rigidity, differentiable algebraic equations, numerical solutions, rigid bodies, Gauss principle, computational methods

We consider the problem of motion of several rigid bodies immersed in a perfect compressible fluid. Using the method of convex integration we establish the existence of infinitely many weak solutions with a priori prescribed motion of rigid bodies. In particular, the dynamics is completely time–reversible at the motion of rigid bodies although the solutions comply with the standard entropy admissibility criterion.

In this study, we examine the dual expression of Valeontis’ concept of parallel p-equidistant ruled surfaces well known in Euclidean 3-space, according to the Study mapping. Furthermore, we show that the dual part of the dual angle on the unit dual sphere corresponds to the p-distance. We call these ruled surfaces we obtained “dual parallel equidistant ruled surfaces” and we briefly denote them with “DPERS”. Furthermore, we find the Blaschke vectors, the Blaschke invariants and the striction curves of these DPERS and we give the relationships between these elements. Moreover, we show the relationships between the Darboux screws, the instantaneous screw axes, the instantaneous dual Pfaff vectors and dual Steiner rotation vectors of these surfaces. Finally, we give an example, which we reinforce this article, and we explain all of these features with the figures on the example. Furthermore, we see that the corresponding dual curves on the dual unit sphere to these DPERS are such that one of them is symmetric with respect to the imaginary symmetry axis of the other.

In this paper we present a simulation of a moving rigid circular body suspended in a cavity containing a rarefied gas. The rigid body moves due the flow developed in the gas by continuous uniform motion of one of the wall of the cavity. The flow in the gas is simulated by solving the Boltzmann equation using DSMC particle method. The motion of the rigid body is governed by Newton–Euler equations, where the force and the torque on the rigid body are computed from the momentum transfer of the gas molecules colliding with the rigid body. On the other hand, the motion of rigid body influences the gas flow in its surroundings. The numerical solutions obtained for the dynamics of the rigid body by solving the Boltzmann equation implementing moment and momentum approaches in the DSMC framework are compared. Furthermore, the numerical solutions obtained by solving the Boltzmann equation implementing momentum approach in DSMC framework are compared with the solutions obtained by solving the Navier–Stokes equations using finite pointset method for small as well as large values of Knudsen numbers.

This paper aims at presenting a method to determine the ”exact” natural frequencies and mode shapes of a hybrid beam composed of multiple elastic beam segments and multiple rigid bodies with each rigid body connected with two adjacent elastic beam segments. Furthermore, each rigid body has its own mass and rotary inertia, and is supported by a translational spring and/or a rotational spring. First, based on the equations of the continuity of deformations and the equilibrium of moments and forces for each of the intermediate rigid bodies and boundary conditions, the coefficient matrices of the entire hybrid beam are derived. The overall coefficient matrix for the entire hybrid beam is obtained using the numerical assembly technique. The exact natural frequencies are determined by equating the determinant of the last overall coefficient matrix to zero. With respect to each of the natural frequencies, one may obtain the associated integration constants from the simultaneous equations. Finally, substituting these integration constants into the displacement functions for all the elastic beam segments and replacing the space occupied by each of the rigid bodies by a straight line, one determines each of the corresponding mode shapes of the hybrid beam. Finally, the influence of materials for the elastic beam segments on natural frequencies and mode shapes of the hybrid beam is studied.

Motion planning for coupled rigid bodies in a horizontal plane is investigated. The rigid bodies are serially connected by passive revolute joints. The dynamic constraints on the system are second-order nonholonomic constraints. We attempted to control those n coupled rigid bodies by the translational acceleration inputs at the first joint. If each rigid body is hinged at the center of percussion, it is possible to compose a positioning trajectory by connecting rotational and translational trajectory segments. Each rigid body can be rotated about its center of percussion one after another. When all of the rigid bodies are aligned on a straight line, they can be translated. The algorithm for positioning is presented. Simulations show that the coupled planar rigid bodies can reach the desired configuration by the constructed inputs.

Motion planning for coupled rigid bodies in a horizontal plane is investigated. The rigid bodies are serially connected by passive revolute joints. The dynamic constraints on the system are second-order nonholonomic constraints. We attempted to control those n coupled rigid bodies by the translational acceleration inputs at the first joint. If each rigid body is hinged at the center of percussion, it is possible to compose a positioning trajectory by connecting rotational and translational trajectories. Each rigid body can be rotated about its center of percussion by turns. When all rigid bodies are aligned on a straight line, they can be translated. The algorithm for positioning is presented. Simulation shows that the coupled planar rigid bodies can reach the desired configuration by the constructed inputs.

The pose of a rigid object is usually regarded as a rigid transformation, described by a translation and a rotation. However, equating the pose space with the space of rigid transformations is in general abusive, as it does not account for objects with proper symmetries -- which are common among man-made objects.In this article, we define pose as a distinguishable static state of an object, and equate a pose with a set of rigid transformations. Based solely on geometric considerations, we propose a frame-invariant metric on the space of possible poses, valid for any physical rigid object, and requiring no arbitrary tuning. This distance can be evaluated efficiently using a representation of poses within an Euclidean space of at most 12 dimensions depending on the object's symmetries. This makes it possible to efficiently perform neighborhood queries such as radius searches or k-nearest neighbor searches within a large set of poses using off-the-shelf methods. Pose averaging considering this metric can similarly be performed easily, using a projection function from the Euclidean space onto the pose space. The practical value of those theoretical developments is illustrated with an application of pose estimation of instances of a 3D rigid object given an input depth map, via a Mean Shift procedure.