Analytical Solution of an Approximate Equation for the Vector of a Rigid Body Finite Rotation and its Application to Construct the Algorithm for Determining the Strapdown INS Orientation

2019 ◽  
Author(s):  
A. V. Molodenkov ◽  
S.E. Perelyaev ◽  
Ya.G. Sapunkov ◽  
T.V. Molodenkova
Keyword(s):  
Analytical Solution ◽  
Rigid Body ◽  
Approximate Equation ◽  
Finite Rotation

The New Analytical Algorithm for Determining the Strapdown INS Orientation

2019 ◽  
Vol 20 (10) ◽  
pp. 624-628 ◽  
Author(s):  
A. V. Molodenkov ◽  
Ya. G. Sapunkov ◽  
T. V. Molodenkova
Keyword(s):  
Angular Velocity ◽  
Analytical Solution ◽  
Rigid Body ◽  
Linear Equation ◽  
Initial Position ◽  
Finite Rotation ◽  
Zero Initial Condition ◽  
Analytical Algorithm ◽  
Subsequent Step ◽  
Regular Character

The analytical solution of an approximate (truncated) equation for the vector of a rigid body finite rotation has made it possible to solve the problem of determining the quaternion of orientation of a rigid body for an arbitrary angular velocity and small angle of rotation of a rigid body with the help of quadratures. Proceeding from this solution, the following approach to the construction of the new analytical algorithm for computation of a rigid body orientation with the use of strapdown INS is proposed: 1) By the set components of the angular velocity of a rigid body on the basis of mutually — unambiguous changes of the variables at each time point, a new angular velocity of a rigid body is calculated; 2) Using the new angular velocity and the initial position of a rigid body, with the help of the quadratures we find the exact solution of an approximate linear equation for the vector of a rigid body finite rotation with a zero initial condition; 3) The value of the quaternion orientation of a rigid body (strapdown INS) is determined by the vector of finite rotation. During construction of the algorithm for strapdown INS orientation at each subsequent step the change of the variables takes into account the previous step of the algorithm in such a way that each time the initial value of the vector of finite rotation of a rigid body will be equal to zero. Since the proposed algorithm for the analytical solution of the approximate linear equation for the vector of finite rotation is exact, it has a regular character for all angular motions of a rigid body).

On the Rigid-Body Motion of Traveling, Sagged, Elastic Cables

2002 ◽  
Author(s):  
Albert C. J. Luo ◽  
Yuefang Wang
Keyword(s):  
Analytical Solution ◽  
Rigid Body ◽  
Elastic Deformation ◽  
Flexible Structures ◽  
Rigid Body Motion ◽  
Body Motion ◽  
Cable Model ◽  
Cable Structures ◽  
Elastic Cables

This paper presents a methodology for modeling very flexible structures. A theory for the dynamics of traveling, arbitrarily sagged, elastic cables is developed for demonstration of this methodology. In this theory, the cable motion is modeled through the rigid-body motion and elastic deformation, and the rigid-body motion of cable configuration is modeled as an inextensible cable model. The dynamic, rigid-body configuration of a cable is a referenced base to describe its elastic deformation motion for any instantaneous moment. In this paper, the analytical solution for the rigid-body motion of the cable under a certain loading is developed as Part I of this investigation. From the dynamical configuration of the rigid-body motion of cable, an elastic motion of nonlinear cables is further investigated in sequel as the part II. This theory can be applied any cable structures and the methodology is useful for the perfectly flexible structures such as membranes.

On the Synthesis of Three-Poses Rigid-Body Guidance Mechanisms

2015 ◽  
Author(s):  
Giorgio Figliolini ◽  
Pierluigi Rea
Keyword(s):  
Rigid Body ◽  
Finite Rotation ◽  
Graphical Approach ◽  
Kinematic Characteristics ◽  
Different Types

A graphical approach that is based on the use of the pole triangle among the three poles of the finite rotation, along with its circumcircle and those of the three image pole triangles, which intersect each other at the orthocenter of the pole triangle, is presented in this paper. This method is applied and parameterized in SolidWorks to synthesize different types of mechanisms for any triple of prescribed positions, as four-bar linkages (4R), slider-crank/rocker mechanisms (3RP), double-sliders (2R2P), guiding mechanisms (RPRP). Finally, the kinematic characteristics of the rigid-body guidance mechanisms for three finitely and infinitesimally separated positions are compared via significant examples.

Static load sharing by tooth pairs in contact in internal involute spur gearing with thin rimmed pinion

2016 ◽  
Vol 230 (4) ◽  
pp. 485-499 ◽  
Author(s):  
Vineet Sahoo ◽  
Rathindranath Maiti
Keyword(s):  
Analytical Solution ◽  
Rigid Body ◽  
Static Load ◽  
Angular Position ◽  
Load Sharing ◽  
The Other ◽  
Contact Pattern ◽  
Contact Deformation ◽  
Angular Rotation ◽  
External Standard

Involute toothed internal−external standard gear sets are modeled for load-sharing by the teeth pairs in mesh along the line of contact. An analytical solution is proposed. Considering the rigid body in rotation, it is assumed that angular rotation of a gear with respect to the other gear due to deformation along the line of contact is equal. The sum of the normal loads in all tooth pairs in contacts, which equals to the total transmitted load, is considered constant. All possible deformations such as, tooth bending deflection, tooth compressive (contact) deformation, tooth foundation deflection and tooth shearing deflection are considered in analyses. Detailed tooth geometries are incorporated in modeling. Ultimately, the map of load sharing by tooth pairs in contacts, at different angular position, over a cycle of similar contact pattern, is established. Finally, considering thin rimmed gears, the effects of the rim thickness on load sharing, which is the aim of the present investigation are analyzed and the results are presented in terms of backup ratios.

Inverse Kinematics of an Untethered Rigid Body Undergoing a Sequence of Forward and Reverse Rotations

2004 ◽  
Vol 126 (5) ◽  
pp. 813-821 ◽  
Author(s):  
Sung K. Koh ◽  
G. K. Ananthasuresh
Keyword(s):  
Analytical Solution ◽  
Rigid Body ◽  
Numerical Solution ◽  
Inverse Kinematics ◽  
Underwater Vehicles ◽  
Numerical Examples ◽  
Graphical Visualization ◽  
Inverse Kinematics Problem ◽  
Closed Form Analytical Solution ◽  
Neutrally Buoyant

A sequence of rotations considered in this paper is a series of rotations of an untethered rigid body about its body-fixed axes such that the rotation about each axis is fully reversed at the end of the sequence. Due to the noncommutative property of finite rigid body rotations, such a sequence can effect nonzero changes in the orientation of the rigid body even though the net rotation about each axis is zero. These sequences are useful for attitude maneuvers of miniature spacecraft that use elastic deformation-based microactuators, or of other airborne or neutrally buoyant underwater vehicles. This paper considers the inverse kinematics problem of determining the angles in a given sequence to achieve a desired change in the orientation. Two types of problems are addressed. For the first problem, where four-rotation sequences are used, an analytical solution is presented and it is shown that a pointing vector attached to the rigid-body can be arbitrarily oriented. In the second problem, six-rotation sequences are used to control all three of the orientation freedoms of the rigid body. Some of the six-rotation sequences can provide any change in orientation while others are limited in their capabilities. A general numerical solution for all types, and a closed-form analytical solution for one type are presented along with the numerical examples and graphical visualization.

Applications of the 4D Geometric Algebra to Dimensional Mobility Criteria of Delassus-Parallelogram and Bennett Paradoxical Linkages

2015 ◽  
Author(s):  
Chung-Ching Lee
Keyword(s):  
Rigid Body ◽  
Geometric Algebra ◽  
Three Dimensional ◽  
Grassmann Algebra ◽  
Body Motion ◽  
Finite Rotation ◽  
Finite Rotations ◽  
Hypercomplex Number ◽  
Geometric Problems ◽  
Methodical Approach

Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming three-dimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to g4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of g4. This article relies on the combined rotation and translation in g4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in g4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassus-parallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.

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