Kinematic Convexity of Rigid Body Displacements

2010 ◽  
Author(s):  
Anurag Purwar ◽  
Jeff Ge
Keyword(s):  
Rigid Body ◽  
Projective Space ◽  
Projective Geometry ◽  
Geometric Analysis ◽  
Image Space ◽  
Geometric Algorithms ◽  
Oriented Projective Geometry ◽  
Swept Volumes ◽  
Rigid Body Motions ◽  
Interference Checking

In this paper, we explore the notion of kinematic convexity for rigid body displacements. Previously, we have shown that when spatial rigid body displacements are represented by dual quaternions, an oriented projective space is better suited for the image space of displacements. Geometric algorithms for rigid body motions become more general and elegant when developed from the perspective of oriented projective geometry. By extending the concept of convexity in affine geometry to oriented projective geometry of the image space of rigid body displacements, we define the concept of kinematic convexity. This concept, apart from being theoretically significant, facilitates localization of a displacements and provides a measure of the kinematic separation useful in collision prediction, interference checking, and geometric analysis of swept volumes.

Clifford algebra of three dimensional geometry

Robotica ◽  
2002 ◽  
Vol 20 (6) ◽  
pp. 687-697 ◽  
Author(s):  
Glen Mullineux
Keyword(s):  
Rigid Body ◽  
Clifford Algebra ◽  
Projective Geometry ◽  
Geometric Algebra ◽  
Three Dimensional ◽  
Basis Element ◽  
Rigid Body Motions ◽  
Robotic Applications

The use of Clifford or geometric algebra for dealing with three dimensional geometry is discussed. One issue is the representation of the rigid body motions of rotations and translations as elements within the algebra. The approach used is to work with projective geometry and choose the square of an additional basis element to be large (infinite). This allows Euclidean points to be represented as vectors in the algebra and transforms on these to be handled using bivectors. This paper looks at the use of Clifford algebra for handling the types of transforms required in robotic applications. A number of example applications are given.

Analytical models for the rigid body motions of skew bridges

1987 ◽  
Vol 15 (8) ◽  
pp. 923-944 ◽  
Author(s):  
Emmanuel A. Maragakis ◽  
Paul C. Jennings
Keyword(s):  
Rigid Body ◽  
Analytical Models ◽  
Skew Bridges ◽  
Rigid Body Motions

MULTIBODY DYNAMIC MODELING AND FLUTTER ANALYSIS OF A FLEXIBLE SLENDER VEHICLE

2012 ◽  
Vol 12 (06) ◽  
pp. 1250049 ◽  
Author(s):  
A. RASTI ◽  
S. A. FAZELZADEH
Keyword(s):  
Differential Equations ◽  
Rigid Body ◽  
Dynamic Modeling ◽  
Equations Of Motion ◽  
The Body ◽  
Elastic Deformations ◽  
Strip Theory ◽  
Multibody Dynamic ◽  
Flutter Analysis ◽  
Rigid Body Motions

In this paper, multibody dynamic modeling and flutter analysis of a flexible slender vehicle are investigated. The method is a comprehensive procedure based on the hybrid equations of motion in terms of quasi-coordinates. The equations consist of ordinary differential equations for the rigid body motions of the vehicle and partial differential equations for the elastic deformations of the flexible components of the vehicle. These equations are naturally nonlinear, but to avoid high nonlinearity of equations the elastic displacements are assumed to be small so that the equations of motion can be linearized. For the aeroelastic analysis a perturbation approach is used, by which the problem is divided into a nonlinear flight dynamics problem for quasi-rigid flight vehicle and a linear extended aeroelasticity problem for the elastic deformations and perturbations in the rigid body motions. In this manner, the trim values that are obtained from the first problem are used as an input to the second problem. The body of the vehicle is modeled with a uniform free–free beam and the aeroelastic forces are derived from the strip theory. The effect of some crucial geometric and physical parameters and the acting forces on the flutter speed and frequency of the vehicle are investigated.

The Use of Projective Geometry on the Mechanism System Motion and Stability of the Special Problems in Judgement

2012 ◽  
Vol 482-484 ◽  
pp. 1041-1044
Author(s):  
Xiao Zhuang Song ◽  
Ming Liang Lu ◽  
Tao Qin
Keyword(s):  
Rigid Body ◽  
Projective Geometry ◽  
Euclidean Geometry ◽  
Specific Problem ◽  
Zero Point ◽  
Rigid Bodies ◽  
Parallel Connection ◽  
Fixed Plane ◽  
Instantaneous Center ◽  
Proof Of Principle

In a principle of kinematics, when a rigid body is motion in a plane, and the fixed plane only the presence of a speed zero point -- the instantaneous center of velocity. In the mechanism of two rigid bodies be connected by two parallel connection links, why can the continuous relative translation? Where is the instantaneous center of velocity? ... ... The traditional Euclidean geometry theory can’t explain these phenomenon, must use projective geometry theory to solve. The actual motion of the mechanism is disproof in Euclidean geometry principle limitation. This paper introduces the required in projective geometry basic proof of principle, and applied to a specific problem.

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