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Freaking Awesome Geometry or Does God have Redundant Manipulators?

ME392-02 Special Topics: Advanced Topics in Robotics and Mechanism Synthesis

(will be offered in Spring 2011)

This is a graduate level course in robotics and mechanism synthesis. The course focuses on covering several topics in kinematics of serial and parallel robots, special methods in kinematics including dual number representations and quaternion methods, introduction to screw-based kinematics and its applications to mechanism analysis and synthesis, line geometry methods and applications in kinematics, open research problems in robotics, mathematical methods for the solution of polynomial systems related to design/analysis problems of mechanisms (e.g. direct kinematics of parallel robots) including homotopy continuation methods and resultant-based methods.

The course is highly recommended for graduate students with some background in robotics and mechanism theory who are interested in expanding their background for research in these areas. It is intended to provide a wider perspective on the mathematical methods and on performance evaluation/optimization of different mechanisms/robots including parallel robots, serial robots, multi-fingered hands, robots with actuation redundancy and with kinematics redundancy (e.g. snakes).

The mathematical background necessary for this course includes linear algebra (you should be familiar with matrix computations, eigenvalues/eigenvectors) and ordinary differential equations (recommended).

Course Administration

This course is open for graduate students interested in strengthening their background in kinematics and robotics. The course grading is based on home works, research projects.

Textbooks

The course will mainly focus on the class notes of the instructor. Most of the topics covered are included in the following recommended text books:

  1. Fundamentals of Robotic Mechanical Systems, by Jorge Angeles, 2nd Edition (3rd edition is expected soon so check if it is available), Springer, 2003.
  2. Advanced Robotics, by Yoshihiko Nakamura, Addison-Wesley Publishing Company, 1991 (this book is not in print anymore, but I will be providing some of my own notes based on this book. You will be able to find some used copies of it online)

There are several other books that I will recommend at the beginning of the class. The above-mentioned books are the most relevant to my class - but they do not cover all the topics that I will discuss.

Subjects covered

The course will cover the following topics based on time availability:

  • Review of rigid body transformations (Homogeneous matrices,  The screw and the screw transformation and the exponential representation of a screw transformation, Transformation verses displacement matrices: the zero-reference method, Calculating the finite displacement screw for a specified motion).
  • Special methods in kinematics (Quaternion algebra, rotation vectors, and applications of dual quaternions for kinematic analysis and synthesis).
  • Introduction to line geometry (Introduction to projective geometry: homogenous representation of points and lines, The Plücker line coordinates, The independent line varieties in plane and space).
  • Introduction to screw theory (Reciprocity and the principle of virtual work, The screw, twist, and wrench systems: the cylindroid, regulus, congruence and linear the complex)
  • Instantaneous kinematics of open and closed kinematic chains (Derivation of the instantaneous kinematics Jacobian based on kinematic constraints, Geometric interpretation of the instantaneous kinematics Jacobian of serial and parallel manipulators & applications for line-based singularity analysis, Input-output relationship in closed-loop mechanisms using auxiliary coordinates)
  • Kinetostatic analysis of serial and parallel robots ( Alternative derivation of the instantaneous kinematics Jacobian using statics, Screw-based formulation of the Jacobian of parallel and serial robots, Stiffness mapping for parallel and serial robots, Dualities between serial and parallel manipulators, Singularity and Performance indices (kinematic conditining indec, manipulability ellipsoid, stiffness ellipsoid, isotropy, condition number, joint range availability))
  • Introduction to redundant manipulators (Singular Value Decomposition, Solution of underdetermined and over-constrained linear system of equations, Task-priority optimization methods for obstacle avoidance, singularity avoidance, and dexterity optimization, Actuation redundancy: optimization of actuator effort, applications to wire-actuated parallel robots and multi-fingered grasping).
  • Introduction to numerical and symbolic methods for the solution of polynomial systems arising in kinematics (Direct kinematics of parallel robots and synthesis of spatial mechanisms: motivating examples from the literature, Homogenization of polynomial systems, Root counts for polynomial systems: the Bezout number and the BKK bounds, Introduction to Homotopy continuation methods, Introduction to the theory of elimination: the Dialytic elimination method, The Sylvester and Bezout resultants, Transforming the resultant formulation to an associated generalized eigenvalue problem.

Examples

The following images show some of the robots/topics we will be focusing on in this class. The course is intentionally designed to cover wider topics, but will provide you a prossibility to get familiar with some of these topics for later research.

RSPR moving

Example 1: a parallel robot is composed of several closed kinematics chains (RSPR3 robot)

Inverse Kinematics RSPRinverse kinematics rspr2

Example 2: eight possible inverse kinematics solutions for a Composite Serial-in-Parallel (CISP) robot RSPR3

line geometry

Example 3: The singularities of the RSPR3 robot and other parallel robots can be explained by associating a physical interpretation to the rows of the jacobian and using line geometry.

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Geometries presented by Chinese scholars for all possible space-time kinematics and their relations

November 26th, 2012

Geometries presented by Chinese scholars for all possible space-time kinematics and their relations
Enlarge

This shows the contraction scheme for the possible kinematics in a more symmetric way. Credit: ©Science China Press

The possible kinematics and their corresponding geometries were once regarded as an already-solved problem. The de Sitter relativity research group formed by researchers from Chinese Academy of Sciences, Tsinghua University, and Beijing Normal University, restudied the problem and showed that additional, previously unknown realizations exist of possible kinematical algebras, each of which has so(3) isotropy and a ten-generators symmetry group. They presented these geometries corresponding to all these realizations and provided a classification in an article, entitled "Geometries for Possible Kinematics", published in the 2012 10th issue ofSCIENCE CHINA [1] .

In the 1960s Bacry and Lévy-Leblond established connections among eleven kinematical algebras of eight types. Each kinematical algebra was supposed to possess (i) an so(3) isotropy, (ii) parity and time-reversal automorphisms, and (iii) a non-compact one-dimensional subgroup generated by each boost. For a long time, it was widely accepted that the accounting for all kinematical algebras satisfying those three conditions had been exhausted. Two years ago, the de Sitter relativity research group showed by using linear combinations of generators that there are 24 kinematical algebras if the third condition is relaxed [2], and all of those kinematical algebras are subalgebras of a 4-dimensional "inertial motion algebra".

Geometries presented by Chinese scholars for all possible space-time kinematics and their relations
Enlarge

This shows the contraction scheme for the geometries for the possible kinematics. Credit: ©Science China Press

In Ref. [1], it was shown that, with the exception of two static algebras, the 22 possible kinematical algebras with so(3) isotropy can be obtained by the Inönü-Wigner contraction from the Riemann, Lobachevsky, de Sitter, and anti-de Sitter algebras (r, l, d±), respectively. The existence of more possible kinematical algebras than obtained by Bacry and Lévy-Leblond arises from taking different realizations of the generators and then performing the contraction under two opposite limits. For example, it is well known that the de Sitter and anti-de Sitter algebras contract to the Poincaré algebra (p) when a certain length parameter tends to infinity. What was overlooked was that when the length parameter tends to zero, [3] these algebras contract to other realizations of the Poincaré algebra, called the second Poincaré algebras and denoted p2±, for brevity. Although these are isomorphic, p and p2± have very different geometrical significance. The 22 possible kinematical algebras are related, as depicted in figure 1.

More importantly, a kinematic action should be established on a suitable geometry so that the geometry is invariant under the transformations generated by the kinematical algebra. In Ref. [1], the geometries for all 22 kinematical algebras were presented using the contraction technique. As arranged in figure 2, there are 45 different 4-dimensional geometries in total. Each geometry is defined on a portion of a 4-dimensional real projective manifold. Among these geometries, some are non-degenerate like the de Sitter and Minkowski geometries, but most are degenerate as for example for the Galilei and Carroll geometries. In these geometries, including the degenerate ones, some have Lorentzian signature that may serve as space-time geometries, and some have Euclidean signature that may serve as Euclidean versions of space-time. Moreover, there are geometries which have a (+, +, -, -)-signature and are interpreted as double-time geometries. Many of the geometries are related by the transformation t < 1/(n2t). When this transformation is regarded as a coordinate transformation, these geometries are not independent from the view of differential geometry.

Geometries presented by Chinese scholars for all possible space-time kinematics and their relations
Enlarge

This shows the geometries for the genuine possible kinematics. Credit: ©Science China Press

The explicit geometric structures show that the requirement that transformations generated by boosts in any given direction form a noncompact subgroup does not guarantee a geometry having Lorentz-like signature. Some geometries satisfying that requirement possess Euclidean signature, whereas others violating that requirement possess Lorentz-like signature. In addition, before the geometry is presented, the isotropy (rotational invariance) of the space is expressed by an so(3) subalgebra. However, although many geometries are invariant under transformations generated by so(3), they might not have spatial isotropy with respect to each point in the manifold.

Therefore, the requirements suitable in identifying genuine possible kinematics have been revised to the following: (1) space is isotropic with respect to any point in the manifold; (2) space-reflection and time-reversal are automorphisms of the kinematical group; and (3) the geometry has Lorentz-like signature. According to those requirements, the genuine possible kinematics from the viewpoint of differential geometry correspond to three relativistic geometries, three absolute-time geometries and three absolute-space geometries (see Table 1).

More information:
1. HUANG CG, TIAN Y, WU XN, XU Z and ZHOU B, Geometries for possible kinematics, SCI CHINA Phys. Mech. Astron. 2012, 55(11), doi: 10.1007/s11433-012-4788-4.

2. GUO HY, HUANG CG, WU HT and ZHOU B, SCIENCE CHINA Phys. Mech. Astron. 2012, Vol 53(4) 591

3. HUANG CG, In Proceedings of the 9th Asia-Pacific International Conference. Singapore ed Luo J et al World Scientific Publishing, 2010. 118; HUANG CG, TIAN Y,WU XN, et al. Chin Phys Lett, 2012, 29: 040303; Commun Theor Phys, 2012, 57: 553.

Provided by Science China Press

Yep!

Adding weight to my ignorance =)

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