scholarly journals Critical configurations of planar robot arms

2013 ◽  
Vol 11 (3) ◽  
Author(s):  
Giorgi Khimshiashvili ◽  
Gaiane Panina ◽  
Dirk Siersma ◽  
Alena Zhukova
Keyword(s):  
Critical Points ◽  
Morse Index ◽  
Robot Arms ◽  
Polygonal Chain ◽  
Oriented Area ◽  
Critical Configurations ◽  
Planar Robot ◽  
Polygonal Chains ◽  
Closed Polygon ◽  
Cyclic Polygon

AbstractIt is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, P can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or robot arms. We introduce the notion of the oriented area for an open polygonal chain, prove that critical points are exactly the cyclic configurations with antipodal endpoints and derive a formula for the Morse index of a critical configuration.

Computer Aided Synthesis of Piecewise Rational Motions for Spherical 2R and 3R Robot Arms

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
Anurag Purwar ◽  
Zhe Jin ◽  
Q. J. Ge
Keyword(s):  
Motion Planning ◽  
Spline Interpolation ◽  
Robot Arm ◽  
Spherical Robot ◽  
Kinematic Constraints ◽  
Robot Arms ◽  
Revolute Joints ◽  
Computer Aided ◽  
Planar Robot ◽  
Circular Interpolation

This paper deals with the problem of synthesizing smooth piecewise rational spherical motions of an object that satisfies the kinematic constraints imposed by a spherical robot arm with revolute joints. This paper brings together the kinematics of spherical robot arms and recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. The kinematic constraints under consideration are workspace related constraints that limit the orientation of the end link of robot arms. This paper extends our previous work on synthesis of rational motions under the kinematic constraints of planar robot arms. Using quaternion kinematics of spherical arms, it is shown that the problem of synthesizing the Cartesian rational motion of a 2R arm can be reduced to that of circular interpolation in two separate planes. Furthermore, the problem of synthesizing the Cartesian rational motion of a spherical 3R arm can be reduced to that of constrained spline interpolation in two separate planes. We present algorithms for the generation of C1 and C2 continuous rational motion of spherical 2R and 3R robot arms.

Computer Aided Synthesis of Piecewise Rational Motions for Planar 2R and 3R Robot Arms

2006 ◽  
Vol 129 (10) ◽  
pp. 1031-1036 ◽  
Author(s):  
Zhe Jin ◽  
Q. J. Ge
Keyword(s):  
Motion Planning ◽  
Spline Interpolation ◽  
Circular Ring ◽  
Robot Arm ◽  
Kinematic Constraints ◽  
Robot Arms ◽  
Revolute Joints ◽  
Computer Aided ◽  
Planar Robot ◽  
Circular Interpolation

This paper deals with the problem of synthesizing piecewise rational motions of an object that satisfies kinematic constraints imposed by a planar robot arm with revolute joints. This paper brings together the kinematics of planar robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Through the use of planar quaternions, it is shown that for the case of a planar 2R arm, the problem of rational motion synthesis can be reduced to that of circular interpolations in two separate planes and that for the case of a planar 3R arm, the problem can be reduced to a combination of circular interpolation in one plane and a constrained spline interpolation in a circular ring on another plane. Due to the limitation of circular interpolation, only C1 continuous rational motions are generated that satisfy the kinematic constraints exactly. For applications that require C2 continuous motions, this paper presents a method for generating C2 continuous motions that approximate the kinematic constraints for planar 2R and 3R robot arms.

scholarly journals Stationary Boltzmann equation: an approach via Morse theory

2021 ◽  
Vol 40 (6) ◽  
pp. 1473-1487
Author(s):  
Rafael Galeano Andrades ◽  
Joel Torres del Valle
Keyword(s):  
Boltzmann Equation ◽  
Critical Points ◽  
Morse Theory ◽  
Morse Index ◽  
Critical Groups

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.

Morse Index of Approximating Periodic Solutions for the Billiard Problem. Application to Existence Results

1998 ◽  
Vol 50 (3) ◽  
pp. 497-524
Author(s):  
Philippe Bolle
Keyword(s):  
Periodic Solutions ◽  
Critical Points ◽  
Morse Index ◽  
Approximate Solutions ◽  
Existence Results ◽  
Lagrangian Systems ◽  
Potential Well ◽  
Regular Solutions ◽  
Open Set

AbstractThis paper deals with periodic solutions for the billiard problem in a bounded open set of ℝN which are limits of regular solutions of Lagrangian systems with a potential well. We give a precise link between the Morse index of approximate solutions (regarded as critical points of Lagrangian functionals) and the properties of the bounce trajectory to which they converge.

Decentralized Adaptive Coordinated Control of Multiple Robot Arms for Constrained Tasks

2006 ◽  
Vol 18 (5) ◽  
pp. 580-588 ◽  
Author(s):  
Haruhisa Kawasaki ◽  
◽  
Rizauddin Bin Ramli ◽  
Satoshi Ueki
Keyword(s):  
Reference Model ◽  
Coordinated Control ◽  
Rolling Contact ◽  
Stability Of Motion ◽  
Robot Arms ◽  
Control Scheme ◽  
Planar Robot ◽  
End Effectors ◽  
Multiple Robot ◽  
Object Reference

The decentralized adaptive coordinated control of multiple robot arms grasping a common object constrained by a known environment involves the analysis of cases of rigid and rolling contact between end-effectors and object. In the proposed controller, the dynamic parameters of both object and robot arms are estimated adaptively. Desired motion of the robot arms is generated by an estimated object reference model. The asymptotic stability of motion is proven by the Lyapunov-like lemma. Experimental results for two planar robot arms with 3 DOF moving a constrained object demonstrate the effectiveness of the control scheme.

Computer Aided Synthesis of Piecewise Rational Motions for Planar 2R and 3R Robot Arms

2006 ◽  
Author(s):  
Zhe Jin ◽  
Q. J. Ge
Keyword(s):  
Spline Interpolation ◽  
Joint Space ◽  
Circular Ring ◽  
Robot Arm ◽  
Kinematic Constraints ◽  
Robot Arms ◽  
Revolute Joints ◽  
Computer Aided ◽  
Planar Robot ◽  
Circular Interpolation

This paper deals with the problem of synthesizing piecewise rational motions of an object that satisfies kinematic constraints imposed by a planar robot arm with revolute joints. The paper brings together the kinematics of planar robot arms and the recently developed freeform rational motions to study the problem of synthesizing constrained rational motions for Cartesian motion planning. Through the use of planar quaternions, it is shown that the problem of synthesizing the Cartesian rational motion of a planar 2R arm can be reduced to that of circular interpolations in two separate planes. Furthermore, the problem of synthesizing the Cartesian rational motion of a planar 3R arm can be reduced to that of circular interpolation in one plane and constrained spline interpolation in a circular ring. Due to the limitation of circular interpolation, only C1 continuous rational motions are generated. For applications that require C2 continuous motions, the paper presents a joint-space based method for generating a C2 continuous motion that approximates a given C1 rational motion of the end link.

scholarly journals Area collapse algorithm computing new curve of 2D geometric objects

2017 ◽  
Vol 66 (1) ◽  
pp. 23-43
Author(s):  
Michał Mateusz Buczek
Keyword(s):  
Input Data ◽  
Laser Scanning ◽  
Medial Axis ◽  
Point Clouds ◽  
Additional Information ◽  
Polygonal Chain ◽  
And Topology ◽  
Geometric Objects ◽  
Linear Plot ◽  
Polygonal Chains

Abstract The processing of cartographic data demands human involvement. Up-to-date algorithms try to automate a part of this process. The goal is to obtain a digital model, or additional information about shape and topology of input geometric objects. A topological skeleton is one of the most important tools in the branch of science called shape analysis. It represents topological and geometrical characteristics of input data. Its plot depends on using algorithms such as medial axis, skeletonization, erosion, thinning, area collapse and many others. Area collapse, also known as dimension change, replaces input data with lower-dimensional geometric objects like, for example, a polygon with a polygonal chain, a line segment with a point. The goal of this paper is to introduce a new algorithm for the automatic calculation of polygonal chains representing a 2D polygon. The output is entirely contained within the area of the input polygon, and it has a linear plot without branches. The computational process is automatic and repeatable. The requirements of input data are discussed. The author analyzes results based on the method of computing ends of output polygonal chains. Additional methods to improve results are explored. The algorithm was tested on real-world cartographic data received from BDOT/GESUT databases, and on point clouds from laser scanning. An implementation for computing hatching of embankment is described.

ADIABATIC SPECTRAL FLOW AND CHANGE OF MORSE INDICES

1994 ◽  
Vol 09 (15) ◽  
pp. 1369-1375
Author(s):  
HIDEKI ONO ◽  
HIROSHI KURATSUJI
Keyword(s):  
Critical Points ◽  
Morse Index ◽  
Theory Model ◽  
Structure Change ◽  
Spectral Flow ◽  
Action Functional ◽  
One Dimensional ◽  
Physical Systems ◽  
Field Action ◽  
Morse Indices

The concept of the Morse index characterizes the behavior near the critical points of the action functional that governs physical systems. We study the specific feature of Morse indices for the case that there are several number of critical points; on each of these critical points the Morse indices are defined. As a concrete example, we consider a model of one-dimensional field theory model. By using the idea of the adiabatic spectral flow, it is shown that the change of the Morse indices is correlated with the structure change inherent in the solutions of field equation that is controlled by the parameters built in the field action.

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