Spherical trigonometry constrained kinematics for a dexterous robotic hand with an articulated palm

Robotica ◽  
2015 ◽  
Vol 34 (12) ◽  
pp. 2788-2805 ◽  
Author(s):  
Evangelos Emmanouil ◽  
Guowu Wei ◽  
Jian S. Dai
Keyword(s):  
Point Clouds ◽  
Joint Space ◽  
Singularity Analysis ◽  
Spherical Triangle ◽  
Design Criteria ◽  
Robotic Hand ◽  
Spherical Trigonometry ◽  
Joint Angles ◽  
Singularity Avoidance ◽  
Spherical Triangles

SUMMARYThis work presents a method based on spherical trigonometry for computing all joint angles of the spherical metamorphic palm. The spherical palm is segmented into spherical triangles which are then solved and combined to fully solve the palm configuration. Further, singularity analysis is investigated with the analysis of each spherical triangle the palm is decomposed. Singularity-avoidance-based design criteria are then presented. Finally, point clouds are generated that represent the joint space of the palm as well as the workspace of the hand with the advantage of an articulated palm is shown.

New Formulae for Combined Spherical Triangles

2018 ◽  
Vol 72 (2) ◽  
pp. 503-512
Author(s):  
Tsung-Hsuan Hsieh ◽  
Shengzheng Wang ◽  
Wei Liu ◽  
Jiansen Zhao
Keyword(s):  
Research Field ◽  
Great Circle ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Calculation Process ◽  
Different Types ◽  
Celestial Bodies ◽  
Spherical Triangles

Spherical trigonometry formulae are widely adopted to solve various navigation problems. However, these formulae only express the relationships between the sides and angles of a single spherical triangle. In fact, many problems may involve different types of spherical shapes. If we can develop the different formulae for specific spherical shapes, it will help us solve these problems directly. Thus, we propose two types of formulae for combined spherical triangles. The first set are the formulae of the divided spherical triangle, and the second set are the formulae of the spherical quadrilateral. By applying the formulae of the divided spherical triangle, waypoints on a great circle track can be obtained directly without finding the initial great circle course angle in advance. By applying the formulae of the spherical quadrilateral, the astronomical vessel position can be yielded directly from two celestial bodies, and the calculation process concept is easier to comprehend. The formulae we propose can not only be directly used to solve corresponding problems, but also expand the spherical trigonometry research field.

scholarly journals Kinematics and Singularity Analysis of a 7-DOF Redundant Manipulator

Sensors ◽  
2021 ◽  
Vol 21 (21) ◽  
pp. 7257
Author(s):  
Xiaohua Shi ◽  
Yu Guo ◽  
Xuechan Chen ◽  
Ziming Chen ◽  
Zhiwei Yang
Keyword(s):  
Matrix Method ◽  
Inverse Kinematics ◽  
Joint Space ◽  
Singularity Analysis ◽  
Robotic Arm ◽  
Redundant Manipulator ◽  
Singularity Avoidance ◽  
Primitive Matrix ◽  
Motion Characteristics ◽  
Kinematic Singularities

A new method of kinematic analysis and singularity analysis is proposed for a 7-DOF redundant manipulator with three consecutive parallel axes. First, the redundancy angle is described according to the self-motion characteristics of the manipulator, the position and orientation of the end-effector are separated, and the inverse kinematics of this manipulator is analyzed by geometric methods with the redundancy angle as a constraint. Then, the Jacobian matrix is established to derive the conditions for the kinematic singularities of the robotic arm by using the primitive matrix method and the block matrix method. Then, the kinematic singularities conditions in the joint space are mapped to the Cartesian space, and the singular configuration is described using the end poses and redundancy angles of the robotic arm, and a singularity avoidance method based on the redundancy angles and end pose is proposed. Finally, the correctness and feasibility of the inverse kinematics algorithm and the singularity avoidance method are verified by simulation examples.

The Modern Approach: Right-Angled Triangles

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Pythagorean Theorem ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
Modern Approach ◽  
Locality Principle ◽  
Spherical Triangles

This chapter discusses the modern approach to solving right-angled triangles. After a brief background on John Napier's trigonometric work, in which he referred mostly to right-angled spherical triangles, the chapter describes the theorems for right triangles. It then considers an oblique triangle split into two right triangles and the ten fundamental identities of a right-angled spherical triangle, how the locality principle can be applied to derive the Pythagorean Theorem, and how to find a ship's direction of travel using the theorem. It also looks at Napier's work on logarithms which was devoted to trigonometry, along with Napier's Rules. The chapter concludes with an overview of “pentagramma mirificum,” a pentagram in spherical trigonometry that was discovered by Napier.

Vector Solutions for Great Circle Navigation

2005 ◽  
Vol 58 (3) ◽  
pp. 451-457 ◽  
Author(s):  
Michael A. Earle
Keyword(s):  
Great Circle ◽  
North Pole ◽  
Spherical Triangle ◽  
Vector Analysis ◽  
Small Computer ◽  
Spherical Trigonometry ◽  
The North ◽  
Document Vector ◽  
Spherical Triangles ◽  
Vector Approach

Traditionally, navigation has been taught with methods employing Napier's rules for spherical triangles while methods derived from vector analysis and calculus appear to have been avoided in the teaching environment. In this document, vector methods are described that allow distance and azimuth at any point on a great circle to be determined. These methods are direct and avoid reliance on the formulae of spherical trigonometry. The vector approach presented here allows waypoints to be established without the need to either ascertain the position of the vertex or select the nearest pole; the method discussed here requires only one spherical triangle having an apex at the North Pole and is also easy to implement on a small computer.

Direct Methods of Sight Reduction: An Historical Review

1982 ◽  
Vol 35 (2) ◽  
pp. 260-273 ◽  
Author(s):  
Charles H. Cotter
Keyword(s):  
Direct Method ◽  
Historical Review ◽  
Direct Methods ◽  
The Other ◽  
Spherical Triangle ◽  
Spherical Trigonometry ◽  
General Terms ◽  
Spherical Triangles ◽  
Two Sides ◽  
Fundamental Formula

In general terms the principal problem in astronomical navigation is the solving of a spherical triangle - the PZX-triangle. The fundamental formula of spherical trigonometry for finding an angle given the three sides of a spherical triangle is the cosine formula. By transposition this formula can be used for finding a side given the opposite angle and the other two sides. Because the cosine formula is not suitable for use with logarithms numerous formulae have been derived from it with the aim of simplifying logarithmic computation. The term ‘direct method’ applies to a method the basis of which is generally the cosine formula or any of its derivatives although some direct methods are based on Napier's Rules for right-angled spherical triangles.

Areas, Angles, and Polyhedra

2017 ◽  
Author(s):  
Glen Van Brummelen
Keyword(s):  
Eighteenth Century ◽  
Unit Sphere ◽  
Regular Polyhedron ◽  
Spherical Triangle ◽  
French Mathematician ◽  
Regular Polyhedra ◽  
Leonhard Euler ◽  
Spherical Triangles

This chapter explains how to find the area of an angle or polyhedron. It begins with a discussion of how to determine the area of a spherical triangle or polygon. The formula for the area of a spherical triangle is named after Albert Girard, a French mathematician who developed a theorem on the areas of spherical triangles, found in his Invention nouvelle. The chapter goes on to consider Euler's polyhedral formula, named after the eighteenth-century mathematician Leonhard Euler, and the geometry of a regular polyhedron. Finally, it describes an approach to finding the proportion of the volume of the unit sphere that the various regular polyhedra occupy.

scholarly journals Fast geometry-based computation of grasping points on three-dimensional point clouds

2019 ◽  
Vol 16 (1) ◽  
pp. 172988141983184 ◽  
Author(s):  
Brayan S Zapata-Impata ◽  
Pablo Gil ◽  
Jorge Pomares ◽  
Fernando Torres
Keyword(s):  
Point Cloud ◽  
Three Dimensional ◽  
Point Clouds ◽  
Service Robots ◽  
Robotic Hand ◽  
Complex Task ◽  
Changing Environments ◽  
Different Shapes ◽  
Points Of View ◽  
Partial View

Industrial and service robots deal with the complex task of grasping objects that have different shapes and which are seen from diverse points of view. In order to autonomously perform grasps, the robot must calculate where to place its robotic hand to ensure that the grasp is stable. We propose a method to find the best pair of grasping points given a three-dimensional point cloud with the partial view of an unknown object. We use a set of straightforward geometric rules to explore the cloud and propose grasping points on the surface of the object. We then adapt the pair of contacts to a multi-fingered hand used in experimentation. We prove that, after performing 500 grasps of different objects, our approach is fast, taking an average of 17.5 ms to propose contacts, while attaining a grasp success rate of 85.5%. Moreover, the method is sufficiently flexible and stable to work with objects in changing environments, such as those confronted by industrial or service robots.

scholarly journals Kinematic analysis of a newly designed cable-driven manipulator

2018 ◽  
Vol 42 (2) ◽  
pp. 125-135 ◽  
Author(s):  
Wei Xu ◽  
Yaoyao Wang ◽  
Surong Jiang ◽  
Jiafeng Yao ◽  
Bai Chen
Keyword(s):  
Trajectory Planning ◽  
Kinematic Analysis ◽  
Joint Space ◽  
Joint Angles ◽  
Coupled Motion ◽  
Cartesian Space ◽  
The Impact ◽  
The Relationship

In this paper, the cable routing configurations for a cable-driven manipulator are introduced, and the impact of motion coupling caused by cable transmission routing of a 2n type cable-driven manipulator is analyzed in detail. Based on different configurations of cable routings, the relationship between variation of joint angles and the geometrical sizes of guide pulleys is established, represented by a matrix for coupled motion. Moreover, based on the effects of the motion coupling of a cable-driven manipulator, we propose the condition for the invariance of orientation, which can be achieved constraining of the geometrical sizes of guide pulleys and driven wheels. In addition, to identify the correctness of analysis for coupled motion, a 3-DOFs planer cable-driven manipulator prototyping model is constructed, and the kinematics and trajectory planning has been solved. Finally, the relationship among actuator space, joint space, and Cartesian space, including the mapping of the motion coupling, is experimentally validated. The property of invariance of orientation is also validated by an experiment.

scholarly journals On the special spherical triangles for physical and cosmological applications

2021 ◽  
pp. 112-114
Author(s):  
Kalimuthu S
Keyword(s):  
Spherical Triangle ◽  
Spherical Triangles ◽  
The Way

It is well known that a spherical triangle of 270 degree triangle is constructible on the surface of a sphere; a globe is a good example. Take a point (A) on the equator, draw a line 1/4 the way around (90 degrees of longitude) on the equator to a new point (B).

scholarly journals Dihedral f-tilings of the sphere by rhombi and triangles

2005 ◽  
Vol Vol. 7 ◽  
Author(s):  
Ana Breda ◽  
Altino F. Santos
Keyword(s):  
Spherical Triangle ◽  
International Audience ◽  
Spherical Triangles

International audience We classify, up to an isomorphism, the class of all dihedral f-tilings of S^2, whose prototiles are a spherical triangle and a spherical rhombus. The equiangular case was considered and classified in Ana M. Breda and Altino F. Santos, Dihedral f-tilings of the sphere by spherical triangles and equiangular well-centered quadrangles. Here we complete the classification considering the case of non-equiangular rhombi.

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