Direct Differential Kinematics of Hybrid-Chain Manipulators Including Singularity and Stability Analyses

1994 ◽  
Vol 116 (2) ◽  
pp. 614-621 ◽  
Author(s):  
Yong-Xian Xu ◽  
D. Kohli ◽  
Tzu-Chen Weng
Keyword(s):  
Stability Index ◽  
Inverse Kinematic ◽  
Degree Of Freedom ◽  
Static Force ◽  
Force Analysis ◽  
Kinematic Singularity ◽  
Numerical Examples ◽  
Transformation Matrices ◽  
Unstable Configuration ◽  
Direct Kinematics

A general formulation for the differential kinematics of hybrid-chain manipulators is developed based on transformation matrices. This formulation leads to velocity and acceleration analyses, as well as to the formation of Jacobians for singularity and unstable configuration analyses. A manipulator consisting of n nonsymmetrical subchains with an arbitrary arrangement of actuators in the subchain is called a hybrid-chain manipulator in this paper. The Jacobian of the manipulator (called here the system Jacobian) is a product of two matrices, namely the Jacobian of a leg and a matrix M containing the inverse of a matrix Dk, called the Jacobian of direct kinematics. The system Jacobian is singular when a leg Jacobian is singular; the resulting singularity is called the inverse kinematic singularity and it occurs at the boundary of inverse kinematic solutions. When the Dk matrix is singular, the M matrix and the system Jacobian do not exist. The singularity due to the singularity of the Dk matrix is the direct kinematic singularity and it provides positions where the manipulator as a whole loses at least one degree of freedom. Here the inputs to the manipulator become dependent on each other and are locked. While at these positions, the platform gains at least one degree of freedom, and becomes statically unstable. The system Jacobian may be used in the static force analysis. A stability index, defined in terms of the condition number of the Dk matrix, is proposed for evaluating the proximity of the configuration to the unstable configuration. Several illustrative numerical examples are presented.

Direct Differential Kinematics of Hybrid-Chain Manipulators Including Singularity and Stability Analyses

1992 ◽  
Author(s):  
Yong-Xian Xu ◽  
Dilip Kohli ◽  
Tzu-Chen Weng
Keyword(s):  
Stability Index ◽  
Inverse Kinematic ◽  
Degree Of Freedom ◽  
Static Force ◽  
Force Analysis ◽  
Kinematic Singularity ◽  
Numerical Examples ◽  
Transformation Matrices ◽  
Unstable Configuration ◽  
Direct Kinematics

Abstract A general formulation for the differential kinematics of hybrid-chain manipulators is developed based on transformation matrices. This formulations leads to velocity and acceleration analyses, as well as to the formation of Jacobians for singularity and unstable configuration analyses. A manipulator consisting of n nonsymmetrical subchains with an arbitrary arrangement of actuators in the subchain is called a hybrid-chain manipulator in this paper. The Jacobian of the manipulator (called here the system Jacobian) is a product of two matrices, namely the Jacobian of a leg and a matrix M containing the inverse of a matrix Dk, called the Jacobian of direct kinematics. The system Jacobian is singular when a leg Jacobian is singular; the resulting singularity is called the inverse kinematic singularity and it occurs at the boundary of inverse kinematic solutions. When the Dk matrix is singular, the M matrix and the system Jacobian do not exist. The singularity due to the singularity of the Dk matrix is the direct kinematic singularity and it provides positions where the manipulator as a whole loses at least one degree of freedom. Here the inputs to the manipulator become dependent on each other and are locked. While at these positions, the platform gains at least one degree of freedom, and becomes statically unstable. The system Jacobian may be used in the static force analysis. A stability index, defined in terms of the condition number of the Dk matrix, is proposed for evaluating the proximity of the configuration to the unstable configuration. Several illustrative numerical examples are presented.

Automated Formation of Closed Form Kinematic Displacement Polynomials for Open Loop Planar and Spatial Chains

1995 ◽  
Vol 117 (1) ◽  
pp. 83-88
Author(s):  
S. C. Jen ◽  
D. Kohli
Keyword(s):  
Closed Form ◽  
Linear Equations ◽  
Numerical Approach ◽  
Inverse Kinematic ◽  
Open Loop ◽  
Hand Position ◽  
Numerical Examples ◽  
Polynomial Type ◽  
Joint Variable ◽  
Direct Kinematics

A new numerical approach for determining inverse kinematic polynomials of manipulators is presented in this paper. Let the inverse kinematic polynomial of a manipulator in one revolute joint variable θi be represented by gnTn + gn-1Tn-1 + gn-2Tn-2 + • + g1T + go = 0. T = tanθi/2 and go, g1...gn are polynomial type functions of hand position variables. The coefficients g are expressed in terms of undetermined coefficients and hand position variables. Then the undetermined coefficients are evaluated by using direct kinematics and the solutions of sets of linear equations, thus determining coefficients g and the inverse kinematic polynomial. The method is general and may be applied for determining inverse kinematic polynomials of any manipulator. However, the number of linear equations required in determining coefficients g become significantly larger as the number of links and the degrees of the manipulator increase. Numerical examples of 2R planar and 3R spatial manipulator are presented for illustration.

scholarly journals Isotropic Design of Tendon-Driven Manipulators

1996 ◽  
Vol 118 (3) ◽  
pp. 360-366 ◽  
Author(s):  
Yeong-Jeong Ou ◽  
Lung-Wen Tsai
Keyword(s):  
Necessary Conditions ◽  
Degree Of Freedom ◽  
Mechanical Transmission ◽  
Force Distribution ◽  
Static Force ◽  
Design Equation ◽  
Force Analysis ◽  
Transmission Characteristics ◽  
Transmission Structure ◽  
Tendon Force

This paper deals with the synthesis of mechanical transmission structures for tendon-driven manipulators. Based on static force analysis, necessary conditions are developed for the synthesis of tendon-driven manipulators with isotropic transmission characteristics. It is shown that an n degree-of-freedom (dof) manipulator will possess the isotropic transmission characteristics, if it satisfies two isotropic conditions. Furthermore, a design equation is derived for the construction of isotropic transmission structure matrices and a three-dof spatial manipulator is synthesized to demonstrate the methodology. It is shown that the isotropic design leads to a more uniform tendon force distribution.

scholarly journals Solution of The Inverse Kinematic Task of A Robot-Arm Based on A Quasi-Differential Fixed-Point Transformation Method

2018 ◽  
Vol 9 (1) ◽  
pp. 71-74
Author(s):  
Tamás Faitli ◽  
József Tar
Keyword(s):  
Fixed Point ◽  
Transformation Method ◽  
Inverse Kinematic ◽  
Point Transformation ◽  
Degree Of Freedom ◽  
Robot Arm ◽  
Transformation Matrices ◽  
Forward Kinematic ◽  
Three Degree Of Freedom

Abstract While the forward kinematic task of robots can be solved easily through homogenous transformation matrices, the inverse kinematic task leads to difficulties as the construction of the system becomes more complex. In this paper, a solution has been worked out for a three Degree-of-Freedom (DOF) robot-arm based on recent research, by the use of a novel, fixed-point transformation based technique.

Kinematics of a New High-Precision Three-Degree-of-Freedom Parallel Manipulator

2006 ◽  
Vol 129 (3) ◽  
pp. 320-325 ◽  
Author(s):  
Farhad Tahmasebi
Keyword(s):  
Power Dissipation ◽  
Inverse Kinematics ◽  
Parallel Manipulator ◽  
Degree Of Freedom ◽  
Base Plane ◽  
Degree Polynomial ◽  
Payload Capacity ◽  
Half Angle ◽  
Three Degree Of Freedom ◽  
Direct Kinematics

Closed-form direct and inverse kinematics of a new three-degree-of-freedom (DOF) parallel manipulator with inextensible limbs and base-mounted actuators are presented. The manipulator has higher resolution and precision than the existing three-DOF mechanisms with extensible limbs. Since all of the manipulator actuators are base mounted, higher payload capacity, smaller actuator sizes, and lower power dissipation can be obtained. The manipulator is suitable for alignment applications where only tip, tilt, and piston motions are significant. The direct kinematics of the manipulator is reduced to solving an eighth-degree polynomial in the square of the tangent of the half-angle between one of the limbs and the base plane. Hence, there are at most 16 assembly configurations for the manipulator. In addition, it is shown that the 16 solutions are eight pairs of reflected configurations with respect to the base plane. Numerical examples for the direct and inverse kinematics of the manipulator are also presented.

Dynamics of a Two-DOF Parallel Pointing Mechanism

2005 ◽  
Author(s):  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Numerical Simulation ◽  
Parallel Mechanism ◽  
Inverse Dynamics ◽  
Kinematic Model ◽  
Degree Of Freedom ◽  
Numerical Examples ◽  
Proposed Model ◽  
Dynamic Analyses ◽  
Moving Platform ◽  
Two Degree Of Freedom

This paper presents kinematic and dynamic analyses of a two-degree-of-freedom pointing parallel mechanism. The mechanism consists of a moving platform, connected to a fixed platform by two legs of type PUS (prismatic-universal-spherical). At first a simplified kinematic model of the pointing mechanism is introduced. Based on this proposed model, the dynamics equations of the system using the Natural Orthogonal Complement method are developed. Numerical examples of the inverse dynamics results are presented by numerical simulation.

Kinematic and Workspace Modelling of a 6-PUS Parallel Mechanism

2018 ◽  
Author(s):  
Jérôme Landuré ◽  
Clément Gosselin
Keyword(s):  
Kinematic Analysis ◽  
Parallel Mechanism ◽  
Inverse Kinematic ◽  
Degree Of Freedom ◽  
Accurate Description ◽  
Transmission Ratio ◽  
Inverse Kinematic Problem ◽  
Jacobian Matrices ◽  
Six Degree Of Freedom

This article presents the kinematic analysis of a six-degree-of-freedom six-legged parallel mechanism of the 6-PUS architecture. The inverse kinematic problem is recalled and the Jacobian matrices are derived. Then, an algorithm for the geometric determination of the workspace is presented, which yields a very fast and accurate description of the workspace of the mechanism. Singular boundaries and a transmission ratio index are then introduced and studied for a set of architectural parameters. The proposed analysis yields conceptual architectures whose properties can be adjusted to fit given applications.

The Kinematics of Spatial Double-Triangular Parallel Manipulators

1995 ◽  
Vol 117 (4) ◽  
pp. 658-661 ◽  
Author(s):  
H. R. Mohammadi Daniali ◽  
P. J. Zsombor-Murray ◽  
J. Angeles
Keyword(s):  
Degrees Of Freedom ◽  
Quaternion Algebra ◽  
Parallel Manipulators ◽  
Numerical Examples ◽  
Six Degrees Of Freedom ◽  
Dual Number ◽  
Direct Kinematics

Two versions of spatial double-triangular mechanisms are introduced, one with three and one with six degrees of freedom. Using dual-number quaternion algebra, a formula for the direct kinematics of these manipulators is derived. Numerical examples are included.

Kinematic Analysis of Spatial Mechanisms by Means of Screw Coordinates. Part 2—Analysis of Spatial Mechanisms

1971 ◽  
Vol 93 (1) ◽  
pp. 67-73 ◽  
Author(s):  
M. S. C. Yuan ◽  
F. Freudenstein ◽  
L. S. Woo
Keyword(s):  
Computer Program ◽  
Kinematic Analysis ◽  
Static Force ◽  
Single Degree Of Freedom ◽  
Numerical Examples ◽  
Spatial Mechanism ◽  
Single Degree ◽  
Spatial Mechanisms ◽  
Torque Analysis ◽  
Basic Concepts

The basic concepts of screw coordinates described in Part I are applied to the numerical kinematic analysis of spatial mechanisms. The techniques are illustrated with reference to the displacement, velocity, and static-force-and-torque analysis of a general, single-degree-of-freedom spatial mechanism: a seven-link mechanism with screw pairs (H)7. By specialization the associated computer program is capable of analyzing many other single-loop spatial mechanisms. Numerical examples illustrate the results.

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