Mobility of Single-Loop Kinematic Mechanisms Under Differential Displacement

2010 ◽  
Vol 132 (4) ◽  
Author(s):  
Paul Milenkovic
Keyword(s):  
Degrees Of Freedom ◽  
Single Loop ◽  
Kinematic Chains ◽  
Lie Bracket ◽  
Numerical Characterization ◽  
Serial Chain ◽  
Kinematic Mechanisms ◽  
Serial Chains ◽  
Analysis Of Motion

Linear analysis of motion screws is a means for determining the mobility of a mechanism composed of the parallel combination of serial kinematic chains. Such a mechanism may have one or more degrees of freedom that vanish after a differential displacement from a reference posture. The Lie product, also called the Lie bracket, is known to give the derivative of a motion screw with respect to the displacement along an upstream screw in a serial chain. Serial chains having motion screws that are closed under the Lie product are known to retain their mobility after differential displacement. For a single-loop mechanism, which is composed of a pair of chains that are not closed under the Lie product, mobility is retained when the Lie closures of those chains are within the span of the union of motion screws of the two chains, a new result determined by applying the Baker–Campbell–Hausdorff expansion to the motion screws of the serial chain. When the Lie products have at most one dimension outside the union span, a second-order expression of mobility reduces to a quadratic form, allowing the numerical characterization of constraint singularities under that condition.

Characterization of Workspaces of Parallel Manipulators

1992 ◽  
Vol 114 (3) ◽  
pp. 368-375 ◽  
Author(s):  
V. Kumar
Keyword(s):  
Parallel Manipulators ◽  
Geometric Optimization ◽  
Reachable Workspace ◽  
Serial Chain ◽  
Kinematic Performance ◽  
Serial Chains ◽  
Manipulator Control ◽  
General Method

The workspaces and kinematic characterization of serial chain manipulator geometries and the geometric optimization have been studied extensively. Much less is known about workspaces for manipulation systems which possess several serial chains arranged in parallel. In this paper, two well known workspaces, the reachable workspace and the dexterous workspace, are investigated for parallel manipulators. A general method for obtaining these workspaces is presented. The existence of numerous special configurations in the workspace present problems in manipulator control. Therefore the controllably dexterous workspace is proposed as a useful measure of kinematic performance. The methodology of delineating the workspaces and its limitations are illustrated with examples.

A Relaxation Method for Simulating the Kinematics of Compound Nonlinear Mechanisms

2005 ◽  
Vol 128 (4) ◽  
pp. 719-728 ◽  
Author(s):  
Hod Lipson
Keyword(s):  
Degrees Of Freedom ◽  
Equations Of Motion ◽  
Relaxation Method ◽  
Relaxation Methods ◽  
Kinematic Chains ◽  
Relaxational Process ◽  
Kinematic Mechanisms ◽  
Robotics Design ◽  
2D And 3D ◽  
Nonlinear Behaviors

This paper describes a relaxation-based method for simulating 2D and 3D compound kinematic mechanisms. The relaxational process iteratively propagates node motions and degrees of freedom throughout a given kinematic mechanism. While relaxation methods were classically used to solve static problems, we show that the propagation of displacements during the calculation process itself reveals the kinematics of the structure. The method is slower than approaches based on solving simultaneous differential equations of motion, but provides several advantages: It achieves a higher level of accuracy, is more robust in handling transient singularities and degeneracies of the mechanism, and can handle more complex compound mechanisms with many links in multiple entangled kinematic chains. It also allows straightforward introduction of linkages with nonlinear behaviors such as wrapping strings, hydraulics, actuators, contacts, and other arbitrary responses. The basic simulation algorithm is presented, and a number of applications are provided including robotics, design, and biomechanics.

Mobility Analysis of Parallel Mechanisms Based on Screw Theory and the Concept of Equivalent Serial Kinematic Chain

2005 ◽  
Author(s):  
Xianwen Kong ◽  
Cle´ment M. Gosselin
Keyword(s):  
Systematic Approach ◽  
Screw Theory ◽  
Kinematic Chain ◽  
Second Step ◽  
Parallel Mechanisms ◽  
Mobility Analysis ◽  
Single Loop ◽  
Kinematic Chains ◽  
Serial Chain ◽  
Parallel Kinematic

This paper presents a systematic approach for the mobility analysis of parallel mechanisms. The method is based on screw theory and the concept of equivalent serial chain. An equivalent serial kinematic chain of a k-legged PKC (parallel kinematic chain) is defined as a serial kinematic chain which has the same twist system and the wrench system as the k-legged PKC. Using the proposed approach, the mobility analysis of a PKC is performed in two steps. The first step is the instantaneous mobility analysis, and the second step is the full-cycle mobility inspection. The first step is dealt with based on screw theory. The second step is performed with the aid of the concept of equivalent serial chain and the types of multi-DOF overconstrained single-loop kinematic chains. The proposed approach is illustrated with several examples.

Kinestatic Analysis of General Parallel Manipulators

1995 ◽  
Vol 117 (4) ◽  
pp. 601-606 ◽  
Author(s):  
Shou-Hung Ling ◽  
Ming Z. Huang
Keyword(s):  
Jacobian Matrix ◽  
General Procedure ◽  
Parallel Manipulators ◽  
General Context ◽  
Chain Geometry ◽  
Parallel Chain ◽  
Serial Chain ◽  
Systematic Formulation ◽  
Serial Chains

Robotic mechanisms in general can be of either serial-chain, parallel-chain, or hybrid (a combination of both parallel and serial chains) geometry. While it can be asserted that kinematic theories and techniques are well established for fully serial-chain manipulators, the same assertion cannot be made when it is considered in the general context. In this article, we present a general procedure for systematic formulation and characterization of the instantaneous kinematics for a robotic mechanism with a general parallel-chain geometry. A kinestatic approach based on screw system theory is adopted in this treatment. The resulting equation is a compact Jacobian matrix of the system which includes attributes from not only the active joints but also the passive constraints. An example has been provided to demonstrate the methodology as well as its theoretical feasibility.

Formulation of Unique Form of Screw Based Jacobian for Lower Mobility Parallel Manipulators

2010 ◽  
Vol 3 (1) ◽  
Author(s):  
Man Bok Hong ◽  
Yong Je Choi
Keyword(s):  
Parallel Manipulator ◽  
Parallel Manipulators ◽  
Unique Form ◽  
Kinematic Chains ◽  
Revolute Joints ◽  
Serial Chain ◽  
Velocity Relation ◽  
Lower Mobility ◽  
Serial Chains ◽  
Inverse Velocity

In this paper, the unique form of the screw based Jacobian is suggested for lower mobility parallel manipulators. Utilizing the concept of the reciprocal Jacobian, the forward statics relation for each of the serial kinematic chains of a parallel manipulator can be first obtained and then used to derive both the forward statics and the inverse velocity relations of the manipulator. The screw based Jacobian of a parallel manipulator can be formulated from the inverse velocity relation in such a way that it consists of the reciprocal Jacobians of the serial kinematic chains. Since any reciprocal Jacobian is unique to the corresponding serial chain, the suggested form of the screw based Jacobian is also determined uniquely to the lower mobility parallel manipulator. Two examples are given to illustrate the proposed method, one for the 3DOF parallel manipulator with three identical prismatic-revolute-spherical joints-serial chains and the other for the 4DOF parallel manipulator with nonidentical serial chains (two spherical-prismatic-spherical- and one revolute-revolute-prismatic-revolute joints-serial chains).

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