每日彙整: 2018-04-24

樹莓派、樹莓派之學習、樹莓派之教育

STEM 隨筆︰古典力學︰運動學【五‧一】

或許當文章順暢說明簡單，人的思維容易被牽著鼻子走乎？

 

故耳有時須跳脫其境旁敲側擊也，如是方可直抵關鍵處呦！

Kinematic chain

In mechanical engineering, a kinematic chain is an assembly of rigid bodies connected by joints to provide constrained (or desired) motion that is the mathematical model for a mechanical system.[1] As in the familiar use of the word chain, the rigid bodies, or links, are constrained by their connections to other links. An example is the simple open chain formed by links connected in series, like the usual chain, which is thekinematic model for a typical robot manipulator.[2]

Mathematical models of the connections, or joints, between two links are termed kinematic pairs. Kinematic pairs model the hinged and sliding joints fundamental to robotics, often called lower pairs and the surface contact joints critical to cams and gearing, called higher pairs. These joints are generally modeled as holonomic constraints. A kinematic diagram is a schematic of the mechanical system that shows the kinematic chain.

The modern use of kinematic chains includes compliance that arises from flexure joints in precision mechanisms, link compliance in compliant mechanisms and micro-electro-mechanical systems, and cable compliance in cable robotic and tensegrity systems.[3] [4]

Mobility formula

The degrees of freedom, or mobility, of a kinematic chain is the number of parameters that define the configuration of the chain.[2][5] A system of n rigid bodies moving in space has 6n degrees of freedom measured relative to a fixed frame. This frame is included in the count of bodies, so that mobility does not depend on link that forms the fixed frame. This means the degree-of-freedom of this system is M = 6(N − 1), whereN = n + 1 is the number of moving bodies plus the fixed body.

Joints that connect bodies impose constraints. Specifically, hinges and sliders each impose five constraints and therefore remove five degrees of freedom. It is convenient to define the number of constraints c that a joint imposes in terms of the joint’s freedom f, where c = 6 − f. In the case of a hinge or slider, which are one degree of freedom joints, have f = 1 and therefore c = 6 − 1 = 5.

The result is that the mobility of a kinematic chain formed from n moving links and j joints each with freedom fi, i = 1, …, j, is given by

\displaystyle M=6n-\sum _{i=1}^{j}(6-f_{i})=6(N-1-j)+\sum _{i=1}^{j}f_{i}

Recall that N includes the fixed link.

Analysis of kinematic chains

The constraint equations of a kinematic chain couple the range of movement allowed at each joint to the dimensions of the links in the chain, and form algebraic equations that are solved to determine the configuration of the chain associated with specific values of input parameters, called degrees of freedom.

The constraint equations for a kinematic chain are obtained using rigid transformations [Z] to characterize the relative movement allowed at each joint and separate rigid transformations [X] to define the dimensions of each link. In the case of a serial open chain, the result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link. A chain of n links connected in series has the kinematic equations,

\displaystyle [T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\cdots [X_{n-1}][Z_{n}],\!

where [T] is the transformation locating the end-link—notice that the chain includes a “zeroth” link consisting of the ground frame to which it is attached. These equations are called the forward kinematics equations of the serial chain.[6]

Kinematic chains of a wide range of complexity are analyzed by equating the kinematics equations of serial chains that form loops within the kinematic chain. These equations are often called loop equations.

The complexity (in terms of calculating the forward and inverse kinematics) of the chain is determined by the following factors:

Explanation

Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion of these independent rigid bodies with kinematic constraints. Kinematic constraints are constraints between rigid bodies that result in the decrease of the degrees of freedom of rigid body system.[5]

The movement of the Boulton & Watt steam engineis studied as a system of rigid bodies connected by joints forming a kinematic chain.

Synthesis of kinematic chains

The constraint equations of a kinematic chain can be used in reverse to determine the dimensions of the links from a specification of the desired movement of the system. This is termed kinematic synthesis.[7]

Perhaps the most developed formulation of kinematic synthesis is for four-bar linkages, which is known as Burmester theory.[8][9][10]

Ferdinand Freudenstein is often called the father of modern kinematics for his contributions to the kinematic synthesis of linkages beginning in the 1950s. His use of the newly developed computer to solve Freudenstein’s equation became the prototype of computer-aided design systems.[7]

This work has been generalized to the synthesis of spherical and spatial mechanisms.[2]

A model of the human skeleton as a kinematic chain allows positioning using forward and inverse kinematics.

 

無論想象力可否聞一知十耶？？

Four-bar linkage

From Wikipedia, the free encyclopedia
 
 A four-bar linkage, also called a four-bar, is the simplest movable closed chain linkage. It consists of four bodies, called bars or links, connected in a loop by four joints. Generally, the joints are configured so the links move in parallel planes, and the assembly is called a planar four-bar linkage. Spherical and spatial four-bar linkages also exist and they are used in practice. [1]

Pumpjacks’ main mechanism is a four-bar linkage

Planar four-bar linkage

Planar four-bar linkages are constructed from four links connected in a loop by four one-degree-of-freedom joints. A joint may be either a revolute, that is a hinged joint, denoted by R, or a prismatic, as sliding joint, denoted by P.

A link connected to ground by a hinged joint is usually called a crank. A link connected to ground by a prismatic joint is called a slider. Sliders are sometimes considered to be cranks that have a hinged pivot at an extremely long distance away perpendicular to the travel of the slider.

The link that connects two cranks is called a floating link or coupler. A coupler that connects a crank and a slider, it is often called a connecting rod.

There are three basic types of planar four-bar linkage depending on the use of revolute or prismatic joints:

  1. Four revolute joints: The planar quadrilateral linkage is formed by four links and four revolute joints, denoted RRRR. It consists of two cranks connected by a coupler.
  2. Three revolute joints and a prismatic joint: The slider-crank linkage is constructed from four links connected by three revolute and one prismatic joint, or RRRP. It can be constructed with crank and a slider connected by the connecting rod. Or it can be constructed as a two cranks with the slider acting as the coupler, known as an inverted slider-crank.
  3. Two revolute joints and two prismatic joints: The double slider is a PRRP linkage.[2] This linkage is constructed by connecting two sliders with a coupler link. If the directions of movement of the two sliders are perpendicular then the trajectories of the points in the coupler are ellipses and the linkage is known as an elliptical trammel, or the Trammel of Archimedes.

Planar four-bar linkages are important mechanisms found in machines. The kinematics and dynamics of planar four-bar linkages are important topics in mechanical engineering.

Planar four-bar linkages can be designed to guide a wide variety of movements.

Coupler curves of a crank-rocker four-bar linkage. Simulation done with MeKin2D.

 

且求自能問又自能答矣！！