kinematic of machine

8/3/2019 Kinematic of Machine

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Rapid Design through Virtual and Physical Prototyping

Carnegie Mellon University

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Introduction to MechanismsYi Zhang

with

Susan Finger

Stephannie BehrensTable of Contents

4 Basic Kinematics of Constrained Rigid Bodies

4.1 Degrees of Freedom of a Rigid Body

4.1.1 Degrees of Freedom of a Rigid Body in a Plane

The degrees of freedom (DOF) of a rigid body is defined as the number of independent movements it has

rigid body in a plane. To determine the DOF of this body we must consider how many distinct ways the b

two dimensional plane such as this computer screen, there are 3 DOF. The bar can be translatedalong ththey axis, and rotatedabout its centroid.
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Figure 4-1 Degrees of freedom of a rigid body in a plane

4.1.2 Degrees of Freedom of a Rigid Body in Space

An unrestrained rigid body in space has six degrees of freedom: three translating motions along thex,y a

rotary motions around thex,y andzaxes respectively.

Figure 4-2 Degrees of freedom of a rigid body in space

4.2 Kinematic Constraints

Two or more rigid bodies in space are collectively called a rigid body system. We can hinder the motion

rigid bodies with kinematic constraints.Kinematic constraints are constraints between rigid bodies that r

the degrees of freedom of rigid body system.

The term kinematic pairs actually refers to kinematic constraints between rigid bodies. The kinematic pa

pairs and higher pairs, depending on how the two bodies are in contact.

4.2.1 Lower Pairs in Planar Mechanisms

There are two kinds of lower pairs in planar mechanisms: revolute pairs andprismatic pairs.

A rigid body in a plane has only three independent motions -- two translational and one rotary -- so introd

pair or a prismatic pair between two rigid bodies removes two degrees of freedom.

Figure 4-3 A planar revolute pair (R-pair)

Figure 4-4 A planar prismatic pair (P-pair)

4.2.2 Lower Pairs in Spatial Mechanisms

There are six kinds of lower pairs under the category ofspatial mechanisms. The types are: spherical pair

pair, revolute pair,prismatic pair, and screw pair.

Figure 4-5 A spherical pair (S-pair)

Aspherical pairkeeps two spherical centers together. Two rigid bodies connected by this constraint will

relatively aroundx,y andzaxes, but there will be no relative translation along any of these axes. Thereforemoves three degrees of freedom in spatial mechanism. DOF = 3.

Figure 4-6 A planar pair (E-pair)
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Aplane pairkeeps the surfaces of two rigid bodies together. To visualize this, imagine a book lying on a

in any direction except off the table. Two rigid bodies connected by this kind of pair will have two indepmotions in the plane, and a rotary motion around the axis that is perpendicular to the plane. Therefore, a p

degrees of freedom in spatial mechanism. In our example, the book would not be able to raise off the tab

table. DOF = 3.

Figure 4-7 A cylindrical pair (C-pair)

A cylindrical pairkeeps two axes of two rigid bodies aligned. Two rigid bodies that are part of this kind

independent translational motion along the axis and a relative rotary motion around the axis. Therefore, a

removes four degrees of freedom from spatial mechanism. DOF = 2.

Figure 4-8 A revolute pair (R-pair)

A revolute pairkeeps the axes of two rigid bodies together. Two rigid bodies constrained by a revolute protary motion around their common axis. Therefore, a revolute pair removes five degrees of freedom in s

= 1.

Figure 4-9 A prismatic pair (P-pair)

Aprismatic pairkeeps two axes of two rigid bodies align and allow no relative rotation. Two rigid bodiekind of constraint will be able to have an independent translational motion along the axis. Therefore, a pr

five degrees of freedom in spatial mechanism. DOF = 1.

Figure 4-10 A screw pair (H-pair)

Thescrew pairkeeps two axes of two rigid bodies aligned and allows a relative screw motion. Two rigid

screw pair a motion which is a composition of a translational motion along the axis and a corresponding

axis. Therefore, a screw pair removes five degrees of freedom in spatial mechanism.

4.3 Constrained Rigid Bodies

Rigid bodies and kinematic constraints are the basic components of mechanisms. A constrained rigid bod

kinematic chain, a mechanism, a structure, or none of these. The influence of kinematic constraints in the

has two intrinsic aspects, which are the geometrical and physical aspects. In other words, we can analyzeconstrained rigid bodies from their geometrical relationships or usingNewton's Second Law.

A mechanism is a constrained rigid body system in which one of the bodies is the frame. The degrees of

when considering a constrained rigid body system that is a mechanism. It is less crucial when the system

does not have definite motion.

Calculating the degrees of freedom of a rigid body system is straight forward. Any unconstrained rigid bo
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freedom in space and three degrees of freedom in a plane. Adding kinematic constraints between rigid bo

correspondingly decrease the degrees of freedom of the rigid body system. We will discuss more on this tmechanisms in the next section.

4.4 Degrees of Freedom of Planar Mechanisms

4.4.1 Gruebler's Equation

The definition of the degrees of freedom of a mechanism is the number of independent relative motions a

For example, Figure 4-11 shows several cases of a rigid body constrained by different kinds of pairs.

Figure 4-11 Rigid bodies constrained by different kinds of planar pairs

In Figure 4-11a, a rigid body is constrained by a revolute pairwhich allows only rotational movement aro

degree of freedom, turning around point A. The two lost degrees of freedom are translational movementsThe only way the rigid body can move is to rotate about the fixed point A.

In Figure 4-11b, a rigid body is constrained by aprismatic pairwhich allows only translational motion. Inone degree of freedom, translating along thex axis. In this example, the body has lost the ability to rotate

cannot move along they axis.

In Figure 4-11c, a rigid body is constrained by a higher pair. It has two degrees of freedom: translating al

and turning about the instantaneous contact point.

In general, a rigid body in a plane has three degrees of freedom. Kinematic pairs are constraints on rigid bdegrees of freedom of a mechanism. Figure 4-11 shows the three kinds of pairs inplanar mechanisms. Th

number of the degrees of freedom. If we create alower pair(Figure 4-11a,b), the degrees of freedom are

if we create a higher pair(Figure 4-11c), the degrees of freedom are reduced to 1.

Figure 4-12 Kinematic Pairs in Planar Mechanisms

Therefore, we can write the following equation:

(4-1)

Where

F = total degrees of freedom in the mechanism

n = number oflinks(including the frame)l = number oflower pairs (one degree of freedom)

h = number ofhigher pairs (two degrees of freedom)

This equation is also known as Gruebler's equation.

Example 1

Look at the transom above the door in Figure 4-13a. The opening and closing mechanism is shown in Fig
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calculate its degree of freedom.

Figure 4-13 Transom mechanism

n = 4 (link 1,3,3 and frame 4), l = 4 (at A, B, C, D), h = 0

(4-2)

Note: D and E function as a same prismatic pair, so they only count as one lower pair.

Example 2

Calculate the degrees of freedom of the mechanisms shown in Figure 4-14b. Figure 4-14a is an applicatio

Figure 4-14 Dump truck

n = 4, l = 4 (at A, B, C, D), h = 0

(4-3)

Example 3

Calculate the degrees of freedom of the mechanisms shown in Figure 4-15.

Figure 4-15 Degrees of freedom calculation

For the mechanism in Figure 4-15a

n = 6, l = 7, h = 0

(4-4)

For the mechanism in Figure 4-15b

n = 4, l = 3, h = 2

(4-5)

Note: The rotation of the roller does not influence the relationship of the input and output motion of the m

freedom of the roller will not be considered; It is called apassive orredundantdegree of freedom. Imagi

welded to link 2 when counting the degrees of freedom for the mechanism.


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4.4.2 Kutzbach Criterion

The number ofdegrees of freedom of a mechanism is also called the mobility of the device. The mobility

parameters (usually pair variables) that must be independently controlled to bring the device into a particKutzbach criterion, which is similar to Gruebler's equation, calculates the mobility.

In order to control a mechanism, the number of independent input motions must equal the number of deg

mechanism. For example, the transom in Figure 4-13a has a single degree of freedom, so it needs one ind

to open or close the window. That is, you just push or pull rod 3 to operate the window.

To see another example, the mechanism inFigure 4-15a also has 1 degree of freedom. If an independent

(e.g., a motor is mounted on joint A to drive link 1), the mechanism will have the a prescribed motion.

4.5 Finite Transformation

Finite transformation is used to describe the motion of a point on rigid body and the motion of the rigid b

4.5.1 Finite Planar Rotational Transformation

Figure 4-16 Point on a planar rigid body rotated through an angle

Suppose that a pointPon a rigid body goes through a rotation describing a circular path fromP1

toP2

aro

coordinate system. We can describe this motion with a rotation operatorR12

:

(4-6)

where

(4-7)

4.5.2 Finite Planar Translational Transformation

Figure 4-17 Point on a planar rigid body translated through a distance

Suppose that a pointPon a rigid body goes through a translation describing a straight path fromP1

toP2

coordinates of (

x, y). We can describe this motion with a translation operatorT12:

(4-8)

where
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(4-9)

4.5.3 Concatenation of Finite Planar Displacements

Figure 4-18 Concatenation of finite planar displacements in spaceSuppose that a pointPon a rigid body goes through a rotation describing a circular path fromP

1toP

2'ar

coordinate system, then a translation describing a straight path fromP2'toP

2. We can represent these two

(4-10)

and

(4-11)

We can concatenate these motions to get

(4-12)

where D12

is theplanar general displacement operator:

(4-13)

4.5.4 Planar Rigid-Body Transformation

We have discussed various transformations to describe the displacements of a point on rigid body. Can thto the displacements of a system of points such as a rigid body?

We used a 3 x 1 homogeneous column matrix to describe a vector representing a single point. A benefici

x 3 translational, rotational, and general displacement matrix operators is that they can easily be programmanipulate a 3 x n matrix of n column vectors representing n points of a rigid body. Since the distance of

body from every other point of the rigid body is constant, the vectors locating each point of a rigid body m

transformation when the rigid body moves and the proper axis, angle, and/or translation is specified to re(Sandor & Erdman 84). For example, the general planar transformation for the three pointsA, B, Con a r

represented by

(4-14)

4.5.5 Spatial Rotational Transformation

We can describe a spatial rotation operator for the rotational transformation of a point about an unit axis

origin of the coordinate system. Suppose the rotational angle of the point about u is , the rotation op

by
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(4-15)

where

ux, u

y, u

zare the othographical projection of the unit axis u onx,y, andzaxes, respectively.

s

= sin

c

= cos

v

= 1 - cos

4.5.6 Spatial Translational Transformation


toP2

coordinates of (

x, y, z), we can describe this motion with a translation operatorT:

(4-16)

4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin

Suppose a pointPon a rigid body rotates with an angular displacement about an unit axis u passing throu

coordinate system at first, and then followed by a translationDu along u. This composition of this rotatiothis translational transformation is a screw motion. Its corresponding matrix operator, thescrew operator

the translation operator in Equation 4-7 and the rotation operator in Equation 4-9.

(4-17)

4.6 Transformation Matrix Between Rigid Bodies

4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies

For a system of rigid bodies, we can establish a local Cartesian coordinate system for each rigid body. Tr

are used to describe the relative motion between rigid bodies.

For example, two rigid bodies in a space each have local coordinate systems x1y

1z

1and x

2y

2z

2. Let pointP

at location (x2, y

2, z

2) in body 2's local coordinate system. To find the location ofPwith respect to body 1

system, we know that that the point x2y

2z

2can be obtained from x

1y

1z

1by combining translationL

x1along

z about z axis. We can derive the transformation matrix as follows:

(4-18)

If rigid body 1 is fixed as a frame, a global coordinate system can be created on this body. Therefore, the

can be used to map the local coordinates of a point into the global coordinates.
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4.6.2 Kinematic Constraints Between Two Rigid Bodies

The transformation matrix above is a specific example for two unconstrained rigid bodies. The transform

the relative position of the two rigid bodies. If we connect two rigid bodies with akinematic constraint, th

will be decreased. In other words, their relative motion will be specified in some extent.

Suppose we constrain the two rigid bodies above with a revolute pairas shown in Figure 4-19. We can st

transformation matrix in the same form asEquation 4-18.

Figure 4-19 Relative position of points on constrained bodies

The difference is that theLx1

is a constant now, because the revolute pair fixes the origin of coordinate sy

to coordinate system x1y

1z

1. However, the rotation z is still a variable. Therefore, kinematic constrai

transformation matrix to some extent.

4.6.3 Denavit-Hartenberg Notation

Denavit-Hartenberg notation (Denavit & Hartenberg 55) is widely used in the transformation of coordina

and robot mechanisms. It can be used to represent the transformation matrix between links as shown in th

Figure 4-20 Denavit-Hartenberg Notation

In this figure,

zi-1

and ziare the axes of two revolute pairs;

iis the included angle of axes x

i-1and x

i;

diis the distance between the origin of the coordinate system x

i-1y

i-1z

i-1and the foot of the common

aiis the distance between two feet of the common perpendicular;

iis the included angle of axes z

i-1and z

i;

The transformation matrix will be T(i-1)i

(4-19)

The above transformation matrix can be denoted as T(ai,

i,

i, d

i) for convenience.

4.6.4 Application of Transformation Matrices to Linkages
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A linkage is composed of several constrained rigid bodies. Like a mechanism, a linkage should have a fra

can be used to derive the kinematic equations of the linkage. If all the links form a closed loop, the concatransformation matrices will be an identity matrix. If the mechanism has n links, we will have:

T12

T23

...T(n-1)n

= I

Table of Contents

Complete Table of Contents

1Introduction to Mechanisms

2Mechanisms and Simple Machines

3More on Machines and Mechanisms4Basic Kinematics of Constrained Rigid Bodies

4.1Degrees of Freedom of a Rigid Body

4.1.1Degrees of Freedom of a Rigid Body in a Plane4.1.2Degrees of Freedom of a Rigid Body in Space

4.2Kinematic Constraints

4.2.1Lower Pairs in Planar Mechanisms4.2.2Lower Pairs in Spatial Mechanisms

4.3Constrained Rigid Bodies

4.4Degrees of Freedom of Planar Mechanisms4.4.1Gruebler's Equation

4.2.24.4.2 Kutzbach Criterion

4.5 4.5Finite Transformation

4.5.1Finite Planar Rotational Transformation

4.5.2Finite Planar Translational Transformation4.5.3Concatenation of Finite Planar Displacements

4.5.4Planar Rigid-Body Transformation4.5.5Spatial Rotational Transformation

4.5.6Spatial Translational Transformation

4.5.7Spatial Translation and Rotation Matrix for Axis Through the Origin4.6Transformation Matrix Between Rigid Bodies

4.6.1Transformation Matrix Between two Arbitray Rigid Bodies

4.6.2Kinematic Constraints Between Two Rigid Bodies

4.6.3Denavit-Hartenberg Notation4.6.4Application of Transformation Matrices to Linkages

5Planar Linkages6Cams7Gears

8Other Mechanisms

IndexReferences

0100090000032a0200000200a20100000000a201000026060f003a03574d46430100000000000100ebf900
http://www.cs.cmu.edu/~rapidproto/mechanisms/tablecontents.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt1.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt1.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt2.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt2.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt3.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt3.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#pairshttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#pairshttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53ABhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53ABhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR54http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR54http://www.cs.cmu.edu/~rapidproto/mechanisms/grueblerhttp://www.cs.cmu.edu/~rapidproto/mechanisms/grueblerhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#kutzbachhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#kutzbachhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.5.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.5.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Chttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Chttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Dhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Dhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ehttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ehttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Fhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Fhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ghttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ghttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.6.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.6.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ihttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ihttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Jhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Jhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt5.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt5.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt6.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt6.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/mech.chpt8.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/mech.chpt8.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/references.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/tablecontents.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt1.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt2.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt3.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR52Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#pairshttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53ABhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR53Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR54http://www.cs.cmu.edu/~rapidproto/mechanisms/grueblerhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#kutzbachhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.5.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ahttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Bhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Chttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Dhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ehttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Fhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ghttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#4.6.1http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Hhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Ihttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#HDR61Jhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt5.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt6.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/mech.chpt8.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/references.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.html


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0005000000020001a001b00000046000000280000001c0000004744494302000000ffffffffffffffff2a01000

00000014000000080000004744494303000000250000000c0000000e000080250000000c0000000e0000800000010000000140000000400000003010800050000000b0200000000050000000c0250005000040000

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50000000902000000020d000000320a0c00000001000400000000005100500020000700040000002d0100

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00000014000000080000004744494303000000250000000c0000000e000080250000000c0000000e00008

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50000000902000000020d000000320a0c00000001000400000000005100500020000700040000002d010030000000000

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0000000000018030000010000006c00000000000000000000001a000000370000000000000000000000d2d4600000100180300001200000002000000000000000000000000000000f6090000e40c0000d800000017
http://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#topmailto:[email protected]:[email protected]:[email protected]://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/index.htmlhttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#tophttp://www.cs.cmu.edu/~rapidproto/mechanisms/chpt4.html#topmailto:[email protected]:[email protected]


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000000000000000000000000000000000000000000000000000000000000000000000000000000000000

00a06211002060110052516032a0621100985f110010000000086111008462110024516032a0621100985

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30000000000

[email protected]

(4-9)4.5.3 Concatenation of Finite Planar Displacements

Figure 4-18 Concatenation of finite planar displacements in space

Suppose that a pointPon a rigid body goes through a rotation describing a circular path fromP1

toP2'ar

coordinate system, then a translation describing a straight path fromP2'toP

2. We can represent these two

(4-10)

and

(4-11)

We can concatenate these motions to get
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(4-12)

where D12

is theplanar general displacement operator:

(4-13)

4.5.4 Planar Rigid-Body Transformation

We have discussed various transformations to describe the displacements of a point on rigid body. Can th

to the displacements of a system of points such as a rigid body?

We used a 3 x 1 homogeneous column matrix to describe a vector representing a single point. A benefici

x 3 translational, rotational, and general displacement matrix operators is that they can easily be program

manipulate a 3 x n matrix of n column vectors representing n points of a rigid body. Since the distance ofbody from every other point of the rigid body is constant, the vectors locating each point of a rigid body m

transformation when the rigid body moves and the proper axis, angle, and/or translation is specified to re(Sandor & Erdman 84). For example, the general planar transformation for the three pointsA, B, Con a rrepresented by

(4-14)

4.5.5 Spatial Rotational Transformation

We can describe a spatial rotation operator for the rotational transformation of a point about an unit axis

origin of the coordinate system. Suppose the rotational angle of the point about u is , the rotation op

by

(4-15)

where

ux, u

y, u

zare the othographical projection of the unit axis u onx,y, andzaxes, respectively.

s

= sin

c

= cos

v

= 1 - cos



toP2

coordinates of (

x, y, z), we can describe this motion with a translation operatorT:
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(4-16)


Suppose a pointP

on a rigid body rotates with an angular displacement about an unit axis u passing throucoordinate system at first, and then followed by a translationDu along u. This composition of this rotatio

this translational transformation is a screw motion. Its corresponding matrix operator, thescrew operator

the translation operator in Equation 4-7 and the rotation operator in Equation 4-9.

(4-17)


4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies

For a system of rigid bodies, we can establish a local Cartesian coordinate system for each rigid body. Trare used to describe the relative motion between rigid bodies.

For example, two rigid bodies in a space each have local coordinate systems x1y

1z

1and x

2y

2z

2. Let pointP

at location (x2, y

2, z

2) in body 2's local coordinate system. To find the location ofPwith respect to body 1

system, we know that that the point x2y

2z

2can be obtained from x

1y

1z

1by combining translationL

x1along

z about z axis. We can derive the transformation matrix as follows:

(4-18)

If rigid body 1 is fixed as a frame, a global coordinate system can be created on this body. Therefore, thecan be used to map the local coordinates of a point into the global coordinates.


The transformation matrix above is a specific example for two unconstrained rigid bodies. The transform

the relative position of the two rigid bodies. If we connect two rigid bodies with akinematic constraint, th

will be decreased. In other words, their relative motion will be specified in some extent.

Suppose we constrain the two rigid bodies above with a revolute pairas shown in Figure 4-19. We can st

transformation matrix in the same form asEquation 4-18.


The difference is that theLx1

is a constant now, because the revolute pair fixes the origin of coordinate sy

to coordinate system x1y

1z

1. However, the rotation z is still a variable. Therefore, kinematic constrai

transformation matrix to some extent.
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Denavit-Hartenberg notation (Denavit & Hartenberg 55) is widely used in the transformation of coordina

and robot mechanisms. It can be used to represent the transformation matrix between links as shown in th

Figure 4-20 Denavit-Hartenberg Notation

In this figure,

zi-1

and ziare the axes of two revolute pairs;

iis the included angle of axes x

i-1and x

i;

diis the distance between the origin of the coordinate system x

i-1y

i-1z

i-1and the foot of the common

ai is the distance between two feet of the common perpendicular;

iis the included angle of axes z

i-1and z

i;


(4-19)

The above transformation matrix can be denoted as T(ai

,i

,i

, di

) for convenience.


A linkage is composed of several constrained rigid bodies. Like a mechanism, a linkage should have a fracan be used to derive the kinematic equations of the linkage. If all the links form a closed loop, the conca

transformation matrices will be an identity matrix. If the mechanism has n links, we will have:

T12

T23

...T(n-1)n

= I

Table of Contents

Complete Table of Contents

1Introduction to Mechanisms

2Mechanisms and Simple Machines3More on Machines and Mechanisms

4Basic Kinematics of Constrained Rigid Bodies

4.1Degrees of Freedom of a Rigid Body
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4.1.1Degrees of Freedom of a Rigid Body in a Plane

4.1.2Degrees of Freedom of a Rigid Body in Space

4.2Kinematic Constraints4.2.1Lower Pairs in Planar Mechanisms

4.2.2Lower Pairs in Spatial Mechanisms

4.3Constrained Rigid Bodies4.4Degrees of Freedom of Planar Mechanisms

4.4.1Gruebler's Equation

4.2.24.4.2 Kutzbach Criterion4.5 4.5Finite Transformation

4.5.1Finite Planar Rotational Transformation

4.5.2Finite Planar Translational Transformation4.5.3Concatenation of Finite Planar Displacements

4.5.4Planar Rigid-Body Transformation

4.5.5Spatial Rotational Transformation

4.5.6Spatial Translational Transformation

4.5.7Spatial Translation and Rotation Matrix for Axis Through the Origin4.6Transformation Matrix Between Rigid Bodies

4.6.1Transformation Matrix Between two Arbitray Rigid Bodies4.6.2Kinematic Constraints Between Two Rigid Bodies

4.6.3Denavit-Hartenberg Notation

4.6.4Application of Transformation Matrices to Linkages5Planar Linkages

6Cams

7Gears

8Other MechanismsIndex

References

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50000000902000000020d000000320a0c00000001000400000000005100500020000700040000002d010030000000000

[email protected]

39-245

Rapid Design through Virtual and Physical Prototypi

Carnegie Mellon University

We can describe a spatial rotation operator for the rotational transformation of a point about an unit axis upassing through the origin of the coordinate system. Suppose the rotational angle of the point about u is ,the rotation operator will be expressed by

(4-15)

where

ux, uy, uz are the othographical projection of the unit axis u on x, y, and z axes, respectively.s = sinc = cosv = 1 - cos


Suppose that a point P on a rigid body goes through a translation describing a straight path from P1 to P2with a change of coordinates of (x, y, z), we can describe this motion with a translation operator T:

(4-16)


Suppose a point P on a rigid body rotates with an angular displacement about an unit axis u passingthrough the origin of the coordinate system at first, and then followed by a translation Du along u. Thiscomposition of this rotational transformation and this translational transformation is a screw motion. Itscorresponding matrix operator, the screw operator, is a concatenation of the translation operator in
mailto:[email protected]:[email protected]:[email protected]://www.cs.cmu.edu/~sfinger/home.htmlhttp://www.cs.cmu.edu/~sfinger/home.htmlhttp://www.cs.cmu.edu/~rapidproto/home.htmlhttp://www.cmu.edu/mailto:[email protected]:[email protected]://www.cs.cmu.edu/~sfinger/home.htmlhttp://www.cs.cmu.edu/~rapidproto/home.htmlhttp://www.cmu.edu/


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Equation 4-7 and the rotation operator in Equation 4-9.

(4-17)

4.6 Transformation Matrix Between Rigid Bodies4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies

For a system of rigid bodies, we can establish a local Cartesian coordinate system for each rigid body.Transformation matrices are used to describe the relative motion between rigid bodies.

For example, two rigid bodies in a space each have local coordinate systems x1y1z1 and x2y2z2. Letpoint P be attached to body 2 at location (x2, y2, z2) in body 2's local coordinate system. To find thelocation of P with respect to body 1's local coordinate system, we know that that the point x2y2z2 can beobtained from x1y1z1 by combining translation Lx1 along the x axis and rotation z about z axis. We canderive the transformation matrix as follows:

(4-18)

If rigid body 1 is fixed as a frame, a global coordinate system can be created on this body. Therefore, theabove transformation can be used to map the local coordinates of a point into the global coordinates.


The transformation matrix above is a specific example for two unconstrained rigid bodies. Thetransformation matrix depends on the relative position of the two rigid bodies. If we connect two rigidbodies with a kinematic constraint, their degrees of freedom will be decreased. In other words, theirrelative motion will be specified in some extent.

Suppose we constrain the two rigid bodies above with a revolute pair as shown in Figure 4-19. We canstill write the transformation matrix in the same form as Equation 4-18.


The difference is that the Lx1 is a constant now, because the revolute pair fixes the origin of coordinate

system x2y2z2 with respect to coordinate system x1y1z1. However, the rotation z is still a variable.Therefore, kinematic constraints specify the transformation matrix to some extent.


Denavit-Hartenberg notation (Denavit & Hartenberg 55) is widely used in the transformation of coordinatesystems of linkages and robot mechanisms. It can be used to represent the transformation matrixbetween links as shown in the Figure 4-20.Figure 4-20 Denavit-Hartenberg Notation

In this figure,

* zi-1 and zi are the axes of two revolute pairs;

* i is the included angle of axes xi-1 and xi;* di is the distance between the origin of the coordinate system xi-1yi-1zi-1 and the foot of the commonperpendicular;

* ai is the distance between two feet of the common perpendicular;* i is the included angle of axes zi-1 and zi;


(4-19)


20/21

The above transformation matrix can be denoted as T(ai, i, i, di) for convenience.


A linkage is composed of several constrained rigid bodies. Like a mechanism, a linkage should have aframe. The matrix method can be used to derive the kinematic equations of the linkage. If all the linksform a closed loop, the concatenation of all of the transformation matrices will be an identity matrix. If themechanism has n links, we will have:

T12T23...T(n-1)n = I

(4-20)

Table of ContentsComplete Table of Contents

1 Introduction to Mechanisms2 Mechanisms and Simple Machines3 More on Machines and Mechanisms4 Basic Kinematics of Constrained Rigid Bodies

4.1 Degrees of Freedom of a Rigid Body

4.1.1Degrees of Freedom of a Rigid Body in a Plane4.1.2 Degrees of Freedom of a Rigid Body in Space

4.2 Kinematic Constraints

4.2.1 Lower Pairs in Planar Mechanisms4.2.2 Lower Pairs in Spatial Mechanisms

4.3 Constrained Rigid Bodies4.4 Degrees of Freedom of Planar Mechanisms

4.4.1 Gruebler's Equation4.2.2 4.4.2 Kutzbach Criterion

4.5 4.5 Finite Transformation

4.5.1 Finite Planar Rotational Transformation4.5.2 Finite Planar Translational Transformation4.5.3 Concatenation of Finite Planar Displacements4.5.4 Planar Rigid-Body Transformation4.5.5 Spatial Rotational Transformation4.5.6 Spatial Translational Transformation4.5.7 Spatial Translation and Rotation Matrix for Axis Through the Origin


4.6.1 Transformation Matrix Between two Arbitray Rigid Bodies4.6.2 Kinematic Constraints Between Two Rigid Bodies4.6.3 Denavit-Hartenberg Notation4.6.4 Application of Transformation Matrices to Linkages

5 Planar Linkages6 Cams


21/21

7 Gears8 Other MechanismsIndexReferences

[email protected]

kinematic of machine

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