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Unit-III: Motion Analysis 1

A

B

C

A'

B'

C'

dx

dy D

UNIT – III

MOTION ANALYSIS

3.0 BASIC TRANSFORMATION

Animations are produced by moving the 'camera' or the objects in a scene

along animation paths. Changes in orientation, size and shape are

accomplished with geometric transformations that alter the coordinate

descriptions of the objects. The basic geometric transformations are

translation, rotation, scaling and reflection.

3.1 TWO DIMENSIONAL TRANSFORMATIONS

Translation: A translation is applied to an object by repositioning it along a

straight line path from one coordinate location to another We translate a

two-dimensional point by adding translation distances, dx and dy, to the

original coordinate position (x,y) to move the point to a new position (x',y')

x' = x + dx y’ = y + dy

The translation distance pair (dx, dy) is called translation vector or shift

vector

.(x,y)

.(x‟,y‟)

Figure: Illustration of translation

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Matrix representation of translation

' ' x yx y x y d d

This allows us to write the two-dimensional translation equations in the

matrix form:

'V V D

In the figure, the triangle ABC is translated by distance vector T

x yD d d and

the triangle A‟B‟C‟ is translated one.

It is important point that the size and shape of the object do not change

after translation.

Scaling

Scaling in 2-Dimensional means, stretching the points in the x-y plane. It

can be accomplished by simple multiplication as follows

' .

' .

x

y

x S x

y S y

Where Sx and Sy represents the scaling coefficients in x and y directions

respectively. Scaling can be expressed in vector form as follows.

'V S V or 0'

0'

x

y

Sx x

Sy y

Where V‟ = New (after scaling) point vector of the object

V = Original point vector of the object

S = Scaling coefficient matrix

NOTE: If the values of scaling factor are greater than 1 then the object is

enlarged and if it is less than 1, it reduces the size of the object. Keeping

value as 1 does not changes the object.






Uniform Scaling: To achieve uniform scaling the values of scaling factor

must be kept equal i.e., x yS S .

Differential Scaling: Unequal or Differential scaling is produced incases

when values for scaling factor are not equal i.e., x yS S .

As per usual phenomenon of scaling, an object moves closer to origin when

the values of scaling factor are less than 1.

Rotation

A two-dimensional rotation is applied to an object by repositioning it along a

circular path in the x-y plane. When we generate a rotation, we get a

rotation angle (θ) and the position about which the object is rotated is

known as rotation point or pivot point. Positive values for the rotation

angle define counter-clockwise rotations about the pivot point and the

negative values rotate objects in the clockwise direction.

Suppose the pivot point be at origin, to understand the relationship between

angular and coordinate points of original and transformed position lets look

at the figure below:






Let a point V(x,y) is rotated to V’(x’,y’) through an angle „θ‟ about the origin.

cos

sin

x r

y r

and

' cos cos cos sin sin

' sin cos sin sin cos

x r r r

y r r r

On simplification, we get

' cos sin

' sin cos

x x y

y x y

In matrix form, this expression can be represented as,

' cos sin

' sin cos

x x

y y

This can also be written as 'V R V where R is the rotational matrix.

3.2 THREE DIMENSIONAL TRANSFORMATIONS

3D transformations are similar to 2D transformations in both definition and

derivation.

Translation: In this case we translate a point V(x,y,z) by (dx, dy, dz) to a

point V‟(x‟,y‟,z‟). This can be expressed in the matrix form as,

'

'

'

x

y

z

x x d

y y d

z z d

Scaling: If „S‟ is the scaling coefficient matrix, then the scaling

transformation in 3D is,

' 0 0

' 0 0

' 0 0

x

y

z

x S x

y S y

z S z






Rotation: The rotation of an object could be about any of the axes.

(1) About Z: When we want to rotate about Z – axis, the Z – coordinate of

the point V(x,y,z) after rotation will not change because the rotation takes

place in the x-y plane. Therefore, the new (x,y) coordinates will be same as

those obtained in case of 2D rotation.

Suppose the location of initial point is V(x,y,z) then, we have the location of

new point V‟(x‟,y‟,z‟) is as follows.

'z z

' cos sin

' sin cos

x x y

y x y

In matrix form, we have

' cos sin 0

' sin cos 0

' 0 0 1

x x

y y

z z

Similarly, we obtain the rotation about x and y axes.

(2) About x:

'

' cos sin

' sin cos

x x

y y z

z y z

Or in the matrix form,

' 1 0 0

' 0 cos sin

' 0 sin cos

x x

y y

z z






(3) About y:

'

' cos sin

' sin cos

y y

z z x

x z x

Or in the matrix form,

' cos 0 sin

' 0 1 0

' sin 0 cos

x x

y y

z z

In general, 'V R V , where R is the rotational matrix in 3D

3.3 HOMOGENEOUS TRANSFORMATION

In case of 2D or 3D geometric transformations, translation involves addition

of matrices, whereas scaling and rotations are performed by their

multiplication. It is however possible to develop a homogeneous

transformation scheme, which require only multiplication of matrices in all

cases. This simplifies transformation process.

In the geometric transformation methods, translation, scaling and rotation

have non-uniform equations as follows.

'

'

'

V V D

V S V

V R V

Where V and V‟ are the positions of the original and new point vectors.

In homogeneous transformation scheme, all transformations have

multiplicative form.

'V H V

Where H is the homogeneous transformation matrix.






1. Translation:

1 0 0

0 1 0

0 0 1

0 0 0 1

x

y

z

d

dH

d

Where [H] is the translation transformation matrix with the translation

values dx, dy and dz with respect to x, y and z-axes respectively.

2. Scaling:

0 0 0

0 0 0

0 0 0

0 0 0 1

x

y

z

S

SH

S

Where [H] is the scaling transformation matrix with the scale values Sx, Sy

and Sz with respect to x, y and z-axes respectively.

3. Rotation:

cos sin 0 0

sin cos 0 0

0 0 1 0

0 0 0 1

zH

1 0 0 0

0 cos sin 0

0 sin cos 0

0 0 0 1

xH

cos 0 sin 0

0 1 0 0

sin 0 cos 0

0 0 0 1

yH

Where [Hx], [Hy] and [Hz] are the rotational transformation matrix when the

rotation angle is θ with respect to x, y and z-axes respectively.






Composition of Transformation

In practice, one-step transformations alone may not be useful and a series

of transformations may have to be applied to an object. The techniques for

combining series of transformations are very useful in such situations. The

process of composition is accomplished by multiplying the [H] matrix of

various transformations. Composition is also referred to as Compounding, or

Concatenation of [H1], [H2] ……….., [Hn].

In general, the composition of transformation is expressed as

1 1' ...........n nV H H H V

Where, „n‟ refers to the nth transformation in sequence.

HOMOGENEOUS COORDINATES

A vector is a quantity that has both magnitude and direction. It is usually

represented by an arrow of length equal to its magnitude and pointing in the

appropriate direction. For vectors in Cartesian coordinates, the basis is a set

of unit vectors directed along the orthogonal x, y and z axes used to

represent this space. Thus a vector v can be written as

v x i y j z k

Where the unit vectors , ,i j k

have been weighted by the appropriate

constants (x, y, z) as shown in the Figure. Unit vectors will be boldface

lowercase letters with a circumflex above them.






k

j

i

Figure: Vector and basis set in Cartesian reference frame

Translational Transformation:

Problem: A point (5,5) lies in a 2-D reference frame. The point has to move

along the line at an angle of 45° for a distance of 10 units. What are the

coordinates of the final position of the point?

Solution: Figure below illustrates the initial and final positions of the point.

Figure: Translational transformation example

.

.

Initial

Point

p

Final

Point

P‟

45°

10

x

y

5

5

x2

y2

(1,0,0)

y - axis

x - axis

z - axis

.

.

.

(0,1,0)

(0,0,1)

(x,y,z)

.

v






The original point may be represented as a vector in homogeneous

coordinates as below.

5

5

0

1

p

Since it is desired to move the point along 45° radial extension, the unit

vector corresponding to this direction is

0.707

0.707

0

1

u

To get a motion of 10 units, it is necessary to multiply the components of „u‟

by 10, excluding scaling factor. The new vector becomes,

7.07

7.07

0

1

u

The point which is initially at (5,5) is shifted to new position along the line

45° by a distance of 10 units using the following matrix manipulation.

1 0 0 7.07 5

0 1 0 7.07 5'

0 0 1 0 0

0 0 0 1 1

p

12.07

12.07'

0

1

p






ROBOT ARM KINEMATICS

Robot arm kinematics deals with analytical study of the geometry of motion

of a robot arm with respect to a fixed reference coordinate system without

regard to the forces/moments that cause the motion.

Position representation: The kinematics of the robot with rotational joints

is more difficult to analyze than the robot with linear joints. Figure below

illustrates the geometric form of the manipulator with rotational joints.

Figure: A two-dimensional 2 dof manipulator

The position of the end of the arm may be represented in a number of ways.

One way is to utilize the two joint angles θ1 and θ2. This is known as the

representation in “joint space” and we may define it as,

1 2,jP

Another way to define the arm position is in “world space”. This involves the

use of Cartesian coordinate system that is external to the robot. The origin

of the Cartesian axis system is often located in the robots base. The end-of-

arm position would be defined in world space as ,wP x y .

In 3-D, it is , ,wP x y z .






Representing an arm‟s position in world space is useful when the robot must

communicate with other machines. These other machines may not have a

detailed understanding of the robots kinematics and so a “neutral”

representation such as the world space must be used. In order to use both

representations, we must be able to transform from one to the other.

When the position and orientation of the end-effector, for a manipulator, are

derived from the given joint angles and link parameters, the scheme is called

the “Forward Kinematics”.

Figure: Forward Kinematics scheme

When the joint angles of the manipulator are derived from the position and

orientation of the end-effector, the scheme is known as the “Inverse /

Reverse kinematics”.

Figure: Inverse Kinematics scheme

Joint Angles

Inverse

Kinematics

Position and Orientation

of end-effector

Link

Parameters

Inputs

Output

Joint Angles

Forward

Kinematics

Position and Orientation

of end-effector

Link

Parameters

Inputs

Output






Forward transformation of a 2 degree-of-freedom arm

We can determine the position of the end of the arm in world space by

defining a vector for link 1 and another for link 2.

1 1 1 1 1

2 2 1 2 2 1 2

cos , 1

cos , sin 2

r L L Sin

r L L

Vector addition of (1) & (2) yields the coordinates x and y of the end of arm

(point Pw) in world space.

1 1 2 1 2

1 1 2 1 2

cos cos 3

sin 4

x L L

y L Sin L

Inverse/Reverse transformation of a 2 degree-of-freedom arm

For the two link manipulator shown in Figure below, there are two possible

configurations for reaching the point (x,y). Some strategy must be developed

to select the appropriate configuration. One approach is that employed in

the control system of Unimate PUMA robot. In the PUMA‟s control language,

there is a set of commands called ABOVE and BELOW that determines

whether the elbow is to make an angle θ2 that is greater than or less than

zero as shown in Figure.

Figure: The arm at point P(x,y) indicating two possible configurations to

achieve the position






For example, let us assume the θ2 is positive as shown.

We can write the equations (3) & (4) as,

1 1 2 1 2 2 1 2

1 1 2 1 2 2 1 2

cos cos cos sin sin

sin cos cos sin

x L L L

y L Sin L L

Squaring both sides and adding the two equations, we get,

2 2 2 2

1 22

1 2

cos 52

x y L L

L L

Figure: Solving for joint angles

Defining α and β as shown in Figure.

2 2

2 2 1

sintan

cos

tan

L

L L

y

x






Using trigonometric identity tan tan

tan1 tan tan

A BA B

A B

2 2 1 2 2

1

2 2 1 2 2

cos sintan tan

cos sin

y L L x L

x L L y L

Knowing the link lengths L1 and L2, we are now able to calculate the

required joint angles to place the arm at a position (x,y) in world space.

Rotation with respect to the fixed frame

Many times it is desired to perform a sequence of rotations, each about a

given fixed coordinate frame, rather than about successive current frame.

X – Y – Z Fixed Angles

In a three dimensional space, a coordinate frame is a set of three orthogonal

right handed axes X, Y, Z called principal axes. The frame is labeled as {x y

z} or {A} or {1}. Le t the fixed frame {A} and moving frame {B} be initially

coincident. Now consider the sequence of rotations as shown in the Figure.

(a) Rotation about fixed X – axis (b) Rotation about fixed Y - axis

XA, XB’

YA

ZA

YB’

ZB’

XA,

XB”

YA

ZA

YB”

ZB”






(c) Rotation about fixed Z - axis

First rotate {B} about XA by angle γ, then rotate about YA by angle β and then

rotate about ZA by angle α. This convention for specifying orientation is

known as XYZ- fixed angle representation because each rotation is specified

about an axis of fixed reference frame.

The three rotations about the three axes in fixed angle rotation when applied

to the end-effector produce the roll, pitch and yaw motions.

The derivation of the equivalent rotation matrix , ,A

BR is straightforward

because all rotations occur about axes of the reference frame.

, ,A

BR = Z XR R R

=

0 0 1 0 0

0 0 1 0 0 1

0 0 1 0 0

c s c s

s c c s

s c s c

cos , sin .where c s etc

It is extremely important to understand the order of rotations used in

equation (1). Thinking in terms of rotations as operators, we have applied

the rotations (from the right) of XR , then YR and then ZR .

XA, XB”’

YA

YB”’ ZB”’

ZA






Multiplying the matrices of equation (1), we get

, , 2A

B

c c c s s s c c s c s s

R s c s s s c c s s c c s

s c s c c

The inverse of the problem, i.e., extracting equivalent X-Y-Z fixed angles

from a rotation matrix is as follows.

Let the equation (2) be expressed as below.

, ,A

BR = 11 12 13

21 22 23

31 32 33

3

r r r

r r r

r r r

From equation (2), cosβ can be computed by taking square root of sum of

squares of r11 and r21.

Then we can solve for β with the arc tangent of (–r31) over the computed

cosine.

Then as long as cβ ≠ 0, we can solve for α by taking the arc tangent of r21/cβ

over r11/cβ and we can solve for γ by taking the arc tangent of r32/cβ over

r33/cβ.

In summary,

2 2

31 11 21

21 11

32 33

tan 2 ,

tan 2 ,

tan 2 ,

A r r r

r rA

c c

r rA

c c

where, Atan2(y, x) is a two argument arc tangent function. Atan2(y, x)

computes 1tany

x






EULER ANGLES

Z – Y – X Euler angles: Let the frame {A} and {B} are initially coincident.

First rotate {B} about ZB by an angle α, then rotate about YB by angle β and

then rotate about XB by an angle γ.

(a) (b)

(c)

In this rotation, each rotation is performed about an axis of the moving

frame {B}, rather than the fixed reference {A}. Such a set of three rotations

are called Euler angles. Note that each rotation takes place about an axis

whose location depends upon the preceding rotations. Because the three

rotations occur about the axes Z, Y and X, we will call this representation as

Z-Y-X Euler angles.

XA, XB’

YA

ZA

YB’

ZB’

XB’

YB’

ZB’ ZB’’

XB’’

YB’’

XB’’’

YB’’’ ZB’’’

ZB’’

XB’’

YB’’






Figure shows the axes of {B} after each Euler angle rotation is applied.

Rotation α about Z causes X to rotate into X‟ and Y to rotate Y‟ and so on.

An additional “prime” gets added to each axis with each rotation.

Z-Y-Z Euler angles: Another possible description of frame {B} is as follows.

Let frame {A} and frame {B} are initially coincident. First rotate {B} about ZB

by angle α, then rotate about YB by an angle β and then rotate about ZB by

an angle γ.

, ,A

BR = Z ZR R R

, ,A

BR =

0 0 0

0 0 1 0 0

0 0 1 0 0 0 1

c s c s c s

s c s c

s c

, ,A

B

c c c s s c c s s c c s

R s c c c s s c s c c s s

s c s s c

The solution for extracting Z-Y-Z Euler angles from a rotation matrix is

stated below.

, ,A

BR =

11 12 13

21 22 23

31 32 33

r r r

r r r

r r r

If 0Sin then,

2 2

31 32 33

23 13

32 31

tan 2 ,

tan 2 ,

tan 2 ,

A r r r

r rA

s s

r rA

s s






Department of Mechanical Engineering SR Engineering College, Warangal

Since the rotations are described relative to the frame which is moving {B},

this is an Euler angle description. Because three rotations occur about the

axes Z, Y and Z, it is called Z-Y-Z Euler angle representation.

Rotation about an arbitrary axis (Equivalent angle axis)

Rotations are not always performed about the principal coordinate axes. Let

a coordinate frame can be rotated about an arbitrary axis „k‟.

Therefore let k = (kx, ky, kz)T, expressed in the frame „A‟ be a unit vector

defining an axis.

Let Rk,θ represents a rotation matrix corresponding to a rotation of angle θ

about axis „k‟.

Rk,θ can be derived by several ways. One simple way is to rotate vector „k‟

into one of the coordinate axis, say ZA, then rotate about ZA by θ and finally

„k‟ back to its original position.

We can rotate k into ZA by first rotating about ZA by –α, then rotating about

YA by –β. Since all the rotations are performed relative to the fixed frame

OXAYAZA the matrix Rk,θ is obtained as

, , , , , ,k Z Y Z Y ZR R R R R R

Rk,θ can be derived by several ways. One simple way is to rotate vector „k‟

into one of the coordinate axis, say ZA, then rotate about ZA by θ and finally

„k‟ back to its original position.

Let Rk,θ represents a rotation matrix corresponding to a rotation of angle θ

about axis „k‟.






When the axis of rotation is chosen as one of the principal axes of {A}, then

the equivalent rotation matrix takes on the familiar form of planar rotations.

,

1 0 0

0 cos sin

0 sin cos

XR

,

cos 0 sin

0 1 0

sin 0 cos

yR

,

cos sin 0

sin cos 0

0 0 1

zR

If the axis of rotation is a general axis, it can be shown that the equivalent

matrix is

2

2

,

2

x x y z x z y

k x y z y y z x

x z y y z x z

k V c k k V k s k k V k s

R k k V k s k V c k k V k s

k k V k s k k V k s k V c

Where cos ,c sin ,s and 1 cosV

2 2 2 2sin cos

y x

x y x y

k k

k k k k

2 2sin cosx y zk k k






Department of Mechanical Engineering SR Engineering College, Warangal

Examples:

Solution:

Given,

10

20

30

vB

.A

v B vA T B

0.866 0.500 0.000 11.0 10

0.500 0.866 0.000 3.0 20

0.000 0.000 1.000 9.0 30

0 0 0 1 1

vA

9.66

19.32

39.00

1

9.66

19.32

39.00

vA





UNIT-IIIManipulator Kinematics



ROBOT ARM KINEMATICS

• The study of motion of joints, links and themanipulator as a whole is called kinematics ofmanipulator.

• Robot arm kinematics deals with analyticalstudy of the geometry of motion of a robotarm with respect to a fixed referencecoordinate system without regard to theforces/moments that cause the motion.



What Is Manipulator Kinematics ?

Open Chain Manipulator Kinematics.

Closed Chain Manipulator Kinematics.



Position representation



• The position of the end of the arm may berepresented in a number of ways.

• Position representation is of two types:

1. Joint space representation.

• It involves utilizing of two joint angles i.e., θ1,θ2

2. World space representation.

• It involves use of Cartesian co-ordinate system i.e.,X,Y.

Position representation



KINEMATIC MODELLING OF THE MANIPULATOR

• Kinematic model describes the spatial position of joints, links and position and orientation of end effector or tool holder.

• The kinematic modeling is classified in to two types.

– Forward kinematic Model or Direct kinematics.

– Inverse kinematic Model or Inverse kinematics.



Forward / Direct Kinematics

• When the position and orientation of the end-effector, for a manipulator, are derived from thegiven joint angles and link parameters, thescheme is called the “Forward / DirectKinematics”.

• The joint variables are the angles between thelinks in the case of revolute or rotational joints,and the link extension in the case of prismaticor sliding joints.



Forward / Direct Kinematics



Inverse / Reverse kinematics

• When the joint angles of the manipulator arederived from the position and orientation ofthe end-effector, the scheme is known as the“Inverse / Reverse kinematics”. .

• i.e., at what angles you should set your jointsin order to achieve that task.



Inverse / Reverse kinematics



Extrinsic rotations / Intrinsic rotations

• Any orientation can be achieved by composing three elemental rotations.

– Pitch,

– Yawn and

– Roll.`



http://en.wikipedia.org/wiki/Euler_angles#Extrinsic_rotations

http://en.wikipedia.org/wiki/Euler_angles#Intrinsic_rotations

• Extrinsic Rotation: The elemental rotationscan either occur about the axes of the fixedcoordinate system (Fixed angles).

• Intrinsic Rotation: The elemental rotationsabout the axes of a rotating coordinatesystem, which is initially aligned with the fixedone, and modifies its orientation after eachelemental rotation (Euler angles).

Extrinsic rotations/Intrinsic rotationssccemechanical.wordpress.com


http://en.wikipedia.org/wiki/Euler_angles#Extrinsic_rotations

http://en.wikipedia.org/wiki/Euler_angles#Intrinsic_rotations

Euler Angle Representation

• The Euler angles are three angles introducedby Leonhard Euler to describe the orientationof a rigid body.

• Euler angles are also used to represent theorientation of a frame of reference (typically, acoordinate system or basis) relative toanother.



http://en.wikipedia.org/wiki/Leonhard_Euler

http://en.wikipedia.org/wiki/Orientation_(geometry)

http://en.wikipedia.org/wiki/Rigid_body

http://en.wikipedia.org/wiki/Frame_of_reference

http://en.wikipedia.org/wiki/Coordinate_system

http://en.wikipedia.org/wiki/Basis_(linear_algebra)

Euler Angle Representation



Euler Angle Representation (Z-Y-X)



Euler Angle Representation (Z-Y-Z)



Rotation with Respect to Fixed Frame (X-Y-Z Fixed Angles)



Joint Link & Labelling



axis ( i – 2 )

Joint ( i -1 )

axis (i – 1)

Joint ( i )axis i

Joint ( i +1 )

zi-1 xi-1

a i-1Common normals

xi

Dd i

A B

Denavit - Hartenberg Parameterssccemechanical.wordpress.com


Denavit - Hartenberg Parameters

• Jacques Denavit (Dr. Esaí alumni) and Richard Hartenbergintroduced this convention in 1955 in order to standardize thecoordinate frames for spatial linkages.

• The Denavit–Hartenberg parameters (also called DH parameters) arethe four parameters associated with a particular convention forattaching reference frames to the links of a spatial kinematic chain,or robot manipulator.

– The first two “ai” and “αi” define the structure of the link, whilethe second two “di” and “θi” determine the position of theneighboring link. Each of the four parameters is defined withrespect to the two joint axes attached to a particular link.

– Where “ai”= link length ; αi”= link twist.

– And “di”= Offset displacement; θi” = Angular displacement.



http://en.wikipedia.org/wiki/Linkage_(mechanical)

http://en.wikipedia.org/wiki/Kinematic_chain

http://en.wikipedia.org/wiki/Robot

axis ( i – 2 )

Joint ( i -1 )

axis (i – 1)

Joint ( i )axis i

Joint ( i +1 )

zi-1 xi-1

a i-1Common normal

xi

Dd i

A B


DESCRIPTION OF JOINTS AND LINKS

• To describe the position and orientation of a link in space, a co ordinate frameis attached to each link namely frame {i} to link i.

• The D-H method uses matrices to describe the relationship among referenceframes attached to various points on the manipulator.

a i



• In general every link of the manipulator is connected totwo other links with joints at either end with theexception to base and end effector or tool holder.

• From a geometric point link defines the relative positionand orientation of joint axes at its two ends.

• For the two axes axis (i – 1) and axis i, their exist amutual perpendicular which gives the shortest distancebetween the two axes. This shortest distance along thecommon normal is defined as the link length (ai ) .

• The angle between the projection of axis (i-1) and axis i,on a plane perpendicular to the common normal AB isknown as Link Twist (αi).

• These link parameters αi, ai are known as linkparameters and are constant for given link.


DESCRIPTION OF JOINTS AND LINKS sccemechanical.wordpress.com


• For the two links connected by revolute or prismatic joint the relativeposition of these links is measured by the displacement at the joint which iseither joint distance or joint angle.

• Joint distance (di) defined as the perpendicular distance between the two

adjacent common normal ai-1 and ai measured along axis i-1 i.e., joint

distance is the translation needed along joint axis(i-1) to make ai-1 intersectwith ai .

• Joint angle (θi) is the angle between the two adjacent common normal ai-1

and ai measured in direction about the axis i-1 i.e., it is the rotation aboutjoint axis i-1 needed to make parallel ai-1 to ai .

• These two parameters di , (θi) are known as joint parameters.

• For revolute joint di is zero and θi varies ; For prismatic joint (θi) is constantand di varies. The varying parameter is called as joint variable. The jointvariable is denoted by “q”

– qi = (θi) if the joint i is revolute : qi = di if the joint i is prismatic.


DESCRIPTION OF JOINTS AND LINKS sccemechanical.wordpress.com


axis ( i – 2 )

Joint ( i -1 )

axis (i – 1)

Joint ( i )axis i

Joint ( i +1 )

zi-1 xi-1

a i-1Common normals

xi

Dd i

A B



ALGORITHM-LINK FRAME ASSIGNMENT

• This algorithm is divided in four parts.

– First segment give steps for labelling scheme.

– Second segment give steps for frame assignment to intermediate links 1 to n-1.

– The third and fourth segment give steps for frame {0} and frame {n} assignment respectively.



ALGORITHM• Step 0 : Identify the joints starting with base and

ending with end effector. Number the links from 0 to n starting from base as “0” and ending with last link as “n”.

• Step 1 : Align axis Zi with axis of joint(i+1) for i=0,1… n+1.

• Step 2 : the xi axis is fixed perpendicular to both zi-1and zi axes and points away from zi-1.– The origin of frame {i} is located at the intersection of

zi and xi axes. Here this may lead to three different situations they are:-



axis ( i – 2 )

Joint ( i -1 )

axis (i – 1)

Joint ( i )axis i

Joint ( i +1 )

zi-1 xi-1

a i-1Common normals

xi

Dd i

A B



• Case1: If zi-1 and zi axes intersect choose the origin at the point of their intersection. The xi axis will be perpendicular to the plane containing zi-1 axes and zi axes. This will give ai to be zero.

• Case2 :if zi-1 and zi axes are parallel or lie in the parallel planes.

– If joint i is revolute xi axis is chosen along that common normal which passes through origin of frame {i-1} this will fix the origin and make di zero.

– If the joint I is prismatic xi axis is arbitrarily chosen as any convenient common normal and the origin is located at the distal end of link i.

• If zi-1 and zi axes coincide the origin lies on the common axis.

– If joint I is revolute origin is located to coincide with origin of frame {i-1} and xi axis coincides with xi-1 axes to cause di to be zero. The origin is located at distal end of link i.

• Step 3 : the yi axis has no choice and is fixed to complete the right handed orthonormal co ordinate frame {i}.



• Step 4: Assigning frame to link 0 the base frame {0}.

• Step 5: Link n, the end effector, frame assignment- frame{n}.



KINEMATIC RELATIONSHIP BETWEEN LINKS

• To find the transformation matrix relating frames attached to the adjacent links consider frame{i-1} and frame{i}.

• The transformation of frame i-1 to i consists of four basic transformations.– A rotation about zi-1 axis by angle θi.

– Translation along Zi-1 axis by distance di.

– Translation by distance ai along xi axis and

– Rotation by an angle αi about xi axis.



axis ( i – 2 )

Joint ( i -1 )

axis (i – 1)

Joint ( i )axis i

Joint ( i +1 )

zi-1 xi-1

a i-1Common normals

xi

Dd i

A B



KINEMATIC RELATIONSHIP BETWEEN LINKS



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