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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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Unit-III: Motion Analysis 2
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.
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Unit-III: Motion Analysis 3
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:
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Unit-III: Motion Analysis 4
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
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Unit-III: Motion Analysis 5
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
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Unit-III: Motion Analysis 6
(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.
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Unit-III: Motion Analysis 7
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.
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Unit-III: Motion Analysis 8
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.
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Unit-III: Motion Analysis 9
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
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Unit-III: Motion Analysis 10
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
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Unit-III: Motion Analysis 11
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 .
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Unit-III: Motion Analysis 12
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
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Unit-III: Motion Analysis 13
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
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Unit-III: Motion Analysis 14
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
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Unit-III: Motion Analysis 15
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”
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Unit-III: Motion Analysis 16
(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
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Unit-III: Motion Analysis 17
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
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Unit-III: Motion Analysis 18
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’’
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Unit-III: Motion Analysis 19
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
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Unit-III: Motion Analysis 20
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‟.
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Unit-III: Motion Analysis 21
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
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Unit-III: Motion Analysis 22
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
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UNIT-IIIManipulator Kinematics
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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.
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What Is Manipulator Kinematics ?
Open Chain Manipulator Kinematics.
Closed Chain Manipulator Kinematics.
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Position representation
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• 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
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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.
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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.
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Forward / Direct Kinematics
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Forward / Direct Kinematics
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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.
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Inverse / Reverse kinematics
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Inverse / Reverse kinematics
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Extrinsic rotations / Intrinsic rotations
• Any orientation can be achieved by composing three elemental rotations.
– Pitch,
– Yawn and
– Roll.`
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• 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).
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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.
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Euler Angle Representation
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Euler Angle Representation (Z-Y-X)
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Euler Angle Representation (Z-Y-Z)
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Rotation with Respect to Fixed Frame (X-Y-Z Fixed Angles)
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Rotation with Respect to Fixed Frame (X-Y-Z Fixed Angles)
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Rotation with Respect to Fixed Frame (X-Y-Z Fixed Angles)
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Rotation with Respect to Fixed Frame (X-Y-Z Fixed Angles)
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Joint Link & Labelling
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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
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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.
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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
Denavit - Hartenberg Parameters
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
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• 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.
Denavit - Hartenberg Parameters
DESCRIPTION OF JOINTS AND LINKS sccemechanical.wordpress.com
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• 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.
Denavit - Hartenberg Parameters
DESCRIPTION OF JOINTS AND LINKS sccemechanical.wordpress.com
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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
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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.
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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:-
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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
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• 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}.
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• Step 4: Assigning frame to link 0 the base frame {0}.
• Step 5: Link n, the end effector, frame assignment- frame{n}.
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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.
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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
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KINEMATIC RELATIONSHIP BETWEEN LINKS
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KINEMATIC RELATIONSHIP BETWEEN LINKS
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