forward kinematics
DESCRIPTION
Introduction to ROBOTICS. Forward Kinematics. University of Bridgeport. Kinematic. Forward (direct) Kinematics Given: The values of the joint variables. Required: The position and the orientation of the end effector. Inverse Kinematics - PowerPoint PPT PresentationTRANSCRIPT
![Page 1: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/1.jpg)
Forward Kinematics
University of Bridgeport
1
Introduction to ROBOTICS
![Page 2: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/2.jpg)
Kinematic• Forward (direct) Kinematics• Given: The values of the joint variables.• Required: The position and the orientation of the end
effector.
• Inverse Kinematics• Given : The position and the orientation of the
end effector.• Required : The values of the joint variables.
2
![Page 3: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/3.jpg)
Why DH notation
• Find the homogeneous transformation H relating the tool frame to the fixed base frame
3
![Page 4: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/4.jpg)
Why DH notation
• A very simple way of modeling robot links and joints that can be used for any kind of robot configuration.
• This technique has became the standard way of representing robots and modeling their motions.
4
![Page 5: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/5.jpg)
DH Techniques
1. Assign a reference frame to each joint (x-axis and z-axis). The D-H representation does not use the y-axis at all.
2. Each homogeneous transformation Ai is represented as a product of four basic transformations
5
![Page 6: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/6.jpg)
DH Techniques
6
• Matrix Ai representing the four movements is found by: four movements
, , , ,i i i ii z z d x a xA Rot Trans Trans Rot
1. Rotation of about current Z axis
2. Translation of d along current Z axis
3. Translation of a along current X axis
4. Rotation of about current X axis
![Page 7: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/7.jpg)
7
i i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
iA
CS
SCxRotRx0
0
001
),(,
100
0
0
),( ,
CS
SC
zRotRz
1000
00
00
0001
1000
0100
0010
001
1000
100
0010
0001
1000
0100
00
00
ii
ii
i
i
ii
ii
i CS
SC
a
d
CS
SC
A
![Page 8: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/8.jpg)
DH Techniques
• The link and joint parameters :
• Link length ai : the offset distance between the Zi-1 and Zi axes along the Xi axis.
• Link offset di the distance from the origin of frame i−1 to the Xi axis along the Zi-1 axis.
8
![Page 9: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/9.jpg)
DH Techniques
9
•Link twist αi :the angle from the Zi-1 axis to the Zi axis about the Xi axis. The positive sense for α is determined from zi-1 and zi by the right-hand rule.
•Joint angle θi the angle between the Xi-1 and Xi axes about the Zi-1 axis.
![Page 10: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/10.jpg)
DH Techniques
• The four parameters: ai: link length, αi: Link twist , di : Link offset and
θi : joint angle.
• The matrix Ai is a function of only a single variable qi , it turns out that three of the above four quantities are constant for a given link, while the fourth parameter is the joint variable.
10
![Page 11: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/11.jpg)
DH Techniques
• With the ith joint, a joint variable is qi associated where
All joints are represented by the z-axis.• If the joint is revolute, the z-axis is in the direction of
rotation as followed by the right hand rule.• If the joint is prismatic, the z-axis for the joint is along
the direction of the liner movement.
11
![Page 12: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/12.jpg)
DH Techniques
3. Combine all transformations, from the first joint (base) to the next until we get to the last joint, to get the robot’s total transformation matrix.
4. From , the position and orientation of the tool frame are calculated.
12
01 2. .......n nT A A A
0nT
![Page 13: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/13.jpg)
DH Techniques
13
![Page 14: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/14.jpg)
DH Techniques
14
![Page 15: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/15.jpg)
DH Techniques
15
![Page 16: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/16.jpg)
DH Techniques
16
i i i i i i i
i i i i i i i
i i i
c -c s s s a c
s c c -s c a s
0 s c d
0 0 0 1
iA
![Page 17: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/17.jpg)
Example I The two links arm
17
• Base frame O0
•All Z ‘s are normal to the page
![Page 18: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/18.jpg)
Example I The two links arm
18
Where (θ1 + θ2 ) denoted by θ12 and
![Page 19: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/19.jpg)
Example 2
19
![Page 20: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/20.jpg)
Example 2
20
![Page 21: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/21.jpg)
Example 3 The three links cylindrical
21
![Page 22: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/22.jpg)
Example 3 The three links cylindrical
22
![Page 23: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/23.jpg)
Example 3 The three links cylindrical
23
![Page 24: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/24.jpg)
Example 3 The three links cylindrical
24
![Page 25: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/25.jpg)
Example 4 Spherical wrist
25
![Page 26: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/26.jpg)
Example 4 Spherical wrist
26
![Page 27: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/27.jpg)
Example 4 Spherical wrist
27
![Page 28: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/28.jpg)
Example 4 Spherical wrist
28
![Page 29: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/29.jpg)
Example 5The three links cylindrical with Spherical wrist
29
![Page 30: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/30.jpg)
Example 5The three links cylindrical with Spherical wrist
30
03T
36T• given by example 2, and given by
example 3.
![Page 31: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/31.jpg)
Example 5The three links cylindrical with Spherical wrist
31
![Page 32: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/32.jpg)
Example 5The three links cylindrical
with Spherical wrist
32
![Page 33: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/33.jpg)
Example 5The three links cylindrical
with Spherical wrist
33
• Forward kinematics:1. The position of the end-effector: (dx ,dy ,dz )
2. The orientation {Roll, Pitch, Yaw }Rotation about X axis{ROLL}
Rotation about fixed Y axis{PITCH}
Rotation about fixed Z axis{YAW}
![Page 34: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/34.jpg)
Roll Pitch Yaw• The rotation matrix for the following
operations:
X
Y
Z
axis{YAW} Zfixedabout Rotation
}axis{PITCH Y fixedabout Rotation
axis{ROLL} Xabout Rotation
34
CCSCS
CSSSCSCSSCCS
SSCSCSSCSSCC
CS
SCCS
SC
xRotyRotzRotR
0
0
001
C0S-
010
S0C
100
0
0
),(),(),(
![Page 35: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/35.jpg)
Example 4The three links cylindrical with Spherical wrist
35
• How to calculate
• Compare the matrix R with Of the matrix
, ,and
06T
C C S S C S S C S C S S
R S C C S S C S C S S S C
S C S C C
31S r 32C S r 21S C r
131( )Sin r 1 32( )
rSin
C
1 21sin ( )
r
C
11 12 13
21 22 23
31 32 33
r r r
r r r
r r r
![Page 36: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/36.jpg)
Module 1RRR:RRR
36
Links α a θ d
1 90 0 * 10
2 0 10 * 0
3 -90 0 * 0
4 90 0 * 10
5 -90 0 * 0
6 0 0 * 0
![Page 37: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/37.jpg)
HW
• From Spong book: page 112• 3.2, 3.3, 3.4, 3.6 , 3.7, 3.8, 3.9, 3.11
• No Class on next Tuesday
37
![Page 38: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/38.jpg)
Module 1
38
1 2 3, ,and • Where are Roll, Pitch, and Yaw
![Page 39: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/39.jpg)
Representing forward kinematics
• Forward kinematics
1
2
3
4
5
6
x
y
z
p
p
p
11 12 13
21 22 23
31 32 33
0 0 0 1
x
y
z
r r r d
r r r dT
r r r d
39
• Transformation Matrix
![Page 40: Forward Kinematics](https://reader033.vdocuments.net/reader033/viewer/2022061312/56815adc550346895dc8a98f/html5/thumbnails/40.jpg)
Remember
• For n joints: we have n+1 links. Link 0 is the base• Joints are numbered from 1 to n• Joint i connects link i − 1 to link i. • Frame i {Xi Yi Zi} is attached to joint i+1.
• So, frame {O0 X0 Y0 Z0}, which is attached to the robot base (inertial frame) “joint 1”.
40