Ch.02 Rotation Kinematics
-
Upload
icarus-can -
Category
Documents
-
view
24 -
download
2
description
Transcript of Ch.02 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
02. Rotation Kinematics
Robotics 2.01 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
โข Consider a rigid body ๐ฉ with a fixed point ๐ถ โน Rotation about the fixed
point ๐ถ is the only possible motion of the body ๐ฉ
โข Consider
- local coordinate frame ๐๐ฅ๐ฆ๐ง
attaches to the rigid body ๐ฉ
- global coordinate frame ๐๐๐๐
โข Determine
- Transformation matrices for
point ๐ท between two coordinates
โข Point ๐ถ of the body ๐ฉ is fixed to the ground ๐ฎ and is the origin of both
coordinate frames
๐ถ
Robotics 2.02 Rotation Kinematics
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Rigid body ๐ฉ rotates ๐ผ degrees about the ๐-axis of the global coordinate
โข ๐ท2 has local coordinate: ๐2๐ต =
๐ฅ2๐ฆ2๐ง2
and global coordinate: ๐2๐บ =
๐2๐2๐2
โน Relation between the two coordinates:
๐2๐2๐2
=๐๐๐ ๐ผ โ๐ ๐๐๐ผ 0๐ ๐๐๐ผ ๐๐๐ ๐ผ 00 0 1
๐ฅ2๐ฆ2๐ง2
or ๐2๐บ = ๐ธ๐,๐ผ ๐2
๐ต (2.1)
where ๐ธ๐,๐ผ is the ๐-rotation matrix
๐ธ๐,๐ผ =๐๐๐ ๐ผ โ๐ ๐๐๐ผ 0๐ ๐๐๐ผ ๐๐๐ ๐ผ 00 0 1
(2.3)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
Robotics 2.03 Rotation Kinematics
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Similarly, rotation ๐ฝ degrees about the ๐-axis, and ฮณ degrees about the
๐-axis of the global frame relate the local and global coordinates of point
๐ท by the following equations
๐๐บ = ๐ธ๐,๐ฝ ๐๐ต (2.4)
๐๐บ = ๐ธ๐,๐พ ๐๐ต (2.5)
Where ๐ธ๐,๐ฝ is the ๐-rotation matrix
๐ธ๐,๐ฝ =๐๐๐ ๐ฝ 0 ๐ ๐๐๐ฝ0 1 0
โ๐ ๐๐๐ฝ 0 ๐๐๐ ๐ฝ (2.6)
and ๐ธ๐,๐พ is the ๐ฟ-rotation matrix
๐ธ๐,๐พ =1 0 00 ๐๐๐ ๐พ โ๐ ๐๐๐พ0 ๐ ๐๐๐พ ๐๐๐ ๐พ
(2.7)
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
Robotics 2.04 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 3 (Successive rotation about global axes)
The corner ๐ท(5,30,10) of the slab rotates 300 about
the ๐-axis, then 300 about the ๐-axis, and 900
about the ๐-axis โน Find the final position of ๐ท ?
The new global position of ๐ท after first rotation
After the second rotation
And after the last rotation
Robotics 2.05 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
The slab and the point ๐ท in first, second, third, and fourth positions
Robotics 2.06 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 4 (Time dependent global rotation)
A rigid body ๐ฉ continuously turns about ๐-axis of ๐ฎ at a rate of 0.3๐๐๐/๐
โน Find the global position and global velocity of the body point ๐ท?
The rotation transformation matrix of the body is
๐ธ๐ต๐บ =
๐๐๐ 0.3๐ก 0 ๐ ๐๐0.3๐ก0 1 0
โ๐ ๐๐0.3๐ก 0 ๐๐๐ 0.3๐ก
Any point of ๐ฉ will move on a circle with radius ๐ = ๐2 + ๐2 parallel
to (๐, ๐)-plane
๐๐๐
=๐๐๐ 0.3๐ก 0 ๐ ๐๐0.3๐ก
0 1 0โ๐ ๐๐0.3๐ก 0 ๐๐๐ 0.3๐ก
๐ฅ๐ฆ๐ง
=๐ฅ๐๐๐ 0.3๐ก + ๐ง๐ ๐๐0.3๐ก
๐ฆ๐ง๐๐๐ 0.3๐ก โ ๐ฅ๐ ๐๐0.3๐ก
๐2 + ๐2 = (๐ฅ๐๐๐ 0.3๐ก + ๐ง๐ ๐๐0.3๐ก)2+(๐ง๐๐๐ 0.3๐ก โ ๐ฅ๐ ๐๐0.3๐ก)2
= ๐ฅ2 + ๐ง2 = ๐ 2
Robotics 2.07 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
After ๐ก = 1๐ , the point ๐๐ต = 1 0 0 ๐ will be seen at
๐๐๐
=cos 0.3 0 sin 0.30 1 0
โ sin 0.3 0 cos 0.3
100
=0.9550
โ0.295
After ๐ก = 2๐ , the point ๐๐ต = 1 0 0 ๐ will be seen at
๐๐๐
=๐๐๐ 0.6 0 sin 0.60 1 0
โ sin 0.6 0 cos 0.6
100
=0.8250
โ0.564
Taking a time derivative of ๐๐ ๐บ = ๐ธ๐,๐ฝ ๐๐
๐ต to get the global velocity
vector of any point ๐ท
๐๐ ๐บ = ๐ธ ๐,๐ฝ ๐๐
๐ต
= 0.3๐ง๐๐๐ 0.3๐ก โ ๐ฅ๐ ๐๐0.3๐ก
0โ๐ฅ๐๐๐ 0.3๐ก โ ๐ง๐ ๐๐0.3๐ก
Robotics 2.08 Rotation Kinematics
I. ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 5 (Global rotation, local position)
A point ๐ท moved to ๐2 ๐บ = [4,3,2]๐ after rotating 600 about ๐-axis
โน Find local position of ๐ท in the local coordinate?
๐2 ๐ต = ๐ธ๐,60
โ1 ๐2 ๐บ
โน
๐ฅ2๐ฆ2๐ง2
=๐๐๐ 60 โ๐ ๐๐60 0๐ ๐๐60 ๐๐๐ 60 00 0 1
โ1 432
=4.5981โ1.9641
2.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
Robotics 2.09 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
II. SUCCESSIVE ROTATION ABOUT GLOBAL CARTESIAN AXES
โข The final global position of a point ๐ท ( ๐๐บ ) in a rigid body ๐ฉ with position
vector ๐๐ต , after a sequence of rotations ๐ธ1, ๐ธ2, ๐ธ3, โฆ , ๐ธ๐ about the
global axes can be found by
๐๐บ = ๐ธ๐ต ๐บ ๐๐ต (2.35)
Where the global rotation matrix
๐ธ๐ต ๐บ = ๐ธ๐ โฆ๐ธ3๐ธ2๐ธ1 (2.36)
A rotation matrix is orthogonal
๐ธ๐ = ๐ธโ1 (2.37)
Robotics 2.10 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
II. SUCCESSIVE ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 6 (Successive global rotation matrix)
The global rotation matrix after a rotation ๐ธ๐,๐ผ followed by ๐ธ๐,๐ฝ and
then ๐ธ๐,๐พ is
๐ธ๐ต ๐บ = ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ (2.38)
=1 0 00 ๐๐๐ ๐พ โ๐ ๐๐๐พ0 ๐ ๐๐๐พ ๐๐๐ ๐พ
๐๐๐ ๐ฝ 0 ๐ ๐๐๐ฝ0 1 0
โ๐ ๐๐๐ฝ 0 ๐๐๐ ๐ฝ
๐๐๐ ๐ผ โ๐ ๐๐๐ผ 0๐ ๐๐๐ผ ๐๐๐ ๐ผ 00 0 1
=
๐๐ผ๐๐ฝ โ๐๐ฝ๐ ๐ผ ๐ ๐ฝ๐๐พ๐ ๐ผ + ๐๐ผ๐ ๐ฝ๐ ๐พ ๐๐ผ๐๐พ โ ๐ ๐ผ๐ ๐ฝ๐ ๐พ โ๐๐ฝ๐ ๐พ๐ ๐ผ๐ ๐พ โ ๐๐ผ๐๐พ๐ ๐ฝ ๐๐ผ๐ ๐พ + ๐๐พ๐ ๐ผ๐ ๐ฝ ๐๐ฝ๐๐พ
Robotics 2.11 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
II. SUCCESSIVE ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 7 (Successive global rotations, global position)
The end point ๐ท of the arm is located at
๐1๐1๐1
=0
๐๐๐๐ ๐๐๐ ๐๐๐
=0
1๐๐๐ 750
1๐ ๐๐750=
0.000.260.97
(2.39)
The rotation matrix to find the new position of
the end point ๐ท: rotate โ290 about ๐-axis, then
300 about ๐-axis, and 1320 about ๐-axis is
๐ธ๐ต๐บ = ๐ธ๐,132๐ธ๐,30๐ธ๐,โ29 =
0.87 โ0.44 โ0.24โ0.33 โ0.15 โ0.930.37 0.89 โ0.27
(2.40)
and
๐2๐2๐2
=0.87 โ0.44 โ0.24โ0.33 โ0.15 โ0.930.37 0.89 โ0.27
0.000.260.97
=โ0.3472โ0.9411โ0.0305
(2.41)
Robotics 2.12 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
II. SUCCESSIVE ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 8 (Twelve independent triple global rotations)
There are 12 different independent combinations of triple rotations about
the global axes which transform a rigid body coordinate frame ๐ฉ from the
coincident position with a global frame ๐ฎ to any final orientation by only
three rotations about the global axes provided that no two
consequence rotations are about the same axis:
๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ
๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ
๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ
Robotics 2.13 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
II. SUCCESSIVE ROTATION ABOUT GLOBAL CARTESIAN AXES
โข Example 9 (Order of rotation, and order of matrix multiplication)
Changing the order of global rotation matrices is equivalent to
changing the order of rotations. The position of a point ๐ท of a rigid
body ๐ฉ is located at ๐๐ = 1 2 3 ๐๐ต
Its global position after rotation 300 about ๐-axis and then 450 about ๐-
axis is at
๐๐๐บ
1 = ๐ธ๐,45๐ธ๐,30 ๐๐ =0.53 โ0.84 0.130.0 0.15 0.99
โ0.85 โ0.52 0.081
123
๐ต =โ0.763.27โ1.64
And if we change the order of rotation then its position would be at:
๐๐๐บ
2 = ๐ธ๐,30๐ธ๐,45 ๐๐ =0.53 0.0 0.85โ0.84 0.15 0.52โ0.13 โ0.99 0.081
123
=3.081.02โ1.86
๐ต
These two final positions of ๐ท are ๐ = ๐๐๐บ
1 โ ๐๐๐บ
2 = 4.456 apart
Robotics 2.14 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
III. GLOBAL ROLL-PITCH-YAW ANGLES
โข The rotation about the ๐-axis of the global coordinate frame is called a
roll, the rotation about the ๐-axis of the global coordinate frame is called
a pitch, and the rotation about the ๐-axis of the global coordinate frame is
called a yaw
โข This figure illustrates 450 roll, pitch, and yaw rotation about the axes of a
global coordinate frame
Robotics 2.15 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
III. GLOBAL ROLL-PITCH-YAW ANGLES
โข Given the roll, pitch, and yaw angles, we can compute the overall
rotation matrix
โข The global roll-pitch-yaw rotation matrix is
๐ธ๐ต = ๐ธ๐,๐พ๐ธ๐,๐ฝ๐ธ๐,๐ผ ๐บ
=
๐๐ฝ๐๐พ โ๐๐ผ๐ ๐พ + ๐๐พ๐ ๐ผ๐ ๐ฝ ๐ ๐ผ๐ ๐พ + ๐๐ผ๐๐พ๐ ๐ฝ๐๐ฝ๐ ๐พ ๐๐ผ๐๐พ + ๐ ๐ผ๐ ๐ฝ๐ ๐พ โ๐๐พ๐ ๐ผ + ๐๐ผ๐ ๐ฝ๐ ๐พโ๐ ๐ฝ ๐๐ฝ๐ ๐ผ ๐๐ผ๐๐ฝ
(2.55)
Robotics 2.16 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
III. GLOBAL ROLL-PITCH-YAW ANGLES
โข Also we are able to compute the equivalent roll, pitch, and yaw angles
when a rotation matrix is given. Suppose that ๐๐๐ indicates the element of
row ๐ and column ๐ of the roll-pitch-yaw rotation matrix (2.55)
โข Then the roll angle is
๐ผ = ๐ก๐๐โ1๐32
๐33
and the pitch angle is
๐ฝ = โ๐ ๐๐โ1 ๐31
and the yaw angle is
๐พ = ๐ก๐๐โ1๐21๐11
Robotics 2.17 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
III. GLOBAL ROLL-PITCH-YAW ANGLES
โข Example 11 (Determination of roll-pitch-yaw angles)
Determine the required roll-pitch-yaw angles to make the ๐ฅ-axis of the
body coordinate ๐ฉ parallel to ๐ = ๐ฐ + 2๐ฑ + 3๐ฒ , while ๐ฆ-axis remains in
(๐, ๐)-plane?
We have
๐ =๐
๐=
1
14๐ฐ +
2
14๐ฑ +
3
14๐ฒ ๐บ
๐ = ๐ฐ โ ๐ ๐ฐ + ๐ฑ โ ๐ ๐ฑ = ๐๐๐ ๐๐ฐ ๐บ + ๐ ๐๐๐๐ฑ
The axes ๐ ๐บ and ๐ ๐บ must be orthogonal, therefore
1/ 14
2/ 14
3/ 14
โ๐๐๐ ๐๐ ๐๐๐0
= 0 โ ๐ = โ26.560
Robotics 2.18 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
III. GLOBAL ROLL-PITCH-YAW ANGLES
Find ๐ ๐บ by a cross product: ๐ = ๐ ร ๐ =
1/ 14
2/ 14
3/ 14
ร0.894โ0.447
0=
0.3580.717โ0.597
๐บ๐บ๐บ
Hence, the transformation matrix ๐ธ๐ต๐บ is
๐ธ๐ต =
๐ฐ โ ๐ ๐ฐ โ ๐ ๐ฐ โ ๐
๐ฑ โ ๐ ๐ฑ โ ๐ ๐ฑ โ ๐
๐ฒ โ ๐ ๐ฒ โ ๐ ๐ฒ โ ๐
=
1/ 14 0.894 0.358
2/ 14 โ0.447 0.717
3/ 14 0 โ0.597
๐บ
Now it is possible to determine the required roll-pitch-yaw angles
๐ผ = ๐ก๐๐โ1๐32๐33
= ๐ก๐๐โ10
โ0.597= 0
๐ฝ = โ๐ ๐๐โ1 ๐31 = โ๐ ๐๐โ13
14โ โ0.93 rad
๐พ = ๐ก๐๐โ1๐21๐11
= ๐ก๐๐โ12/ 14
1/ 14โ 1.1071 rad
Robotics 2.19 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IV. ROTATION ABOUT LOCAL CARTESIAN AXES
โข The rigid body ๐ฉ undergoes a rotation ๐ about the ๐ง-axis of its local
coordinate frame
โข Coordinates of any point ๐ท of the rigid body in local and global
coordinates frame are related by the following equation
๐2 = ๐จ๐ง,๐ ๐2๐บ๐ต
The vector ๐2 = ๐2 ๐2 ๐2๐๐บ and
๐2 = ๐ฅ2 ๐ฆ2 ๐ง2 ๐๐ต are the position
vectors of the point in local and
global frames respectively
๐จ๐ง,๐ is ๐-rotation matrix
๐จ๐ง,๐ =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 00 0 1
Robotics 2.20 Rotation Kinematics
0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IV. ROTATION ABOUT LOCAL CARTESIAN AXES
โข Similarly, rotation ๐ about the ๐ฆ-axis and rotation ๐ about the ๐ฅ-axis are
described by the ๐-rotation matrix ๐จ๐ฆ,๐ and the ๐-rotation matrix ๐จ๐ฅ,๐
respectively
๐จ๐ฆ,๐ =๐๐๐ ๐ 0 โ๐ ๐๐๐0 1 0
๐ ๐๐๐ 0 ๐๐๐ ๐
๐จ๐ฅ,๐ =1 0 00 ๐๐๐ ๐ ๐ ๐๐๐0 โ๐ ๐๐๐ ๐๐๐ ๐
Robotics 2.21 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IV. ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 12 (Local rotation, local position)
If a local coordinate frame ๐๐ฅ๐ฆ๐ง has been rotated 600 about the ๐ง-axis
and a point ๐ท in the global coordinate frame ๐๐๐๐ is at 4 3 2
Its coordinates in the local coordinate frame ๐๐ฅ๐ฆ๐ง are
๐ฅ๐ฆ๐ง
=๐๐๐ 60 ๐ ๐๐60 0โ๐ ๐๐60 ๐๐๐ 60 0
0 0 1
432
=4.60โ1.972.0
Robotics 2.22 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
Robotics 2.23 Rotation Kinematics
IV. ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 13 (Local rotation, global position)
If a local coordinate frame ๐๐ฅ๐ฆ๐ง has been rotated 600 about the ๐ง-axis
and a point ๐ท in the local coordinate frame ๐๐ฅ๐ฆ๐ง is at 4 3 2
Its position in the global coordinate frame ๐๐๐๐ is at
๐๐๐
=๐๐๐ 60 ๐ ๐๐60 0โ๐ ๐๐60 ๐๐๐ 60 0
0 0 1
๐ 432
=โ0.604.962.0
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IV. ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 14 (Successive local rotation, global position)
The arm shown in Fig 2.11 has two actuators. The first actuator rotates
the arm โ900 about ๐ฆ-axis and then the second actuator rotates the arm
900 about ๐ฅ-axis. If the end point ๐ท is at
๐๐ = 9.5 โ10.1 10.1 ๐๐ต
Then its position in the global coordinate frame is at
๐2 = ๐จ๐ฅ,90๐จ๐ฆ,โ90โ1
๐๐ = ๐จ๐ฆ,โ90โ1 ๐จ๐ฅ,90
โ1 ๐๐๐ต๐ต๐บ
= ๐จ๐ฆ,โ90๐ ๐จ๐ฅ,90
๐ ๐๐๐ต =
10.1โ10.19.5
Robotics 2.24 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
V. SUCCESSIVE ROTATION ABOUT LOCAL CARTESIAN AXES
โข The final global position of a point ๐ท in a rigid body ๐ฉ with position
vector ๐, after some rotation ๐จ1, ๐จ2, ๐จ3, โฏ , ๐จ๐ about the local axes, can
be found by
๐๐ต = ๐จ๐บ ๐๐บ๐ต
where
๐จ๐บ = ๐จ๐โฏ๐จ3๐จ2๐จ1๐ต
โข ๐จ๐บ๐ต is called the local rotation matrix and it maps the global
coordinates to their corresponding local coordinates
โข Rotation about the local coordinate axis is conceptually interesting
because in a sequence of rotations, each rotation is about one of the axes
of the local coordinate frame, which has been moved to its new global
position during the last rotation
Robotics 2.25 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
V. SUCCESSIVE ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 15 (Successive local rotation, local position)
A local coordinate frame ๐ฉ(๐๐ฅ๐ฆ๐ง) that initially is coincident with a
global coordinate frame ๐ฎ(๐๐๐๐) undergoes a rotation ๐ = 300 about
the ๐ง-axis, then ๐ = 300 about the ๐ฅ-axis, and then ๐ = 300 about the
๐ฆ-axis.
The local rotation matrix is
๐จ๐บ = ๐จ๐ฆ,30๐จ๐ฅ,30๐จ๐ง,30 =0.63 0.65 โ0.43โ0.43 0.75 0.500.65 โ0.125 0.75
๐ต
The global coordinate of ๐ท is ๐ = 5, ๐ = 30, ๐ = 10, so the coordinate
of ๐ท in the local frame is
๐ฅ๐ฆ๐ง
=0.63 0.65 โ0.43โ0.43 0.75 0.500.65 โ0.125 0.75
53010
=18.3525.357.0
Robotics 2.26 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
V. SUCCESSIVE ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 16 (Successive local rotation)
The rotation matrix for a body point ๐ท(๐ฅ, ๐ฆ, ๐ง) after rotation ๐จ๐ง,๐
followed by ๐จ๐ฅ,๐ and ๐จ๐ฆ,๐ is
๐จ๐บ๐ต = ๐จ๐ฆ,๐๐จ๐ฅ,๐๐จ๐ง,๐
=
๐๐๐๐ โ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ + ๐๐๐ ๐๐ ๐ โ๐๐๐ ๐โ๐๐๐ ๐ ๐๐๐๐ ๐ ๐
๐๐๐ ๐ + ๐ ๐๐๐๐ ๐ ๐ ๐๐ ๐ โ ๐๐๐ ๐๐๐ ๐๐๐๐
(2.97)
Robotics 2.27 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
V. SUCCESSIVE ROTATION ABOUT LOCAL CARTESIAN AXES
โข Example 17 (Twelve independent triple local rotations)
Any 2 independent orthogonal coordinate frames with a common origin
can be related by a sequence of three rotations about the local coordinate
axes, where no two successive rotations may be about the same axis. In
general, there are 12 different independent combinations of triple
rotation about local axes
1 โ ๐จ๐ฅ,๐๐จ๐ฆ,๐๐จ๐ง,๐ 5 โ ๐จ๐ฆ,๐๐จ๐ฅ,๐๐จ๐ง,๐ 9 โ ๐จ๐ง,๐๐จ๐ฅ,๐๐จ๐ง,๐
2 โ ๐จ๐ฆ,๐๐จ๐ง,๐๐จ๐ฅ,๐ 6 โ ๐จ๐ฅ,๐๐จ๐ง,๐๐จ๐ฆ,๐ 10 โ ๐จ๐ฅ,๐๐จ๐ง,๐๐จ๐ฅ,๐
3 โ ๐จ๐ง,๐๐จ๐ฅ,๐๐จ๐ฆ,๐ 7 โ ๐จ๐ฅ,๐๐จ๐ฆ,๐๐จ๐ฅ,๐ 11 โ ๐จ๐ฆ,๐๐จ๐ฅ,๐๐จ๐ฆ,๐
4 โ ๐จ๐ง,๐๐จ๐ฆ,๐๐จ๐ฅ,๐ 8 โ ๐จ๐ฆ,๐๐จ๐ง,๐๐จ๐ฆ,๐ 12 โ ๐จ๐ง,๐๐จ๐ฆ,๐๐จ๐ง,๐
Robotics 2.28 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข The rotation about the ๐ -axis of the global coordinate is called
precession, the rotation about the ๐ฅ-axis of the local coordinate is called
nutation, and the rotation about the ๐ง-axis of the local coordinate is
called spin
โข The precession-nutation-spin rotation angles are also called Euler angles
โข Euler angles rotation matrix has many application in rigid body
kinematics
โข To find Euler angles rotation matrix to go from the global frame
๐บ(๐๐๐๐) to the final body frame ๐ต(๐๐ฅ๐ฆ๐ง), we employ a body frame
๐ตโฒ(๐๐ฅโฒ๐ฆโฒ๐งโฒ) that shown in Fig 2.12 that before the first rotation coincides
with the global frame
Robotics 2.29 Rotation Kinematics
VI. EULER ANGLES
โข Let there be at first a rotation ๐ about the ๐งโฒ-axis. Because ๐-axis and
๐งโฒ-axis are coincident, we have
๐ = ๐จ๐บ ๐๐บ๐ตโฒ๐ตโฒ
๐จ๐บ = ๐จ๐ง,๐ =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 00 0 1
๐ตโฒ
โข Next we consider the ๐ฉโฒ(๐๐ฅโฒ๐ฆโฒ๐งโฒ) frame as a new fixed global frame and
introduce a new body frame ๐ฉโฒโฒ(๐๐ฅโฒโฒ๐ฆโฒโฒ๐งโฒโฒ). Before the second rotation,
the two frame coincide
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
Robotics 2.30 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข Then, we execute a ๐ rotation about ๐ฅโฒโฒ-axis as shown in Fig 2.13. The
transformation between ๐ฉโฒ(๐๐ฅโฒ๐ฆโฒ๐งโฒ) and ๐ฉโฒโฒ(๐๐ฅโฒโฒ๐ฆโฒโฒ๐งโฒโฒ) is
๐ = ๐จ๐ตโฒ ๐๐ตโฒ๐ตโฒโฒ๐ตโฒโฒ
๐จ๐ตโฒ = ๐จ๐ฅ,๐ =1 0 00 ๐๐๐ ๐ ๐ ๐๐๐0 โ๐ ๐๐๐ ๐๐๐ ๐
๐ตโฒโฒ
โข Finally, we consider the ๐ฉโฒโฒ(๐๐ฅโฒโฒ๐ฆโฒโฒ๐งโฒโฒ) frame as a new fixed global
frame and consider the final body frame ๐ฉ(๐๐ฅ๐ฆ๐ง) to coincide with ๐ฉโฒโฒ before the third rotation
Robotics 2.31 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข We now execute a ๐ rotation about the ๐งโฒโฒ-axis as shown in Fig 2.14. The
transformation between ๐ฉโฒโฒ(๐๐ฅโฒโฒ๐ฆโฒโฒ๐งโฒโฒ) and ๐ฉ(๐๐ฅ๐ฆ๐ง) is
๐ = ๐จ๐ตโฒโฒ ๐๐ตโฒโฒ๐ต๐ต
๐จ๐ตโฒโฒ = ๐จ๐ง,๐ =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 00 0 1
๐ต
By the rule of composition of rotation, the
transformation from ๐ฎ(๐๐๐๐) to ๐ฉ(๐๐ฅ๐ฆ๐ง) is
๐๐ต = ๐จ๐บ ๐๐บ๐ต
๐จ๐บ = ๐จ๐ง,๐๐จ๐ฅ,๐๐จ๐ง,๐๐ต
=
๐๐๐๐ โ ๐๐๐ ๐๐ ๐ ๐๐๐ ๐ + ๐๐๐๐๐ ๐ ๐ ๐๐ ๐โ๐๐๐ ๐ โ ๐๐๐๐๐ ๐ โ๐ ๐๐ ๐ + ๐๐๐๐๐๐ ๐ ๐๐๐
๐ ๐๐ ๐ โ๐๐๐ ๐ ๐๐
Robotics 2.32 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข Example 18 (Euler angle rotation matrix)
The Euler or precession-nutation-spin rotation matrix for ๐ = 79.150,
๐ = 41.410, and ๐ = โ40.70 would be found by substituting ๐, ๐ and ๐
in equation (2.106)
๐จ๐บ = ๐จ๐ง,โ40.7๐จ๐ฅ,41.41๐จ๐ง,79.15๐ต
=0.63 0.65 โ0.43โ0.43 0.75 0.500.65 โ0.125 0.75
Robotics 2.33 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข Example 19 (Euler angles of a local rotation matrix)
The local rotation matrix after rotation 300 about the ๐ง-axis, then rotation
300 about the ๐ฅ-axis, and then 300 about the ๐ฆ-axis is
๐จ๐บ = ๐จ๐ฆ,30๐จ๐ฅ,30๐จ๐ง,30๐ต
=0.63 0.65 โ0.43โ0.43 0.75 0.500.65 โ0.125 0.75
The Euler angles of the corresponding rotation matrix are
๐ = ๐๐๐ โ1 ๐33 = ๐๐๐ โ1 0.75 = 41.410
๐ = โ๐ก๐๐โ1๐31๐32
= โ๐ก๐๐โ10.65
โ0.125= 79.150
๐ = ๐ก๐๐โ1๐13๐23
= ๐ก๐๐โ1โ0.43
0.50= โ40.70
Robotics 2.34 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VI. EULER ANGLES
โข Example 20 (Relative rotation matrix of two bodies)
Consider a rigid body ๐ฉ1 with an orientation matrix ๐จ๐บ๐ต1 made by Euler
angles ๐ = 300, ๐ = โ450, ๐ = 600, and another rigid body ๐ฉ2 having
๐ = 100, ๐ = 250, ๐ = โ150, the individual rotation matrices are
๐จ๐บ = ๐จ๐ง,60๐จ๐ฅ,โ45๐จ๐ง,30 ๐ต1 ๐จ๐บ = ๐จ๐ง,โ15๐จ๐ฅ,25๐จ๐ง,10
๐ต2
=0.127 0.78 โ0.612โ0.927 โ0.127 โ0.354โ0.354 0.612 0.707
=0.992 โ0.0633 โ0.1090.103 0.907 0.4080.0734 โ0.416 0.906
The relative rotation matrix to map ๐ฉ2 to ๐ฉ1 may be found by
๐จ๐ต2 = ๐จ๐บ ๐จ๐ต2 =0.992 0.103 0.0734
โ0.0633 0.907 โ0.416โ0.109 0.408 0.906
๐บ๐ต1๐ต1
Robotics 2.35 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VII. LOCAL ROLL-PITCH-YAW ANGLES
โข Rotation about the ๐ฅ-axis of the local frame is called roll or bank, rotation
about ๐ฆ-axis of the local frame is called pitch or attitude, and rotation about
the ๐ง-axis of the local frame is called yaw, spin or heading
โข The local roll-pitch-yaw rotation matrix is
๐จ๐บ = ๐จ๐ง,๐๐จ๐ฆ,๐๐จ๐ฅ,๐ =๐ต
๐๐๐๐ ๐๐๐ ๐ + ๐ ๐๐๐๐ ๐ ๐ ๐๐ ๐ โ ๐๐๐ ๐๐๐โ๐๐๐ ๐ ๐๐๐๐ โ ๐ ๐๐ ๐๐ ๐ ๐๐๐ ๐ + ๐๐๐ ๐๐ ๐๐ ๐ โ๐๐๐ ๐ ๐๐๐๐
Note the difference between roll-pitch-yaw
and Euler angles, although we show both
utilizing ๐, ๐ and ๐
Robotics 2.36 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VII. LOCAL ROLL-PITCH-YAW ANGLES
โข Example 28 (Angular velocity and local roll-pitch-yaw rate)
Using the roll-pitch-yaw frequencies, the angular velocity of a body ๐ฉ
with respect to the global reference frame is
๐๐ต = ๐๐ฅ๐ + ๐๐ฆ๐ + ๐๐ง๐ ๐บ
= ๐ ๐ ๐ + ๐ ๐ ๐ + ๐ ๐ ๐
The roll unit vector ๐ ๐ = 1 0 0 ๐ transforms to the body frame after
rotation ๐ and then rotation ๐
๐ ๐ = ๐จ๐ง,๐๐จ๐ฆ,๐
100
=๐๐๐ ๐๐๐๐ ๐โ๐๐๐ ๐๐ ๐๐๐
๐ ๐๐๐
๐ต
The pitch unit vector ๐ ๐ = 0 1 0 ๐ transforms to the body frame
after rotation ๐
Robotics 2.37 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VII. LOCAL ROLL-PITCH-YAW ANGLES
๐ ๐ = ๐จ๐ง,๐
010
=๐ ๐๐๐๐๐๐ ๐0
๐ต
The yaw unit vector ๐ ๐ = 0 0 1 ๐ is already along the local ๐ง-axis
Hence, ๐๐ต๐บ can be expressed in body frame ๐๐ฅ๐ฆ๐ง as
๐๐ต =๐๐๐ ๐๐๐๐ ๐ ๐ ๐๐๐ 0โ๐๐๐ ๐๐ ๐๐๐ ๐๐๐ ๐ 0
๐ ๐๐๐ 0 1
๐
๐
๐ ๐บ๐ต
And therefore, ๐๐ต๐บ in global frame ๐๐๐๐ is
๐๐ต =1 0 ๐ ๐๐๐0 ๐๐๐ ๐ โ๐๐๐ ๐๐ ๐๐๐0 ๐ ๐๐๐ ๐๐๐ ๐๐๐๐ ๐
๐
๐
๐ ๐บ๐บ
Robotics 2.38 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VIII. LOCAL AXES VERSUS GLOBAL AXES ROTATION
โข The global rotation matrix ๐ธ๐ต๐บ is equal to the inverse of the local
rotation matrix ๐จ๐บ๐ต and vice versa
๐ธ๐ต๐บ = ๐จ๐บ
โ1๐ต , ๐จ๐บ๐ต = ๐ธ๐ต
โ1๐บ
where
๐ธ๐ต๐บ = ๐จ1
โ1๐จ2โ1๐จ3
โ1โฏ๐จ๐โ1
๐จ๐บ๐ต = ๐ธ1
โ1๐ธ2โ1๐ธ3
โ1โฏ๐ธ๐โ1
โข Also, pre-multiplication of the global rotation matrix is equal to post-
multiplication of the local rotation matrix
Robotics 2.39 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VIII. LOCAL AXES VERSUS GLOBAL AXES ROTATION
โข Example 29 (Global position and post-multiplication of rotation matrix)
The local position of a point ๐ท after rotation is at ๐ = 1 2 3 ๐๐ต . If
the local rotation matrix to transform ๐๐บ to ๐๐ต is given as
๐จ๐ง,๐ =๐๐๐ ๐ ๐ ๐๐๐ 0โ๐ ๐๐๐ ๐๐๐ ๐ 00 0 1
=๐๐๐ 30 ๐ ๐๐30 0โ๐ ๐๐30 ๐๐๐ 30 0
0 0 1
๐ต
Then we may find the global position vector ๐๐บ by post-multiplication
๐จ๐ง,๐๐ต by the local position vector ๐๐๐ต
๐๐ =๐บ ๐๐ ๐จ๐ง,๐๐ต๐ต = 1 2 3
๐๐๐ 30 ๐ ๐๐30 0โ๐ ๐๐30 ๐๐๐ 30 0
0 0 1
= โ0.13 2.23 3.0
Robotics 2.40 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
VIII. LOCAL AXES VERSUS GLOBAL AXES ROTATION
โข Example 29 (Global position and post-multiplication of rotation matrix)
Instead of pre-multiplication of ๐จ๐ง,๐โ1๐ต by ๐๐ต
๐๐บ = ๐จ๐ง,๐โ1๐ต ๐๐ต
=๐๐๐ 30 โ๐ ๐๐30 0๐ ๐๐30 ๐๐๐ 30 00 0 1
123
=โ0.132.233
Robotics 2.41 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
โข Consider a general situation in which two coordinate frames, ๐ฎ(๐๐๐๐) and ๐ฉ(๐๐ฅ๐ฆ๐ง) with a common origin ๐ถ, are employed to express the
components of a vector ๐. There is always a transformation matrix ๐น๐ต๐บ
to map the components of ๐ from the reference frame ๐ฉ(๐๐ฅ๐ฆ๐ง) to the
other reference frame ๐ฎ(๐๐๐๐)
๐๐บ = ๐น๐ต ๐๐ต๐บ
โข In addition, the inverse map, ๐๐ต = ๐น๐ตโ1 ๐๐บ๐บ , can be done by ๐น๐บ
๐ต
๐๐ต = ๐น๐บ ๐๐บ๐ต
where
๐น๐ต๐บ = ๐น๐บ
๐ต = 1
and
๐น๐บ๐ต = ๐น๐ต
โ1 = ๐น๐ต๐๐บ๐บ
Robotics 2.42 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
โข Example 30 (Elements of transformation matrix)
The position vector ๐ of a point ๐ท may be expressed in terms of its
components with respect to either ๐ฎ(๐๐๐๐) or ๐ฉ(๐๐ฅ๐ฆ๐ง) frames. If
๐ = 100๐ฐ โ 50๐ฑ + 150๐ฒ ๐บ โ Find components of ๐ in the ๐๐ฅ๐ฆ๐ง frame?
The ๐ฅ-axis lie in the ๐๐ plane at 400 from
the ๐-axis, and the angle between ๐ฆ and ๐ is
600. Therefore
๐น๐บ =
๐๐๐ 40 0 ๐ ๐๐40๐ โ ๐ฐ ๐๐๐ 60 ๐ โ ๐ฒ
๐ โ ๐ฐ ๐ โ ๐ฑ ๐ โ ๐ฒ
๐ต
=
0.766 0 0.643๐ โ ๐ฐ 0.5 ๐ โ ๐ฒ
๐ โ ๐ฐ ๐ โ ๐ฑ ๐ โ ๐ฒ
Robotics 2.43 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
And by using ๐น๐บ ๐น๐ต = ๐น๐บ ๐น๐บ๐ = ๐ฐ๐ต๐ต๐บ๐ต
0.766 0 0.643๐21 0.5 ๐23๐31 ๐32 ๐33
0.766 ๐21 ๐310 0.5 ๐32
0.643 ๐23 ๐33
=1 0 00 1 00 0 1
We obtain a set of equations to find the missing elements. Solving these
equations provides the following transformation matrix
๐น๐บ =0.766 0 0.6430.557 0.5 โ0.663โ0.322 0.866 0.383
๐ต
Then we can find the components of ๐๐ต
๐๐ต = ๐น๐บ ๐ =0.766 0 0.6430.557 0.5 โ0.663โ0.322 0.866 0.383
100โ50150
=173.05โ68.75โ18.05
๐บ๐ต
Robotics 2.44 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
โข Example 31 (Global position, using ๐๐ต and ๐น๐บ๐ต )
The position vector ๐ of a point ๐ท in local frame is
๐ = 100๐ โ 50๐ + 150๐ ๐ต
The transformation matrix to map ๐๐บ to ๐๐ต
๐น๐บ =0.766 0 0.6430.557 0.5 โ0.663โ0.322 0.866 0.383
๐ต
Then the components of ๐๐บ in ๐ฎ(๐๐๐๐) would be
๐๐บ = ๐น๐ต ๐ = ๐น๐บ๐ ๐๐ต๐ต๐ต๐บ
=0.766 0.557 โ0.3220 0.5 0.866
0.643 โ0.663 0.383
100โ50150
=0.45104.9154.9
Robotics 2.45 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
โข Example 32 (Two points transformation matrix)
The global position vector of two points, ๐ท1 and ๐ท2, of a rigid body ๐ฉ are
๐๐1 =1.0771.3652.666
๐บ ๐๐2 =โ0.4732.239โ0.959
๐บ
The origin of the body ๐ฉ(๐๐ฅ๐ฆ๐ง) is fixed on the origin of ๐ฎ(๐๐๐๐), and
the points ๐ท1 and ๐ท2 are lying on the local ๐ฅ-axis and ๐ฆ-axis respectively.
To find ๐น๐ต๐บ , we use the local unit vectors ๐ ๐บ and ๐ ๐บ
๐ =๐๐1
๐บ
๐๐1๐บ
=0.3380.4290.838
๐บ ๐ =๐๐2
๐บ
๐๐2๐บ
=โ0.1910.902โ0.387
๐บ
Robotics 2.46 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
IX. GENERAL TRANSFORMATION
โข Example 32 (Two points transformation matrix)
We can calculate ๐ ๐บ as follow
๐ = ๐ ร ๐ ๐บ๐บ๐บ =โ0.922โ0.0290.387
Hence, the transformation matrix using the coordinates of two points
๐๐1๐บ and ๐๐2
๐บ would be
๐น๐ต = ๐ ๐บ ๐ ๐บ ๐ ๐บ๐บ
=0.338 โ0.191 โ0.9220.429 0.902 โ0.0290.838 โ0.387 0.387
Robotics 2.47 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
X. ACTIVE AND PASSIVE TRANSFORMATION
โข Rotation of a local frame when the position vector ๐๐บ of a point ๐ท is
fixed in global frame and does not rotate with the local frame, is called
passive transformation
โข Rotation of a local frame when the position vector ๐๐ต of a point ๐ท is
fixed in the local frame and rotate with the local frame, is called active
transformation
โข The passive and active transformation are mathematically equivalent
โข The rotation matrix for a rotated frame and rotated vector (active
transformation) is the same as the rotation matrix for a rotated frame and
fixed vector (passive transformation)
Robotics 2.48 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
X. ACTIVE AND PASSIVE TRANSFORMATION
โข Example 39 (Active and passive rotation about ๐-axis)
Consider a local and global frames ๐ฉ and ๐ฎ that are coincident. A body
point ๐ท is at ๐๐ต
๐๐ต =121
A rotation of 900 about ๐-axis will move the point to ๐๐บ
๐ = ๐น๐,90 ๐๐ต๐บ
=
1 0 0
0 ๐๐๐ ๐
2โ๐ ๐๐
๐
2
0 ๐ ๐๐๐
2๐๐๐
๐
2
121
=1โ12
Robotics 2.49 Rotation Kinematics
HCM City Univ. of Technology, Faculty of Mechanical Engineering Phung Tri Cong
X. ACTIVE AND PASSIVE TRANSFORMATION
โข Example 39 (Active and passive rotation about ๐-axis)
Now assume that ๐ท is fixed in ๐ฎ. When ๐ฉ rotates 900 about ๐-axis, the
coordinates of ๐ท in the local frame will change such that
๐๐ต = ๐น๐,โ90 ๐๐บ
=
1 0 0
0 ๐๐๐ โ๐
2โ๐ ๐๐
โ๐
2
0 ๐ ๐๐โ๐
2๐๐๐
โ๐
2
121
=11โ2
Robotics 2.50 Rotation Kinematics