Các hệ tọa độ trong robot

Kỹ thuật robot (robotics) là kỹ thuật liên ngành bao gồm kỹ thuật cơ khí, kỹ thuật điện điện tử, kỹ thuật máy tính, nội dung của kỹ thuật robot bao gồm nghiên cứu, thiết kế, chế tạo, vận hành và áp dụng robot. Kể từ giữa thế kỷ 20 cho đến nay, robot ngày càng đóng vai trò quan trọng trong mọi hoạt động của con người như sản xuất, dịch vụ, y tế, nghiên cứu khoa học, quốc phòng…Chương này trình bày về lịch sử robot, cấu tạo và phân loại robot, các lĩnh vực áp dụng của robot, cung cấp các khái niệm cơ bản sẽ được nghiên cứu chi tiết trong các chương sau

describe the position and

orientation of this frame

with respect to some

reference coordinate

system

 C.B Pham 3-2

Description of a position

Once a coordinate system is established, any point in the universe can be located with a 3  1 position vector

 C.B Pham 3-3

and frames

 C.B Pham 3-4

Description of an orientation

-The complete location of the hand is still not specified until its orientation is also given

- A coordinate system (B) has

been attached to the body in a

known way A description of {B}

relative to (A) now suffices to give

the orientation of the body

 C.B Pham 3-5

Note: Rotation matrix are simply the projections of a frame

onto another frame

 C.B Pham 3-6

Example:

 C.B Pham 3-7

Note: In a rotation matrix, rows / columns are orthogornal unit vectors

and frames

 C.B Pham 3-9

and frames

 C.B Pham 3-10

Description of a frame

A frame is a coordinate system where, in addition to the orientation, we give a position vector which locates its origin relative to some other embedding frame

 C.B Pham 3-11

- It is concerned how to express the same quantity in terms

of various reference coordinate systems

• Mappings involving translated frames

 C.B Pham 3-12

• Mappings involving rotated frames

 C.B Pham 3-13

Example: Determine AP Given that a frame {B} is rotated

relative to frame {A} about 𝑍 by 30 degrees, with:

Rotation matrix

 C.B Pham 3-14

• Mappings involving general frames

01

P P

 C.B Pham 3-16

Example: A frame {B} is rotated relative to frame {A} about

𝑍 by 300, translated 10 units in 𝑋 𝐴, 5 units in 𝑌 𝐴 Determine

AP, given BP = [3.0 7.0 0.0]T

 C.B Pham 3-17

transformation

The same mathematical forms used to map points between frames can also be interpreted as operators that translate points, rotate vectors, or do both

• Translational operators

A translation moves a

point in space a finite

distance along a given

vector direction

 C.B Pham 3-18

To write this translation operation as a matrix operator:

The DQ operator may be thought

of as a homogeneous transform

of a special simple form

where qx, qy, and qz are the components of the translation

vector 𝑄

where q is the signed magnitude of the translation along the vector direction 𝑄

 C.B Pham 3-19

• Rotational operators

Another interpretation of a rotation matrix is as a rotational operator that operates on a vector AP1 and changes that vector to a new vector, AP2, by means of a rotation, R

transformation

In this notation, "RK()" is a rotational operator that performs a rotation about the axis direction 𝐾 by  degrees This operator can be written as a homogeneous transform whose position-vector part is zero

 C.B Pham 3-20

 C.B Pham 3-21

Example: Given AP1 = [0.0 2.0 0.0]T Compute AP2 obtained

by rotating AP1 about 𝑍 by 30 degrees

transformation

0 0

BORG

A A

As a general tool to represent

frames, a homogeneous transform

has been introduced That is a 4 x 4

matrix containing orientation and

position information

There are three interpretations of

this homogeneous transform:

BORG

A CORG

B A B

B C

A B A

C

P P

R R

R T

BORG

A A

B A

B

P

R T

We have:

0 0

BORG

A T A B

T A B B

A

P R

R T

Example: Frame {B} is

rotated relative to frame

{A} about 𝑍 by 300, and

translated 4 units in 𝑋 𝐴

and 3 units in 𝑌 𝐴

 C.B Pham 3-27

Determine:

Solution:

 C.B Pham 3-28

Transform equations

 C.B Pham 3-30

Example: Consider two rotations, one about X by 300, and one about Z by 300

 C.B Pham 3-31

Note:

 C.B Pham 3-32

• Rotate {B} first about by an angle  (roll)

• Then, rotate {B} about by an angle  (pitch)

• Finally, rotate {B} about an angle  (yaw)

 C.B Pham 3-33

 C.B Pham 3-34

• Rotate {B} first about by an angle 

• Then, rotate {B} about by an angle 

• Finally, rotate {B} about by an angle Xˆ B

B

Zˆ

B

YˆStart with the frame {B} coincident with a known reference frame {A}

 C.B Pham 3-35

 C.B Pham 3-36

If

If

 C.B Pham 3-37

Start with the frame

{B} coincident with a

known frame {A}; then

rotate {B} about the

 C.B Pham 3-39

Example: A frame {B) is described as initially coincident with {A} We then rotate {B} about the vector (passing through the origin) by an amount 

= 30 degrees Give the frame description of {B}

T A

]0.0 0.707

7070

0[

ˆ 

K

 C.B Pham 3-40

A frame {B} is described as

initially coincident with {A)

Then {B} is rotated about the

]0.0 0.707

7070

0[

ˆ 

K

Give the frame

description of {B}

 C.B Pham 3-41

Solution: define two new frames {A’} and {B’} so that their origins are at AP = [1.0 2.0 3.0]T and

 C.B Pham 3-42

We have

 C.B Pham 3-43

The term line vector refers to a vector that is dependent on its line of action, along with direction and magnitude, for causing its effects

 C.B Pham 3-44

A free vector refers to a vector that may be positioned anywhere in space without loss or change of meaning, provided that magnitude and direction are preserved

𝑉

𝐴 = 𝑅𝐵𝐴 𝐵𝑉