1. Kinematics 제어시스템 이론 및 실습 2014.05.20 조현우
※참고: John J. Graig, “Introduction to robotics: Mechanics and control,” 3rd Ed. Persion.

2. A position in 3-dimensional Space
Orthonormal basis: 1 2
Cf. Orthogonal basis: Only Satisfies 1
Vector space: Set of all linear combinations of the basis
The basis span the vector space

3. A position in 3-dimensional Cartesian coordinate
Basis of Cartesian coordinate:
A position in frame A: an element of the vector space

4. Relationship under coordinate rotation
Coordinate frame A
Coordinate frame B
Rotational relationship
Their origins coincide with each other

5. Relationship under coordinate rotation
Coordinate frame B

6. Relationship under coordinate rotation
Coordinate frame A
※ The Position is the same!!!
Only the reference is changed to frame A
Position w.r.t frame A
Position w.r.t frame B

7. Relationship under coordinate rotation
The relationship between the coordinates of
and
Linear Transformation
Rotation matrix
3 x 3

8. Relationship under translation
Coordinate frame A
Coordinate frame B
translation

9. Relationship under translation

10. Combination
Step 1: Express
w.r.t frame A By
Step 2: Translate

11. Coordinate Transformation Matrix
The same position

12. Coordinate Transformation Matrix

13. Coordinate Transformation Matrix
Coordinate w.r.t. B  Coordinate w.r.t. A
Coordinate of
w.r.t frame A
Coordinate of
w.r.t frame A
Coordinate of
w.r.t frame A

14. Coordinate Transformation Matrix
<Example>
(0,1)
Basis 2
Basis 1
Frame B
(1,0)
Frame A

15. Coordinate Transformation Matrix
<Example>
(0,1)
Basis 2
Basis 1
Frame B
(1,0)
Frame A

16. Coordinate Transformation from frame B to frame A
Augmented Matrix Form

17. Coordinate Transformation from frame B to frame A
describe frame B relative to frame A
maps 
Multiplication
Find position C in frame C w.r.t frame A: Frame C  B  A
Inversion
We know
Find
(a)
(b)

18. Euler Angle Roll Rotation about Rotation about Pitch Rotation about
Yaw
Rotation about

19. Manipulator Kinematics
-Kinematics: The science of motion that threats the subject without regard to the force that cause the motion
-Dynamics: The relationships between the motions and the forces (or torques)
-Forward Kinematics: The static geometrical problem of computing the position and orientation of the end-effector (tool frame) relative to the base frame, given a set of joint variables (joint angles or link offset)

20. Denavit-Hartenberg notation
-Degree of freedom: The number of independent parameters requires to specify the position & orientation  The number of joints of manipulator
-Link: a rigid body that defines the relationship between neighboring joint axes of manipulator
-Joint axis i: a line in space about which link i rotates relative to link i-1
-Link Length: the distance from to measured along
-Link Twist: the angle from to measured about
-Link Offset: the distance from to measured along
-Joint Angle: the angle from to measured about
Link i-1
Link i

21. Denavit-Hartenberg notation
(link length) is positive value
(link twist, link offset, joint angle) are signed quantities
is variable if joint i is prismatic
is variable if joint i is revolute
Prismatic joint
Revolute joint

22. Kinematic Example
All revolute joint manipulator i 1 L2 L1 2 3 L2 L1

23. Kinematic Example L1 L2 i 1 L1 2 90 3 L2

24. Manipulator Kinematics
1. Construct the transform defining frame {i} relative to the frame {i-1}
Cf)
{i-1}
Rotate around
{R}
Translate along
{Q}
Intermediate frames: {R}, {Q}, {P}
Rotate around
{P}
Translate along
{i}
2. Concatenating link transformations

25. Manipulator Kinematics

26. Manipulator Kinematics
Base frame {B}
Station frame {S}
Wrist frame {W}
Tool frame {T}
Goal frame {G}
Other names
Frame {0}
Frame {N}
Description
 Nonmoving part of the robot
 Base of the manipulator
 Task frame
 World frame
 Universe frame
 Last link of manipulator
 End of any tool origin between the finger tips
 {T} should coincide with {G} at the end of motion