Robot에 대한 기초적인 내용으로 대부분의 자료는 정리가 잘 된 "MATLAB 때려잡기" 블로그를 참조하고 제가 부분적으로 이론적인 내용은 추가 원문은 http://www.matlabinuse.com/Mastering_MATLAB/

1. Basics of Robotics
“MATLAB 때려잡기”
www.matlabinuse.com/Mastering_MATLAB/

2. From Actuator to Work space(Cartesian Space)

3.  2 DOF SCARA Robot
Kinematics

4. Kinematics consider only geometric relationship !
 2 DOF SCARA Robot
Kinematics

5. Kinematics
 2 DOF SCARA Robot
Work Space

6. Kinematics
 2 DOF SCARA Robot
𝑒 𝑥
′
𝑒 𝑦
′
𝑝0
0 0 1
𝑝0

7. Kinematics
 2 DOF SCARA Robot

8. Kinematics
 2 DOF SCARA Robot

9. ① : Reference frame
② : Arm Local frame
③ : End Local frame
Kinematics
 2 DOF SCARA Robot

10. For a series of arms
Motion of nth robot arm can be described with Reference Frame
Kinematics
 2 DOF SCARA Robot

11. 𝜃1=45°
𝜃2=45°
𝜃3 = −45°
𝑙1 = 1
𝑙2 = 1
𝑙3 = 1
𝑒 𝑥
′
𝑒 𝑥
′′
1
0
𝑇 =
𝑐𝜃1 −𝑠𝜃1 0
𝑠𝜃1 𝑐𝜃1 0
0 0 1
1 0 1
0 1 0
0 0 1
=
𝑐𝜃1 −𝑠𝜃1 𝑐𝜃1
𝑠𝜃1 𝑐𝜃1 𝑠𝜃1
0 0 1
2
0
𝑇 = 1
0
𝑇
𝑐𝜃2 −𝑠𝜃2 0
𝑠𝜃2 𝑐𝜃2 0
0 0 1
1 0 1
0 1 0
0 0 1
3
0
𝑇 = 2
0
𝑇
𝑐𝜃3 −𝑠𝜃3 0
𝑠𝜃3 𝑐𝜃3 0
0 0 1
1 0 1
0 1 0
0 0 1
Kinematics
 2 DOF SCARA Robot

12. Calculate Joint angle for a given coordinate values of End effector
Inverse Kinematics
 2 DOF SCARA Robot

13. Inverse Kinematics
 2 DOF SCARA Robot

14. Inverse Kinematics
 2 DOF SCARA Robot

15. This shows more than one Joint angle sets, which satisfy the given
coord. Values of End effector
Inverse Kinematics
 2 DOF SCARA Robot

16. Inverse Kinematics
 2 DOF SCARA Robot

17. Velocity of End effector
Inverse Kinematics
 2 DOF SCARA Robot

18. Velocity of End effector
Inverse Kinematics
 2 DOF SCARA Robot

19. Inverse Kinematics
 6 DOF Robot

20. Position Jacobian : get from Homogeneous Transformation Matrix
Orientation Jacobian : get a last row of Rotation matrix
Inverse Kinematics
 6 DOF Robot

21. Inverse Kinematics
 Generalized IK using Jacobian
 Piecewise Linearization

22. 𝑥1, 𝑦1
𝑥2, 𝑦2
Inverse Kinematics
 Generalized IK using Jacobian
 Piecewise Linearization
- Not only the velocity of joint angles and end effector, but
also the position of them can be estimated using Jacobian
- Jacobian is effective under the condition that angular and
positional motions are small  Piecewise linearization

23. Inverse Kinematics
 Generalized IK using Jacobian
 Piecewise Linearization

24. Importance of Jacobian
 Kinematics (mapping of changes from joint to task space)
• Inverse kinematics control
• Resolve redundancy problems
• Express contact constraints
 Statics (and later also dynamics)
• Principle of virtual work
 Variations in work must cancel for all virtual displacement
 Internal forces of ideal joint don’t contribute

25. Singularities
A singularity is a joint-space configuration such that is column-
rank deficient
• the Jacobian becomes badly conditioned
• small desired velocities produce high joint velocities
Use a damped version of the Moore-Penrose pseudo inverse
Minimize norm of joint angular velocity

26. Redundancy
A kinematic structure is redundant if the dimension of the task-space is
smaller than the dimension of the joint-space
E.g. the human arm has 7DoF (three in the shoulder, one in the
elbow, and three in the wrist)
Many solutions per problem.
Which one to pick ?

27. Min Norm Null space : internal motion,
not effective to the motion of
end effector
Arbitrary
x
Span
Min Norm Null space
Redundancy
r : rank of JE

28. From Actuator to Work space(Cartesian Space)

29. Dynamics

30. Dynamics

31. M (mass + inertia), V (centrifugal + Coriolis),
G (gravity)
Dynamics

32.  M (mass + inertia)
Dynamics

33.  V (centrifugal + Coriolis)
Dynamics

34.  G (gravity)
Dynamics

35. Inverse Dynamics Control
• Model based Torque estimation
• In case of no modeling errors,
• the desired dynamics can be perfectly prescribed
Model
Can achieve great performance…
But requires accurate modeling
𝜏

36. Inverse Dynamics Control
• In real world, modeling errors are always present
• Small error due to modeling error can be compensated

37. Path and Trajectory Planning
Trajectory considers not only the path from A to B
but also the time, velocity, etc

38. Path and Trajectory Planning

39. Path and Trajectory Planning
 After generation of trajectory of end effector at work space(Cartesian space),
the trajectories of joints can be calculated using inverse kinematics
 3rd order polynomial is sufficient if position and velocity are considered
 5th order polynomial is needed if acceleration are included

40. Path and Trajectory Planning
- 5th or polynomial  6 unknowns  6 equations are needed
- Can get a unique solution for a given 6 initial and terminal conditions

41. Path and Trajectory Planning
S=𝑃𝐴 ► 𝐴 = 𝑃−1 𝑆