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Basics of Robotics
“MATLAB 때려잡기”
www.matlabinuse.com/Mastering_MATLAB/
From Actuator to Work space(Cartesian Space)
 2 DOF SCARA Robot
Kinematics
Kinematics consider only geometric relationship !
 2 DOF SCARA Robot
Kinematics
Kinematics
 2 DOF SCARA Robot
Work Space
Kinematics
 2 DOF SCARA Robot
𝑒 𝑥
′
𝑒 𝑦
′
𝑝0
0 0 1
𝑝0
Kinematics
 2 DOF SCARA Robot
Kinematics
 2 DOF SCARA Robot
① : Reference frame
② : Arm Local frame
③ : End Local frame
Kinematics
 2 DOF SCARA Robot
For a series of arms
Motion of nth robot arm can be described with Reference Frame
Kinematics
 2 DOF SCARA Robot
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
Calculate Joint angle for a given coordinate values of End effector
Inverse Kinematics
 2 DOF SCARA Robot
Inverse Kinematics
 2 DOF SCARA Robot
Inverse Kinematics
 2 DOF SCARA Robot
This shows more than one Joint angle sets, which satisfy the given
coord. Values of End effector
Inverse Kinematics
 2 DOF SCARA Robot
Inverse Kinematics
 2 DOF SCARA Robot
Velocity of End effector
Inverse Kinematics
 2 DOF SCARA Robot
Velocity of End effector
Inverse Kinematics
 2 DOF SCARA Robot
Inverse Kinematics
 6 DOF Robot
Position Jacobian : get from Homogeneous Transformation Matrix
Orientation Jacobian : get a last row of Rotation matrix
Inverse Kinematics
 6 DOF Robot
Inverse Kinematics
 Generalized IK using Jacobian
 Piecewise Linearization
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
Inverse Kinematics
 Generalized IK using Jacobian
 Piecewise Linearization
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
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
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 ?
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
From Actuator to Work space(Cartesian Space)
Dynamics
Dynamics
M (mass + inertia), V (centrifugal + Coriolis),
G (gravity)
Dynamics
 M (mass + inertia)
Dynamics
 V (centrifugal + Coriolis)
Dynamics
 G (gravity)
Dynamics
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
Inverse Dynamics Control
• In real world, modeling errors are always present
• Small error due to modeling error can be compensated
Path and Trajectory Planning
Trajectory considers not only the path from A to B
but also the time, velocity, etc
Path and Trajectory Planning
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
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
Path and Trajectory Planning
S=𝑃𝐴 ► 𝐴 = 𝑃−1 𝑆

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Basics of Robotics

  • 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
  • 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)
  • 31. M (mass + inertia), V (centrifugal + Coriolis), G (gravity) Dynamics
  • 32.  M (mass + inertia) Dynamics
  • 33.  V (centrifugal + Coriolis) 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
  • 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