This document discusses inverse kinematics, which is finding the joint parameters of a robot given the desired end effector position and orientation. There are three main solutions to inverse kinematics problems: geometric, algebraic, and numerical. Geometric methods use geometry to directly calculate joint angles. Algebraic methods use transformation matrices from forward kinematics. Numerical methods use iterative techniques like pseudo-inverse or Jacobian transpose to approximate solutions.

1. Inverse
Kinematics
Introduction to Robotics
3 May 2013
120CPG04
Julie Kim

2. Inverse Kinematics
• To find the joint parameters, given the end
effector position and orientation.
• Multiple Solution
http://www.cescg.org/CESCG-2000/RFilkorn/Image9.jpg
Base
End Effector
Target
Base
End Effector
Θ?
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3. Inverse Kinematics
http://projects.csail.mit.edu/pr2/wiki/images/c/c9/Pick.png
http://freesdk.crydev.net/download/attachments/1966279/character_editor_footstep_entry.png?version=1&modificationDate=1330049372000
http://gamma.cs.unc.edu/CA/sitterson1000_wide.png
Interactive
manipulation
Animation
control
Collision
avoidance.
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4. IK Solutions
• Geometric solution
• Algebraic solution
• Numerical Solution
– Pseudo-inverse using Singular Value
Decomposition
– Jacobian Transpose
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5. Geometric Solution
• Using geometric
knowledge to decide
values of joint
parameters
http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%282011%29_files/
ik_5_degrees.jpg
*Obtain angles A1, A2, A3
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6. Geometric Solution
http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%282011%29_files/
ik_3_degrees.jpghttp://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%
282011%29_files/ik_2_degrees.jpg
When L12 > L1 + L2,
There is no solution
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7. Geometric Solution
http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%282011%29_files/
ik_base_rotate.jpg
http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%282011%29_files/
ik_3_degrees.jpg
5 DOF 3 DOF
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8. Geometric Solution
http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entries/2011/2/12_Robot_Arm_with_Inverse_Kinematics_%282011%29_files/
ik_3_degrees.jpg
Java Applet Implementation
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9. Algebraic Solution
• Simple Idea from
Forward Kinematics
*Obtain angles A1, A2, A3
Transformation
composition
matrix
by Forward Kinematics
Transformation
composition
matrix
with Target position
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10. CONSTRAINTS…
http://4.bp.blogspot.com/-K-GEwtABXa8/UQyK6uaJLRI/AAAAAAAADx0/r-AeKGUurNQ/s1600/c435173_m.jpg
Geometric
Algebraic For small kinematics… max 6 DOF
Only for a special set of kinematics
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11. Numerical Solution
Iterative
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12. Numerical Solution 1
Pseudo-Inverse
Singular Value Decomposition
http://www.cs.carleton.edu/cs_comps/0607/recommend/recommender/images/svd1.png
Singular value
Diagonal Matrix
Orthogonal
Matrix
Orthogonal
Matrix
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13. Numerical Solution 1
• Pseudo Inverse Matrix
• 𝑆+
– Zero entries in diagonal position are replaced with its
corresponding
– To be invertible with approximation
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14. http://dimg.donga.com/wps/DKBNEWS/IMAGE/2013/04/26/54722372.2.jpg
Expensive calculation SVD…
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15. Numerical Solution 2
• Jacobian Transpose
http://postfiles4.naver.net/20111209_99/porche72_1323412564964NBxMp_PNG/%A4%B7_(3).png?type=w3
F1
F2
F1+F2=0
Virtual Work Principal
Static Equilibrium
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16. Numerical Solution 2
Virtual Work Principal
Static Equilibrium
What I want to obtain 16
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17. http://www.clker.com/cliparts/n/I/L/N/I/P/question-and-answer-md.pn
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18. http://24.media.tumblr.com/8PysozIOn9crkrcz42KmAjkf_400.jpg
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19. References
• Inverse Kinematics
– http://www.eng.utah.edu/~cs5310/chapter5.pdf
– http://www.cescg.org/CESCG-2000/RFilkorn/
• Gradient Following
– http://freespace.virgin.net/hugo.elias/models/m_ik2.htm
• Robot Arm with Inverse Kinematics
– http://www.maquinapensante.com/robots/Maquina_Pensante/Blog/Entri
es/2011/2/12_Robot_Arm_with_Inverse_Kinematics_(2011).html
– http://www.youtube.com/watch?v=wGlOD6Saa4s
• Numerical Methods for Inverse Kinematics
– http://njoubert.com/teaching/cs184_fa08/section/sec13inversekinematic
sSOL.pdf
– http://www.cs.berkeley.edu/~sequin/CS184/TOPICS/Kinematics/Kin_f1.
html
– http://www.cescg.org/CESCG-2002/LBarinka/paper.pdf
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