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Animation CS 551 / 651

Animation CS 551 / 651. Kinematics Lecture 09. Sarcos Humanoid. Kinematics. The study of object movements irrespective of their speed or style of movement. Degrees of Freedom (DOFs). The variables that affect an object’s orientation How many degrees of freedom when flying?.

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Animation CS 551 / 651

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  1. AnimationCS 551 / 651 Kinematics Lecture 09 Sarcos Humanoid

  2. Kinematics • The study of object movements irrespective of their speed or style of movement

  3. Degrees of Freedom(DOFs) • The variables that affect an object’s orientation • How many degrees of freedom when flying? • So the kinematics of this airplane permit movement anywhere in three dimensions • Six • x, y, and z positions • roll, pitch, and yaw

  4. Degrees of Freedom • How about this robot arm? • Six again • 2-base, 1-shoulder, 1-elbow, 2-wrist

  5. Configuration Space • The set of all possible positions (defined by kinematics) an object can attain

  6. Work Space vs. Configuration Space • Work space • The space in which the object exists • Dimensionality • R3 for most things, R2 for planar arms • Configuration space • The space that defines the possible object configurations • Degrees of Freedom • The number of parameters that necessary and sufficient to define position in configuration

  7. More examples • A point on a plane • A point in space • A point moving on a line in space

  8. A matter of control • If your animation adds energy at a particular DOF, that is a controlled DOF Low DOF, high control High DOF, no control

  9. Hierarchical Kinematic Modeling • A family of parent-child spatial relationships are functionally defined • Moon/Earth/Sun movements • Articulations of a humanoid • Limb connectivity is built into model (joints) and animation is easier

  10. Robot Parts/Terms • Links • End effector • Frame • Revolute Joint • Prismatic Joint

  11. More Complex Joints • 3 DOF joints • Gimbal • Spherical • 2 DOF joints • Universal

  12. Hierarchy Representation • Model bodies (links) as nodes of a tree • All body frames are local (relative to parent) • Transformations affecting root affect all children • Transformations affecting any node affect all its children ROOT

  13. Forward vs. Inverse Kinematics • Forward Kinematics • Compute configuration (pose) given individual DOF values • Good for simulation • Inverse Kinematics • Compute individual DOF values that result in specified end effector position • Good for control

  14. Forward Kinematics • Traverse kinematic tree and propagate transformations downward • Use stack • Compose parent transformation with child’s • Pop stack when leaf is reached

  15. Denavit-Hartenberg (DH) Notation • A kinematic representation (convention) inherited from robotics

  16. Z-axis aligned with joint

  17. X-axis aligned with outgoing limb

  18. Y-axis is orthogonal

  19. Joints are numbered to represent hierarchy Ui-1 is parent of Ui

  20. Parameter ai-1 is outgoinglimb length of joint Ui-1

  21. Joint angle, qi, is rotation of xi-1 about zi-1 relative to xi

  22. Link twist, ai-1, is the rotation of ith z-axis about xi-1-axis relative to z-axis of i-1th frame

  23. Link offset, di-1, specifies the distance along the zi-1-axis (rotated by ai-1) of the ith frame from the i-1th x-axis to the ith x-axis

  24. Inverse Kinematics (IK) • Given end effector position, compute required joint angles • In simple case, analytic solution exists • Use trig, geometry, and algebra to solve

  25. ? End Effector Base What is Inverse Kinematics? • Forward Kinematics

  26. End Effector Base What is Inverse Kinematics? • Inverse Kinematics

  27. ? End Effector Base What does look like?

  28. Infinite number of solutions ! Solution to • Our example Number of equations : 2 Unknown variables : 3

  29. Our example • System DOF = 3 • End Effector DOF = 2 Redundancy • System DOF > End Effector DOF

  30. Analytic solution of 2-link inverse kinematics x2 y2 (x,y) O2 2 y0 y1 a2 2 x1  a1 O1  1 x0 O0

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