Kinematics

1. Kinematics Jehee Lee Seoul National University

2. Kinematics • How to animate skeletons (articulated figures) • Kinematics is the study of motion without regard to the forces that caused it

3. Hierarchical Models • Tree structure of joints and links • The root link can be chosen arbitrarily • Joints • Revolute (hinge) joint allows rotation about a fixed axis • Prismatic joint allows translation along a line • Ball-and-socket joint allows rotation about an arbitrary axis

4. Human Joints • Human joints are actually much more complicated

5. Forward and Inverse Kinematics Forward Kinematics Inverse Kinematics

6. Forward Kinematics: A Simple Example • A simple robot arm in 2-dimensional space • 2 revolute joints • Joint angles are known • Compute the position of the end-effector

7. 3D Position and Orientation • The position and orientation of an object is represented as a rigid transformation • Vector & 3x3 Matrix • 4x4 Matrix • Vector & Quaternion

8. Forward Kinematics: A Simple Example • Forward kinematics map as a coordinate transformation • The body local coordinate system of the end-effector was initially coincide with the global coordinate system • Forward kinematics map transforms the position and orientation of the end-effector according to joint angles

9. A Chain of Transformations

10. Thinking of Transformations • In a view of body-attached coordinate system

11. Thinking of Transformations • In a view of body-attached coordinate system

12. Thinking of Transformations • In a view of body-attached coordinate system

13. Thinking of Transformations • In a view of body-attached coordinate system

14. Thinking of Transformations • In a view of body-attached coordinate system

15. Thinking of Transformations • In a view of global coordinate system

16. Thinking of Transformations • In a view of global coordinate system

17. Thinking of Transformations • In a view of global coordinate system

18. Thinking of Transformations • In a view of global coordinate system

19. Thinking of Transformations • In a view of global coordinate system

20. How to Handle Ball-and-Socket Joints ? • Three revolute joints whose axes intersect at a point (equivalent to Euler angles), or • 3D rotation about an arbitrary axis

21. Floating Base • The position and orientation of the root segment are added

22. 1st link transformation 1st joint transformation The position and orientation of the root segment Joint & Link Transformations • Each segment has its own coordinate frame • Forward kinematics map is an alternating multiple of • Joint transformations : represents joint movement • Link transformations : defines a frame relative to its parent

23. Joint & Link Transformations • Both are rigid transformations in general • Joint transformations may include translation • Human joints are not ideal hinges • Link transformations may include rotation • Some links are twisted

24. Denavit-Hartenberg Notation

25. Link Transformations • How do you compute the link transform for the ith joint if you know the position and orientation of the ith joint as well as its parent’s position and orientation at the neutral pose ?

26. Representing Hierarchical Models • A tree structure • A node contains a joint transformation • A arc contains a link transformation

27. Depth-First Tree Traversal • Draw graphics needs to compute the position and orientation of all links • OpenGL’s matrix stack is useful

28. Summary • Kinematics is the study of motion of articulated figures • Kinematics does not consider physics (forces, mass, …) • Forward kinematics is straightforward • Forward kinematics map can be considered as a coordinate transformation • The next lecture: How to solve the inverse problem

29. Programming Assignment #1 • Create a tree-structured hierarchical model of human figures • You have to compute the forward kinematics map of your model • You may use a matrix stack to build the forward kinematics map incrementally • The model may be displayed using three-dimensional primitives such as polygons, boxes, cylinders, and spheres • The joints are either revolute or ball-and-sockets

30. Programming Assignment #1 • Animate the model to demonstrate the hierarchical structure • For example, use sin/cos functions to make it walk and run • Implement a scene 3D viewer • You should be able to change the viewpoint (scene rotation/translation) • The viewer will be used in all the other programming assignment • Submit a report of at most three A4 pages • Describe the tree-structure, joints, and geometric primitives • Describe how the joints are animated