CS-184: Computer Graphics. Today. Forward kinematics Inverse kinematics. Wednesday, November 12, Pin joints Ball joints Prismatic joints

Transcription

1 CS-184: Computer Graphics Lecture #18: Forward and Prof. James O Brien University of California, Berkeley V2008-F Today Forward kinematics Inverse kinematics Pin joints Ball joints Prismatic joints 2 2

2 Forward Kinematics Articulated skeleton Topology (what s connected to what) Geometric relations from joints Independent of display geometry Tree structure Loop joints break tree-ness 3 3 Root body Forward Kinematics Position set by global transformation Root joint Position Rotation Other bodies relative to root Inboard toward the root Outboard away from root 4 4

3 A joint Forward Kinematics Joint s inboard body Joint s outboard body 5 5 A body Forward Kinematics Body s inboard joint Body s outboard joint May have several outboard joints 6 6

4 A body Forward Kinematics Body s inboard joint Body s outboard joint May have several outboard joints Body s parent Body s child May have several children 7 7 Forward Kinematics Interior joints Typically not 6 DOF joints Pin - rotate about one axis Ball - arbitrary rotation Prism - translation along one axis 8 8

5 Pin Joints Forward Kinematics Translate inboard joint to local origin Apply rotation about axis Translate origin to location of joint on outboard body 9 9 Ball Joints Forward Kinematics Translate inboard joint to local origin Apply rotation about arbitrary axis Translate origin to location of joint on outboard body 10 10

6 Prismatic Joints Forward Kinematics Translate inboard joint to local origin Translate along axis Translate origin to location of joint on outboard body Forward Kinematics Composite transformations up the hierarchy 12 12

7 Forward Kinematics Composite transformations up the hierarchy Forward Kinematics Composite transformations up the hierarchy 14 14

8 Forward Kinematics Composite transformations up the hierarchy Forward Kinematics Composite transformations up the hierarchy 16 16

9 Given Root transformation Initial configuration Desired end point location Find Interior parameter settings Egon Pasztor 18 18

10 A simple two segment arm in 2D Warning: Z!up Coordinate System Direct IK: solve for the parameters 20 20

11 Why is the problem hard? Multiple solutions separated in configuration space Why is the problem hard? Multiple solutions connected in configuration space 22 22

12 Why is the problem hard? Solutions may not always exist Numerical Solution Start in some initial configuration Define an error metric (e.g. goal pos - current pos) Compute Jacobian of error w.r.t. inputs Apply Newton s method (or other procedure) Iterate

13 Recall simple two segment arm: Warning: Z!up Coordinate System We can write of the derivatives 26 26

14 Direction in Config. Space The Jacobian (of p w.r.t.!) Example for two segment arm 28 28

15 The Jacobian (of p w.r.t.!) Solving for and 30 30

16 Solving for and Is the Jacobian invertible? Problems Jacobian may (will!) not always be invertible Use pseudo inverse (SVD) Robust iterative method Jacobian is not constant acobian is not constant Nonlinear optimization, but problem is (mostly) well behaved 32 32

17 More complex systems More complex joints (prism and ball) More links Other criteria (COM or height) Hard constraints (joint limits) Multiple criteria and multiple chains Some issues How to pick from multiple solutions? Robustness when no solutions Contradictory solutions Smooth interpolation Interpolation aware of constraints 34 34

18 Prism Joints } } Ball Joints 36 36

19 Ball Joints (moving axis) { That is the Jacobian for this joint Ball Joints (fixed axis) { That is the Jacobian for this joint 38 38

20 Many links / joints Need a generic method for building Jacobian 2a 2b Can t just concatenate individual matrices 2a 2b

21 Transformation from body to world Rotation from body to world Need to transform Jacobians to common coordinate system (WORLD) 2a 2b b 42 42

22 Note: Each row in the above should be transposed Suggested Reading Advanced Animation and Rendering Techniques by Watt and Watt Chapters 15 and