Kinematics Intro a branch of dynamics that deals

Kinematics Intro : a branch of dynamics that deals with aspects of motion apart from considerations of mass and force Merriam-Webster

Animation • Scripting Kinematics (specify poses without regard to forces) • Procedural • Particle Systems (Reeves ‘ 83) • Behavioral Animation (Reynolds ‘ 87) • Intelligent Systems (game/strategy/AI) • Physically Based Modeling (water, gravity) • Input Driven (face animation) • Algorithmic (e. g. Plant growth)

Labanotation (from dance); similar to Jack’s talk about motion indexing - • where are body parts relative to each other? • put together in a series of key frames

Two Ways • Forward Kinematics: apply transformation to each node to determine pose. • Reverse Kinematics: given desired position of effectors, find necessary joint angles to achieve pose. • Specify pose by moving limbs, then calculate transformations Which is associated with motion capture?

Hierarchical Kinematic Modeling • Closely related to robotics L. Elbow Right Arm R. Elbow Left Arm L. Leg R. Leg L. Knee L. Wrist L. fingers L. Ankle R. Knee R. Ankle R. Wrist R. fingers

Using the Hierarchical Model • Node x has matrix for transformation • Used by all nodes in the tree • Recursively walk tree: matrix for node applied to everything in subtree (depth first or breadth first? ) • Joints may have one or more degrees of freedom – Rotate about i, j, k, translate

Small Example L L 0 L 1 L 2 0 To T 0 L 1 T 1 Lo L 2 T 2 L 1 T 1 T 2 • Each transformation is relative to member above it in hierarchy • 1 st matrix: T 0, Vertex V 0 in L 0 is moved V 0’ = T 0 V 0 • 2 nd Matrix: T 0 T 1 • 3 rd Matrix: T 0 T 1 T 2 • What if there was rotations? Be careful of order!!

Joints 1. Revolute joints - allow you to rotate 2. Prismatic joints (pistons)

Who (in the &!*~^) is Denavit. Hartenberg • We like to define points in a local coordinate frame • Need a deterministic method for converting from coordinates in one frame to another • Especially, parent to child in a hierarchy

Denavit-Hartenberg Notation • For 1 axis of rotation • Link length: distance down the x axis from one joint to the next • Joint angle: Xi+1 Zi Linki Өi+1 Xi

Joint Parameters • Joint angle Өi: angle between xi-1 and xi about zi • Link length ai: distance from zi to z. I+1 along xi • Link offset di : distance from xi-1 to xi along z • Link twist i: angle between zi and z. I+1 about xi Here, d is zero, and alpha is zero Zi+1 Xi+1 Zi ai Өi+1 Xi

Non-Planar Joints Zi Zi+1 Zi d. I+1 Өi+1