Kinematics Intro

1
Kinematics Intro

• a branch of dynamics that deals with aspects of
  motion apart from considerations of mass and
  forceMerriam-Webster

2
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)

3
Labanotation

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

4
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?
5
Hierarchical Kinematic Modeling

• Closely related to robotics

RightArm
LeftArm
L.Elbow
R.Elbow
L. Leg
R. Leg
R. Knee
L. Knee
R. Ankle
L. Ankle
R. Wrist
L. Wrist
R. fingers
L. fingers
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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

7
Small Example
L0
L2
L2
L1
L0
L1
To
T1
T2
Lo
L1
L1
T0
T2
T1

• Each transformation is relative to member above
  it in hierarchy
• 1st matrix T0, Vertex V0 in L0 is moved V0
  T0V0
• 2nd Matrix T0 T1
• 3rd Matrix T0 T1 T2
• What if there was rotations? Be careful of order!!

8
Joints

• Revolute joints - allow you to rotate
• Prismatic joints (pistons)

9
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

10
Denavit-Hartenberg Notation

• For 1 axis of rotation
• Link length distance down the x axis from one
  joint to the next
• Joint angle

Xi1
Zi1
Zi
Linki
?i1
Xi
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Joint Parameters

• Joint angle ?i angle between xi-1 and xi about
  zi
• Link length ai distance from zi to zI1 along xi
• Link offset di distance from xi-1 to xi along z
• Link twist i angle between zi and zI1
  about xi

Here, d is zero, and alpha is zero
Xi1
Zi1
Zi
ai
?i1
Xi
12
Non-Planar Joints
Zi
Zi1
Zi
dI1
?i1