681 - Introduction to Computer Graphics

1
Forward Kinematics
2
Specifying a Pose
3
Forward Kinematics
Hierarchical model - joints and links
Joints - rotational or prismatic
Joints - 1, 2, or 3 Degree of Freedom
Links - displayable objects
Pose - setting parameters for all joint DoFs
Pose Vector - a complete set of joint parameters
4
Animate a Pose
Interpolate pose vectors
5
Joint Representation
Degree of Freedom
DOF
Joint Limits
6
Joint Representation
Multiple
Degrees of Freedom
Joint Limits
gimbal lock
Axis-angle quaternions
7
Joint Representation
Note
User interface representation may not be the
same used for internal representation and
operations
8
Link and Node Representation
transformation
transformation
data
geometry
9
How to Represent Articulators
Joint-Link Hierarchies
Define Link data so its point of rotation is at
the origin
Rotate the link
Position it relative to its parent link in the
hierarchy
Put it through all the transformations of its
parent link
10
Tree Representation
Transformations
Joint
Arc
NodePtr
Transformations
Node
DataPtr
Link
LinkPtr
11
Tree Traversal
M I
M T0
M T0T1.1
M T0T1.1T1.2
T0
M T0T1.1
M T0
T1.1
T2.1
M T0T2.1
T2.2
T1.2
M T0T2.1T2.2
12
Articulated Figure Representation
13
Link and Node Representation
Transformation (fixed)
Transformation (changeable)
transformation
data
geometry
14
Tree Traversal
M I
T0
M T0
T1.1
M T0T1.1R1.1
T2.1
R2.1
R1.1
M T0T1.1R1.1T1.2R1.2
M T0T1.1R1.1
T2.2
T1.2
M T0
R2.2
R1.2
M T0T2.1R2.1
M T0T2.1T2.2R2.2
15
Tree Traversal
T0
How do you represent more than one degree of
freedom at a joint?
T1.1
T2.1
R2.1
R1.1
T2.2
T1.2

1. Zero length segments Between 1 DoF joints
2. Multiple DoF transformation at joint

R2.2
R1.2
16
Joint Transformations
Rotation representation matrix? Euler angles?
Quaternions?
Impose joint limits on rotations? Need complex
functions to model human figure limits
Could have translational joints, e.g.,
telescoping legs
Pose vector vector of all joint angles q1 q2 q3
qn Interpolate between poses gimbal lock
possible if 3 DoF joints
17
Forward Kinematics
Define Joint-Link Hierarchy
Define sequence of keyposes with corresponding
times
For given time t, use keyposes to interpolate
pose at time t
Traverse tree hierarchy using pose vector to
supply angles
18
Forward Kinematics
19
Use key-values
With key-poses - entire pose vector is specified
Better to allow independent keys for each
articulation variable (or avar)
Sometimes called track-based
frames