Class #15
Introduction / Review

What is one application for Linked systems
Learning Outline:
Forward Kinematics:
joint representation, tree traversal, pose vectors
Learning Material:
Concept
Hierarchical Modeling:
 Defined as: the enforcement of connectivity (or
relative placement) constraints among objects
organized in a tree like structure.
Graphic
Articulated
Figure
Vs. Hierarchy
[Fig 4.3 (part 1 &
2)]
 One example is a model (human) with joints of
the limbs that can be manipulated to produce a
figure with moving appendages. This figure is
considered to be articulated.
 Movement of the appendages is articulation
 Animation of hierarchies mostly comes from the
field of robotics:
 Links are the rigid objects forming
connections between the Joints
 A Frame is the local coordinate system with
each joint.
Link
Joint
Frame (x, y, z)
 Animation concerned primarily with revolute
joints
 pinned together at a fixed point
 which allow motion (rotation) in one direction
(axis), are said to have one degree of
freedom (DOF)
 Other joints include:
 Prismatic – one DOF
 Planar – two DOF
 Ball-and-socket - three DOF
Revolute joint
Representing Hierarchical Models (Tree structure):
 Nodes (link or object part) are connected by Arcs
(joints). Initially its counterintuitive, but consider
that an object part (node) can have multiple joints
(arcs) attached to it.
Fig 4.3
(complete)
 Root node is the highest node in the tree, and
corresponds to the root object. Root arc
represents the global transformation applied to
the root node, to position it in the global
coordinate system.
Prismatic joint
Planar joint
Ball-and-socket
joint
 Leaf node is at the bottom of the tree; has no
arcs below it in the tree.
 When considering the relationship between any
two nodes in the tree, the node closer to the root
node would be referred to as the Parent node,
and the node farther from the root node would be
referred to as the Child node.
 An arc represents the transformation required to
correctly position the child node relative to the
parent node.
Object local
coord. system,
trans. to relative
position.
[Fig. 4.5 & 4.6]
Eq. 4.3
 This transformation is applied to the rest of the
linkage down the hierarchy. Each object can be
transformed to their final position by
concatenating the transformations higher up the
tree.
With revolute joints, the rotation is applied first, then
the transformation.
Object local
coord. system,
rotated, then
trans. to relative
position.
[Fig. 4.7 & 4.8]
Eq. 4.5
Traversal of the tree by a depth-first pattern:
 from root node to leaf node
Tree traversal
 then back up to an unexplored arc
↰
 then down the arc
(repeat)
Forward Kinematics:
 To animate the linkage, the rotation parameters at
the joints are manipulated (the changeable
matrices associated with the tree arcs and
parameterized by joint angles).
 A completed set of rotation parameters is a pose.
 A pose is specified by a vector (the pose vector)
consisting of one angle for each joint.
 Positioning a figure by specifying all the joint
rotations is called Forward Kinematics. Joint
rotations can be interpolated between key
positions.
Sample Forward
Kinematic
animation.
Denavit-Hartenberg Notation (DH):
Is a particular way of describing the relationship of a
parent coordinate frame to a child coordinate frame.
 Link Offset
 Joint Angle
 Link length
 Link Twist
Fig. 4.11 &
Table 4.1
Fig. 4.12 &
Table 4.2
To determine a point’s coordinates (Vi+1 in joint i+1)
in terms of joint i, the transformation is described by
matrix M (i+1 into i): Child into Parent. The inverse
can convert Parent into Child or down the hierarchy.
Eq. 4.6
Ball-and-Socket Joints:
 Are modeled as 3 one DOF joints with zero-length
links.