Kinematics
Building an animated character
Rigging
The process of
preparing a character model for animation
,
including setting up an
underlying skeleton
, complete with
constraints
, controllers and kinematic systems, and linking it
to the
mesh
of the character model.
Character Rigging
Skeleton
Skin
Facial Expressions
Muscles
Secondary motion: fat, hair, clothing
Building an animated character
Skeleton
An underlying network of bones used to define and
control the motion of a model during character
animation.
Moving a bone causes the mesh of the model to move
and deform.
Skinning
The process of binding the surface of a model to the
underlying skeleton during character rigging.
Articulated Figures
What is an articulated figure?
A set of rigid objects connected by joints
Individual joints are linked together in a parent

child
hierarchy
Each object has a joint at one end where any child
bones may be attached
The skeleton
Articulated Figures
main figure is described in terms of a global frame of
reference
each individual joint is assigned its own separate
local co

ordinate frame of reference
This coordinate system is with respect to it’s parent.
Can concatenate transformation matrices
Articulated Figures
Degrees of Freedom (DOFs)
The variables that affect an object’s orientation
How many degrees of
freedom when flying?
•
Six
•
x, y, and z positions
•
roll, pitch, and yaw
•
So the kinematics
of this airplane
permit movement
anywhere in three
dimensions
Degrees of Freedom
How about this robot arm?
•
Six again
•
2

base, 1

shoulder, 1

elbow, 2

wrist
Hierarchical Models
Tree structure of joints and links
The root link can be chosen arbitrarily
Joints
Revolute (hinge) joint allows rotation about a fixed axis
Prismatic joint allows translation along a line
Ball

and

socket joint allows rotation about an arbitrary
axis
More Complex Joints
Hinge1 (1 DOF)
Ball & Socket (3 DOF)
Slider (1 DOF)
Hinge2 (2 DOF)
Prismatic and Rotoide (2 DOF)
More Complex Joints
3 DOF joints
Gimbal
Spherical
2 DOF joints
Universal
Human Joints
Human joints are actually much more complicated
Tree structure
Tree structure
Tree structure
Relative movement
Relative movement
Tree structure
Tree structure
Kinematics (
운동학
)
How to animate skeletons (articulated figures)
Kinematics
is the study of motion without regard to
the forces that caused it
운동학과
동력학
⡋楮(浡瑩t猠☠䑹湡浩捳)
Kinematics
is that branch of physics which
involves the description of motion,
without
examining the forces
which produce the motion.
Dynamics
, on the other hand,
involves an
examination of
both a description of motion and
the forces
which produce it.
Kinematics
The study of motion without regard to the forces that
cause it.
Forward Kinematics
Compute configuration (pose) given individual DOF values
•
Good for simulation
Inverse Kinematics
Compute individual DOF values that result in specified end
effector position
•
Good for control
Forward Kinematics (FK)
Traverse kinematic tree and propagate
transformations downward
Use stack
Compose parent transformation with child’s
Pop stack when leaf is reached
Forward Kinematics
Inverse Kinematics (IK)
Given
end effector
position, compute required
joint angles
In simple case, analytic solution exists
Use trig, geometry, and algebra to solve
If simple enough => analytic solution
Else => numeric iterative solution
End Effectors
End effectors
Term, borrowed from robotics, that describes the end of
a jointed link
Also can be described as the bottom node in a hierarchy
Motion spaces
Joint space
Multidimensional space of joint angles
Dimensionality = degrees of freedom
End effector space
Multidimensional space of end effectors
Dimensionality = number of end effectors
Essentially described in world coords
Forward & Inverse Kinematics
Forward Kinematics
Define values for joint angles
Determines positions of end effectors
X = f (θ
)
Inverse Kinematics
Define positions of end effectors
Determine joint angles to make it so
θ = f

1
(X)
Forward & Inverse Kinematics
What is Inverse Kinematics?
Forward Kinematics
Base
1
2
θ
End Effector
3
?
What is Inverse Kinematics?
Inverse Kinematics
Base
1
2
3
End Effector
What does look like?
)
sin(
)
sin(
)
sin(
)
cos(
)
cos(
)
cos(
3
3
2
2
1
1
3
3
2
2
1
1
l
l
l
y
l
l
l
x
?
Base
1
2
End Effector
3
1
l
2
l
3
l
Solution to
Our example
)
sin(
)
sin(
)
sin(
)
cos(
)
cos(
)
cos(
3
3
2
2
1
1
3
3
2
2
1
1
l
l
l
y
l
l
l
x
Number of equations : 2
Unknown variables : 3
Infinite number of solutions !
Inverse Kinematics
Goal directed motion
Reach over and grab that thing!
Easier to specify
Harder to compute
Borrowed from the robotics world
Inverse Kinematics
The problem:
Given the position/orientation of an end

effector
Find the set of joint angle settings
Note that there may be 0, 1, or many solutions.
Overconstrained
–
no solution exists
Underconstrained
–
many solutions exist
Failures of simple IK
Multiple Solutions
Failures of simple IK
Infinite solutions
Failures of simple IK
Solutions may not exist
Forward & Inverse Kinematics
Summary
Kinematics is the study of motion of articulated
figures
Kinematics does not consider physics (forces, mass, …)
Forward kinematics is straightforward
Forward kinematics map can be considered as a
coordinate transformation
Inverse usually requires a numerical solution
Easier to specify
Harder to compute
Demo
Demo
http://www.youtube.com/watch?v=

jVVHDHgOvw
http://www.youtube.com/watch?v=HOxrDCKilr8
http://www.youtube.com/watch?v=tOAFCvSnNDc
http://www.youtube.com/watch?v=jUcoP8BsHvc
http://www.youtube.com/watch?v=CWLw20LUsmk
http://www.youtube.com/watch?v=OprkS

GtKy0
http://www.youtube.com/watch?v=d6ToR0YCARU
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