# Kinematics

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14 Νοε 2013 (πριν από 4 χρόνια και 6 μήνες)

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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

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

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

-
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