Kinematics

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14 Νοε 2013 (πριν από 3 χρόνια και 10 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


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