kinematics & dynamics - hallymgraphics.hallym.ac.kr/teach/2005/cga/src/10kindy.pdf · 2008. 9....

Computer Graphics

Sun-Jeong Kim http://www.hallym.ac.kr/~sunkim/teach/2005/cga

Kinematics & Dynamics

December 07, 2005


Overview

Kinematicsconsidering only motion

determined by positions, velocities, accelerations

Dynamicsconsidering underlying forces

computing motion from initial conditions and physics


Example: 2-Link Structure

Two links connected by rotational joints

“End-Effector”

X=(x, y)

(0, 0)


Forward Kinematics (1)

Specifying joint angles: Θ1 and Θ2

Computing positions of end-effector: X

X=(x, y)

(0, 0)

( ) ( )( )2121121211 sinsin ,coscosX Θ+Θ+ΘΘ+Θ+Θ= llll



Specifying joint motions by spline curves

X=(x, y)

(0, 0)



Specifying joint motions by initial conditions and velocities

X=(x, y)

(0, 0) ( ) ( )

1.02.1

2500600

21

21

−=Θ

=Θ

°=Θ°=Θ

dt

d

dt

d


Example: 2-Link Structure

End-effector positions target positions

“End-Effector”

X=(x, y)

(0, 0)


Inverse Kinematics (1)

Specifying end-effector positions: X

Computing joint angles: Θ1 and Θ2

X=(x, y)

(0, 0)

( ) ( )( ) ( )xllyl

yllxl

ll

llyx

22122

221221

21

22

21

221

2

cossin

cossin

2cos

Θ++ΘΘ++Θ−

=Θ

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−+=Θ −



Specifying end-effector positions by splinecurves

X=(x, y)

(0, 0)



Problem for more complex structuresusually under-defined system of equations

multiple solutions

X=(x, y)

(0, 0) Three unknowns: Θ1, Θ2, Θ3Two equations: x, y



Solution for more complex structuresfinding the best solution (ex. minimizing energy in motion)

non-linear optimization

X=(x, y)

(0, 0)


Summary

Forward kinematicsspecifying conditions (joint angles)

computing positions of end-effectors

Inverse kinematics“goal-directed” motion

specifying goal positions of end effectors

computing conditions required to achieve goals

Inverse kinematics provides easier specification for many animation tasks, but it is computationally more difficult

Inverse kinematics provides easier specification for many animation tasks, but it is computationally more difficult


Overview

Kinematicsconsidering only motion

determined by positions, velocities, accelerations

Dynamicsconsidering underlying forces

computing motion from initial conditions and physics


Dynamics

Simulation of physics insuring realism of motion


Space Time Constraints (1)

Specifying constraintscharacter’s physical structure

ex) articulated figure

character’s actionex) jumping from here to there within time t

other physical structuresex) floor to push off and land

how to perform the motionex) minimizing energy

Computing the “best” physical motionsatisfying constraints



Example: particle with jet propulsionx(t): position of particle at time t

f(t): force of jet propulsion at time t

particle’s equation of motion:

moving from a to b within t0 to t1 with minimum jet fuel:

0=−−′′ mgfxm

( ) ( ) ( ) btxatxdttft

t==∫ 10

2 and subject to Minimizing

1

0



Discretizing time steps

bxaxfh

mgfh

xxxxm

h

xxxx

h

xxx

ii

iiii

iii

ii

==

=−−⎟⎠⎞

⎜⎝⎛ +−

=′′

+−=′′

−=′

∑

−+

−+

−

10

2

211

211

1

and subject to Minimizing

02

2



Solving with iterative optimization methods



Advantagesnot having to specify details of physically realistic motion with spline curves

easy to vary motions due to new parameters and/or new constraints

Challengesspecifying constraints and objective functions

avoiding local minima during optimization


Adapting Motion (1)

Original Jump

Heavier Base


Adapting Motion (2)

Hurdle


Adapting Motion (3)

Ski Jump


Editing Motion

Original Adapted


Morphing Motion

The female character morphs into a smaller character during her spine


Other Physical Simulations

Rigid bodies

Soft bodies

Cloth

Liquids

Gases

Etc. Cloth

Hair

Smoke

Fluid


Summary

Kinematicsforward kinematics

specifying joints (hard)computing end-effectors (easy)

inverse kinematicsspecifying end-effectors (easier)solving joints (harder)

Dynamicsspace-time constraints

specifying structures & constraints (easiest)solving motion (hardest)

other physical simulations

kinematics & dynamics - hallymgraphics.hallym.ac.kr/teach/2005/cga/src/10kindy.pdf · 2008. 9....

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