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Kinematics &Dynamics

Thomas Funkhouser

Princeton University

C0S 426, Fall 2000

Overview

Kinematics Considers only motion Determined by positions, velocities, accelerations

Dynamics Considers underlying forces Compute motion from initial conditions and physics

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Example: 2-Link Structure

Two links connected by rotational joints

1

2

X = (x,y)

2

1

(0,0)

End-Effector

Forward Kinematics

Animator specifies joint angles: 1 and 2 Computer finds positions of end-effector: X

))sin(sin),cos(cos( 2121121211 ++++= llllX

1

2

X = (x,y)

2

1

(0,0)

3

Forward Kinematics

Joint motions can be specified by spline curves

1

2

X = (x,y)

2

1

(0,0)

2

1

t

Forward Kinematics

Joint motions can be specified by initial conditionsand velocities

1

2

X = (x,y)

2

1

(0,0)

1.02.1

250)0(60)0(

21

21

==

==

dt

d

dt

d

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Example: 2-Link Structure

What if animator knows position of end-effector

1

2

X = (x,y)

2

1

(0,0)

End-Effector

Inverse Kinematics

Animator specifies end-effector positions: X

Computer finds joint angles: 1 and 2:

xllyl

yllxl

))cos(())sin((

))cos(()sin((

22122

221221 ++

++=

1

2

X = (x,y)2

1

(0,0)

+= 21

22

21

221

2 cos ll

llxx

2

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Inverse Kinematics

End-effector postions can be specified by splinecurves

1

2

X = (x,y)

2

1

(0,0)

y

x

t

Inverse Kinematics

Problem for more complex structures System of equations is usually under-defined Multiple solutions

1

2

2

1

(0,0)

X = (x,y)

3

3

Three unknowns: 1, 2 , 3Two equations: x, y

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Inverse Kinematics

Solution for more complex structures: Find best solution (e.g., minimize energy in motion) Non-linear optimization

1

2

2

1

(0,0)

X = (x,y)

3

3

Inverse Kinematics

Fujito, Milliron, Ngan, & SanockiPrinceton University

Ballboy

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Summary of Kinematics

Forward kinematics Specify conditions (joint angles)

Compute positions of end-effectors

Inverse kinematics Goal-directed motion Specify goal positions of end effectors

Compute 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

Kinematics Considers only motion Determined by positions, velocities, accelerations

Dynamics Considers underlying forces Compute motion from initial conditions and physics

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Dynamics

Simulation of physics insures realism of motion

Lasseter 87

Spacetime Constraints

Animator specifies constraints: What the characters physical structure is

e.g., articulated figure What the character has to do

e.g., jump from here to there within time t What other physical structures are present

e.g., floor to push off and land How the motion should be performed

e.g., minimize energy

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Spacetime Constraints

Computer finds the best physical motionsatisfying constraints

Example: particle with jet propulsion x(t) is position of particle at time t f(t) is force of jet propulsion at time t Particles equation of motion is:

Suppose we want to move from a to b within t0 to t1with minimum jet fuel:

0 = mgfmx

dttft

t1

0

2)(Minimize subject to x(t0)=a and x(t1)=bWitkin & Kass 88

Spacetime Constraints

Discretize time steps:

02

2

11 =

+= + mgf

h

xxxxm i

iiii

2i

ifhMinimize subject to x0=a and x1=b

211

1

2

h

xxxx

h

xxx

iiii

iii

+

+=

=

Witkin & Kass 88

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Spacetime Constraints

Solve withiterativeoptimizationmethods

Witkin & Kass 88

Spacetime Constraints

Advantages: Free animator from having to specify details of

physically realistic motion with spline curves Easy to vary motions due to new parameters

and/or new constraints

Challenges: Specifying constraints and objective functions Avoiding local minima during optimization

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Spacetime Constraints

Adapting motion:

Witkin & Kass 88

Heavier Base

Original Jump

Spacetime Constraints

Adapting motion:

Witkin & Kass 88

Hurdle

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Spacetime Constraints

Adapting motion:

Witkin & Kass 88

Ski Jump

Spacetime Constraints

Editing motion:

Li et al. 99

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Spacetime Constraints

Morphing motion:

Gleicher 98

Spacetime Constraints

Advantages: Free animator from having to specify details of

physically realistic motion with spline curves Easy to vary motions due to new parameters

and/or new constraints

Challenges: Specifying constraints and objective functions! Avoiding local minima during optimization

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Dynamics

Other physical simulations:" Rigid bodies# Soft bodies$ Cloth% Liquids& Gases' etc.

Hot Gases(Foster & Metaxas 97)

Cloth(Baraff & Witkin 98)

Summary

Kinematics( Forward kinematics

Animator specifies joints (hard) Compute end-effectors (easy - assn 5!)

) Inverse kinematics Animator specifies end-effectors (easier) Solve for joints (harder)

Dynamics* Space-time constraints

Animator specifies structures & constraints (easiest) Solve for motion (hardest)

+ Also other physical simulations