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Transcript of Kinematics & Dynamics - Princeton · PDF file7 Summary of Kinematics • Forward...

  • 1

    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

  • 2

    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)

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    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

  • 7

    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

  • 8

    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

  • 12

    Spacetime Constraints

    Adapting motion:

    Witkin & Kass 88

    Ski Jump

    Spacetime Constraints

    Editing motion:

    Li et al. 99

  • 13

    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

  • 14

    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