Animation CS 551 651

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AnimationCS 551 / 651

• Kinematics
• Lecture 09

Sarcos Humanoid
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Kinematics

• The study of object movements irrespective of
  their speed or style of movement

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Degrees of Freedom(DOFs)

• The variables that affect an objects orientation
• How many degrees of freedom when flying?

• So the kinematics of this airplane permit
  movement anywhere in three dimensions

• Six
• x, y, and z positions
• roll, pitch, and yaw

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Degrees of Freedom

• How about this robot arm?

• Six again
• 2-base, 1-shoulder, 1-elbow, 2-wrist

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Configuration Space

• The set of all possible positions (defined by
  kinematics) an object can attain

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Work Space vs. Configuration Space

• Work space
• The space in which the object exists
• Dimensionality
• R3 for most things, R2 for planar arms
• Configuration space
• The space that defines the possible object
  configurations
• Degrees of Freedom
• The number of parameters that necessary and
  sufficient to define position in configuration

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More examples

• A point on a plane
• A point in space
• A point moving on a line in space

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A matter of control

• If your animation adds energy at a particular
  DOF, that is a controlled DOF

High DOF, no control
Low DOF, high control
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Hierarchical Kinematic Modeling

• A family of parent-child spatial relationships
  are functionally defined
• Moon/Earth/Sun movements
• Articulations of a humanoid

• Limb connectivity is built into model (joints)
  and animation is easier

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Robot Parts/Terms

• Links
• End effector
• Frame
• Revolute Joint
• Prismatic Joint

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More Complex Joints

• 3 DOF joints
• Gimbal
• Spherical (doesnt possess singularity)
• 2 DOF joints
• Universal

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Hierarchy Representation

• Model bodies (links) as nodes of a tree
• All body frames are local (relative to parent)
• Transformations affecting root affect all
  children
• Transformations affecting any node affect all its
  children

ROOT
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Forward vs. Inverse Kinematics

• 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

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Forward Kinematics

• Traverse kinematic tree and propagate
  transformations downward
• Use stack
• Compose parent transformation with childs
• Pop stack when leaf is reached

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Denavit-Hartenberg (DH) Notation

• A kinematic representation (convention) inherited
  from robotics

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Z-axis aligned with joint
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X-axis aligned with outgoing limb
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Y-axis is orthogonal
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Joints are numbered to represent hierarchy Ui-1
is parent of Ui
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Parameter ai-1 is outgoinglimb length of joint
Ui-1
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Joint angle, qi, is rotation of xi-1 about zi-1
relative to xi
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Link twist, ai-1, is the rotation of ith z-axis
about xi-1-axis relative to z-axis of i-1th frame
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Link offset, di-1, specifies the distance along
the zi-1-axis (rotated by ai-1) of the ith frame
from the i-1th x-axis to the ith x-axis
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Inverse Kinematics (IK)

• Given end effector position, compute required
  joint angles
• In simple case, analytic solution exists
• Use trig, geometry, and algebra to solve

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What is Inverse Kinematics?

• Forward Kinematics

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What is Inverse Kinematics?

• Inverse Kinematics

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What does look like?
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Solution to

• Our example

Number of equation 2
Unknown variables 3
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Redundancy

• System DOF gt End Effector DOF

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• Analytic solution of 2-link inverse kinematics

x2
y2
(x,y)
O2
?2
y0
y1
a2
?2
x1
?
a1
O1
?
?1
x0
O0
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Failures of simple IK

• Multiple Solutions

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Failures of simple IK

• Infinite solutions

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Failures of simple IK

• Solutions may not exist

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Iterative IK Solutions

• Frequently analytic solution is infeasible
• Use Jacobian
• Derivative of function output relative to each of
  its inputs
• If y is function of three inputs and one output

• Represent Jacobian, J(X), as a 1x3 matrix of
  partial derivatives

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Jacobian

• In another situation, end effector has 6 DOFs
  and robotic arm has 6 DOFs
• f(x1, , x6) (x, y, z, r, p, y)
• Therefore J(X) 6x6 matrix

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Jacobian

• Relates velocities in parameter space to
  velocities of outputs
• If we know Ycurrent and Ydesired, then we
  subtract to compute Ydot
• Invert Jacobian and solve for Xdot

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Turn to PDF slides

• Slides from OBrien and Forsyth
• CS 294-3 Computer GraphicsStanfordFall 2001

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Differential Kinematics

• Is J always invertible? No!
• Remedy Pseudo Inverse

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Null space

• The null space of J is the set of vectors which
  have no influence on the constraints
• The pseudoinverse provides an operator which
  projects any vector to the null space of J

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Utility of Null Space

• The null space can be used to reach secondary
  goals
• Or to find comfortable positions

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Calculating Pseudo Inverse

• Singular Value Decomposition

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Redundancy

• A redundant system has infinite number of
  solutions
• Human skeleton has 70 DOF
• Ultra-super redundant
• How to solve highly redundant system?

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Redundancy Is Bad

• Multiple choices for one goal
• What happens if we pick any of them?

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Redundancy Is Good

• We can exploit redundancy
• Additional objective
• Minimal Change
• Similarity to Given Example
• Naturalness

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Naturalness

• Based on observation of natural human posture
• Neurophysiological experiments

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Conflict Between Goals
ee 2
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base
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Conflict Between Goals
Goal 1
ee 2
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base
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Conflict Between Goals
Goal 2
ee 2
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base
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Conflict Between Goals
Goal 2
Goal 1
ee 2
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base
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Conflict Between Goals
Goal 2
Goal 1
ee 2
ee 1
base