CPSC 441: Forward kinematics and inverse kinematics

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CPSC 441 Forward kinematics and inverse
kinematics

• Jinxiang Chai

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Outline

• Kinematics
• Forward kinematics
• Inverse kinematics

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Kinematics

• The study of movement without the consideration
  of the masses or forces that bring about the
  motion

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Degrees of freedom (Dofs)

• The set of independent displacements that specify
  an objects pose

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Degrees of freedom (Dofs)

• The set of independent displacements that specify
  an objects pose
• How many degrees of freedom when flying?

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Degrees of freedom (Dofs)

• The set of independent displacements that specify
  an objects pose
• 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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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 are necessary and
  sufficient to define position in configuration

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Forward vs. Inverse Kinematics

• Forward Kinematics
• Compute configuration (pose) given individual DOF
  values
• Inverse Kinematics
• Compute individual DOF values that result in
  specified end effector position

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Example Two-link Structure

• Two links connected by rotational joints

?2
l2
l1
?1
X(x,y)
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Example two-link structure

• Animator specifies the joint angles ?1?2
• Computer finds the position of end-effector x

?2
l2
l1
?1
X(x,y)
(0,0)
xf(?1, ?2)
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Example two-link structure

• Animator specifies the joint angles ?1?2
• Computer finds the position of end-effector x

?2
?2
l2
l1
?1
X(x,y)
(0,0)
x (l1cos?1l2cos(?2 ?2) y l1sin?1l2sin(?2
?2))
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Forward kinematics

• Create an animation by specifying the joint angle
  trajectories

?2
?2
l2
l1
?1
X(x,y)
(0,0)
?1
?2
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Inverse kinematics

• What if an animator specifies position of
  end-effector?

?2
?2
l2
l1
End Effector
?1
X(x,y)
(0,0)
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Inverse kinematics

• Animator specifies the position of end-effector
  x
• Computer finds joint angles ?1?2

?2
?2
l2
l1
End Effector
?1
X(x,y)
(0,0)
(?1, ?2)f-1(x)
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Inverse kinematics

• Animator specifies the position of end-effector
  x
• Computer finds joint angles ?1?2

?2
?2
l2
l1
End Effector
?1
X(x,y)
(0,0)
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Why Inverse Kinematics?

• Basic tools in character animation
• - key frame generation
• - animation control
• - interactive manipulation
• Computer vision (vision based mocap)
• Robotics
• Bioninfomatics (Protein Inverse Kinematics)

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Inverse kinematics

• 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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Inverse kinematics

• Analytical solution only works for a fairly
  simple structure
• Iterative approach needed for a complex structure

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Iterative approach

• Inverse kinematics can be formulated as an
  optimization problem

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Iterative approaches

• Find the joint angles ? that minimizes the
  distance between the character position and user
  specified position

?2
?2
l2
l1
?1
C(Cx,Cy)
(0,0)
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Iterative approaches

• Find the joint angles ? that minimizes the
  distance between the character position and user
  specified position

?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Iterative approach

• Mathematically, we can formulate this as an
  optimization problem
• The above problem can be solved by many
  optimization algorithm
• - Steepest descent
• - Gauss-newton
• - Levenberg-marquardt, etc

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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

How can we decide the amount of update?
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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

Known!
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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

Taylor series expansion
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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

Taylor series expansion
rearrange
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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

Taylor series expansion
rearrange
This is a quadratic function of
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Gauss-newton approach

• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

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Gauss-newton approach

• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b
J
??
Linear equation!
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Gauss-newton approach

• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b
J
??
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Gauss-newton approach

• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b
J
??
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Jacobian matrix
Dofs (N)

• Jacobian is a M by N matrix that relates
  differential changes of ? to changes of C
• Jacobian maps the velocity in joint space to
  velocities in Cartesian space
• Jacobian depends on current state

The number of constraints (M)
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Jacobian matrix

• Jacobian maps the velocity in joint space to
  velocities in Cartesian space

(x,y)
?2
?1
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Jacobian matrix

• Jacobian maps the velocity in joint space to
  velocities in Cartesian space

(x,y)
A small change of ?1 and ?2 results in how much
change of end-effector position (x,y)
?2
?1
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Jacobian matrix

• Jacobian maps the velocity in joint space to
  velocities in Cartesian space

(x,y)
A small change of ?1 and ?2 results in how much
change of end-effector position (x,y)
?2
?1
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
f2
f1
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Jacobian matrix 2D-link structure
?2
?2
l2
l1
?1
C(c1,c2)
(0,0)
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Gauss-newton approach

• Step 1 initialize the joint angles with
• Step 2 update the joint angles

Step size specified by the user
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More Complex Characters?

• A 2D lamp with 6 degrees of freedom

lower arm
middle arm
Upper arm
base
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More Complex Characters?

• A 2D lamp with 6 degrees of freedom

lower arm
middle arm
If you do inverse kinematics with position of
this point Whats the size of Jacobian matrix?
Upper arm
base
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More Complex Characters?

• A 2D lamp with 6 degrees of freedom

lower arm
middle arm
If you do inverse kinematics with position of
this point Whats the size of Jacobian
matrix? 2-by-6 matrix
Upper arm
base
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Human Characters
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Inverse kinematics

• Analytical solution only works for a fairly
  simple structure
• Iterative approach needed for a complex structure

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Inverse kinematics

• Is the solution unique?
• Is there always a good solution?

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Ambiguity of IK

• Multiple solutions

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Ambiguity of IK

• Infinite solutions

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Failures of IK

• Solution may not exist

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0

Minimize the difference between the solution and
a reference pose
Satisfy the constraints
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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0
• Naturalness g(?) (particularly for human
  characters)

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0
• Naturalness g(?) (particularly for human
  characters)

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Inverse kinematics

• Generally ill-posed problem when the Dofs is
  higher than the number of constraints
• Additional objective
• Minimal Change from a reference pose ?0
• Naturalness g(?) (particularly for human
  characters)

How natural is the solution pose?
Satisfy the constraints
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Interactive Human Character Posing

• video