CSCE 689: Forward Kinematics and Inverse Kinematics

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CSCE 689: Forward Kinematics and Inverse Kinematics. Jinxiang Chai. Outline. Kinematics Forward kinematics Inverse kinematics. Kinematics. The study of movement without the consideration of the masses or forces that bring about the motion. Degrees of freedom (Dofs).

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### CSCE 689: Forward Kinematics and Inverse Kinematics

Jinxiang Chai

Outline

Kinematics

• Forward kinematics
• Inverse kinematics
Kinematics

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

Degrees of freedom (Dofs)

The set of independent displacements that specify an object’s pose

Degrees of freedom (Dofs)

The set of independent displacements that specify an object’s pose

• How many degrees of freedom when flying?
Degrees of freedom (Dofs)

The set of independent displacements that specify an object’s 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
Degrees of Freedom
• Six again
• 2-base, 1-shoulder, 1-elbow, 2-wrist
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
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

End Effector

Base

• Two links connected by rotational joints

θ2

l2

l1

θ1

X=(x,y)

End Effector

Base

• Animator specifies the joint angles: θ1θ2
• Computer finds the position of end-effector: x

θ2

l2

l1

θ1

X=(x,y)

(0,0)

x=f(θ1, θ2)

End Effector

Base

• Animator specifies the joint angles: θ1θ2
• Computer finds the position of end-effector: x

θ2

θ2

l2

l1

θ1

X=(x,y)

(0,0)

x = (l1cosθ1+l2cos(θ2+ θ2)

y = l1sinθ1+l2sin(θ2+ θ2))

End Effector

Base

Forward kinematics
• Create an animation by specifying the joint angle trajectories

θ2

θ2

l2

l1

θ1

X=(x,y)

(0,0)

θ1

θ2

Base

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

θ2

θ2

l2

l1

End Effector

θ1

X=(x,y)

(0,0)

Base

Inverse kinematics
• Animator specifies the position of end-effector: x
• Computer finds joint angles: θ1θ2

θ2

θ2

l2

l1

End Effector

θ1

X=(x,y)

(0,0)

(θ1, θ2)=f-1(x)

Base

Inverse kinematics
• Animator specifies the position of end-effector: x
• Computer finds joint angles: θ1θ2

θ2

θ2

l2

l1

End Effector

θ1

X=(x,y)

(0,0)

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)
Inverse kinematics
• Given end effector position, compute required joint angles
• In simple case, analytic solution exists
• Use trig, geometry, and algebra to solve
Inverse kinematics
• Analytical solution only works for a fairly simple structure
• Iterative approach needed for a complex structure
Iterative approach
• Inverse kinematics can be formulated as an optimization problem

Base

Iterative approaches

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

θ2

θ2

l2

l1

θ1

C=(Cx,Cy)

(0,0)

Base

Iterative approaches

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

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Iterative approach

Mathematically, we can formulate this as an optimization problem:

The above problem can be solved by many nonlinear optimization algorithms:

- Steepest descent

- Gauss-newton

- Levenberg-marquardt, etc

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

How can we decide the amount of update?

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Known!

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Taylor series expansion

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Taylor series expansion

rearrange

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Taylor series expansion

rearrange

Can you solve this optimization problem?

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

Gauss-newton approach
• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero
Gauss-newton approach
• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b

J

Δθ

Linear equation!

Gauss-newton approach
• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b

J

Δθ

Gauss-newton approach
• Optimizing an quadratic function is easy
• It has an optimal value when the gradient is zero

b

J

Δθ

Jacobian matrix

Dofs (N)

• Jacobian is a Mby Nmatrix that relates differential changes of θ 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)

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

(x,y)

θ2

θ1

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

(x,y)

A small change of θ1and θ2results in how much change of end-effector position (x,y)

θ2

θ1

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

(x,y)

A small change of θ1and θ2results in how much change of end-effector position (x,y)

θ2

θ1

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

f2

f1

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Base

θ2

θ2

l2

l1

θ1

C=(c1,c2)

(0,0)

Gauss-newton approach

Step 1: initialize the joint angles with

Step 2: update the joint angles:

Step size: specified by the user

Another Example

lower arm

middle arm

Upper arm

base

How to compute forward kinematics?

A 2D lamp with 6 degrees of freedom

Another Example

lower arm

middle arm

Upper arm

base

A 2D lamp with 6 degrees of freedom

Another Example

base

A 2D lamp with 6 degrees of freedom

Another Example

Upper arm

base

A 2D lamp with 6 degrees of freedom

Another Example

middle arm

Upper arm

base

A 2D lamp with 6 degrees of freedom

Another Example

lower arm

middle arm

Upper arm

base

A 2D lamp with 6 degrees of freedom

More Complex Characters?

lower arm

middle arm

Upper arm

base

A 2D lamp with 6 degrees of freedom

More Complex Characters?

lower arm

middle arm

If you do inverse kinematics with position of this point

What’s the size of Jacobian matrix?

Upper arm

base

A 2D lamp with 6 degrees of freedom

More Complex Characters?

lower arm

middle arm

If you do inverse kinematics with position of this point

What’s the size of Jacobian matrix?

2-by-6 matrix

Upper arm

base

A 2D lamp with 6 degrees of freedom

Inverse kinematics
• Analytical solution only works for a fairly simple structure
• Iterative approach needed for a complex structure
Inverse kinematics
• Is the solution unique?
• Is there always a good solution?
Ambiguity of IK
• Multiple solutions
Ambiguity of IK
• Infinite solutions
Failures of IK
• Solution may not exist
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0

Minimize the difference between the solution and a reference pose

Satisfy the constraints

Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0
• Naturalness g(θ) (particularly for human characters)
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0
• Naturalness g(θ) (particularly for human characters)
Inverse kinematics
• Generally ill-posed problem when the Dofs is higher than the number of constraints
• Minimal Change from a reference pose θ0
• Naturalness g(θ) (particularly for human characters)

How natural is the solution pose?

Satisfy the constraints