m. zareinejad 1. 2 grounded interfaces very similar to robots need kinematics ––––––...
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Haptics and Virtual Reality
M. Zareinejad
Lecture 6:
Haptic interface kinematics
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Grounded interfaces
Very similar to robotsNeed Kinematics–
–
–
Determine endpoint positionCalculate velocitiesCalculate force-torque relationships
Sometimes need DynamicsIf you are weak on these topics, you may wantto check out Introduction to robotics:mechanics and control by Craig (1989).
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Kinematics
o Think of a manipulator/
interface as a set of bodies
connected by a chain of joints
o Revolute most common for
haptic interfaces
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Orientation of a Body
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Homogenous Transforms
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Homogenous Transforms
Translation
Rotation
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Kinematics
Moving between Joint Space and Cartesian Space
Forward kinematics: based on joint angles, calculateend-effector position
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Joint variables
Be careful how you define joint positions
Absolute Relative
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Absolute forward kinematics
x = L1cos(1) + L2cos(2)
y = L1sin(1) + L2sin(2)
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Relative forward kinematics
x = L1cos(1) + L2cos(1+ 2)
y = L1sin(1) + L2sin(1+ 2)
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D-H Parameters
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D-H Parameters
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Inverse Kinematics
Using the end-effector position, calculate thejoint angles
Not used often in haptics– But useful for calibration
There can be:–
–
–
No solution (workspace issue)
1 solution
More than 1 solution
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Example
Two possible solutions
Two approaches:–
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algebraic method
geometric method
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Example
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Velocity
Recall that the forward kinematics tells us theend-effector position based on joint positions
How do we calculate velocity?
Use a matrix called the Jacobian
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Formulating the Jacobian
Use the chain rule:
Take partial derivatives:
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Assemble in a Matrix
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Singularities
Most devices will have values offor which the Jacobian is singular
This means that the device has lost one ormore degrees of freedom in Cartesian Space
Two kinds:–
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Workspace boundary
Workspace interior
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Singularity Math
If the matrix is invertible, then it is non-singular
Can check invertibility of J by taking thedeterminant of J. If it is equal to 0, then it issingular
Can use this method to check which values of θwill cause singularities
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Calculating Singularities
Simplify text: sin(1+2)=s12
For what values of 1 and 2 does this equalzero?
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Even more useful….
The Jacobian can be used to relate jointtorques to end-effector forces:
Why is this important for haptic displays?
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How do you get this equation?
Principle of virtual work– States that changing the
coordinate frame doesnot change the total workof a system
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Dynamic
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Homework #1
SensAble’s Phantoms Quanser Pantograph