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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–

–

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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 anglesNot used often in haptics– But useful for calibration

There can be:–

–

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No solution (workspace issue)1 solutionMore than 1 solution

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Example

Two possible solutionsTwo approaches:–

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algebraic methodgeometric method

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Example

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Velocity

Recall that the forward kinematics tells us theend-effector position based on joint positionsHow 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 singularThis means that the device has lost one ormore degrees of freedom in Cartesian SpaceTwo kinds:–

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Workspace boundaryWorkspace 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