1
Haptics and Virtual Reality
M. Zareinejad
Lecture 6:
Haptic interface kinematics
2
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).
3
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
4
Orientation of a Body
5
Homogenous Transforms
6
Homogenous Transforms
Translation
Rotation
7
Kinematics
Moving between Joint Space and Cartesian Space
Forward kinematics: based on joint angles, calculateend-effector position
8
Joint variables
Be careful how you define joint positions
Absolute Relative
9
Absolute forward kinematics
x = L1cos(1) + L2cos(2)
y = L1sin(1) + L2sin(2)
10
Relative forward kinematics
x = L1cos(1) + L2cos(1+ 2)
y = L1sin(1) + L2sin(1+ 2)
11
D-H Parameters
12
D-H Parameters
13
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
14
Example
Two possible solutions
Two approaches:–
–
algebraic method
geometric method
15
Example
16
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
17
Formulating the Jacobian
Use the chain rule:
Take partial derivatives:
18
Assemble in a Matrix
19
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:–
–
Workspace boundary
Workspace interior
20
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
21
Calculating Singularities
Simplify text: sin(1+2)=s12
For what values of 1 and 2 does this equalzero?
22
Even more useful….
The Jacobian can be used to relate jointtorques to end-effector forces:
Why is this important for haptic displays?
23
How do you get this equation?
Principle of virtual work– States that changing the
coordinate frame doesnot change the total workof a system
24
Dynamic
25
Homework #1
SensAble’s Phantoms Quanser Pantograph