Joint%20Velocity%20and%20the%20Jacobian

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Joint Velocity and the Jacobian
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Chapter Objectives

• By the end of the Chapter, you should be able to
• Characterize frame velocity
• Compute linear and rotational velocity
• Compute Jacobian and robot singularities
• Bibliography
• Craigs book
• Handout

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Velocity of a Point
The position of a point in frame B in terms of
frame A is
Point velocity in A derivative with respect to
time

• When differentiating, two frames come into play
• The frame with respect to which we differentiate
• The frame in which result is expressed, e.g.

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Rotational Velocity

• Suppose now that B is rotating w.r.t. A

Differentiating
A trick to get an economic representation
It can be shown that
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Rotational Velocity (cont.)

• We write

Wedge operator
Where
Rotational Velocity
A cool expression of velocity due to time
varying rotation
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Linear Rotational Velocity

• If we have simultaneous time varying rotation
  translat.

Using Homogeneous Coordinates, we can show that
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Velocity Propagation
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Velocity Propagation (cont.)

• Rotational velocities may be added as vectors

Where
Also
With respect to the linear velocity
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An Example
V3
L2
L1
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The Jacobian

• Jacobian Multidimensional Derivative
• Example
• 6 functions fi , i1,,6
• 6 variables xi , i1,,6
• Write y1 f1(x1, x2, , x6)
• y2 f2(x1, x2, , x6)
• ? ?
• y6 f6(x1, x2, , x6)
• In vector form
• Y F(X)

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The Jacobian (cont.)

• Taking derivatives

Jacobian

• Dividing by ?t on both sides

• The Jacobian is a time varying transformation
  mapping velocities to velocities

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Jacobian for a Manipulator

• Robot kinematics give
• frame of EE F(joint variables)
• Using the Jacobian
• velocity of EE J?joint variable derivatives
• If all joints rotational, and calling

then we write
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Singularities of a Robot

• If J is invertible, we can compute joint
  velocities given Cartesian velocities

• Important relationship shows how to design
  joint velocities to achieve Cartesian ones
• Most robots have joint values for which J is
  non-invertible
• Such points are called singularities of the
  robot.

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Singularities (cont.)

• Two classes of singularities
• Workspace boundary singularities
• Workspace interior singularities
• Robot in singular configuration it has lost one
  or more degrees of freedom in Cartesian space