Topic 3: Velocity

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1. Velocity 2. Jacobian Matrix Concept 3.
Singularity Concept 4. Static forces in
manipulators
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Velocity
They are vector and represented in frame i.
Velocity of the origin of frame i
Angular velocity of link i
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Velocity
Viewpoint angular velocity link i1 can be
viewed as experiencing two consecutive rotations
and
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Velocity
Likewise, the velocity of the origin of frame
i1 can be viewed as superposition of the
velocity of the origin of frame i and angular
velocity of link i, namely we have
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Velocity
When the joint i1 is prismatic one, then we have
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Singularity

• Observation the relation between the joint
  velocity and the end-effectors velocity is
  linear. This means that the relation can be
  written as a matrix. V J
  , where J Jacobian matrix.
• A reasonable question Is this J-matrix
  invertible? or Non-singularity?
• Singularity give V, at certain configuration,
  joint velocity may not be existed.

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• At all the workspace boundary, there exist
  singularities.
• Singularities may take place in the interior of
  workspace.
• When a manipulator is in a singular
  configuration, it has lost one or more degrees of
  freedom of motion, viewed from the Cartesian
  space.
• Loss of degree of freedom along a particular
  motion direction implies that whatever joint
  velocities are, it is impossible to move the
  end-effector along that direction in the
  Cartesian space.

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Example Suppose that a Jacobian is shown in the
following
To find the singular points of a
manipulator. Solution To let DET of Jacobina is
equal to zero. We can get the following
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The physical meaning of the singularities in this
example Singularity exists when is 0 or 180
The arm is stretched straight out. In
this Configuration, motion of the end-effector is
possible only along one Cartesian direction (the
one perpendicular to the arm.
0
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The arm is folded completely back on itself.
There is only one direction of motion in this
case as well.
180
Static forces in manipulators
We consider how forces and moments propagate
from one link to the next. Typically, the robot
is pushing on something in the environment with
the chains free end (the end-effector) or is
perhaps supporting a load at the hand. We wish to
solve for joint torques needed to support a
static load acting at the end-effector.
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Figure below should define notations
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Summing the forces and setting them equal to
zero Summing torques about the origin of frame
i and setting them equal to zero If we start
with a description of the force and moment
applied by the hand, we can calculate the force
and moment applied by each link working from the
last link down to the base, link 0.
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We write the two equations above as such In
order to write these equations in terms of forces
and moments defined within their own link frames,
we transform with the rotation matrix describing
frame I1 relative to frame I. This leads to
the following equations.
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Suppose we know the load at the end-effector.
What torques are needed at the joints in order to
balance the reaction forces and moments acting on
the links. The joint torque can be calculated by
the following equation
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Jacobians in the force domain

• According to virtual work principle
• where

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It is noted that the Jacobian matrix is
frame-dependent and in particular, it depends on
the frame, with respect to which, the vectors at
both sides of the equation are written.