Differential Kinematics and Statics

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Differential Kinematics and Statics Ref
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Incremental Motion

• What small (incremental) motions at the
  end-effector (Dx, Dy, Dz) result from small
  motions of the joints (Dq1, Dq2, , Dqn )?
• Alternatively, what velocities at the
  end-effector (vx, vy, vz) result from velocities
  at the joints (w1, w2, wn)?

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Some Definitions

• Linear Velocity The instantaneous rate-of-change
  in linear position of a point relative to some
  frame.
• v(vx, vy, vz)T
• Angular Velocity The instantaneous
  rate-of-change in the orientation of one frame
  relative to another.
• Angular Velocity depends on the way to represent
  orientation (Euler Angles, Rotation Matrix, etc.)
• Angular Velocity Vector and the Angular Velocity
  Matrix.

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Some Definitions

• Angular Velocity Vector A vector whose direction
  is the instantaneous axis of rotation of one
  frame relative to another and whose magnitude is
  the rate of rotation about that axis.

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Free Vector

• Linear velocity are insensitive to shifts in
  origin but are sensitive to orientation.

D
x
x
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Free Vector

• Angular velocity are insensitive to shifts in
  origin but are sensitive to orientation.

D
x
x
x
x
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Velocity Frames

• frame of reference this is the frame used to
  measure the objects velocity
• frame of representation. this is the frame in
  which the velocity is expressed.

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Y0
R
y0
x2
y2
v
a2
Y1
q2
X1
a1
v
v
q1
0
X0
v
x0
0
Figure 2.13 Two-Link Planar Robot
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End-effector velocity for w1
Y0
y0
v
r0n
v
v
w1
0
X0
v
x0
0
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End-effector velocity for w2
Y0
y0
v
r1n
w2
v
v
0
X0
v
x0
0
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Two-Link Planar Robot

• Direct kinematics equation

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Incremental Motion

• taking derivatives of the position equation
  w.r.t. time we have
• note that

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Incremental Motion

• written in the more common matrix form,
• or in terms of incremental motion,

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Differential Kinematics

• Find the relationship between the joint
  velocities and the end-effector linear and
  angular velocities.

Linear velocity Angular velocity
for a revolute joint
for a prismatic joint
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Differential Kinematics

• Differential kinematics equation
• Geometric Jacobian

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Relationship with T(q)

• Direct kinematics equation
• Linear velocity
• Angular velocity?

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Vector (Cross) Product

• Vector product of x and y
• Skew-symmetric matrix

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Vector (Cross) Product

• Skew-symmetric matrix

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Derivative of a Rotation Matrix
define
S(t) is skew-symmetric
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Interpretation of S(t)
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Interpretation of S(t)
Given R(t)
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Example 3.1 Rotation about Z