Homework

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Homework 4, problem 1.
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Homework 4, problem 1.
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Frames 0 and 1 share a common origin.
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Frames 0 and 1 share a common origin.
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Z axes are permanently aligned in 0 and 1 frames.
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Angle between the two X axes is q1.
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Angle between the two X axes is q1.
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Note angle between X direction of 0 frame and Y
direction of 1 frame is p/2q1
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Simplifying trig identity.
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Simplifying trig identity.
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Consider next the homogeneous transformation
matrix between the 1 and 2 frames.
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The direction cosine matrix is the same as
before, replacing q1 with q2
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The two origins this time are separated by L1
along the X1 direction.
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The two origins this time are separated by L1
along the X1 direction.
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Note that regardless of how the robot may move
point P is permanently a distance L2 along the X2
direction.
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Note that regardless of how the robot may move
point P is permanently a distance L2 along the X2
direction.
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Note that regardless of how the robot may move
point P is permanently a distance L2 along the X2
direction.
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Problem 3.
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Find the 2-4 element of the inverse of this
matrix.
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The 2-4 element relates to the relative position
of the AB origins.
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Reverse sign.
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Find the 2-4 element of the inverse of this
matrix.
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Find the 2-4 element of the inverse of this
matrix.
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Note that if you applyMatlab, for example
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Note that if you applyMatlab, for example
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Which is right?
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Original T matrix not quite a true homog. trans.
matrix.
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For example, rows of rotation matrix dont sum
sq. to 1.0.
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Hence the inverse isnt exactly a h.t. matrix
either.
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From last time Homogeneous transformation matrix
for the blue link.
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Since the two frames share their origin, a1d20
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Since the two frames share their origin, a1d20
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Recall the one-one element of the h.t. matrix
for link i-1 using the Denavit- Hartenberg convn.
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Now the one-two element of the h.t. matrix
for link i-1 using the Denavit- Hartenberg convn.
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The component of Yi that lies in the plane of
Zi-1 Yi-1 is the component that is
perpendicular to Xi-1.
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The component of Yi that lies in the plane of
Zi-1 Yi-1 is the component that is
perpendicular to Xi-1.
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The component of Yi that lies in the plane of
Zi-1 Yi-1 is the component that is
perpendicular to Xi-1.
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The component of Yi that lies in the plane of
Zi-1 Yi-1 is the component that is
perpendicular to Xi-1.
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But what about a1?
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But what about a1?
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But what about a1?
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Homogeneous transformation matrix for blue link.
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Consider next the yellow link.
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For the yellow member, link i-1 is link 2.
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Note that the q2 and q3 axes are parallel.
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Distance between parallel axes is a2.
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To determine d3 we locate the origin of the
reference frame that moves with the orange member
...
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... that being the 3-frame.
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The origin of the 3 frame must lie along the axis
of rotation of q3.
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The origin of the 3 frame must lie along the axis
of rotation of q3.
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The origin of the 3 frame must lie along the axis
of rotation of q3.
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Consider a magnification of the orange link.
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Wrist center
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Axis of rotation of q4.
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Axis of rotation of q4.
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a3
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d3
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What about a2 ?
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What about a2 ?
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Consecutive Z axes are parallel therefore, a2 0
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Note that in the position shown, q30.
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Note that in the position shown, q30.
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Note that in the position shown, q30.
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Note that in the position shown, q30.
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Homogeneous transformation matrix for the yellow
link for a20.