Inverse kinematics for our 2link robot'

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Inverse kinematics for our 2-link robot.
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The goal is to place P on the target
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The goal is to place P on the target
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Let xtargetx, ytargety
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Note that there is a quadrant ambiguity for q2.
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Note that there is a quadrant ambiguity for q2.
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Choice of sign determines elbow-up or elbow-down
option.
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A two-argument function Resolves quadrant
ambiguity.
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sin positive and cos positive.
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sin positive and cos negative.
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sin negative and cos positive.
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sin negative and cos negative.
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c12 c1c2-s1s2
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Then
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Then
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Then
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Then
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Then
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Inverse kinematics for a digging task.
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We seek a kinematic solution for a portion of the
motion of a nonholonomic system
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one that will duplicate the effect of the
following holonomic sequence.
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one that will duplicate the effect of the
following holonomic sequence.
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Inverse kinematics for a digging task.
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Let point P track the target path.
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Let point P track the target path.
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Let point P track the target path.
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At the same time, keep the knife edge aligned
with the tangent to the target path.
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This is a three-degree-of-freedom task.
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We seek, at each juncture along the length of the
trajectory to control 2 components of position of
point P ...
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and one component of end-of-arm-tool
orientation.
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We have three degrees of robot freedom in order
to realize these three objectives.
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We have three degrees of robot freedom in order
to realize these three objectives.
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We have three degrees of robot freedom in order
to realize these three objectives.
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We have three degrees of robot freedom in order
to realize these three objectives.
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We assume that the
Although q3 does entail rotation, it is not a
revolute joint in the sense that we have
considered such degrees of freedom in Craig.
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We assume that the
Nevertheless in this limited context (of
simultaneous action of both wheels with no slip
on a flat surface) we could create a T(q1,q2,q3)
matrix to describe the forward kinematics of the
blue shovel
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We assume that the
although it could not be made to fit the
Denavit Hartenberg convention, for example.
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We assume that the
Rather than do that as the basis for our inverse
kinematics, we instead will work with a direct
model for the kinematics of P, as per HW1.
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This is a planar task, and we assume that
previous action of the base has been conducted in
order to place the robot in the desired (near-)
vertical plane.
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We will take x0, y0 to be the juncture at which
P would begin its action of traversing the red
dashed line.
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We will take x0, y0 to be the juncture at which
P would begin its action of traversing the red
dashed line.
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The coefficients Co C1 C2 of the path of the
line y(x)CoC1xC2x2 have been chosen in order
to produce a particular scoop volume.
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Forward kinematics of OPX OPY and f.
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f
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q1
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q2
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f -q1 q2 - Y
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f -q1 q2 - Y
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OPY yp(q1,q2) L1 sin(q1) L2 sin(q1-q2)
y0
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OPY yp(q1,q2) L1 sin(q1) L2 sin(q1-q2)
y0
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OPX xp(q1,q2,q3) L1 cos(q1) L2
cos(q1-q2) Rq3 x0
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OPX xp(q1,q2,q3) L1 cos(q1) L2
cos(q1-q2) Rq3 x0
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OPX xp(q1,q2,q3) L1 cos(q1) L2
cos(q1-q2) Rq3 x0
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OPX xp(q1,q2,q3) L1 cos(q1) L2
cos(q1-q2) Rq3 x0
x0 is known
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x0 is known, together with y0
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Also, L1 L2 and Y are known
x0 is known, together with y0
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Note that, as things are defined, Co will always
be zero.
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y(x)C1xC2x2
Note further that, as shown, C1 is negative, and
C2 positive.
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y(x)C1xC2x2
Take xfinal to be the nonzero x such that
y(xfinal)0.
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y(x)C1xC2x2
Take xfinal to be the nonzero x such that
y(xfinal)0, i.e. xfinal-C1/C2
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y(x)C1xC2x2
Prob. 1a. For any x, 0ltxltxfinal, determine the
target angle f as defined herein. Be sure to
avoid quadrant ambiguities by using the
dual-argument inverse tangent of Craig. Generate
results in terms of C1 C2
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y(x)C1xC2x2
Prob. 1b. For any x, 0ltxltxfinal, determine the
robot angles q1 q2 as defined herein. Do this in
radians and in terms of given yo L1 L2 Y C1 C2.
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y(x)C1xC2x2
Prob. 1c. For any x, 0ltxltxfinal, determine the
robot angle q3 as defined herein. Do this in
radians and in terms of given xo yo L1 L2 Y C1 C2.
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y(x)C1xC2x2
Prob. 1d. For all x, 0ltxltxfinal, given R150mm
L1300mm L2450mm Y1.0 radian C1-1 x018mm
y0150mm C2(500mm)-1, use your
inverse-kinematics relations of 1a-1c to make a
single plot superimposing each of the three robot
angles as functions of x.