Use of the Jacobian for laserspot convergence

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Use of the Jacobian for laser-spot convergence

• Spot Target

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Use of the Jacobian for laser-spot convergence

• Spot Target

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Suppose you wanted to use the experience of this
procedure in order to update or improve locally
your approximation to J.
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Recall the table constructed for HW4.
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Recall the table constructed for HW4.
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As new samples are acquired, we can apply locally
more highly weighted input.
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As new samples are acquired, we can apply locally
more highly weighted input.
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As new samples are acquired, we can apply locally
more highly weighted input.
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Due to linearity of the ri in the estimation
parameters b, this updating process can be
streamlined.
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Due to linearity of the ri in the estimation
parameters b, this updating process can be
streamlined.
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As well see, this streamlined procedure does not
require retention in memory of individual batch
data.
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This rapid, recursive b-updating ability is not
too consequential w.r.t. our immediate problem of
laser-spot convergence.
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The reason has to do with the surface/slope
discontinuities of objects on which the
laser-spot falls.
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The reason has to do with the surface/slope
discontinuities of objects on which the
laser-spot falls.
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The reason has to do with the surface/slope
discontinuities of objects on which the
laser-spot falls.
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Moreover, the ability to see/pan/tilt
sequentially and quickly obviates the need for a
highly precise Jacobian J.
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With CSM, the estimated nonlinear
parameters C1-C6, however, change very
slowly/smoothly.
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This fact allows us to exploit the more local
samples and command finite
robot-joint rotations in a way that consumates
the maneuver with very high precision.
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There is a very interesting variation on our
laser-spot-convergence problem
that also has this property of slow, continuous
change of the linear parameters.
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There is a very interesting variation on our
laser-spot-convergence problem
that also has this property of slow, continuous
change of the linear parameters.
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Suppose that, instead of a laser pointer,
the camera itself is placed upon a pan/tilt unit.
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Suppose that our objective is to move the target
to the center of camera space.
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Suppose that our objective is to move the target
to the center of camera space.
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Suppose that our objective is to move the target
to the center of camera space.
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Suppose that our objective is to move the target
to the center of camera space.
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Here, it definitely pays to update the Jacobian
en route to the terminus.
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Here, it definitely pays to update the Jacobian
en route to the terminus.
And the update can be accomplished identically
with our earlier example.
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Here, it definitely pays to update the Jacobian
en route to the terminus.
And the update can be accomplished identically
with our earlier example.
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The model being once again linear, we can apply
the streamlined procedure which does not require
retention in memory of individual batch data.
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Suppose we had a slightly harder problem.
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Suppose we had a slightly harder problem.
Suppose the target object, the one we wish to
draw into the center of camera space, is itself
moving.
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Even if the camera remained stationary, i.e. no
pan/tilt,
the target body would move in camera space.
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If the camera does pan/tilt then the movement of
the target body in camera space becomes a
consequence of both the autonomous physical
movement of the body, and the camera pan/tilt.
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Using two cameras to center a body simultaneously
in two images was one
thought behind active vision widely
researched in the early through mid 1990s.
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A number of startup companies refined and
marketed this kind of dual pan/tilt/etc.
platform for use with active vision to guide
robots, and for other purposes.
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Even in the presence of a moving target, we could
still use some reasonable Jacobian
to try to keep the target in the center of the
image.
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But is there any way to use our observations to
improve upon our Jacobian J?
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But is there any way to use our observations to
improve upon our Jacobian J?
Or, maybe even better still, to try to predict
the autonomous movement of the target?
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But is there any way to use our observations to
improve upon our Jacobian J?
Or, maybe even better still, to try to predict
the autonomous movement of the target?
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Consider a simple model of how the
feature X moves autonomously in camera space.
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In other words we assume that the camera-space
velocity is constant.
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We could approximate vxc vyc using just two
consecutive samples.
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Substituting the known Dt together with Dxc Dyc
into the above equation leaves us with an
approximation to vxc vyc.
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Suppose we did this, and suppose the time
interval to the next sample is this same Dt.
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Suppose we did this, and suppose the time
interval to the next sample is this same Dt.
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This would be our prediction.
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Not too good, and it gets worse as the pairs move
on.
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It can be improved, however, if we can reduce the
interval Dt somewhat.
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But what if we wanted to use redundant data,
acquired over the course of our experiment?
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But what if we wanted to use redundant data,
acquired over the course of our experiment?
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This too could be achieved by way of
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This too could be achieved by way of
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This too could be achieved by way of
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This too could be achieved by way of
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This too could be achieved by way of
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This too could be achieved by way of
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This too could be achieved by way of
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Depending upon the speed of the autonomous motion
of the target, it may be prudent to set W in such
a way as to slow updates in J and speed updates
in v.
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Depending upon the speed of the autonomous motion
of the target, it may be prudent to set W in such
a way as to slow updates in J and speed updates
in v.
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Depending upon the speed of the autonomous motion
of the target, it may be prudent to set W in such
a way as to slow updates in J and speed updates
in v.
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Depending upon the speed of the autonomous motion
of the target, it may be prudent to set W in such
a way as to slow updates in J and speed updates
in v.
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Depending upon the speed of the autonomous motion
of the target, it may be prudent to set W in such
a way as to slow updates in J and speed updates
in v.
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The ability to apply new data or observations
in this way is a feature of the Kalman Filter.
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Among its virtues the KF allows for specification
of parameters that result in relatively slow
updates of the Jacobian elements, the first four
elements of b
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Among its virtues the KF allows for specification
of parameters that result in relatively slow
updates of the Jacobian elements, the first four
elements of b
while at the same time allowing for a much more
sensitive and responsive updating of the
camera-space velocity components.
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Such sensitivity is clearly needed in this case
where the camera-space velocity components are
changing fast relative to the sampling frequency.
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Such sensitivity is clearly needed in this case
where the camera-space velocity components are
changing fast relative to the sampling frequency.
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Such sensitivity is clearly needed in this case
where the camera-space velocity components are
changing fast relative to the sampling frequency.
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Such sensitivity is clearly needed in this case
where the camera-space velocity components are
changing fast relative to the sampling frequency.
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Such sensitivity is clearly needed in this case
where the camera-space velocity components are
changing fast relative to the sampling frequency.
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In addition, the KF allows the user to specify
the extent of confidence in any initial guess of
the J and v elements.
Thus, if initial confidence in (say) J is low,
the KF will initially adjust these based upon
incoming data relatively rapidly.
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The Extended Kalman Filter which allows for
approximation of the KF for cases where
estimation parameters appear nonlinearly is the
basis for our wheelchair example.
Both the KF and EKF are recursive they do not
require batch retention of all observations
that are used to factor in to current estimates,
as we will see later.
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Meanwhile, the way we have implemented CSM is
batch (i.e. is not recursive and not based on
KF or EKF algorithms.)
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Meanwhile, the way we have implemented CSM is
batch (i.e. is not recursive and not based on
KF or EKF algorithms.)
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Meanwhile, the way we have implemented CSM is
batch (i.e. is not recursive and not based on
KF or EKF algorithms.)
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Meanwhile, the way we have implemented CSM is
batch (i.e. is not recursive and not based on
KF or EKF algorithms.)
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Meanwhile, the way we have implemented CSM is
batch (i.e. is not recursive and not based on
KF or EKF algorithms.)
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Recall the nominal kinematics gx gy gz for
this robot.
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Recall the nominal kinematics gx gy gz for
this robot.
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Recall the nominal kinematics gx gy gz for
this robot.
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What would gx gy gz be for the holonomic part
of our robot?
Ackn B. Marek.
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What would gx gy gz be for the holonomic part
of our robot?
Ackn B. Marek.
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Direction cosine matrix between 1 and 2 frames
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Relative displacement of origin of 2 frame w.r.t.
origin of 1 frame referred to the 1 frame.
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Cascading
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After multiplication/simplification
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Consider point P fixed to the blue member.