Compare the forward kinematics of a nonholonomic robot to those of a holonomic robot
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Compare the forward kinematics of a nonholonomic robot to those of a holonomic robot. Note that our textbook Craig will provide a very useful and general way to develop the forward kinematics of a Holonomic robot. E P A P Recall, that a holonomic robot will (kinematically) “repeat”.

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Slide2 l.jpg

Note that our textbook Craig will provide those of a holonomic robot.

a very useful and general way to

develop the forward kinematics of a

Holonomic robot.


Slide3 l.jpg

E those of a holonomic robot.P

AP


Recall that a holonomic robot will kinematically repeat l.jpg
Recall, that a those of a holonomic robot.holonomic robot will (kinematically) “repeat”.


Recall that a holonomic robot will kinematically repeat5 l.jpg
Recall, that a those of a holonomic robot.holonomic robot will (kinematically) “repeat”.

  • In general, by returning the internal, servo-controlled angles to earlier, taught values, you will bring about a return of the borne load to its corresponding location in space.


Recall that a holonomic robot will kinematically repeat6 l.jpg
Recall, that a those of a holonomic robot.holonomic robot will (kinematically) “repeat”.


Slide7 l.jpg
However there is a subtle distinction between holonomic robots and nonholonomic robots that can be appreciated with this animation.


Slide8 l.jpg
The nonholonomic robot will not repeat by merely returning the internal angles of rotation to their previous values.



Slide10 l.jpg

Let’s consider a “bird’s eye view” of the axis that connects the wheelchair’s two wheels.


Bird s eye view of axle l.jpg

Bird’s eye view of axle. connects the wheelchair’s two wheels.


Bird s eye view of axle12 l.jpg

Bird’s eye view of axle. connects the wheelchair’s two wheels.













Slide25 l.jpg

The wheelchair can get away with a offer some distinct three-degree-of-freedom movement using just two servomechanisms, one for each wheel, precisely because it is a nonholonomic system.


Slide26 l.jpg
How will we control such a robot if teach/repeat, the standard for holonomic robots, cannot be applied?


One possibility is to track a line in the floor l.jpg
One possibility is to track a line in the floor. standard for holonomic robots, cannot be applied?


One possibility is to track a line in the floor but this can get messy l.jpg
One possibility is to track a line in the floor. But this standard for holonomic robots, cannot be applied?can get messy!


Also tracking of a physical line doesn t permit deviation from the path in the event of an obstacle l.jpg
Also tracking of a physical line doesn’t permit deviation from the path in the event of an obstacle.


And line tracking doesn t permit pivoting in tight spaces l.jpg
And line tracking doesn’t permit pivoting in tight spaces. from the path in the event of an obstacle.


You could track a wall instead l.jpg
You could track a from the path in the event of an obstacle.wall instead.

  • Many early developments of this kind.

  • Shakey

  • The Kent floor-cleaning robot


Slide32 l.jpg
Alternatively, create sensor-based autonomy and reasoning, such as “simultaneous localization and mapping”, SLAM.


Slide33 l.jpg

So far SLAM methods have not produced much fruit. Hard to achieve absent humans’ ability of object recognition. The “correspondence problem”.


Random motion l.jpg
Random motion. achieve absent humans’ ability of object recognition. The “correspondence problem”.


Intelligent random motion l.jpg
Intelligent achieve absent humans’ ability of object recognition. The “correspondence problem”. random motion.


Slide36 l.jpg

Compare Jacobians for our two two-degree-of-freedom robots. achieve absent humans’ ability of object recognition. The “correspondence problem”.


Slide46 l.jpg

E achieve absent humans’ ability of object recognition. The “correspondence problem”.P

AP


Z y x euler angles l.jpg

Z-Y-X Euler Angles achieve absent humans’ ability of object recognition. The “correspondence problem”.


Z y x euler angles48 l.jpg

Z-Y-X Euler Angles achieve absent humans’ ability of object recognition. The “correspondence problem”.

- Just three numbers are needed to specify the orientation of one set of axes relative to another.


Z y x euler angles49 l.jpg

Z-Y-X Euler Angles achieve absent humans’ ability of object recognition. The “correspondence problem”.

Just three numbers are needed to specify the orientation of one set of axes relative to another.

One possible set of these numbers is the Z-Y-X Euler angles


Consider the a and b frames shown below l.jpg
Consider the {A} and {B} frames shown below. achieve absent humans’ ability of object recognition. The “correspondence problem”.


Slide51 l.jpg

How can we define just three quantities from which we can express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Beginning with the a frame rotate a positive a about the z a axis l.jpg
Beginning with the {A} frame, rotate a positive express all nine elements of the rotation matrix that defines the relative orientations of these frames?a about the ZA axis.


Call this new frame b l.jpg
Call this new frame {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Note the rotation matrix between a and b l.jpg
Note the rotation matrix between {A} and {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Note the rotation matrix between a and b55 l.jpg
Note the rotation matrix between {A} and {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Note the rotation matrix between a and b56 l.jpg
Note the rotation matrix between {A} and {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Note the rotation matrix between a and b57 l.jpg
Note the rotation matrix between {A} and {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Note the rotation matrix between a and b58 l.jpg
Note the rotation matrix between {A} and {B’} express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Next consider just the intermediate b frame l.jpg
Next consider just the intermediate {B’} frame. express all nine elements of the rotation matrix that defines the relative orientations of these frames?


Consider a positive rotation b about the y b axis l.jpg
Consider a positive rotation express all nine elements of the rotation matrix that defines the relative orientations of these frames?b about the YB’ axis.


Suppose the second rotation b had instead occurred about the original y a axis l.jpg
Suppose the second rotation express all nine elements of the rotation matrix that defines the relative orientations of these frames?b had instead occurred about the original YA axis?


Suppose the second rotation b had instead occurred about the original y a axis62 l.jpg
Suppose the second rotation express all nine elements of the rotation matrix that defines the relative orientations of these frames?b had instead occurred about the original YA axis?




Take the last rotation g to be about the x b axis l.jpg
… take the last rotation original Yg to be about the XB” axis.


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