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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.

### Z-Y-X Euler Angles

### Z-Y-X Euler Angles

Note that our textbook Craig will provide

a very useful and general way to

develop the forward kinematics of a

Holonomic robot.

EP

AP

Recall, that a 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.

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

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

What is the nature of the forward kinematics of this nonholonomic robot?

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

Development of nonholonomic equations for numerical integration.

Imperfect tracking won’t affect the holonomic robot’s terminal pose.

Imperfect tracking won’t affect the holonomic robot’s terminal pose.

Imperfect tracking won’t affect the holonomic robot’s terminal pose.

Imperfect tracking won’t affect the holonomic robot’s terminal pose.

But imperfect tracking has a cumulative effect on the nonholonomic robot.

But imperfect tracking has a cumulative effect on the nonholonomic robot.

But imperfect tracking has a cumulative effect on the nonholonomic robot.

But imperfect tracking has a cumulative effect on the nonholonomic robot.

For all their disadvantages, nonholonomic, wheeled robots offer some distinct advantages.

For all their disadvantages, nonholonomic, wheeled robots offer some distinct advantages.

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

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. But this can get messy!

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

You could track a wall instead.

- Many early developments of this kind.
- Shakey
- The Kent floor-cleaning robot

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

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

EP

AP

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

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

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?

Suppose the second rotation b had instead occurred about the original YA axis?

Suppose the second rotation b had instead occurred about the original YA axis?

Suppose the second rotation had instead occurred about the original YA axis?

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