The document discusses nonholonomic constraints and their application to wheeled mobile robots. It provides definitions of key concepts like degrees of freedom, kinematic constraints, and controllability in relation to nonholonomic systems. Examples of nonholonomic systems include unicycles. The document also references control methods for nonholonomic systems, noting that smooth feedback cannot stabilize them at a point.

1. Behzad Samadi, Research Engineer, Maplesoft

2. Pioneer 2 DX
Lexus Parking Assist
LAAS-CNRS h2
Beam
Rattleback

3. Pure rolling, no slip
Nonholonomic constraint:
Any configuration can be reached
Three Degrees of Freedom
Refs. [1] and [2]

4. Generalized coordinates:
Generalized velocities:
Geometric constraint:
Kinematic constraints:
Pfaffian kinematic constraints:
‘s are independent.
Ref. [1]

5. Kinematic constraints:
These constraints can be integrated to
geometric constraints:
One Degree of Freedom
Ref. [2]

6. Given a set of kinematic constraints, how can
we tell if they are nonholomic?
Nonholonomic constraints allow accessing all
configurations (do not decrease Degrees of
Freedom).
It is a controllability question.
Ref. [1]

7. For a system with the following constraints,
the kinematic model is:
Ref. [1]

8. Nonholonomic constraint:
Kinematic model:
is the driving input and is the steering
input.
Ref. [1]

9. It can be shown that a system with the
following kinematic model,
is nonholonomic if the set
is not closed with respect to Lie bracket:
Ref. [1]

10. Let . A filtration
generated by is the sequence where:
The involutive closure of is where is
the smallest number such that:
If , there are geometric
constraints and nonholonomic
constraints.
If , the system is controllable and
fully nonholonomic.
Ref. [1]

11. Therefore unicycle is a nonholonomic
system.
Ref. [1]

12. Brockett Theorem (1983): If the system
is locally asymptotically stabilizable at with
smooth state feedback , then the
image of the map contains
some neighborhood of (a necessary
condition)
Nonholonomic systems cannot be stabilized
at a point by smooth feedback.
There is no time invariant linear controller for
stabilizing a nonholonomic system!
Ref. [1]

13. Dynamic model:
Using the kinematic model, we have:
Ref. [1]

14. Choose:
Then, we have:
Ref. [1]

15. Reference model:
where the timing law is :

16. Dynamic Model:
Therefore:

17. Let us:
To find consider:
Control law:

18. Vector Control:
Control law:

19. 1. G. Oriolo, Control of Nonholonomic Systems, Lecture Notes,
http://www.dis.uniroma1.it/~oriolo/cns/cns_slides.pdf
2. M. Manson, Nonholonomic Constraint, Lecture Notes,
http://www.cs.rpi.edu/~trink/Courses/RobotManipulation/lectures/lecture
5.pdf
3. G. Oriolo, Wheeled Mobile Robots: Modeling, Planning and Control, 2010
SIDRA Doctoral School on Robotics, http://bertinoro2010.dii.unisi.it

20. ありがとうございます
Thank you