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