1. M. Lopez-Martinez, J.A. Acosta and
J.M. Cano
In Proceedings of ….. Conference 2008
- 6
-

2. J. Á. Acosta was born in Huelva, Spain. He obtained both the Servo-
Electrical and Mechanical Engineering degrees at the University of Huelva,
and the Electrical Engineering degree at the University of Seville, Spain.
Ph.D. degree in 2004, in the Department of Systems Engineering and
Automatic Control at the University of Seville.
Ph.D. European Award in 2005.
Outstanding Paper Award in the IEEE Transactions on Automatic Control
journal in 2006.
In 1999 he joined as Research Assistant in that Department, where he is
currently Professor and Researcher member of the Automatic and Robotics
Institute. He also has been Visitor in the Laboratoire des Signaux &
Systèmes (CNRS, France) repeatedly since 2005 up to today and, Academic
Visitor Researching in the Electrical & Electronic Engineering Department as
member of the Control & Power Group at Imperial College London in 2008
and 2009.
His research interests are in the fields of Nonlinear Control of Dynamical
Systems with emphasis on Electro-Mechanical and Robotic systems. He is
author of a number of research publications on Non-linear Control System
Theory in Internationals Conferences and Scientific Journals.

3. Sliding Mode Control
Feedback Linearization
Control Lyapunov Function

4. The sliding condition:

5. Spong has shown that all underactuated system
M. W. Spong where and
Director of
Center for
can be globally partially linearized using a ch
Autonomous
Engineering
Systems and
Robotics at the where
University of and .
Illinois
at Urbana-
References:
Champaign
M. W. Spong, “Energy based control of a class of underactuated mechanical systems”,

6. References:
M. W. Spong, “Energy based control of a class of underactuated mechanical systems”,

7. Remarks:
 fully linearized system (using a change of control) 
Impossible
 partially linearized system (q2 is transformed into a double
integrator)  Possible
 the new control u appears in the both nonlinear (internal
dynamics) (q1, p1)
subsystem and linear (q2, p2) subsystem
 the system has m-vector relative degree (2,…,2)T with
respect to the output q2
 this procedure is called collocated partial linearization
References:
 suitable “Energy based control ofcontroller design mechanical systems”,
M. W. Spong, for energy based a class of underactuated

8. The proposed “constructive” output: redesign:
Output
(the corresponding output to the
passive output for the storage
Defining the free smooth function
function)
(decreasing of storage function of zero dynamics)
Consider
The external controller:
Let Constructive linearizing law:
reduces the input-output map to
References: J. A. Acosta and M. Lopez-Martinez “Constructive Feedback
Linearization of Mechanical Systems with Friction and Underactuation Degree

11. Dynamic equations: Find the linearizing
law from
The partially feedback-linearized system: control input:
The total
Constructive sliding surface:
with

14. 2 dof. Underactuated Mechanical Systems
Collocated Partial Feedback Linearization
Design the Fictitious Output
(using the concept of passive systems)
Constructive Design of Sliding Surface
Feedback Linearization Controller + Sliding Mode Co