1. Session : FrB03 Mechanical Systems
IEEE Multi-Conference on Systems and Control
October , Antibes, France
IEEE MSC 2014
Session : FrB03 Mechanical Systems
External Disturbance Rejection in IDA-PBC Controller for Underactuated Mechanical Systems : From Theory to Real Time Experiments
Nahla Khraief Haddada , Ahmed Chemorib , Safya Belghitha
a National School of Engineers of Tunis, University of Tunis El Manar,
BP.37, le Belvédère, 1002 Tunis, Tunisia
b Laboratory of Informatics, Robotics and Microelectronics of Montpellier
LIRMM, University of Montpellier 2 - CNRS
161, rue Ada 34095
Montpellier, France
Note that the uniform bounds for the system’s state and control signals are expressed in terms of the impulse response of proper BIBO-stable transfer functions, which
correspond to the L1-norms of the underlying systems. Consequently, the corresponding control architectures are referred to as L1 adaptive controllers.
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2. Outline of the presentation
Background on IDA-PBC for mechanical systems
Problem of disturbed underactuated mechanical systems
Problem formulation
Main result
Application : Inertia wheel inverted pendulum (IWIP)
Inertia wheel control systems
IWIP : Description and modeling
IDA-PBC controller
Simulation results
Real-time experimental results
Experimental test bed
Scenario 1 : Nominal case
Scenario 2 : Punctual external disturbances
Scenario 3 : Persistent external disturbances
Conclusion & future work

3. Background on IDA-PBC for mechanical systems
Disturbed systems
Application
Experiments
Conclusion
Background on IDA-PBC for mechanical systems
IDA-PBC : Interconnection and Damping Assignment - Passivity Based Control

4. Background on IDA-PBC for mechanical systems
Disturbed systems
Application
Experiments
Conclusion
Background on IDA-PBC for mechanical systems
Mechanical systems : Equation of motion (Lagrange formulation)
The total energy can be written as :
: is the generalized position
: is potential energy
: is the momenta
: is the inertia matrix
Equation of motion (Port-Hamiltonian formulation) with no natural damping
: is defined by the manner in which the control enters into the system
: is the control input
If  fully actuated mechanical systems
If  underactuated mechanical systems

5. Background on IDA-PBC for mechanical systems
Disturbed systems
Application
Experiments
Conclusion
Background on IDA-PBC for mechanical systems
IDA-PBC is based on two steps [Ortega et al ] :
Energy shaping : modify the total energy function to assign the desired equilibrium
Damping injection : to achieve asymptotic stability
Consider the desired energy function (in closed loop) :
: is desired potential energy function  isolated equilibrium
: is the closed-loop inertia matrix
To achieve energy shaping
To inject damping

6. Problem of disturbed underactuated mechanical systems
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Problem of disturbed underactuated mechanical systems
Problem formulation
Main result

7. IDA-PBC for disturbed underactuaed mechanical systems
Disturbed systems
Application
Experiments
Conclusion
IDA-PBC for disturbed underactuaed mechanical systems
Problem Formulation :
Consider the case of disturbed underactuated mechanical systems
Matched disturbances Unmatched disturbances
Objective : Find the sufficient conditions on the bounds of the disturbances that the IDA-PBC controller is able to reject and keep the equilibrium asymptotically stable despite these disturbances
The two cases of matched and unmatched disturbances are treated separately

8. IDA-PBC for disturbed underactuaed mechanical systems
Disturbed systems
Application
Experiments
Conclusion
IDA-PBC for disturbed underactuaed mechanical systems
Main result :
Proposition 1 : Case of matched disturbances
Proposition 2 : Case of unmatched disturbances
Details proofs : see the paper

9. Application : Inertia wheel inverted pendulum (IWIP)
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Application : Inertia wheel inverted pendulum (IWIP)
Inertia wheel controlled systems
IWIP : Description and modeling
IDA-PBC controller
Simulation results

10. Inertia wheel controlled systems : Basic principle
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Inertia wheel controlled systems : Basic principle
The effect of a torque (e.g. gravity / excitation moments)
Causes a variation in the spin axis
Reaction of a spinning wheel to compensate this effect
Provides an effective means of motion control and balance (inertia wheel)
Two early examples of application :
Murata boy
Single gimbal active stabilizer unit
With 40 inch diameter & 4.5 inch thick
Flywheel operated at rpm
The Schilovski Gyrocar (1914)
Twin type active stabilizer system (3000 rpm)
40 feet long and weighted 22 tons
Developed primary for military applications
Brennan monorail (1924)

11. Inertia wheel controlled systems : Application examples
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Inertia wheel controlled systems : Application examples
[Townsend et al 2007]
Robotics
Marine systems
Aerospace
Gyrostabilizer
Applications
Monorails
Underwater vehicles
Cars
Academic
IKURA AUV
ECP 750
Two-wheel gyro car

12. Our system : Inertia Wheel Inverted Pendulum (IWIP)
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Our system : Inertia Wheel Inverted Pendulum (IWIP)
Inclinometer
Pendulum body
Inertia wheel
Active articulation
Passive articulation
Frame

13. The actuator is controlled to produce a torque on the inertia wheel
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : How does it work ?
The actuator is controlled to produce a torque on the inertia wheel
Torque can produce an acceleration of the rotating wheel
Thanks to the dynamic coupling, a torque acting on the passive joint is generated
This passive joint can be controlled through the acceleration of the inertia wheel G2 G1 O
Rotation G2 G1 O A B
Initial mechanical model
Equivalent mechanical model

14. Three moments are acting on the passive joint :
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : How does it work ?
Three moments are acting on the passive joint :
Moment relative to the force :
Moment du to the gravity force :
One moment (torque of actuator) is acting on the active joint (inertia wheel) O
Rotation 1
Rotation 2

15. IWIP : Dynamic modeling
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : Dynamic modeling
Generalized coordinates :
Propose to use the formalism of Lagrange :
The application of Lagrange principle gives :
Can be rewritten in the following Hamilton’s equations of motion :

16. Consider the following IDA-PBC controller [Ortega et al 2002] :
Disturbed systems
Application
Experiments
Conclusion
IWIP : The controller
Consider the following IDA-PBC controller [Ortega et al 2002] :
Can be rewritten in terms of q and p as :
With :

17. IWIP : Simulation results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : Simulation results
Control in nominal case

18. IWIP : Simulation results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : Simulation results
Control in case with matched disturbances

19. IWIP : Simulation results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
IWIP : Simulation results
Control in case with unmatched disturbances

20. Real-time experimental results 3 experimental scenarios
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Real-time experimental results
3 experimental scenarios
Nominal case
Punctual external disturbances
Persistent external disturbances

21. Description of the experimental testbed
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Description of the experimental testbed
encoder
Micro strain FAS-G
Maxon EC-Powermax 30 (DC Brushless)
Control PC
Power supply (12V)
Speed variator
Inclinometer
Pendulum body
Inertia wheel
Input/output card
Mechanical part
Electric/electronic part

22. Real-time experimental results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Real-time experimental results
No external disturbances
Scenario 1 : Control in nominal case

23. Real-time experimental results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Real-time experimental results
Apply some external collisions with the body of the pendulum
Scenario 2 : Punctual external disturbances

24. Real-time experimental results
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Real-time experimental results
Hitch small mass up to the body of the pendulum
Scenario 3 : Persistent external disturbances

25. Conclusion & future work
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Conclusion & future work

26. Conclusion & future work
IDA-PBC
Disturbed systems
Application
Experiments
Conclusion
Conclusion & future work
Problem: Disturbance rejection in control of underactuated mechanical systems
Proposed solution : IDA-PBC for stabilisation
Two types of disturbances : matched and unmatched
Sufficient stability conditions : despite the considered disturbances
Validation:
Case study : Inertia wheel inverted pendulum
First validation in simulation
Real-time experiments (nominal and disturbed cases)
Future work:
Case of more complex underactuated mechanical systems
An adaptive extension  Compensate uncertainties on the parameters