A. CHEMORI M. ALAMIR C.CANUDAS

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Workshop on Humanoid and Legged Robots
HLR
Some developments on under-actuated robotic
systems at LAG Application to Rabbit
A. CHEMORI M. ALAMIR
C.CANUDAS
Laboratoire d'Automatique de Grenoble. UMR
5528 BP46,Domaine Univesitaire, 38402 Saint
Martin d'Hères, France. Ahmed.Chemori(Mazen.Alami
r,Carlos.Canudas-de-Wit)_at_inpg.fr
October 14-15, 2004 Université de Metz, Metz ,
France
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Outline of the presentation

• Control approach 1 Nonlinear predictive
  control scheme

• The dynamic walking control problem
  formulation
• Summary of the approach the basic idea
• The stability issue
• Some simulation results

• Control approach 2 Orbital stabilization
  scheme

• Principal of virtual constraints
• Virtual limit system
• The control design
• Some simulation results

• Concluding remarks

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Appr1 The dynamic walking control problem
formulation

• Objective stable dynamic walking control of
  the biped
• Assumptions walking in the sagittal plan, on
  a level surface, without obstacles

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Appr1 Summary of the approach the basic idea
7 d.o.f Nonlinear Dynamic
Single support assumption
Holonomic constraints
Independent coordinates
Dependent coordinates
Under-actuation
actuated coordinates
Unactuated coordinates
Directly controlled variables
Indirectly controlled variables
Coupling
Zero dynamic trajectories
Parameterized trajectories
Optimization
Prediction
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Appr1 Summary of the approach the basic idea
impact
impact
. . . . . .
. . . . . .
indirectly controlled variables just after the
impact
Prediction Impact map
Computation of the optimization parameter
optimization
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Appr1 The stability issue
since
then the overall stability depends on the
stability of the sequence
using and the impact map leads to
the prediction over gives
In multi-step (k0 step) form
A key quantity in stability analysis, see the
following
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Appr1 The stability issue
Proposition If for some
there exist and such that

Example of typical situation convergence to a
neighborhood of a limit cycle
bisector
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Appr1 Some simulation results
Actuated
Control forces
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Appr1 Some simulation results
Stability analysis
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Appr1 Some simulation results
Unactuated
Phase portrait
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• Control approach 1 Nonlinear predictive
  control scheme

• The dynamic walking control problem
  formulation
• Summary of the approach the basic idea
• The stability issue
• Some simulation results

• Control approach 2 Orbital stabilization
  scheme

• Principal of virtual constraints
• Virtual limit system
• The control design
• Some simulation results

• Concluding remarks

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Appr2 Principal of virtual constraints
Balancing in under-actuated systems
Create a sustained oscillation for the complete
set of n-coordinated in under-actuated
mechanical systems (high-dimensional inverted
pendulums).

• Learn how to create stable orbits in
  under-actuated systems
• General (constructive) method for pendular
  systems

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Appr2 Principal of virtual constraints
Other examples
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Appr2 Virtual limit system
Generalized inverted pendulum
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Appr2 Virtual limit system
General properties of virtual limit system
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Appr2 The control design
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Appr2 Summary of the main result
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Appr2 Summary of the main result
Control bloc diagram
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Appr2 Some simulation results
Evolution of the Rabbit restricted to 3 links
Equilibrium position (center)
Motion over a period
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Appr2 Some simulation results
Simulated closed-loop responses
motion of the unactuated angle
Behavior of
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Concluding remarks
- Two proposed control approaches -
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Thank you for your attention