On sufficient conditions to keep differential flatness under the addition of new inputs

Abstract

It is well known that a controllable nonlinear system will retain its controllabality when new actuator inputs are added to it. In this article, we ask the question if a system, linearisable by static or dynamic feedback, will retain this property when new actuator inputs are added to it. Alternatively, a system may be linearisable after removing one or more inputs from it. This question is important in the design of robotic systems from the perspective of trajectory planning and control, specially if they are under-actuated. The goals of this article are as follows: (i) using counter examples, we first show that feedback linearisability may not be preserved when new inputs are added to a robotic system, (ii) sufficient conditions are determined when a system will retain this property under the addition of new inputs. The theory is illustrated through some examples from the robotics field.

Keywords:

• feedback linearisability
• differential flatness
• under-actuated robots

Acknowledgements

The authors wish to thank PhD student Vivek Sangwan for fruitful discussions on the planar chain example. We used this example to first realise that feedback linearisation property could be lost by adding inputs. We also want to thank Technical University of Catalonia for its mobility program that provided travel support to Dr Franch to visit the University of Delaware. We also gratefully acknowledge the research support from the WCU (World Class University) program through the Korea Science and Engineering Foundation funded by the Ministry of Education, Science and Technology (No. R32-2008-000-10022-0).