Continuity of the stack of tasks under discrete scheduling operations

1. Continuity of the stack of tasks under discrete scheduling operations Francois Keith(CNRS-UM2 LIRMM, France – CNRS-AIST JRL, UMI3218/CRT Japan) Pierre-Brice Wieber (INRIA Rhône-Alpes, France) Nicolas Mansard (CNRS-LAAS, France) Abderrahmane Kheddar(CNRS-UM2 LIRMM, France – CNRS-AIST JRL, UMI3218/CRT Japan)

2. Realization of a robotic mission • Explicit trajectory • Continuous control law • Lacks of reactivity • Implicit trajectory • Based on task function • Easy on-line adaptation to environment changes

3. - + Sensor Sensor Task Function • Defined by three elements: • A task space (error between current and desired sensor values) • A reference behavior of the error • A Jacobian • Regulation of the error [Samson91]

4. eLeft Arm eRight Arm Realization of a set of tasks • Slacked hierarchy • Task weighting • User-defined coefficients • Blur motion • Strict hierarchy (Stack of tasks) • Task realized in the null space left by the higher priority ones. [Salini10] (low priority) eHead eWalk [Slotine91] (high priority)

5. Stack of Tasks • Pseudo-inverse approach not suited • Discontinuities near singularities • Use of damped inverse • Continuous control law for a fixed set of tasks

6. eLeft Arm eRight Arm Event related discontinuities • Discontinuity due to • Additional control value • Change of null space for lower priority tasks Smoothing methods • Additional control  insertion gain • Null space  ? eHead eWalk

7. Swap-based approach • Operation between neighbouring tasks • Insertion and removal only at the end of the stack • Pairewise swaps eHead eLeft Arm eLeft Arm eHead eLeft Arm eLeft Arm eRight Arm eRight Arm eRight Arm eRight Arm eHead eWalk eWalk eWalk eWalk

8. Swap process • 2-task layer (without damping) Swap phase eLeft Arm eLeft Arm eLeft Arm eRight Arm era eHead eHead eHead eRight Arm eWalk eWalk eWalk 0+ 1-

9. Swap process • 2-task layer (without damping) • At the limits ( ), the two following problems are equivalent. • (control law during the swap) • (control law corresponding to a strict hierarchy) • Continuity at the limits if there is no damping process. [Van Loan 84]

10. Swap and damping • Introduction of discontinuities at the limits eLeft Arm eRight Arm eHead eWalk

11. Swap and damping • Introduction of discontinuities at the limits eLeft Arm eRight Arm eHead eWalk

12. Swap and damping • Introduction of discontinuities at the limits when eLeft Arm eRight Arm eLeft Arm 3 tasks layer  eHead eHead eWalk eWalk

13. Swap by linear interpolation • External merge eLeft Arm eHead era era eHead eWalk • Flaws • Additional computation cost • No optimization-based formulation • Continuity of the control law during the events

14. Experiments • Simulation of a task sequence. • Insertion and removal of 3 tasks sharing several dofs Control law with damping Control law with damping and smoothing

15. Experiments • Simulation of a task sequence. • Insertion and removal of 3 tasks sharing several dofs • Continuous evolution of the control law • Events delayed

16. Experiments Unnoticeable differences... But better tracking results

17. Conclusion and Perspectives • Conclusion + Continuous control law during the events • Time consuming • Compromise between reactivity and continuity. • Perspective • Dynamic inverse control • Test stability issues

18. Realization of a robotic mission • Explicit trajectory • Continuous control law • Lacks of reactivity • Implicit trajectory • Based on task function • Easy on-line adaptation to environment changes • Possible discontinuity of the control law