1. Virtual Gravity Control for Swing-Up pendulum K.Furuta *, S.Suzuki ** and K.Azuma * * Department of Computers and Systems Engineering, TDU, Saitama Japan ** Frontier R&D Center, Tokyo DenkiUniversity(TDU),Saitama Japan. 1. Introduction 2. Virtual gravity drive for swing-up 3.Adaptive VS-differentiator 4. Simulation and Experiment 5. Conclusion Outline

2. 1. Introduction Adaptive VS-differentiator Levant’s approach with fixed parameters →Adaptive tuning of parameters. Virtual gravity control The direction of virtual gravity is controlled Both swing-up and stabilization on inverted position is realized by one control law. Exact differentiator for any nonlinear system

3. ・ nonlinear mechanical system ・ under normal(vertical) gravity ・ with input 2. Virtual gravity drive plant reference-model ・ mechanical structure is the same as a plant ・ under virtual gravity (the direction is ) ・ with no input (1) (2)

4. If holds,, then above Eq.(4) corresponds to a dynamics of a reference-model:Eq.(2). VG-input ・ subtraction of gravity terms of each models:Eq.(1)(2) Virtual gravity input(VG-input) Assuming that… is closed to sufficiently plant with VG-input (3) (4)

5. Inverted position is unstable f pivot -mg (virtual gravity) +mg (normal gravity link This can be seemed as stable equilibrium in the upside- down virtual gravity world. Swing-up motion from pending position to inverted one and stabilization near upper position can be realized by only one control low. Swing up control of 1-link pendulum By VG-input the link moves as in inverted gravity field.

6. mg l l f q -mg ・ one-link without a moving base ・ the pivot is permitted to move only in the horizontal direction plant reference-model ・ under vertical gravity ・ with external force (input) ・ under opposite gravity ・ free motion (no input) VG-input More modified (on next sheet) (5) (6) from Eq.(5) (6) (7)

7. ・ Small attenuation of the oscillation near inverted pose by Eq.(7) ・ Enhancement of the stability near upper position :tuning parameters :minimum value to avoid 0 division Modified VG-input Closed loop system (8)

8. ・ Calculate torque which yields target force at the pivot of a link. ( for application of a rotational type pendulum) Generation of target acceleration In the case of 1-lik Furuta pendulum. plant Target force to the pivot of a link Target acceleration Torque to generate the target accel. (K.J.Astrom, K.Furuta,1999)

9. Let input signal (to the differentiator) as 3. Adaptive VS-Differentiator base signal having a derivative with Lipschitz’s constant C(>0) noise and be measurable locally and bounded. assumption differentiator : output of differentiator. : state of the differentiator

10. Applying a modified 2-sliding algorithm(Levant,1993) to keep,obtain. tuning parameters VS differentiator is estimated derivative of sufficient condition for convergence

11. Theorem of convergence(continuous) Discrete VS- differentiator There exists such a constant b>0 dependent on and that after a finite time next inequality holds. The convergence is ensured as followings for some constant a>0 which is dependent on and Let be measured with sampling interval, and be successive measurement times and current time.

12. Derivative,then transform… estimated tuning parameters Adaptive VS-differentiator Assume an existence of true values here, adjustment term (mentioned later) Objective: estimation of

13. Lyapunov candidate adaptive law adjustment term Tuning parameters are estimated on line

14. simulation result of Adaptive VS differentiator :output of diffentiator Quick & exact estimation is realized

15. 4-1 Simulation result Inverted position Swing up and stabilization are established Virtual gravity swing up control with Adaptive VS-differentiator

16. Estimated Velocity Real velocity Real velocity Estimated Velocity Estimation has done quick and exactly

17. Experimental result (additional) experimental result of Virtual Gravity Swing Up Control without Adaptive VS-differentiator Swing up is realized

18. 5. Conclusion ・ The scheme of virtual gravity control is suggested. ・ An application for swing-up control of pendulum is demonstrated. ・ Adaptive exact differentiator technique is introduced. ・ These efficiency are confirmed by simulation and experiment.