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

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

・ 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)

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)

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.

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)

・ 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)

・ 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)

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

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

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.

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

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

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

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

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

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

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.