Embedding Nonsmooth Systems in Digital Controllers. Ryo Kikuuwe

Transcription

1 Embedding Nonsmooth Systems in Digital Controllers Ryo Kikuuwe

2 2 Nonsmooth Systems? Systems subject to discontinuous dierential equations. A good example is... Coulomb riction (orce) M v v (velocity) Mathematically very cumbersome. but very amiliar in our daily lives. and useul or some control applications

3 3 Today's Talk 1. about natural nonsmooth systems 2. about artiicial nonsmooth systems

4 4 Simulation o Natural Nonsmooth Systems: Systems Subject to Coulomb Friction

5 5 Motivation: Request rom Automobile Industry robotic device (with actuators) orce sensor Robotic assistance or manual ine positioning in assembling. Problem: What kind o resistive orces are appropriate or enhancing the perormance?

6 6 Haptic Feedback To realize a prescribed position-orce relation at the interace between a device and an operator contacting therewith. Applications: Robotic assistance o manual tasks Virtual reality (involving haptics) surgery simulators, computer games, CAD interaces,...

7 7 Idea: Coulomb Friction It cancels external orces below static riction level. (Thus, eective or ine positioning.) Most amiliar dynamics in our daily lives. orce v velocity v? But producing such orce-velocity relation by digital control is very challenging!!

8 8 Why Is Coulomb Friction Problematic? In discrete time, the orce is determined according to the relative velocity. Due to the latency, chattering (repeated zero-velocity crossings) occurs. sensor velocity device orce actuator? velocity orce computer kinetic riction kinetic riction v static riction Idea: put a delayless eedback in the loop to remove chattering.

9 9 Delayless Feedback Including Signum? x + y where equivalent x y where (unit saturation unction)

10 10 Solution Rigid, discontinuous riction between objects: diicult to simulate. input u output Put a delayless eedback as a virtual viscoelastic element. + input velocity v output orce

11 Derivation o Simulation Algorithm + input u [Kikuuwe et al.: IEEE Trans. on Robotics, 2006] step 1: continuous-time representation dierential algebraic equations 11 output step 2: discrete-time representation backward Euler; algebraic equations output: orce input: velocity u e K B step 3: Algorithm (discrete-time) solution to the above

12 12 Applications clamping riction orce on the needle shat

13 13 Implementation o Artiicial Nonsmooth Systems: Proxy-based Sliding Mode Control

14 14 Motivation Coulomb Friction What happens? (orce) (orce) v (velocity) p + Hv (position + velocity multiplied by something) Physical characteristic that does not exist in nature but can be produced by digital control Sliding Mode Control?

15 15 Sliding Mode Control A nonsmooth control scheme. The actuator orce is discontinuously switched on a sliding surace in the state space. velocityv sliding surace Simplest Example: positionp position p F : upper bound o orce H : time constant exponentially converges to the target position with time constant H timet In discrete time, however, there is chattering due to the discontinuity!!

16 [Kikuuwe & Fujimoto, ICRA2006] Proxy-based Sliding Mode Control target position proxy controlled object sliding mode controller PID controller p p d q step 1: continuous-time representation dierential algebraic equations step 2: discrete-time representation algebraic equations step 3: Discrete-time representation solution to the above 16 external orces

17 17 Beneit o PSMC target position proxy controlled object sliding mode controller PID controller p external orces p d q PSMC is as accurate as but much saer than PID!! Beore saturation, equivalent to PID control. ast response to small positional error Ater saturation, SMC-like response is dominant moderate recovery rom large positional error without overshoots. v p slow ast p slow time ast t

18 18 PID-C vs PSMC Response to Step Input Response to Large Disturbance PSMC is as accurate as but saer than PID

19 Voice o Users Voice o Users [Van Damme et al., Vrije Univ. Brussel 19

20 20 Conclusions Introduced an algorithm or producing Coulomb riction characteristics in discretetime controller. Introduced a new position control scheme, "Proxy-based sliding mode control." Future (Current Ongoing) Projects Utilization o riction algorithms Driving simulator with realistic steering resistance. Extension o PSMC or saer, human-riendly robots. Low-cost robot equipped with standardized gear transmissions.