Embedding
Nonsmooth Systems
in
Digital Controllers
Ryo Kikuuwe
2
Nonsmooth Systems?

Systems subject to discontinuous differential
equations. A good example is ...

Coulomb friction
f
f (force)
M
v
v (velocity)
Mathematically very cumbersome.
 but very familiar in our daily lives.
 and useful for some control applications

3
Today's Talk
1. about natural nonsmooth systems
2. about artificial nonsmooth systems
4
Simulation of
Natural Nonsmooth Systems:
Systems Subject to
Coulomb Friction
5
Motivation: Request from Automobile
Industry
robotic device
(with actuators)
force sensor
Robotic assistance for manual fine
positioning in assembling.
 Problem: What kind of resistive forces are
appropriate for enhancing the performance ?

6
Haptic Feedback
To realize a prescribed position-force relation
at the interface between a device and an
operator contacting therewith.
 Applications:



Robotic assistance of manual tasks
Virtual reality (involving haptics)

surgery simulators, computer games, CAD interfaces, ...
7
Idea: “Coulomb Friction”
It cancels external forces below static friction
level. (Thus, effective for fine positioning.)
 Most familiar dynamics in our daily lives.

force
f
?
velocity
v

v
f
But producing such force-velocity
relation by digital control is very
challenging!!
8
Why Is Coulomb Friction Problematic?


In discrete time, the force is determined according to
the relative velocity.
Due to the latency, chattering (repeated zero-velocity
crossings) occurs.
computer
velocity
sensor
velocity
device
force
actuator

?
f
kinetic friction
force
kinetic friction
v
static friction
Idea: put a delayless feedback in the loop to
remove chattering.
9
Delayless Feedback Including Signum?
x+ ¡
where
y
equivalent
x
where
y
(unit saturation function)
10
Solution

Rigid, discontinuous friction between objects:
difficult to simulate.
input u
output f

Put a delayless feedback as a virtual
viscoelastic element.
+
input velocity v
output force f
[Kikuuwe et al.: IEEE Trans. on Robotics, 2006]
Derivation of Simulation Algorithm

+
input u ¡
step 1: continuous-time representation


output f
differential algebraic equations
step 2: discrete-time representation

backward Euler; algebraic equations
input: velocity u
e
output:
force f
K
B

step 3: Algorithm (discrete-time)

solution to the above
11
12
Applications
clamping
friction force on
the needle shaft
13
Implementation of
Artificial Nonsmooth Systems:
“Proxy-based Sliding Mode Control”
14
Motivation

Coulomb Friction
What happens?
f (force)
f (force)
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
Sliding Mode Control

A “nonsmooth” control scheme.

The actuator force is “discontinuously” switched
on a “sliding surface” in the state space.
velocity v
sliding surface

position
p
F : upper bound of force
 H : time constant
exponentially converges to the target
position with time constant H


position p
time t
Simplest Example:

In discrete time, however, there is
chattering due to the discontinuity!!
16
[Kikuuwe & Fujimoto, ICRA2006]
“Proxy-based Sliding Mode Control”

target
position
f
f

sliding mode
controller

step 2: discrete-time representation

PID
controller
f
controlled
object
differential algebraic equations
pd
proxy
f
step 1: continuous-time representation
algebraic equations
q

step 3: Discrete-time representation

p
external forces
solution to the above
17
Benefit of PSMC

target
position
f
f
pd

sliding mode
controller
proxy
f
PSMC is as accurate as but
much safer than PID!!
PID
controller
f
controlled
object
q
p
external forces
Before saturation, equivalent to
PID control.


fast response to small positional error
After saturation, SMC-like
response is dominant

moderate recovery from large
positional error without overshoots.
p
v
slow
fast
p
slow
fast
time t
18
PID-C vs PSMC
Response to Step Input

Response to Large Disturbance
PSMC is as accurate as but safer than PID
19
Voice of Users
[Van Damme et al., 2007]
@ Vrije Univ. Brussel
20
Conclusions
Introduced an algorithm for producing
Coulomb friction characteristics in discretetime controller.
 Introduced a new position control scheme,
"Proxy-based sliding mode control."

Future (Current Ongoing) Projects

Utilization of friction algorithms


Driving simulator with realistic steering resistance.
Extension of PSMC for safer, human-friendly robots.

Low-cost robot equipped with standardized gear
transmissions.