Department of Applied Mechanics – Budapest University of Technology and Economics

Dynamics of Dynamics of rehabilitation robots –rehabilitation robots – shake hands or hold shake hands or hold

hands?hands?G. Stépán, L. Kovács

Department of Applied MechanicsBudapest University of Technology and Economics

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Position control

1 DoF models x

Blue trajectories:Q = 0

Pink trajectories: Q = – Px – Dx

.

Department of Applied Mechanics – Budapest University of Technology and Economics

Force control

Desired contact force:Fd = kyd ;

Sensed force: Fs = ky

Control force: Q = – P(Fd – Fs) – DFs + Fs or d

.

Department of Applied Mechanics – Budapest University of Technology and Economics

Stabilization (balancing)

Control force:Q = – Px – Dx

Special case of force control: with k < 0

.

Department of Applied Mechanics – Budapest University of Technology and Economics

Modelling digital control

Special cases of force control:- position control with zero stiffness (k = 0)- stabilization with negative stiffness (k < 0)

Digital effects:- quantization in time: sampling – linear - quantization in space: round-off errors at ADA converters – non-linear

Department of Applied Mechanics – Budapest University of Technology and Economics

Alice’s Adventures in Wonderland

Lewis Carroll (1899)

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Motivation

- Polishing turbine blade(Newcastle/Parsons robot)

- Rehabilitation robotics(human/machine contact)

- Coupling force control (CFC)(between truck and trailer)*

- Electronic brake force control(added to ABS systems)** Knorr-Bremse

Department of Applied Mechanics – Budapest University of Technology and Economics

Motivation

- Polishing turbine blade(Newcastle/Parsons robot)

- Rehabilitation robotics(human/machine contact)

- Coupling force control (CFC)(between truck and trailer)*

- Electronic brake force control(added to ABS systems)** Knorr-Bremse

Department of Applied Mechanics – Budapest University of Technology and Economics

Motivation

- Polishing turbine blade(Newcastle/Parsons robot)

- Rehabilitation robotics(human/machine contact)

- Coupling force control (CFC)(between truck and trailer)*

- Electronic brake force control(added to ABS systems)** Knorr-Bremse

Department of Applied Mechanics – Budapest University of Technology and Economics

Motivation

- Polishing turbine blade(Newcastle/Parsons robot)

- Rehabilitation robotics(human/machine contact)

- Coupling force control (CFC)(between truck and trailer)*

- Electronic brake force control(added to ABS systems)** Knorr-Bremse

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Turbine blade polishing

Department of Applied Mechanics – Budapest University of Technology and Economics

Mechanical model of polishing

mr = 2500 [kg] br = 32 [Ns/mm] C = 150 [N]

ms = 0.95 [kg] bs = 2 [Ns/m(!)] ks = 45 [Ns/mm]

me = 4.43 [kg] be = 3 [Ns/m(!)] ks = 13 [Ns/mm]

Fd = 50 [N]

Department of Applied Mechanics – Budapest University of Technology and Economics

Experimental stability chart

Department of Applied Mechanics – Budapest University of Technology and Economics

Self-excited oscillation – Hopf bifurcation

Department of Applied Mechanics – Budapest University of Technology and Economics

Time-history and spectrum

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control- Motivation- Turbine blade polishing- Human force control- Rehabilitation robotics- 1 DoF model of force control- Digital effect: sampling- Theoretical and experimental stability charts- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Human force control

“Good memory causes trouble”

Meeting others on narrow corridors oscillations

caused by the delay of our reflexes

Why do we “shake” hands?

Department of Applied Mechanics – Budapest University of Technology and Economics

Hemingway, E., The Old Man and the Sea (1952)

Santiago plays arm-wrestling in a pub of Casablanca

Department of Applied Mechanics – Budapest University of Technology and Economics

Human-robotic force control

Dextercartoon

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Rehabilitation roboticsIRB 1400 H

IRB 140

Couch

Touch screen

Control panelOperationmode selector

Safety switch

Etc.

Instrumented orthosis

Frame

Stack lights

Start pedal

Department of Applied Mechanics – Budapest University of Technology and Economics

The safety relaxer

Department of Applied Mechanics – Budapest University of Technology and Economics

The instrumented orthosis

Robot

Safety relaxer mechanism (SRM)

6 DOF force/torque transducer #1Quick changer #1

6 DOF force/torque transducer #2

Quick changer #2

Orthosis shell

Handle (the so-called safeball)

Teaching-in device

Department of Applied Mechanics – Budapest University of Technology and Economics

Mechanical model of relaxer

Department of Applied Mechanics – Budapest University of Technology and Economics

Force control during teaching-in

Sampling time at outer loop with 60 [ms]

Sampling time at force sensor with t 4 [ms]

Fd

0Fe,n xd,n

xn-1

xd,n

DSP (or PC)

xn-L Fm,n

force controlrobot -

controllerrobot arm +

teaching-in device

force / torqueacquisition

deadtime 400ms !

Department of Applied Mechanics – Budapest University of Technology and Economics

Delay and vibrations

Department of Applied Mechanics – Budapest University of Technology and Economics

Delay and vibrations

Department of Applied Mechanics – Budapest University of Technology and Economics

Delay and vibrations

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

1 DoF model of force control

Equation of motion:

Equilibrium:

Force error: (Craig ’86)

Stability for ,

)(sgn

)(

))(()()( or tyC

F

tky

FtkyPtkytmy

d

d

.. .

kFy dd /)1/(or/ PCPCF

0 Pkxmx)()( txyty d ..0P

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Modelling sampling

Time delay andzero-order-holder(ZOH)

Dimensionless time/tT

Department of Applied Mechanics – Budapest University of Technology and Economics

Modelling sampling

Sampling time is , the j th sampling instant is tj = j

Natural frequency:

Sampling frequency: dim.less time: T = t/Dimensionless equations of motion:

x(j), x’(j) B1,B2

),[),()()( jjjdj ttttkyFtkyPtQ

)2/(/)2/( mkf nn /1sf

)1()1()()()()( 22 jxPTxTx nn )1,[ jjT

)()()( TxTxTx ph

)1()1()sin()cos( 21 jxPtBtB nn

Department of Applied Mechanics – Budapest University of Technology and Economics

Contents- Introduction: force control

- Motivation

- Turbine blade polishing

- Human force control

- Rehabilitation robotics

- 1 DoF model of force control

- Digital effect: sampling

- Theoretical and experimental stability charts

- Conclusion: bifurcations

Department of Applied Mechanics – Budapest University of Technology and Economics

Stability of digital force control

stability

Parameters: and

jj Azz 1

)(

)(

)1(

jx

jx

jxjz

)cos()sin()sin()1(

)sin()cos())cos(1)(1(

0101

nnnnn

nnn

P

Pn

A

0)det( AI

13,2,1

snn ff /)2/()( P

Department of Applied Mechanics – Budapest University of Technology and Economics

Checking |1,2,3|<1 algebraically

Routh-Hurwitz

Department of Applied Mechanics – Budapest University of Technology and Economics

Stability chart of force control

Vibration frequency: 0 < f < fs/2

Maximumgain:

Minimumforce error:

CF )3/2(min,

5.1max P

Department of Applied Mechanics – Budapest University of Technology and Economics

Effect of viscous damping

Damping ratio : 0.04 and 0.5

)1()1()(

)()()()(2)(2

2

jxP

TxTxTx

n

nn

Department of Applied Mechanics – Budapest University of Technology and Economics

Experimental verification

Department of Applied Mechanics – Budapest University of Technology and Economics

Experimental verification

Damping ratio = 0.04 P

[ms]

Department of Applied Mechanics – Budapest University of Technology and Economics

Differential gain D in force control

Sampling at the force sensor with t = q :

)1()()1()1()(

)()()(2

2

jxDjxP

TxTx

nn

n

)()()()( jjdj tkytykDFtkyPtQ

q

qjxjxjx

)1()1()1(

jjTj qjxjxjxjx Azzz 1)1()1()()(

Department of Applied Mechanics – Budapest University of Technology and Economics

Stability chart and bifurcations

Dimensionless differential gain Dn = 0.1

Department of Applied Mechanics – Budapest University of Technology and Economics

Stability chart and bifurcations

Dimensionless differential gain Dn = 1.0

Department of Applied Mechanics – Budapest University of Technology and Economics

Conclusions

- All the 3 kinds of co-dimension 1 bifurcations arise in digital force control(Neimark-Sacker, flip, fold)

- Application of differential gain leads to loss of stable parameter regions

- Force derivative signal can be filtered with the help of sampling, but stability properties do not improve

- Do not use differential gain in force control

Department of Applied Mechanics – Budapest University of Technology and Economics