time delay systems in robotics - gipsa-lab.fr€¦ · human/environment dynamics largely unknown...

Time Delay Systems in Robotics

Sandra Hirche

Institute for Information-Oriented ControlTechnische Universitat Munchen

34. Intnl Summer School of Automatic Control, Grenoble, July 1-5, 2013

www.itr.ei.tum.de

http://www.itr.ei.tum.de

Contributors

Telerobotics: Dr. Tilemachos Matiakis Dr. Iason Vittorias Dr. Markus Rank

Networked visual servoing: Dr. Kolja Kuhnlenz Dr. Chih-Chung Chen Dr. Haiyan Wu

... and a big thanks to Sebastian Erhart for helping inpreparing the material!!

introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics2

Robot systems with time delay

Telerobotics Cooperative robot control Networked and embedded robot control


Teleoperated robonaut R2 at ISS

http://robonaut.jsc.nasa.gov

diagnosis and maintenance in space


Teleoperated underwater vehicle

ROV Jason (http://noaa.gov)

search and rescue, ocean floor sampling, monitoringintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

5

Minimal invasive surgery

da Vinci Surgical System (http://intuitivesurgical.com)


Distributed (haptic) virtual environments

CAVE Virtual Reality (http://mechdyne.com/cave.aspx)

education, training, e.g. surgical, skill transfer, entertainment


Telerobotics

Tsm

Master/

b

bFh xm

Fs Envir.

Human operator

Slave/

Tms

Human

Nonlinear or linearized master/slave dynamics Human/environment dynamics largely unknown Communication delay (wired/wireless, satellite, underwater)

Achieve tracking (xs xm, Fm Fs)


Roadtrains - vehicle platooning

Scoop project (http://scoop.md.kth.se)

fuel (CO2) reduction, congestion reductionintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

10

UAV formation

http://dronesintourism.ch

disaster relief, search & rescue, environment monitoringintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

11

Spaceflight formation

ESAs cluster satellite

separated spacecraft interferometry, microsatellite clustersintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

12

Cooperative robot manipulation


smart factories, search and rescue, constructionintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

13

Underwater robotic sensor networks

http://bluefinrobotics.com

environment monitoring, surveillance, disaster reliefintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

14

Robotic camera networks

T. Matsuyama - Cooperative Distributed Vision

environment monitoring, surveillance, disaster relief


Cooperative robot control

T12 T21

b

bqd, qd q1,q1

qd, qdq2,q2

Robot 1

Robot 2

Nonlinear manipulator dynamics for robot i Additional coupling conditions in cooperative manipulation Communication delay (wireless), information processing delay

Achieve synchronization (q1 q2) and tracking (qi 0)


Cooperative robot control as MAS

many robotic agents, e.g. mobile sensor network

linear 1st order dynamics

x =

x1

.

.

.xN

=

1 0

...

0 1

u1

.

.

.uN

linear 2nd order dynamics

Agent x

Comm.

Network

Tij

dynamics

Robots

u topology

information topology

Laplacian L = [lij ], lij =

deg(vi) if i = j,

1 if i 6= j and vi adj. to vj ,

0 otherwise.

1 2

43

L =

3 1 1 1

1 2 0 1

1 0 1 0

1 1 0 2

Achieve consensus on state (xi xj) or formation


Robot control in smart environments


service robotics, smart factories/ manufacturing


Robot embedded control system

CAN bus (ISO 11898)

internal control system in robots and vehicles x-by-wire technology


Networked and embedded robot control

Tc

Dynamics

Controller

T1

Perception

Robots

CPU

y

u

TN

...

yd

+

Multiple sensors connected via bus and/or wireless to internalor external CPU

Communication (and computation delays) T1, . . . , TN

Achieve tracking (y yd)


Take home message

time delay in many robot control problems: telerobotics,cooperative robot control, networked/embedded robot control

time delay often by communication (partially by computation)

General system structure:

T1 T2

1

2b

br1 y1

y2 r2

1,2 : Dynamical systems

T1, T2 : Delay operator

r1, r2 : External/reference signal

y1, y2 : System output


Overview

Introduction to robot systems with time delay

How to model communication time delay?

Telerobotics as time delay problem

Telerobotics as time delay problem (2)

Time delay problems in robotics - beyond telerobotics


Control over communication networks

ZOH

Actuator

Sensor

Data / packet processing

Data / packetprocessing

C

P

wk ek uk

yky

k

u(t)

y(t)

Continuous timeDiscrete time

Communication system

Controller

External application data

shared access mechanism time delay?


Communication protocols along OSI

1

2

3

4

5

6

7

Layers

Application

Presentation

Session

Transport

Network

Data link

Physical

applicationoriented

transportoriented

applicationhttpftp

TCP/IP

Transport TCP, UDP

IP

Network

Ethernet

Token ring

Example

Internet


Induced communication time delay

Application layer

Data Link Layer

Physical Layer

Node A

Application layer

Data Link Layer

Physical Layer

Node B

Communication network

pre

mac

tx

post

Communication starting from Node A Node B: pre-processing delay: pre medium access delay: mac

transmission delay: tx post-processing delay: post

End-to-end delay = pre + mac + tx + post


What happens on the data link layer?

Data link layer is concerned with local delivery of data framesbetween adjacent network nodes:

Medium Access Control (MAC) sublayer: regulating accessto shared transmission channel

Logical Link Control (LLC) sublayer: multiplexing ofprotocols, flow control, error control

The MAC sublayer design is crucial for real-time capabilities (latency, reliability) scalability energy efficiency

There exist a multiplicity of MAC protocols with strong impact oncommunication delay via mac!


MAC protocols

Contention-free Contention-based Hybrid protocols

Fixed assignment Dynamic assignment

FDMA

TDMA

CDMA

Polling

Token passing

Reservation-based

Z-MAC

FlexRay

ALOHA

CSMA

MACA

MACAW


Contention-based MAC protocol ALOHA

when data arrive sender transmits receiver sends ACK if packet received if no ACK received within timeout (collision) then retransmitafter random delay


ALOHA throughput-delay characteristics

[Yang and Yum 2003]

finite moments of delay distribution only in certain throughputregimes [Yang and Yum 2003]

delay depends on max. number of retransmission trials rmax


Carrier Sense Multiple Access (CSMA)

Contention-based MAC protocol: p-persistent CSMA

1. if medium idle, transmit with probability p or delaytransmission with probability (1 p) for one time unit(= max. propagation delay)

2. if medium busy, wait until it becomes idle and go to 1.

3. if transmission delayed by one time unit, go to 1.

Limit cases: non-persistent: if medium busy, delay transmission by random 1-persistent: if medium busy, wait until it is idle & transmit


Throughput of p-persistent CSMA

ensure that expected number of nodes starting to transmitwhen medium becomes idle: #nodes p < 1

1-persistent CSMA most greedy: low delay & efficiency non-persistent CSMA least greedy: high delay & efficiency p-persistent CSMA: delay and efficiency adjustable

Throughput

Packets waiting to be sent (including retransmissions)

[Tanenbaum 2002]


CSMA throughput-delay characteristics

max. throughput dependson parameter p

high throughput for low pimplies high delay (manypackets waiting to be sent)

[Kleinrock and Tobagi 1975]Assumptions for simulation:

1. positive packet acknowledgment mechanism

2. randomly delayed retransmission in case of failure

3. average retransmission delay >> packet transmission time

4. interarrival times of packet start & packet retransmissions are independent


Improvements of pure CSMA

Collision detection (CD) continue listening while transmitting, send a jam signal andstop transmission if collision detected

schedule re-transmission after random time, e.g. binaryexponential backoff time: after c collisions, choose backofftime out of [0, 2c 1] times the contention window

not feasable for wireless networks (hidden node problem)

Collision avoidance (CA) suitable for wireless networks if medium idle, wait random backoff time from contentionwindow until sending


CSMA/CA delay characteristics

[Wang, Vuran, and Goddard 2012]

increased end-to-end delay with multi-hop communication end-to-end delay configuration dependent


Applications of CSMA

Ethernet (IEEE 802.3) 1-persistent CSMA/CD

low non-deterministic delay at low network load many collisions and no delay bounds at high network load

Wireless Ethernet (IEEE 802.11) p-persistent CSMA/CA

positive ACK mechanism, retransmission in case of failure delay also depends on environment, potentially many retrans-missions large non-deterministic delay even at low net-work load possible

at high network load additionally long waiting time beforetransmission trial large non-deterministic delay


Take home message

Contention-based protocols:

trade-off between throughput and delay soa Internet protocols induce non-deterministic delay delay characteristics depends on concurrent traffic from communication community typically averaging delaymodels over many similar traffic flows, Poisson assumption fortraffic generation Right models for control design?

Other contention-based MAC layer principles?


Contention-based MAC protocol CAN

node node

node

node

CA

N-B

us

line

bit-rate up to 1 Mbit/s (< 40m, ISO 11898) serial communication network message prioritization by bit arbitration

node start transmitting if network idle assigned priority to each node to resolvepacket collision

node with high priority is guaranteed toobtain access to network

Controller Area Network (CAN) bit arbitration

deterministic and bounded delay for highest priority node non-deterministic traffic-dependent delay for lower prios comparatively low throughput


Contention-free MAC protocol TDMA

often used in embeddedcontrol systems

Time division multiple access (TDMA)

deterministic constant time delay requires time synchronization of nodes no scalability: only limited number of nodes online reconfiguration difficult


Application of TDMA: WirelessHART

First international standard for wireless communication in processautomation (IEC 62591)

TDMA-based transmission channel hopping adresses persistent noise sources central network manager maintains routes and schedules security mechanisms to encrypt communication

http://www.emersonprocess.com


Application of hybrid MAC protocol: FlexRay

Distributed implementation of embedded control applications with guaranteed signal latency (upper bound) operational robustness

is achieved by hybrid communication strategy

Communication is based on TDMA (static segment) andprioritization (dynamic segment)

t

Static segment

Dynamic segment

Symbol window

NIT

Time frame


Take home message

Contention-free/hybrid protocols:

low and guaranteed delay but low scalability suitable only for communication in local proprietarynetworks, not over large distances


Communication time delay classification

Deterministic delay with contention-free protocols

TDMA (e.g. WirelessHART)

Token-ring

Non-deterministic delay with contention-based protocols

CSMA/CD (e.g. Ethernet)

CSMA/CA (e.g. wireless Ethernet)

bit arbitration (e.g. CAN)- highest priority: fixed time-delay (real-time compatible)- lower priority: random


Modeling communication time delay

Important: cross-layer model of delay (all protocol layers)

constant time delay (t) = T = constant (known or unknown)

varying delay with upper bound (t) T

random time delay:

Markov process of different discrete time delays 1, . . . , n time-delay as i.i.d. sequence bounded or unbounded moments

out-of-order arrival > 1


Summary delay models

contention-based protocols suitable for flexible, long-rangecommunication but non-deterministic delay

tuning of the protocol can change delay characteristicssignificantly

more detailed delay models desirable (also predictive models) computation delay underexplored


Overview

Introduction to robot systems with time delay

How to model communication time delay?

Telerobotics as time delay problemIntroduction to teleroboticsPassivity and stabilityPassivity of teleoperation system w/o time delayTeleoperation with constant time delay

Telerobotics as time delay problem (2)

Time delay problems in robotics - beyond telerobotics


Multimodal telepresence

tele-maintenance and -diagnosis tele-assembly in nano/macro environments minimal invasive and tele-surgery rapid prototyping, x-by-wire applications


Telerobotics challenges

control loop closed over communication network stability human should feel like directly interacting transparency


Instability with time delay

Facts even small time delay may destabilize the system performance degradation by time delay


How to get here?

Networked haptic telepresence experiment with 278ms time delay

[Peer et al. 2008]


Bilateral teleoperation control approaches

passivity-based scattering/wave variable [Anderson and Spong 1989; Niemeyer

and Slotine 1991]

port-Hamiltonians [Stramigioli et al. 2002] Llewelyn [Hashtrudi-Zaad and Salcudean 1999] time-domain [Ryu, Kim, and Hannaford 2004] PD-type/Lyapunov-Krasowskii [Lee and Spong 2006]


Bilateral teleoperation control approaches

passivity-based scattering/wave variable [Anderson and Spong 1989; Niemeyer

and Slotine 1991]

port-Hamiltonians [Stramigioli et al. 2002] Llewelyn [Hashtrudi-Zaad and Salcudean 1999] time-domain [Ryu, Kim, and Hannaford 2004] PD-type/Lyapunov-Krasowskii [Lee and Spong 2006]

robust control [Leung, Francis, and Apkarian 1995] model-mediated [Mitra and Niemeyer 2008] adaptive, switching [Zhu and Salcudean 1999] predictive methods [Munir and Book 2001; Sheng and Spong 2004]

General question: model-based vs.model-free

depends on assumptions/knowledge of human/environment trade-off transparency vs. robust stability certificates


Excursus: passivity and stability (1)

Consider square system

x = f(x, u)y = h(x)

with state x Rn, input u Rp, output y Rp

f : Rn Rp Rn locally Lipschitz and f(0, 0) = 0 h : Rn Rp continuous and h(0) = 0



Definition (Passivity [Sepulchre, Jankovic, and Kokotovic 1997])

is passive if there exists a positive semidefinite functionS(x) : Rn R (storage function) s.t. for each admissible u andeach t 0

S(x(t)) S(x(0))

t

0uT y d

Physical interpretation:internal energy increase external energy supply

Differential form: is passive if there exists a positive semidefiniteC1 function S(x) s.t. S(x) uT y (x, u).



is strictly passive if S(x) uT y D(x) for some positivedefinite function D(x)

lossless if S(x) = uT y input-feedforward strictly passive (IFP) ifS(x) uT y uT(u) where uT(u) > 0 for some function and u 6= 0 (often (u) = u)

output-feedback strictly passive (OFP) ifS(x) uT y yT(y) where yT(y) > 0 for some function and y 6= 0 (often (y) = y)

for LTI systems with transfer function G(s) passivity impliesRe{G(j)} 0

time delay G(s) = esT is not passive



Consider passive systems 1 with S1(x1) t0 e

T1 y1 d and 2

with S2(x2) t0 e

T2 y2 d (S1(x1(0)) = S2(x2(0)) = 0).

Compositional properties of passive systems

Negative feedback interconnection of 1 and 2 is passivewith S(x) = S1(x1) + S2(x2)

t0 r

Ty d , where x = [x1, x2]T ,

r = [r1, r2]T , and y = [y1, y2]

T .

1

2

e1

e2b

br1 y1

y2 r2

enables modular designintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

55


Definition (Finite gain L2 stability [Khalil 2002] )

is finite gain L2-stable if there exist a positive semidefinite func-tion S : Rn R+ and a scalar constant > 0 s.t. for each ad-missible u and each t 0

S(x(t)) S(x(0))

t

02uTu yT y d.

The smallest possible is the L2-gain of . If 1 then hasthe small gain property.

let u() L2 and x(0) = 0: ||y||2L2

2||u||2L2

L2 inputs cause L2 outputs ratio of output/input energies bounded from above by 2

L2 gain of constant time delay operator is T = 1



L2-Stability of interconnected passive systems [Khalil 2002]

Consider IF-OFP passive systems 1 with 1, 1 and 2 with 2, 2.The negative feedback interconnection of 1 and 2 is finite gainL2-stable if 2 + 1 > 0 and 1 + 2 > 0.

L2-stability of interconnected L2-stable systems

Consider L2-stable systems 1 and 2. The negative feedbackinterconnection of 1 and 2 is finite gain L2-stable if 12 < 1.

1

2

e1

e2b

br1 y1

y2 r2



Asymptotic stability of origin from L2-stability

Assume is finite gain L2-stable. If is zero state detectable,then the origin is asymptotically stable.

Asymptotic stability of origin from passivity

Assume strictly output passive. If is zero state detectable,then the origin is asymptotically stable.

Zero state detectability: y(t) = 0 implies x(t) 0 as t 0.

In general: zero dynamics determines state stability.



Definition (Scattering operator)

of system is defined as (y u) = s(y + u).

Theorem (passivity small gain property)

is passive iff the scattering operator s has a L2-gain s 1.

Scattering operator of

u+ y

y u

u

yb

b


Take home message

passivity is a powerful analysis tool for uncertain systems non-linear systems modular and large-scale interconnected systems

equivalence of passivity to small gain property via scatteringoperator

finite gain L2 stability and asymptotic stability from passivityunder certain conditions

Passivity of telerobotic sub-systems?


Model of telerobotic subsystems

Manipulator dynamics for master and slave robot with gravitycompensation and external (Cartesian) forces

M(q)q + C(q, q)q = JT fe =

w inertia M(q) Rnn, Coriolis/Centrifugal forces C(q, q)q Rn

Properties of Lagrangian dynamic structure

1. M() =M()T 0 and m > 0 s.t. mI M(q)

2. M 2C is skew-symmetric under appropriate definition of C


Passive subsystems for master/slave (1)

Manipulator dynamics is passive (lossless)

with I/O pair torque / joint velocity q and storage function

S =1

2qTM(q)q

Proof:

S = qTM(q)q + 12 qT M(q)q

= qT + qT (12M C)q (existence of M(q)1)

= qT (skew-symmetry of 12M C)


Passive subsystems for master/slave (2)

Now rigid body dynamics in task coordinates

(q)x+ (q, q)x = JT fe = f

with (q) = JTMJ1 and (q, q) = JT (C MJ)J1

Manipulator dynamics in task space is passive (lossless)

with I/O pair force f / velocity x and storage function

S =1

2xT(q)x

Proof:

S = xT f xT ( JT (MJ + 12M)J1)x ( existence of 1)

= xT f ( skew-symmetry)


Passive subsystems for human & environment

What we know about the human: human arm dynamics difficult to model trained human behaves passive [Hogan 1989] human arm can be approximated by second order dynamics [Tsuji

et al. 1995; Rahman, Ikeura, and Mizutani 1999] task-dependent damping, e.g. 16Ns/m in [Buerger and Hogan 2006]

What we assume about the environment: environment is largely unknown environment is passive with I/O pair velocity/force friction effects typically render it strictly passive


Subsystem TO/environment is passive

Subsystem teleoperator & environment is passive

with I/O pairs desired TO endeffector velocity xds and coordinatingforce fc and storage function Se + Ss + SPID (negative feedbackinterconnection of passive subsystems).

xdsxs

PD controlmanipulatordynamics environment

b

bfc

passive

passive

passive

fe

holds in generalized coordinates with I/O pair qds and JT fc


Subsystem human/HSI is passive

System composed from Human and HSI is passive

with I/O pairs desired master force fdm and HSI/human endeffec-tor velocity x and storage function Sh + Sm (negative feedbackinterconnection of passive subsystems).

xm

humanmanipulatordynamics

b

fm

passivepassive

fdm

passive

holds in generalized coordinates with I/O pair JT fdm and qmintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics

66

Passivity of telerobotic system

Without time delay T1 = T2 = 0 overall system is passive

with I/O pairs force fh and velocity xm, storage functionSh + Sm + SPID + Ss + Se (negative feedback interconnection ofpassive subsystems).

xm

human HSI

b

fh

passivepassive

fdm

passive

PID TO

passive

environment

T1

T2

xds xs

fefc

b

b

fh


Take home message

Lagrangian nonlinear dynamics for telerobotic subsystems passivity of the subsystems including uncertain human andenvironment

passivity of the interconnected overall system stability of the overall system

What about time delay?


Telerobotic system with time delay (1)


From observation: unstable even with very small time delay values.


Telerobotic system with time delay (2)

Passivity with time delay

Overall system not passive as phase of Ti as .

T1 T2

1

2b

br1 y1

y2 r2



Closed control loop withconstant time delay T

Open loop:GOL(s) = G(s)e

sT

101

100

101

20

0

20

40

|GO

L| [d

B]

101

100

101

200

100

0

[

]

T = 500 ms

T = 100 ms

unstable stableG(s) =

10

s + 1

Theorem (linear time invariant system)

Stability if gain margin A < 1 (A = |GOL| at = 180).

stability for arbitrary constant time delay iff |G| < 1, > 0


totmovie.aviMedia File (video/avi)

Passivity-based stabilization

Theorem ([Anderson and Spong 1989; Niemeyer and Slotine 1991] )

Stability for arbitrary large constant time delay with(i) velocity-force-architecture, and(ii) scattering transformation:

u = (2b)12 (f + bx); v = (2b)

12 (f bx),

with wave impedance b > 0.

transformation

scattering

men

t

mation

scatteringTO

hum

an

envi

ron

transforHSI

xhul ur

vl vr

T1

T2

xt

fefdh


Small gain interpretation

HSI/human, TO/environment (strictly) passive scattering operator (y u) = S(y + u)

Stability for arbitrarily large constant time delay - Why?

passive L2-gain S 1 constant time delay T1 = T2 = 1 open loop r l T1 T2 < 1 L2-stability for closed loop

scatteringtransformation

scatteringtransformation

TO

small gain loop

hum

an

envi

ron

men

t

HSI

ur

vl vr

xhul xt

f dh

T1

T2fe

rl

passivity: t

0uT ()y() d V (x(t)) V (x(0)); L2-stability:

Take home message

passive subsystems become small gain via scatteringtransformation

using small gain property of constant time delay L2 stabilitycan be ensured

passivity of the interconnected overall system stability of the overall system

Can we do better while keeping the passivity formulation?


Relaxing passivity (losslessness)

passivity-based approaches very successful due to robustness but conservative

IdeaRelax passivity conservatism by incorporating approximate knowl-edge on dissipation properties

human dynamics should be modeled with large uncertainty high inter-subject variability should be allowed

Steps towards this direction: robust control for LTI-systems [Buerger and Hogan 2006] generalized scattering transformation with impedancecontrolled or PD-controlled HSI/TO [I. Vittorias 2010b]


Impedance controlled manipulator (1)

Desired properties: avoid large impact forces due to geometric uncertainties compliant behavior of manipulator for stable contact

Desired impedance

Md(xd x) +Dd(xd x) +Kd(xd x) = fe,

desired motion xd(t), inertia Md, damping Dd, stiffness Kd.

Parameter choice: Md and Kd for low contact forces Md and Kd for good motion tracking damping Dd to shape transient behaviors

trade-off between contact forces and position accuracy


Impedance controlled manipulator (2)

Remember manipulator dynamics in task coordinates (w gravity)

(q)x+ (q, q)x+ gx(q) = JT fe

Design of impedance control law in 2 steps

1. feedback linearization in Cartesian space = JT (u+ (q, q)x+ gx(q) + fe) results in x = u

2. impose desired impedance through choosingu = xd +M

1d (Dd(xd x) +Kd(xd x) fe)

resulting control law in joint coordinates

= M(q)J1(xd J q +M1

d (Dd(xd x) +Kd(xd x)))+C(q, q)q + g(q) (M(q)J1M1d J

T )fe


Impedance-controlled manipulator (3)

Consider impedance-controlled manipulator with resulting dynamics

Md(xd x) +Dd(xd x) +Kd(xd x) = fe

Impedance-controlled manipulator is strictly passive

with storage function S = 12xTMdx+

12x

TKdx OFP( = min(Dd)) with input fe and output x = xd x IFP( = min(Dd)) with input x and output fe

Proof:

S = xT fe +xT (DdxKdx) + x

TKdx= xT fe x

TDdx xT fe min(Dd)x

Tx

[I. Vittorias 2010a]


Strict passivity of telerobotic subsystems (1)

modularity: interconnection of IF-OFP systems is IF-OFP compensation: lack of passivity excess passivity

Environ.

Slave

Velocity-force-architecture and impedance-controlled manipulators: master OFP(m) & human IFP(h) l OFP(l = m + h) slave IFP(s) & env. OFP(e) r IFP(r=min(s, s + e))



Velocity-force-architecture and PD-controlled manipulators master OFP(m) & human IFP(h) environment IFP(e) & slave OFP(s) & controller IFP(Dc)



Velocity-force-architecture and PD-controlled manipulators master OFP(m) & human IFP(h) environment IFP(e) & slave OFP(s) & controller IFP(Dc)

Result: Interconnection of strictly passive systems

l is OFP(l = m + h) r is IFP(r = d

PDmin)


Take home message

subsystems including human typically exhibit strictly passivebehavior

behavior of TO/environment and human/HSI then alsostrictly passive

How can we use this additional knowledge to improveconservatism of standard scattering transformation?


Generalized scattering transformation (1)

System:p: xp = fp(xp, up), yp = hp(xp, up)c: xc = fc(xc, e), yc = hc(xc, e)unknown time delay T1, T2 = const.

Network

up

ype

w

yc

uc

c

T2

T1

p

Theorem ([Willems 1972])

Assume p and c (Q,S,R)-dissipative with[

Qp SpSTp Rp

]

+

[

Rc ScSTc Qc

]

= Pp + Pc 0.

Closed-loop system with T1 = T2 = 0 is L2-stable

Example IF-OFP subsystems:

L2-stability for T1 = T2 = 0 if p + c > 0 and p + c > 0


Excursus: (Q, S,R)-dissipative systems

(QSR)-dissipative systems

: x = f(x, u), y = h(x, u), x n, u p, y q dissipative if

V (x(t)) V (x(0)) t0

[

uT yT]

P

[

uy

]

d, P =

[

Q SST R

]

.

Input-feedforward output-feedback passive systems (IF-OFP)Q = I,R = I, S = I, , , = 12 , u, y

p

V (x(t)) V (x(0)) t0 u

T y uTu yTy d

Passive systemsQ = R = 0, S = 12I V (x(t)) V (x(0))

t0 u

T y d L2-stable systems: Q = I, S = 0, R =

2I


Generalized scattering transformation (2)

Theorem ([T. Matiakis 2009])

Closed-loop system is L2-gain-stable independently of constanttime delay T1, T2 6= 0 if existsM and diagonal satisfying

=MTM Pp 0 and Pc + Pp+ 0.

M Rnn exists if p is scalar feedback-stabilizable choose M s.t. = 0 nominal stability reserve! solution of M via LMIs, in some cases analytic

Pp+

vr

ur

Pc

e

ul

vl

yp

yc up

uc

c M1 M p

T2

T1


GST for IF-OFP subsystems (1)

Consider p, c IF-OFP

Pi =

[

i12

12 i

]

,

i, i , i {p, c} T2e

ul

vl vr

ur

yp

yc upT1

w uc

Mc M1 p

p + c > 0 & c + p > 0 L2-stability for T1 = T2 = 0 polar decomposition M = RB (R() SO(2), B invertible)

Generalized Scattering Transformation

M = R B =

[

cos I sin I sin I cos I

] [

b11I 00 b22I

]

with tuning variables , b11, b22.



Environ.

Slave

Result for x-f -architecture [I. Vittorias 2010a]

Delay-independent finite L2-gain stability if [l, r] given by cot 2l =

b11b22l with sin(l) cos(l) l sin

2(l) > 0

cot 2r = b22b11r with sin(r) cos(r) r cos

2(r) > 0

where B = diag{bii}.

Remark: Standard scattering transformation is special casel = r = 0 l = r = 45

, b11 = b12 , b22 = b

12



Consider p, c IF-OFP

Pi =

[

i12

12 i

]

,

i, i , i {p, c} T2e

ul

vl vr

ur

yp

yc upT1

w uc

Mc M1 p

p + c > 0 & c + p > 0 L2-stability for T1 = T2 = 0 polar decomposition M = RB (R() SO(2), B invertible)

Construction of M [Hirche, Matiakis, and Buss 2009]

delay-independent L2-stability if [l, r] we can analytically compute l from c, c, p, p, b11, b22 we can also analyticaly compute such that = 0 b11, b22 are free tuning parameters for performance


GST for x-f -architecture

Observe (compared to standard scattering)

more freedom of design due to [l, r] standard scattering transformation is special case:l = r = 0 l = r = 45

, b11 = b12 , b22 = b

12


Intuitive interpretation

rotation of generalized input-output-cones proof exploits small-gain-property of time delay operator

ul

vl

yp

yc up

vr

ur

M

uc

yc

z

0

delay M1

ee

ul

vl

yp

yc up

vr

ur

0

0

0

0

plantsector sector

0

0

sector

0

time

up

yp

z

vr

ur

vl

ul

p T1 T2

ucuc

cM MM1 M1c

T2

T1

p

T2

T1

p

sector


Take home message

L2 stability with generalized scattering transformation more freedom for design with consideration of strict passivity

Does it pay off in performance?


Comparison with standard small gain

0 2 40

0.5

1

Pos

ition

0 2 40

0.5

1

Pos

ition

Zeit [s]

T1 = T2 = 400 ms

T1 = T2 = 100 ms

Transformation

StandardSmallGain

Robot position control

Transformation approach stable for arbitrary T1, T2 stationary control error = 0 performance slightly as T1, T2

Standard small gain stable for arbitrary T1, T2 stationary control error > 0 performance strongly w T1, T2

Result [Matiakis, Hirche, and Buss 2009]

Transformation approach superior


Comparison with standard scattering(1)

101

100

101

102

103

50

0

50

100

Ma

gn

itu

de

(d

B)

101

100

101

102

103

100

50

0

Ph

ase

(d

eg

)

Frequency (rad/sec)

environment

standard scattering transformation

environment

generalized scattering

transformation

standard scattering transformation

generalized scattering

transformation

Spring-damper environment

Ze(s) =300s + 30

slave IFP withr = s = 30

stable if [45, 89],choose = 89

T1 + T2 = 100ms

Result: Substantially improved transparency [I. Vittorias 2010a]

Displayed stiffness kh = 166N/m closer to environment stiffnesske = 300N/m than with standard scattering kh = 36N/m


Comparison with standard scattering (2)

Human damping boundknown, spring environment

Ze(s) =300s

Human OFP withl = h = 30

stable if [2, 45] T1 + T2 = 100ms

GST-based design increases performance

Displayed stiffness kh = 109N/m for = 10 and with standard

scattering just kh = 34N/m


Experimental comparison T1 = T2 = 50ms (1)

Scattering Transformation

0 5 10

40

20

0

Forc

e

Z (

N)

Time (sec)

contact phase

60

Generalized Scattering Transformation

0 5 1060

40

20

0

Fo

rce

Z

(N

)Time (sec)

Master

Slave

contact phase

Result [I. Vittorias 2010b]

Higher displayed force with GST for similar motion


Experimental comparison T1 = T2 = 50ms (2)

Disp. Stiffness Disp. Damping

Environment (ideal) 1400 N/m 10 Ns/mST 367 N/m 68 Ns/m

GST=11 492 N/m 12 Ns/m

Result [I. Vittorias 2010b]

Substantially improved transparency with generalized scattering


Scattering trafo: related work & extensions

Challenge Contributions

time delay Spong+ - varying gains, passive position(varying) Niemeyer+ - wave integral transmission

Munir+ - prediction based

packet loss Yokokohji+ - energy controlStramigioli+ - sampled data, port HamiltonianSpong+ - passive interpolationHirche+ - passive extrapolation

data compression Hirche+ - passive deadband control

based on the small gain property of inner loop can be used with generalized scattering transformation


Varying delay: violation of small gain condition

T1(t) instead of constant T1 (for T2 analogous) assumption T1 T1,max < 1 (causality)

Fact

2T1

= 11T1,max

> 1 (if there exist time intervals with T1 > 0)

stability condition violated!

Proof.

ur,t2 =

t

0

u2l ( T1())d mit = T1()

=tT1(t)

T1(0)

11T1()

u2l ()d t

0

11T1()

u2l ()d

11T1,max

ul,t2 ul, t


Varying delay: stability by time-varying gain

Introduce time-varying gain urulf1(t)T1(t)

Theorem ([Lozano, Chopra, and Spong 2002])

Stability for varying time delay with f1(t) =

1 T1(t).

Proof.

ur,t2 =

t

0

f21 ()u2l ( T1()) d mit = T1()

=tT1(t)

T1(0)

1T11T1

u2l () d =tT1(t)

0

u2l ()d

ul,t2 ul, t

f1T1 = 1 stability by small gain theorem


GST with varying delay and packet loss

robust stability & transparency [Hirche, Matiakis, and Buss 2009]

L2-stability for arbitrary large, constant time delay same stability reserve as for nominal case T1 = T2 = 0

Unknown variable time delay [Matiakis, Hirche, and Buss 2008]

has L2-gain d = (1 d)1/2 where T d upper bounded

stability if = MTM Pp 0 and Pc +Pp++M

TT (d1 , d2)(d1 , d2)M 0 satisfied

Unknown packet loss [Matiakis 2009]

assume reconstruction operator is L2-stable stability result same as for time-varying time delay


Take home message

significant performance improvement with generalizedscattering transformation

validated in simulations and experiments displayed impedance closer to environment impedancecompared to standard scattering transformation

But how close is good enough?


Networked haptic telepresence challenges

control loop closed over communication network stability human should feel like directly interacting transparency


Perceived transparency

Transparency [Lawrence 1993]

if displayed impedance Zh = environment impedance Ze

with time delay & packet loss not achievable idea: consider perceptual limits in analysis and control design

Perceived transparency [Hirche and Buss 2012]

if Zh (Ze , Ze +), with determined by perception threshold


Psychophysics

Human cannot perceive arbitrarily small stimulus differences.

Webers law

I

I= constant = JND

where I stimulus intensity, I just noticeable absolute difference

JNDs determined in psychophysical experiments e.g. for parameters of mechanical impedance:

JND stiffness = (23 3)% [Jones and Hunter 1990] JND viscosity = (34 5)% [Jones and Hunter 1993] JND inertia = (21 3.5)% [Tan et al. 1995]


Transparency with constant time delay

Assumptions: constant time delay, scattering transf., Ze linear time-invariant

Result: Zh with Pade approximation time delay, T = T1 + T2

Zh(s) = bZe(s) + b+ (Ze(s) b) e

sT

Ze(s) + b (Ze(s) b) esT b

2Ze(s) + bTs

2b+ TsZe(s)

Summary results

inertia displayed in free space mit T stiff wall displayed softer with T maximum displayable stiffness mit T displayed stiffness difference mit T


Perception-oriented design

Example stiff wall (spring characteristics):

Ze = ke/s, ke > 0 Zh kh/s mit1/kh = 1/ke + T/2b

k

e

= 100 N/m

T = 200 ms

Z

e

[s1]

Z

h

kh

sj

Z

(

j

!

)

j

d

B

Transparency: kh = ke

b not realizable

Perceived transparency: kh ke(1 JND, 1 + JND)

b >1 JND

JND

Tke2

realizable, validated in user studies


Transparency with packet loss

Zeroing - no stiffness displayed Energy supervised HLS/zeroing - reduced stiffness displayed

k

e

= 200N/m

P

l

= 2%

P

l

= 20%

P

l

= 80%

b

e

= 1Ns/m

P

l

= 20%

T = 1 ms

,

Zeroing Energy supervised HLS/zeroing

Amplitude response for perceived and environment impedance

packet rate 1000Hzspringdamperenvironment

j

Z

h

j

[

d

B

! [Hz

Z

e

Z

e

! [Hz

Method: Monte Carlo simulations for different packet loss probabilitiesPl,

mean frequency response for perceived impedance from cross correlation


Perceived transparency with packet loss

transparency degradation not perceivable for quite largepacket loss probabilities

bound not invariant to env. properties (and sampling rate)

0 10 20 30 40 50 60 70 80 90

60

80

100

120

140

160

Pl [%]

k h [N

/m]

Loss not perceivable

23% (JND for stiffness)

Energy supervised HLS/zeroing


Transparent design for random delay & loss

Assumptions: Time delay with probability density function p(T ),communication network induced packet loss with probability P komml

Dejitter buffer: Trade-off between higher constant delay T andadditional packet loss P dejitterl through discarding packets

p(T )

P

dejitter

l

T

T

T

min

GoalDisplayed impedance environment impedance


Transparent design for random delay & loss

Example: Environment = stiff wall Ze =kes (P

komml = 0)

Displayed stiffness with time delay and packet loss Dejitter buffer: T = 90ms packet loss P dejitterl = 48%

020

4060

80100

0

100

200

300

4000

10

20

30

40

50

60

Pl [%] T [ms]

k h [N

/m]

0 10 20 30 40 50 60 70 80 90

50

100

150

200

250

300

350

400

Pl [%]

T [m

s]k

h

= 30N/m

k

h

= 5N/m

k

h

= 20N/m

k

h

= 10N/m

k

e

= 200N/m b

e

= 1Ns/m

P

l

(T

)

T

min

= 70ms

p(T )

Environment:Energy supervised HLS/zeroing

maximal perceivable stiffnessOptimum with respect

mapPoisson distributionwith

for

50

150100

200

020

4060

80100 50

100

150

0 20 40 60P

l

[%

P

l

[%

k

h

[

N

/

m

T

[ms

T

[

m

s

validated in objective experiments and human user studies


Take home message

influence of time delay on displayed impedance quantifiable perceived transparency is the important measure includehuman perceptual limits for transparency evaluation

extendable to randomly varying delay and loss


Summary telerobotics

passivity-based schemes popular because of uncertain &nonlinear human and environment dynamics

generalized scattering transformation for L2 stability withunknown but constant time delay

improves performance compared to standard scattering extensions for varying delay and packet loss exist also suitable for other networked robot control problems performance analysis with human perception model


Example: Networked visual servo control

Distrib

ute

d

senso

rdata

Control s

ignal

Task

Robot

Distributed sensors

Process node

Network

Overall system model in sample data formulation

feedback-linearized manipulator x(t) = Ax(t)Bu(tk) camera connected with networked computation units random delay from communication and computationk = (

sck +

cck ) +

ck modeled as i.i.d. sequence


Networked visual servo control (2)

ZOH Plant

Controller

1h

cc

k

c

k

sc

kk !!"

cc

k

c

k

sc

k

3h

2h)(tx

Approach [Wu et al. 2013]

delay-dependent switching control for mean exp. stability via stochastic jump system analysis and Lyapunov-Krasowskii novel communication protocol for vision-based control apps error-dependent data rate scheduling for fair network load



7DoF manipulator tracks object on other 7DoF manipulator high-speed camera (1000Hz, 640x480x8) 2 computation units with GPUs



0 2 4 6 8 1030

35

40

45

time [s]

timec

ost [

ms]

(a)

0 2 4 6 8 10

20

40

60

time [s]

num

ber

(b)

40 50 60 70 80 90 1000

500

1000

1500

[ms]

appe

ar ti

mes

(c)

Computation delaydepends on numberof extracted features

view angles illumination noise


Take home message

Networked visual servo control:

low performance mostly due to computation delay distributed (cloud) computing of state estimates improvesperformance

requires novel realtime capable communication protocols trade-off between good control performance and high networktraffic

data rate scheduling for best trade-off [Chen, Molin, and Hirche2009]


Cooperative robot control - overview

Lagrangian dynamicsM(qi)qi + C(qi, qi)qi + g(qi) = i

constant, heterogeneous,asymmetric time delayTij = const. 6= Tji = const.

T12 T21

b

bqd, qd q1,q

qd, qdq2,q2

Robot 1

Robot 2

Selected approaches and results

passivity-based analysis asymptotic synchronizationas e.g. in [Chopra and Spong 2006]

contraction analysis asymptotic contraction/sync.as e.g. in [Wang and Slotine 2006; Chung and Slotine 2007]


Multi-agent networks - overview

integrator dynamics foragents xi = ui

constant, heterogeneous,symmetric time delayTij = Tji = const.

Agent x

Comm.

Network

Tij

dynamics

Robots

u topology

Selected approaches and results

frequency domain analysis asymptotic consensus forTij

time delay systems in robotics - gipsa-lab.fr€¦ · human/environment dynamics largely unknown...

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