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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
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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
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Robot systems with time delay
Telerobotics Cooperative robot control Networked and embedded robot control
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics3
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Teleoperated robonaut R2 at ISS
http://robonaut.jsc.nasa.gov
diagnosis and maintenance in space
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics4
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Teleoperated underwater vehicle
ROV Jason (http://noaa.gov)
search and rescue, ocean floor sampling, monitoringintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
5
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Minimal invasive surgery
da Vinci Surgical System (http://intuitivesurgical.com)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics6
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Distributed (haptic) virtual environments
CAVE Virtual Reality (http://mechdyne.com/cave.aspx)
education, training, e.g. surgical, skill transfer, entertainment
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics7
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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)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics8
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Robot systems with time delay
Telerobotics Cooperative robot control Networked and embedded robot control
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics9
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Roadtrains - vehicle platooning
Scoop project (http://scoop.md.kth.se)
fuel (CO2) reduction, congestion reductionintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
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UAV formation
http://dronesintourism.ch
disaster relief, search & rescue, environment monitoringintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
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Spaceflight formation
ESAs cluster satellite
separated spacecraft interferometry, microsatellite clustersintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
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Cooperative robot manipulation
http://www.itr.ei.tum.de
smart factories, search and rescue, constructionintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
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Underwater robotic sensor networks
http://bluefinrobotics.com
environment monitoring, surveillance, disaster reliefintroduction delay model telerobotics (1) telerobotics (2) beyond telerobotics
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Robotic camera networks
T. Matsuyama - Cooperative Distributed Vision
environment monitoring, surveillance, disaster relief
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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)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics16
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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
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Robot systems with time delay
Telerobotics Cooperative robot control Networked and embedded robot control
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Robot control in smart environments
http://www.itr.ei.tum.de
service robotics, smart factories/ manufacturing
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Robot embedded control system
CAN bus (ISO 11898)
internal control system in robots and vehicles x-by-wire technology
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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)
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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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics22
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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
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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?
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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
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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
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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!
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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
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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
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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
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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
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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]
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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
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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
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CSMA/CA delay characteristics
[Wang, Vuran, and Goddard 2012]
increased end-to-end delay with multi-hop communication end-to-end delay configuration dependent
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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
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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?
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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
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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
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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
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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
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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
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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
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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
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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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics45
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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
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Multimodal telepresence
tele-maintenance and -diagnosis tele-assembly in nano/macro environments minimal invasive and tele-surgery rapid prototyping, x-by-wire applications
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Telerobotics challenges
control loop closed over communication network stability human should feel like directly interacting transparency
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Instability with time delay
Facts even small time delay may destabilize the system performance degradation by time delay
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How to get here?
Networked haptic telepresence experiment with 278ms time delay
[Peer et al. 2008]
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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]
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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
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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
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Excursus: passivity and stability (2)
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).
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics53
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Excursus: passivity and stability (3)
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
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Excursus: passivity and stability (4)
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
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Excursus: passivity and stability (5)
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
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Excursus: passivity and stability (6)
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
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Excursus: passivity and stability (7)
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.
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Excursus: passivity and stability (8)
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
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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?
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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
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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)
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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)
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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
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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
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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
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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
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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?
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Telerobotic system with time delay (1)
Instability with time delay
From observation: unstable even with very small time delay values.
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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
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Instability with time delay
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
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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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics72
-
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?
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics74
-
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]
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics75
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics76
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics77
-
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]
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics78
-
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))
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics79
-
Strict passivity of telerobotic subsystems (3)
Velocity-force-architecture and PD-controlled manipulators master OFP(m) & human IFP(h) environment IFP(e) & slave OFP(s) & controller IFP(Dc)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics80
-
Strict passivity of telerobotic subsystems (3)
Velocity-force-architecture and PD-controlled manipulators master OFP(m) & human IFP(h) environment IFP(e) & slave OFP(s) & controller IFP(Dc)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics80
-
Strict passivity of telerobotic subsystems (3)
Velocity-force-architecture and PD-controlled manipulators master OFP(m) & human IFP(h) environment IFP(e) & slave OFP(s) & controller IFP(Dc)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics80
-
Strict passivity of telerobotic subsystems (3)
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)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics80
-
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?
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics81
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics82
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics83
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics84
-
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.
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics85
-
GST for IF-OFP subsystems (2)
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics86
-
GST for IF-OFP subsystems (3)
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics87
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics88
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics89
-
Take home message
L2 stability with generalized scattering transformation more freedom for design with consideration of strict passivity
Does it pay off in performance?
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics90
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics91
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics92
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics93
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics94
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics95
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics96
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics97
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics98
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics99
-
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?
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics100
-
Networked haptic telepresence challenges
control loop closed over communication network stability human should feel like directly interacting transparency
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics101
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics102
-
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]
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics103
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics104
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics105
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics106
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics107
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics108
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics109
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics110
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics111
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics112
-
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics113
-
Networked visual servo control (3)
7DoF manipulator tracks object on other 7DoF manipulator high-speed camera (1000Hz, 640x480x8) 2 computation units with GPUs
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics114
-
Networked visual servo control (3)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics115
-
Networked visual servo control (4)
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
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics116
-
Networked visual servo control (5)
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics117
-
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]
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics118
-
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]
introduction delay model telerobotics (1) telerobotics (2) beyond telerobotics119
-
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
top related