UTA Research Institute (UTARI)The University of Texas at Arlington, USA
F.L. Lewis
Optimal Control Introduction
Supported byNSF, ARO, AFOSR
He who exerts his mind to the utmost knows nature’s pattern.
The way of learning is none other than finding the lost mind.
Meng Tz500 BC
Man’s task is to understand patterns innature and society.
Mencius
Optimal Control
Reinforcement learning
Control of Robotic Devices
Adaptive Learning of Human-Robot Interfaces
Dynamic Games for HRI
Importance of Feedback Control
Darwin- FB and natural selectionVolterra- FB and fish population balanceAdam Smith- FB and international economyJames Watt- FB and the steam engineFB and cell homeostasis
The resources available to most species for their survival are meager and limited
Nature uses Optimal control
SYSTEMoutputsAlfred North Whitehead
Von BertalanffySystems Theory 1920s
inputs
processing
Relevance- Machine Feedback Control
qd
qr1 qr2
AzEl
barrel flexiblemodes qf
compliant coupling
moving tank platform
turret with backlashand compliant drive train
terrain andvehicle vibrationdisturbances d(t)
Barrel tipposition
qd
qr1 qr2
AzEl
barrel flexiblemodes qf
compliant coupling
moving tank platform
turret with backlashand compliant drive train
terrain andvehicle vibrationdisturbances d(t)
Barrel tipposition
Vehicle mass m
ParallelDamper
mc
activedamping
uc(if used)
kc cc
vibratory modesqf(t)
forward speedy(t)
vertical motionz(t)
surface roughness(t)
k c
w(t)
Series Damper+
suspension+
wheel
Single-Wheel/Terrain System with Nonlinearities
High-Speed Precision Motion Control with unmodeled dynamics, vibration suppression, disturbance rejection, friction compensation, deadzone/backlash control
VehicleSuspension
IndustrialMachines
Military LandSystems
Aerospace
plant
controlu(t)
outputy(t)
controller
systemidentifier estimated
output
)(ˆ ty
identificationerror
desiredoutput
)(tyd
plant
controlu(t)
outputy(t)
controllerdesiredoutput
)(tyd
trackingerror
plant
controlu(t)
outputy(t)
controller #1
controller #2
desiredoutput
)(tyd
trackingerror
Indirect Scheme
Fixed Controller Topologies
Direct Scheme
Feedback/FeedforwardScheme
Cell Homeostasis The individual cell is a complex feedback control system. It pumps ions across the cell membrane to maintain homeostatis, and has only limited energy to do so.
Cellular Metabolism
Permeability control of the cell membrane
http://www.accessexcellence.org/RC/VL/GG/index.html
Optimality in Biological Systems
Optimality in Control Systems DesignR. Kalman 1960
Rocket Orbit Injection
http://microsat.sm.bmstu.ru/e-library/Launch/Dnepr_GEO.pdf
FmmmF
rwvv
mF
rrvw
wr
cos
sin22
ObjectivesGet to orbit in minimum timeUse minimum fuel
Dynamics
Optimal Control is Effective for:Aircraft AutopilotsVehicle engine controlAerospace VehiclesShip ControlIndustrial Process ControlRobot Control
Optimal Control:Minimum timeMinimum fuelMinimum energyConstrained control
Optimal control solutions are found byOffline solution of Matrix Design equationsA full dynamical model of the system is needed
Optimality and Games
PBPBRQPAPA TT 10
1 Tu R B Px Kx
BuAxx ( ( )) ( ) ( ) ( )T T T
t
V x t x Qx u Ru d x t Px t
Full system dynamics must be knownOff‐line solution
0 ( , , ) 2 2 ( )T
T T T T T T TV VH x u V x Qx u Ru x x Qx u Ru x P Ax Bu x Qx u Rux x
Optimal Control: Linear Quadratic Regulator (LQR)
System
Performance Index
Leibniz’s formula‐
Optimal Control is SVFB
Algebraic Riccati equation
( , , ) 2( ) 0T T Td V dH x u Ax Bu Px x Qx u Rudu x du
Stationarity Condition
2 0TRu B Px
( ) ( ) ( ) ( )T T T T T T T TdV x Qx u Ru x Px x Px x Px x Px Ax Bu Px x P Ax Budt
( , , )VH x ux
Hamiltonian function
Differential equivalent to PI is the Bellman equation
PBPBRQPAPA TT 10
1 Tu R B Px Kx
BuAxx
( ( )) ( ) ( ) ( )T T Tt
V x t x Qx u Ru d x t Px t
Full system dynamics must be knownOff‐line solutionCannot change performance objectives
Optimal Control: Linear Quadratic Regulator
System model
Performance Function
Optimal Control is
Algebraic Riccati equation
MATLAB Control Systems Toolbox
[K,P]=lqr(A,B,Q,R)
PBPBRQPAPA TT 10
1 Tu R B Px Kx
BuAxx
( ( )) ( ) ( ) ( )T T Tt
V x t x Qx u Ru d x t Px t
Full system dynamics must be knownOff‐line solution
0 ( , , ) 2 2 ( )T
T T T T T T TV VH x u V x Qx u Ru x x Qx u Ru x P Ax Bu x Qx u Rux x
Optimal Control: Linear Quadratic Regulator
0 ( ) ( )T TA BK P P A BK Q K RK
u Kx
System
Cost
Leibniz’s formula‐ Differential equivalent is the Bellman equation
Given any stabilizing FB policy
The cost value is found by solving Lyapunov equation
Optimal Control is
Algebraic Riccati equation
( , , ) 2( ) 0T T Td V dH x u Ax Bu Px x Qx u Rudu x du
Many Successful Design Applications of Optimal Control
PBPBRQPAPA TT 10
1 Tu R B Px Kx
BuAxx
( ( )) ( ) ( , ) ( ) ( )T T Tt t
V x t x Qx u Ru d r x u d x t Px t
Full system dynamics must be knownOff‐line solution
Optimal Control: Linear Quadratic Regulator
System
Cost function
Optimal Control is
Algebraic Riccati equation
MATLAB Control Systems Toolbox
Chop off Tail of cost function
( ( )) ( , ) ( ( ))t T
t
V x t r x u d V x t T
( ( )) ( , ) ( , )t T
t t T
V x t r x u d r x u d
Bellman Equation
1 Tu R B Px Kx Update Control using Hamiltonian Physics
Reinforcement Learning‐ Policy Iteration
Online Learning Feedback ControlSynthesis of Robot Control and Biologically Inspired Learning
Neural Network Properties
Learning
Recall
Function approximation
Generalization
Classification
Association
Pattern recognition
Clustering
Robustness to single node failure
Repair and reconfiguration
Nervous system cell. http://www.sirinet.net/~jgjohnso/index.html
Dynamic Neural Networks and Control Learning
Robot System[I]
Robust ControlTerm
qqd
v(t)
e
Outer Tracking Loop
f(x)
rKv
Nonlinear Inner Learning Loop
^
Feedforward Loop
Tunable weights
dq
Theorem 1 (NN Weight Tuning for Stability)
Let the desired trajectory )(tqd and its derivatives be bounded. Let the initial tracking error be within a certain allowable set U . Let MZ be a known upper bound on the Frobenius norm of the unknown ideal weights Z . Take the control input as
vrKxVW vTT )ˆ(ˆ with rZZKtv MFZ )()( .
Let weight tuning be provided by
WrFxrVFrFW TTT ˆˆ'ˆˆˆ
, VrGrWGxVTT ˆ)ˆ'ˆ(ˆ
with any constant matrices 0,0 TT GGFF , and scalar tuning parameter 0 . Initialize the weight estimates as randomVW ˆ,0ˆ .
Then the filtered tracking error )(tr and NN weight estimates VW ˆ,ˆ are uniformly ultimately bounded. Moreover, arbitrarily small tracking error may be achieved by selecting large control gains vK .
NEW NN Tuning Laws for Control Learning
Force Control
Flexible pointing systemsSimis labs, Inc. and US Army
Vehicle active suspensionLeo Davis Technol.
SBIR Contracts
Learning NN Controller
Applications at Boeing Defense Space & Security
Kevin Wise and Eugene Lavretsky
Highly reliable adaptive uncertainty approximation compensators for flight control applications:
unmanned aircraft – Phantom Ray
Tailkit adaptive control systems for Joint Direct Attack Munition (JDAM) munitions:
Mk-82, Mk-84, and Laser-guided variantsFielded for defense in Iraq and Afghanistan.
BUT-Nature is More than StableNature is OPTIMAL and conserves energy
We want to Learn optimal control solutions Online in real-time Using adaptive control techniquesWithout knowing the full dynamics
For Adaptive Man/Machine systems-
Reinforcement Learning
Every living organism improves its control actions based on rewards received from the environment
The resources available to living organisms are usually meager.Nature uses optimal control.
1. Apply a control. Evaluate the benefit of that control.2. Improve the control policy.
RL finds optimal policies by evaluating the effects of suboptimal policies
Optimality in Biological Systems
Different methods of learning
SystemAdaptiveLearning system
ControlInputs
outputs
environmentTuneactor
Reinforcementsignal
Actor
Critic
Desiredperformance
Reinforcement learningIvan Pavlov 1890s
Actor-Critic Learning
We want OPTIMAL performance- ADP- Approximate Dynamic Programming
Sutton & Barto book
Adaptive Critic structure
Slow learning Improvement loop
Reinforcement learning
Fast inner control loops
PBPBRQPAPA TT 10
1 Tu R B Px Kx
BuAxx
( ( )) ( ) ( , ) ( ) ( )T T Tt t
V x t x Qx u Ru d r x u d x t Px t
Full system dynamics must be knownOff‐line solution
Optimal Control: Linear Quadratic Regulator
System
Cost function
Optimal Control is
Algebraic Riccati equation
MATLAB Control Systems Toolbox
Chop off Tail of cost function
( ( )) ( , ) ( ( ))t T
t
V x t r x u d V x t T
( ( )) ( , ) ( , )t T
t t T
V x t r x u d r x u d
Bellman Equation
1 Tu R B Px Kx Update Control using Hamiltonian Physics
Reinforcement Learning‐ Policy Iteration
CT Policy Iteration – How to implement online?Linear Systems Quadratic Cost‐ LQR
( ) ( ) ( )( ) ( ) ( ) ( )t T
T T T Tk k k k
tx t P x t x Q K RK x d x t T P x t T
Policy evaluation‐ solve IRL Bellman Equation
( ) ( ) ( ) ( ) ( )( ) ( )t T
T T T Tk k k k
tx t P x t x t T P x t T x Q K RK x d
( ( )) ( ) ( )TV x t x t Px tValue function is quadratic
( ) ( ) ( ) ( ) ( ) ( )t T
T T T Tk k k k
tp t p x t x t T x Q K RK x d
Reinforcement on time interval [t, t+T]regression vector
Same form as standard System ID problems
Solve using RLS or batch LS
Union of Reinforcement Learning and Adaptive Control
Value Function Approximation
1. Select initial control policy
2. Find associated cost
3. Improve control 11T
k kK R B P
( ) ( ) ( ) ( ) ( ) ( , )t T
T T Tk k k
tp x t x t T x Q K RK x d t t T
Bellman EquationSolves Lyapunov eq. without knowing dynamics
t t+T
observe x(t)
observe x(t+T)
apply uk(t)=Kkx(t)
observe cost integral
update P
do RLS until convergence to Pk
update control gain to Kk+1
A is not needed anywhere
This is a data-based approach that uses measurements of x(t), u(t)Instead of the plant dynamical model.
Integral Reinforcement Learning (IRL)
Data set at time [t,t+T)
( ), ( , ), ( )x t t t T x t T
Motor control 200 Hz
Oscillation is a fundamental property of neural tissue
Brain has multiple adaptive clocks with different timescales
theta rhythm, Hippocampus, Thalamus, 4-10 Hzsensory processing, memory and voluntary control of movement.
gamma rhythms 30-100 Hz, hippocampus and neocortex high cognitive activity.
• consolidation of memory• spatial mapping of the environment – place cells
The high frequency processing is due to the large amounts of sensorial data to be processed
Spinal cord
D. Vrabie, F. Lewis, and Dan Levine- RL for Continuous-Time Systems
Doya, Kimura, Kawato 2001
Limbic system
Cerebral cortexMotor areas
ThalamusBasal ganglia
Cerebellum
Brainstem
Spinal cord
Interoceptivereceptors
Exteroceptivereceptors
Muscle contraction and movement
Summary of Motor Control in the Human Nervous System
reflex
Supervisedlearning
ReinforcementLearning- dopamine
(eye movement)inf. olive
Hippocampus
Unsupervisedlearning
Limbic System
Motor control 200 Hz
theta rhythms 4-10 Hz
picture by E. StinguD. Vrabie
Memoryfunctions
Long term
Short term
Hierarchy of multiple parallel loops
gamma rhythms 30-100 Hz,
Adaptive Critic structure
Theta waves 4-8 Hz
Reinforcement learning
Motor control 200 Hz
F.L. Lewis and D. Vrabie,“Reinforcement learning andadaptive dynamic programmingfor feedback control,” IEEECircuits & Systems Magazine,Invited Feature Article, pp. 32-50, Third Quarter 2009.
IEEE Control Systems Magazine“Reinforcement learning andfeedback Control,” Dec. 2012
D. Vrabie, K. Vamvoudakis, and F.L. Lewis, Optimal Adaptive Control and Differential Games by Reinforcement Learning Principles, IET Press, 2012, to appear.
Books
F.L. Lewis, D. Vrabie, and V. Syrmos, Optimal Control, third edition, John Wiley and Sons, New York, 2012.
New Chapters on:Reinforcement LearningDifferential Games
Direct Optimal Adaptive Controller
A hybrid continuous/discrete dynamic controller whose internal state is the observed cost over the interval
Draguna Vrabie
DynamicControlSystemw/ MEMORY
xu
V
ZOH T
x A x B u System
T Tx Q x u R u
Critic
ActorL
T T
Run RLS or use batch L.S.To identify value of current control
Update FB gain afterCritic has converged
Reinforcement interval T can be selected on line on the fly – can change
Solves Riccati Equation Online without knowing A matrix
Optimal Control Design Allows a Lot of Design Freedom
( ( )) ( , )t
V x t r x u d
Tailor controls design by choosing utility
Minimum EnergyMinimum FuelMinimum TimeControl Actuator Constraints
Optimal Control for Constrained Input Systems
This is a quasi-norm
2
0
2 ( )u
Tq
u d
Weaker than a norm –homogeneity property is replaced by the weaker symmetry property
qqxx
(Used by Lyshevsky for H2 control)
Control constrained by saturation function tanh(p)
p
1
-1
0 0
( , ) ( ) 2 ( )u
TJ u d Q x d dt
Encode constraint into Value function
Then Is BOUNDED1 ( )T Vu R g xx
Murad Abu-Khalaf
-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6State Evolution for both controlllers
x1
x2
Switching Surface methodNonquadratic functionals method
10 0
tanh( ) 2 ( )u
TTV x Q x Rd dt
Near Minimum-Time ControlEncode into Value Function
Murad Abu-Khalaf
Force ControlIntelligent Automation, Inc.
Flexible pointing systemsSimis labs, Inc. and US Army
Vehicle active suspensionLeo Davis Technol.
SBIR ContractsSBA Tibbets Award 1996 (led by TMAC)
Learning NN Controller
A. Yesildirek and F.L. Lewis, "Method for feedback linearization of neural networks and
neural network incorporating same," U.S. Patent 5,943,660, awarded 24 August 1999.
S. Jagannathan and F.L. Lewis, "Discrete-time tuning of neural network controllers for
nonlinear dynamical systems," U.S. Patent 6,064,997, awarded 16 May 2000.
R. Selmic, F.L. Lewis, A.J. Calise, and M.B. McFarland, "Backlash Compensation Using
Neural Network," U.S. Patent 6,611,823, awarded 26 Aug. 2003.
J. Campos and F.L. Lewis, "Method for Backlash Compensation Using Discrete-Time
Neural Networks," U.S. Patent 7,080,055, awarded July 2006.
Patents
K. Vamvoudakis, D. Vrabie, and F.L. Lewis, “Control methodology for online adaptation to optimal feedback controller using integral reinforcement learning,” provisional patent, filed March 2012.
Applications of our Algorithms at Boeing Defense Space & Security
Kevin Wise and Eugene Lavretsky
Highly reliable adaptive uncertainty approximation compensators for flight control applications:
Unmanned aircraft – Phantom Ray
Tailkit adaptive control systems for Joint Direct Attack Munition (JDAM) munitions:
Mk-82, Mk-84, and Laser-guided variants
Fielded for defense in Iraq and Afghanistan.Saved Lives.
optimal engine controllers based on RL for Caterpillar, Ford (Zetec engine), and GM
Applications of Our Algorithms to Auto Engine Control
Student S. Jagannathan11 US patents
8-10% improvement in fuel efficiency and a drastic reduction in NOx (90%), HC (30%) and CO (15%) by operating with adaptive exhaust gas recirculation.
Savings to Caterpillar were over $65,000 per component.
A Q-Learning Based Adaptive Optimal Controller Implementation for a Humanoid
Robot Arm
Said Ghani Khan1, Guido Herrmann1, Tony Pipe1, Chris Melhuish1
Bristol Robotics LaboratoryUniversity of Bristol and University of the West of England, Bristol, UK
Conference on Decision and Control (CDC) 2011, Orlando, 11 December 2011
The cost function is modified to include constraints
Introducing Constraints
The new cost function....
qL is the joint limit.
BRL BERT II ARM Bristol Robotics Lab
Natural Human-Like Motion
An Approximate Dynamic Programming Based Controllerfor an Underactuated 6DoF Quadrotor
Emanuel Stingu Frank Lewis
™
Automation & Robotics Research InstituteUniversity of Texas at ArlingtonARRI
Supported byARO grant W91NF-05-1-0314NSF grant ECCS-0801330
3 control loops
The quadrotor has 17 states and only 4 control inputs, thus it is very under-actuated.Three control loops with dynamic inversion are used to generate the 4 control signals.
x
y
zamountof tilt
yaw
directionof tilt
Momentum theory applied to the propeller
Translationand yaw
Controller
AttitudeController
Motor andPropellerController
Referencetrajectory
Position and velocity errors
Desired attitude
Desired forces
Motor commands
Dynamic inversionaccelerations to attitude
Dynamic inversionrotation accelerations to forces
Altitude control: thrust force
Dynamic inversionthrust forces to motor voltages
Using standard controller
UsingReinforcement learningcontroller
Potential New Applications in Prosthetics
With Sesh Commuri, Oklahoma UniversityAnd Muthu Wijesundara
Adaptive Reinforcement Learning of Human/Prosthetic Ankle Interactions
Current methods use stored gait information
What about Adaptive Human-Robot Interfaces (HRI)?
Adaptive Learning
HRI
Different usersDifferent robotic devices
Interface learns so as to improve Human/Robot team performance
Tune parameters such as stiffness, speed of response, dynamic motion
There are 3 componentsHumanRobotThe Interface
They should all be intelligent
We want verifiable performance and straightforward tuning algorithms
Several Approaches to Adaptive HRI1. Learning of Coordinate Transforms
2. Dynamic Motion Primitives to Modify Task Parameters
3. Adaptive Impedance Control
Current and Expected Funding
F.L. Lewis, “Adaptive Dynamic Programming for Real-Time Cooperative Multi-Player Games and Graphical Games,” NSF Grant, $272,000 for 3 years, July 2011.
D. Popa, Z. Celik-Butler, D. Butler, and F.L. Lewis, “NRI: Multi-Modal Skin and Garments for Healthcare and Home Robots,” NSF Grant, $1.3M for 4 years, Sept. 2012.
F.L. Lewis, “Games and Learning for Cooperative Nonlinear Systems and Internal Structure of Coalitions on Graphs, TARDEC/NAC grant, $81,000 for 1 year, Jan. 2013 expected.IF NO FISCAL CLIFF!
1d dq J y
Robot Inverse kinematics
( )dy t ( )dq t
( )dy t Robot Inverse Jacobian
Hard to find AccuratelyNeed dynamics model of robotDoes not generalize to different robots
Learning of Coordinate TransformsUser coordinates Robot coordinates
User coordinates Robot coordinates
Standard robot controller
( )dy t Robot Inverse Jacobian
Hard to find accuratelyNeed dynamics model of robotDoes not generalize to different robots
1. Learning of Coordinate Transforms
Usercoordinates Robot
coordinates
Standard robot controller
Learning the Interface MappingWhat is it exactly?
u y
f uWhat can we do to get f(u)?
The simplest way is to obtain a set of inputs and a set of outputs and calculate the relationship (Curve Fitting)
uy
Curve Fitting Algorithm f
x
0 0( )
M P
i i ij ii j
y f u w w u
Static nonlinear map
Dan PopaReinforcement Learning
NN Training
Log sigmoid function is used as a neuron activation function . 1
1 ex
Train using MATLAB NN Toolbox very easily if input/output pairs are known.
Dan Popa
Static Mapping Approach
f u
f(u) is a coordinate transformation
Found by training neural network with known input/output pairs Dan Popa
Reinforcement Learning
uy
Curve Fitting Algorithm f u
What if we cannot specify the desired output trajectory directly ?
“Learning by interacting with an environment”Reinforcement LearningReinforcement Learning
With RL we do not have to specify a desired trajectory.
Instead, a Reward Function is used
Dan Popa
How to find the Reward Function?
Model of Task
A task is a profile in velocity/force spaceDifferent Curve for Different Tasks
2. Learn and Modify Task Parameters
20( ) ( ) ( )
T Tx Dx Ks g x K g x KW V ss s
Dynamic Motion Primitives for Characterizing the Task
Learning the task:Kinesthetic demonstration to learn D, K, initial weightsReward-based learning to tune the weights
Phase variable s makes DMP independent of time
Tunable NN weights
Different NN weights give different task trajectories
• Build an adaptive interface system that allows a single operator to manipulate a robots with multiple degrees of freedom and/or multiple robots• The interface system should be intuitive, easy to use and can be learned quickly , and should be able to apply with different robot configurations
Robot System
Operator
Performance Evaluation
Output
Interface Systeminput
Reinforcement Learning for Human-Robot InterfacesDan Popa
What to learn?How to evaluate Performance?
Adaptive Impedance Control
3. Adaptive Impedance Control
ImpedanceModel
Inner loop FB controllers
Human intentRobot performance
Performanceevaluation
Tuning
Tune interface so robot learns to control the human model to improve task performance
Model of Human Behavior
For a given task, a skilled human operator adapts to learn-
1. An inverse model of the robot system to cancel nonlinearities
2. A feedforward predictive control based on the task
Frequency Crossover Theory -
Human operator adapts to make the overall closed-loop man-machine system
look like a high bandwidth frequency response
And remain invariant to a wide range of task variations and changes in operating condition
Suzuki & Furuta 2012
For a given task, after learning, the human model has three parts1. Time delay due to vision processing and reaction time2. A first-order filter due to neuromuscular dynamics3. A PD controller in the cerebellum
Adaptive Impedance Control as a 2-Player Game
InterfaceParameters
Human intentRobot performance
Critic Agent forPerformance Evaluation
Tuning
A 2-Player Dynamic Game
Player 1- the human
Player 2- the Interface!
Sun Tz bin fa孙子兵法
Graphical Coalitional Games
500 BC
Optimal Control is Effective for:Aircraft AutopilotsVehicle engine controlAerospace VehiclesShip ControlIndustrial Process ControlRobot Control
Multi-player Games Occur in:EconomicsControl Theory disturbance rejectionTeam gamesInternational politicsSports strategy
Optimality and Games
Actor‐Critic Game structure ‐ three time scales
x
u2 1 0;x Ax B u B w x
System
Controller/Player 1
w( 1)T i kuu B P x
11T i
ww B P x
Interface/Player 2
CriticLearning procedure
ˆ ˆ, if =1
ˆ ˆ ˆ ˆ, if >1
T T T
T T
V x C Cx u u i
V w w u u i
T T
1 2
1iuP( 1) 1 ( 1)i k i i k
u u uP P Z
V
iuP
x
( ) 1 ( )i k i i ku u uP P Z
The cloud-capped towers, the gorgeous palaces,The solemn temples, the great globe itself,Yea, all which it inherit, shall dissolve,And, like this insubstantial pageant faded,Leave not a rack behind.
We are such stuff as dreams are made on, and our little life is rounded with a sleep.
Our revels now are ended. These our actors, As I foretold you, were all spirits, and Are melted into air, into thin air.
Prospero, in The Tempest, act 4, sc. 1, l. 152-6, Shakespeare
The way that can be told is not the Constant WayThe name that can be named is not the Constant Name
For nameless is the true wayBeyond the myriad experiences of the world
To experience without intention is to sense the world
All experience is an archwherethrough gleams that untravelled landwhose margins fade forever as we move
Dao ke dao feichang daoMing ke ming feichang ming
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