Linguistic Control of Mobile Robots

1
Linguistic Control of Mobile Robots
Magnus Egerstedt Electrical and Computer
Engineering Georgia Institute of Technology

• How should tokenized, symbolic instructions be
• interpreted when controlling mobile robots?
• Linguistic Control
• Graph Models (FRF-Automata)
• Complexity
• Coding
• Hybrid Models
• Open Issues

2

• Really, what I would like to talk about is

Theorem The Fundamental Theorem of
Robotics
3

• Really, what I would like to talk about is

Theorem Theorema Egregium Robotica
4

• Really, what I would like to talk about is

Theorem Theorema Egregium Robotica
?
5
Research Taxonomy

• By scanning the Proceedings from the last few
  ICRAs
• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.
• A fundamental theorem of robotics should capture
  all of these areas.
• Claim Linguistic control provides a tool for
  achieving this

6
Main Questions

• What is a linguistic control procedure?
• What is the complexity of a given control
  procedure?
• How should mobile robots be instructed so that
  this complexity is minimized?
• How should the instructions be represented in
  order to minimize the communications?

• A control procedure specifies the evolution of a
  dynamical system in terms of open-loop control,
  closed-loop control and temporal duration

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FRF Automata

• Linguistic The control procedure is finitely
  describable. (Initially, let the system evolve on
  a digraph.)
• The input symbols should capture both open-loop
  and closed-loop components as well as the
  interrupts, i.e. the state should be advanced
  using a constant input symbol until an interrupt
  is triggered.
• Let the input be s(u,k,x), where
• The input alphabet
• is finite

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FRF Automata

• Definition (X,S,Y,d,g) with
• is said to be a free-running, feedback
  automaton. Transitions are generated according
  to
  
  
  
  
  
  
  
• The complexity of the instruction should
  correspond to how many bits of information are
  needed for coding the control procedure
• A control sequence s over S has description
  length

Egerstedt, Brockett. Feedback Can Reduce the
Specification Complexity of Motor Programs. CDC
2001, IEEE Transactions on Automatic Control,
2002.
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Specification Complexity
Definition The task of driving A between
x0 and x1 has specification complexity
where s is the shortest word that achieves
this
C(A,x0,x1)L(s,S)
A larger input set might result in shorter input
strings but still produce higher complexity
instructions. In particular How does
C(ACL,x0,x1) compare to C(AOL,x0,x1)?
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Feedback vs. Open-Loop Control

• The benefits of feedback can be understood by
• The Black argument for reducing drift in
  high-gain amplifiers
• The stochastic disturbance rejection argument
• The game theoretic argument for enforcing saddle
  point conditions ( )
• The complexity argument?

• We should expect that feedback is to prefer when
  observer-based stabilizing controllers can be
  designed
• By observability we understand that there exists
  a feedback controller such that it is possible to
  uniquely determine the state of the system (after
  a while)

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How Do We Give Instructions?

• We can draw inspiration from the way we give
  instructions
• Follow the highway until exit 5. Then make a
  right
• Sail west until you see the shore. Aim towards
  land
• Cook for 5 minutes until golden brown. Serve
  after it has cooled down
• It seems like we combine long open-loop paths
  with short closed-loop paths

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Subset Observers

• If we use observer-based feedback the complexity
  depends on the cardinality of the state space of
  the observer automaton
• Construct observers on reduced parts of the state
  space

Lemma Let Xf be an observable subset,
ballistically reachable from x0. If xf is
control-invariantly reachable in Xf then xf
can be reached from x0 using only one
instruction
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Main Theorem
Theorem Given assumptions about ballistic
reachability and observable subsets
Egerstedt, Brockett. Feedback Can Reduce the
Specification Complexity of Motor Programs. CDC
2001, IEEE Transactions on Automatic Control,
2002.
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Navigation Using Landmarks

The complexity of the instruction is reduced if
we use many, easily detected landmarks. Consiste
nt with current practice in robotics.
Egerstedt. Some Complexity Aspects of the
Control of Mobile Robots. ACC 2002.
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Sensor Selection

• card(S) depends exponentially on card(Y), i.e. by
  choosing sensors with different resolutions we
  should change the specification complexity
• Laser scanners Large Y
• Sonars Smaller Y

• Is there an optimal choice of sensors relative
• to a given task and a given automaton?

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k-Sensors on a Bounded Lattice
Assume the automaton is defined on a bounded
d-dimensional lattice. Assume that the states
of distance k from the current state can be
observed (k-sensor). If the goal-state is
unknown then the state space has to be
traversed in a systematic way.
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Lattice Automata
Lemma The goal state can be reached using
a total of closed-loop instructions.

Proposition C(k)log2(card(S)) is convex in k
(over R)
Egerstedt. Some Complexity Aspects of the
Control of Mobile Robots. ACC 2002.
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Sensor Selection
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The Fundamental Theorem of Robotics

• What should such a result include?
• Sensor selection (Y)
• Actuator design (U)
• Control law (s)
• The specification complexity is
• i.e. C(A,x0,xf) provides a unified tool for
  addressing these questions
• Further robotics applications
• Teleoperation
• Multi-agent coordination
• Communications in embedded systems
• Minimum attention control
• Control driven coding theory

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Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.

21
Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.

22
Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.

23
Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.

24
Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.

25
Research Taxonomy

• Electro-mechanical design of robot platforms,
  sensors, and actuators
• The development of rational methods for
  collecting, compressing, and analyzing sensory
  data
• Deliberative, high-level mission planning
• Control design for different classes of dynamical
  systems and
• The construction of appropriate software
  architectures and data structures for internal
  representation.
• We have the tool for asking such a unified
  question!

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Coding Theory

• is only correct if each
  input symbol is equally likely
• The entropy of S is given by
• Shannons source coding theorem

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Coding Theory

• A probability distribution over S is needed for
  optimal coding of the control procedures
• Existing multi-modal systems
• Theoretical mode synthesis results
• Learning techniques
• Mode identification
• Control designs and coding strategies are
  interlinked

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Mode Identification
Courtesy of Tucker Balch, http//www.cc.gatech.edu
/tucker/
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Mode Identification
Courtesy of Jessica Hodgins, http//www-2.cs.cmu.e
du/jkh/
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Mode Identification

• Typically, a string of outputs (possibly also
  inputs) is observed
• How can the mode structure be recovered?
• Minimize the number of modes consistent with the
  data

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Dynamic Programming
Dynamic programming can be used for finding the
smallest number of consistent modes in steps
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Conclusions So Far

• Complexity
• The role of feedback and open loop control can be
  understood from a complexity point of view.
• A formal framework for capturing this distinction
  is presented - FRF-Automata
• Applications and implications
• Minimum attention control, decentralized control,
  hybrid and embedded control.
• Robotics How should robots be programmed?
  Teleoperation.
• Fundamental Theorem of Robotics
• Sensor Selection (How choose Y?)
• Actuator Selection (How choose U?)
• Control Design (How pick s in S?)
• These questions can all be answered within one
  unified framework!
• Coding
• By establishing a probability distribution over S
  optimal coding strategies can be produced
• Continuous devices? Hybrid models?

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Trigger Based Hybrid Systems

• Brocketts type-B hybrid system
• where v(1), v(2), is a string of tokenized
  inputs
• It should be possible to model linguistic control
  procedures on this form

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Motion Description Languages

• Let U,Y be finite. A motion description language
  (MDL) is given by a subset of all the finite
  length words over S, e.g. s(u1,k1,x1),,(uq,kq,xq
  )
• The hybrid system evolves according to
• where Ti is the time at which xi changes from 0
  to 1

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Specification Complexity

• The specification complexity can be defined as
• and complexity results can be derived for
  continuous machines in a manner analogous to the
  FRF-automata
• Mode design for such systems is an open question
• How about mode learning?

Egerstedt. On the Specification Complexity of
Linguistic Control Procedures. International
Journal of Hybrid Systems, 2002. Egerstedt.
Linguistic Control of Mobile Robots. IROS 2001.
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Soccer Robots
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Soccer Robots
Egerstedt. Learning to Linguistically Control
Mobile Robots. IEEE Transactions on Robotics and
Automation. Submitted.
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Conclusions

• Complexity FRF-Automata and Trigger-Based Hybrid
  Systems
• Applications and Implications
• Minimum attention control, decentralized control,
  hybrid and embedded control, robotics.
• Fundamental Theorem of Robotics
• Coding
• By establishing a probability distribution over S
  optimal coding strategies can be produced

Much interesting work remains to be done in
this intersection between control theory and
information theory!