Dynamical Systems, Stochastic Processes, and Probabilistic Robotics
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Transcript of Dynamical Systems, Stochastic Processes, and Probabilistic Robotics
Dynamical Systems, Stochastic Processes,
and Probabilistic Robotics
David Rosen
Goals Overview of some of
the big ideas in autonomous systems
Theme: Dynamical and stochastic systems lie at the intersection of mathematics and engineering
ZOMG ROBOTS!!!
Actually, no universally accepted definition For this talk:
◦ Sensing (what’s going on?)◦ Decision (what to do?)◦ Planning (how to do it?)◦ Actuation & control (follow the plan)
What is a robot?
Dynamical SystemsHow do we begin to think about this problem?
General Mathematical Framework: Dynamical Systems
General Mathematical Framework: Dynamical Systems
Sensing and Estimation
What’s going on?
Sensing and Estimation How do we know
what the state is at a given time?
Generally, we have some sensors:
◦ Laser rangefinders◦ GPS◦ Vision systems◦ etc…
Great!
Well, not quite…◦ In general, can’t measure all state variables
directly. Instead, an observation function H : M → O maps the current state x to some manifold O of outputs that can be directly measured
◦ Usually, dim O < dim M ◦ Given some observation z = H(x), can’t determine x
!
Sensing and Estimation
Sensing and Estimation Maybe we can use the
system dynamics (f ) together with multiple observations?
Observability: Is it possible to determine the state of the system given a finite-time sequence of observations?◦ “Virtual” sensors!
Detectability (weaker): Are all of the unobservable modes of the system stable?
Sensing and Estimation What about noise?
In general, uncorrected/unmodeled error accumulates over time.
Stochastic processes: nondeterministic dynamical systems that evolve according to probability distributions.
New model:
for randomly distributed variables wt and vt .
We assume that xt conditionally depends only upon xt-1 and the control ut (completeness):
Stochastic processes that satisfy this condition are called Markov chains.
Sensing and Estimation
Similarly, we assume that the measurement zt conditionally depends only upon the current state xt :
Sensing and Estimation
Thus, we get a sequence of states and observations like this:
This is called the hidden Markov model (HMM).
Sensing and Estimation
How can we estimate the state of a HMM at a given time?
Any ideas?
Sensing and Estimation
Hint: How might we obtainfrom ?
Sensing and Estimation
Bayes’ Rule
Bayes’ Rule
Bayes’ Rule
Punchline: If we regard probabilities in the Bayesian sense, then Bayes’ Rule provides a way to optimally update beliefs in response to new data. This is called Bayesian inference.
It also leads to recursive Bayesian estimation.
Define
Then by conditional independence in the Markov chain:
and by Bayes’ rule:
Recursive Bayesian Estimation: The Bayes Filter
Recursive Bayesian Estimation: The Bayes Filter
This shows how to compute given only and the control input .
Recursive filter!
Initialize the filter with initial belief
Recursion step:
◦ Propagate:
◦ Update:
Recursive Bayesian Estimation: The Bayes Filter
Recursive Bayesian Estimation: The Bayes Filter
Benefits of recursion:◦ Don’t need to
remember observations◦ Online implementation◦ Efficient!
Applications:◦ Guidance◦ Aerospace tracking◦ Autonomous mapping
(e.g., SLAM)◦ System identification◦ etc…
Example: Missile guidance This clip was
reportedly sampled from an Air Force training video on missile guidance, circa 1955.
It is factually correct.
See also:◦ Turboencabulator◦ Unobtainium
Rudolf Kalman
Trajectory GenerationHow do we identify trajectories of the system with desirable properties?
Recap: Control Systems
Controllability: given two arbitrary specified states p and q, does there exist a finite-time admissible control u that can drive the system from p to q ?
Reachability: Given an initial state p, what other states can be reached from p along system trajectories in a given length of time?
Stabilizability: Given an arbitrary state p, does there exist an admissible control u that can stabilize the system at p ?
Key questions for trajectory generation
Several methods for generating trajectories ◦ Rote playback◦ Online synthesis from libraries of moves◦ etc…
Optimal control: Minimize a cost functional
amongst all controls whose trajectories have prescribed initial and final states x0 and x1.
Trajectory Generation
Provides a set of necessary conditions satisfied by any optimal trajectory.
Can often be used to identify optimal controls of a system.
The Pontryagin Maximum Principle
Lev Pontryagin
The Pontryagin Maximum Principle
The Pontryagin Maximum Principle
Can also derive versions of the PMP for:
State-constrained control
Non-autonomous (i.e., time-dependent) dynamics.
etc…
The Pontryagin Maximum Principle
Nothing Could Possibly Go Wrong…
Trajectory FollowingHow can we regulate autonomous systems?
The problem
Real-world systems suffer from noise, perturbations
If the underlying system is unstable, even small perturbations can drive the system off of the desired trajectory.
Example: Pendulum on a Cart
We have a desired trajectory that we would like to follow, called the reference.
At each time t, we can estimate the actual state of the system .
In general there is some nonzero error
at each time t.
What to do?
What to do? Maybe we can find some
rule for setting the control input u (t ) at each time t as a function of the error e (t ) such that the system is stabilized?
In that case, we have a feedback control law:
Many varieties of feedback controllers:
Proportional-integral-derivative (PID) control
Fuzzy logic control Machine learning Model adaptive control Robust control H∞ control etc…
Trajectory Following: Feedback Control
We started with what (at least conceptually) were very basic problems from engineeringe.g.,
make do this this
The Moral of the Story
and ended up investigating all of this:
Dynamical systems Stochastic processes Markov chains The hidden Markov model Bayesian inference Recursive Bayesian estimation The Pontryagin Maximum Principle Feedback stabilization
and this is just the introduction!
The Moral of the Story
Questions?