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Summary for the course Human Robot Interaction (0HM280)
Summary of all required literature for the course human robot interaction (0HM280)
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0HM280 - Human Robot InteractionProbabilistic Robotics
Chapter 1: Basics
Robotics: science of perceiving and manipulating physical world through computer-controlled
devices.
Robots have to be able to accommodate the enormous uncertainty that exists in the physical world.
There are a number of factors that contribute to a robot’s uncertainty:
- Robot environments are inherently unpredictable, particularly when in proximity of people.
- Sensors are limited in what they can perceive. The limitations arise from:
o Range and solution of a sensor is subject to physical limitations.
o Subject to noise which perturbs sensor measurements in unpredictable ways
o Sensors can break
- Robot actuation involves motors that are, at least to some extent, unpredictable. Uncertainty
arises from effects as control noise, wear-and-tear, and mechanical failure.
- All internal models of the world are approximate, they are abstractions of the real world.
- Algorithmic approximations: robots are real time systems. This limits the amount of
computations that can be carried out.
Level of uncertainty depends on the application domain.
Probabilistic robotics: pays tribute to the uncertainty in robot perception and action. The key is to
represent uncertainty explicitly using the calculus of probability theory.
Robot localization: problem of estimating a robot’s coordinates relative to an external reference
frame. A robot has a map of its environment, but to localize itself relative to this it needs to consult
sensor data.
Global localization: a robot is placed somewhere in a known environment and has to localize itself
from scratch. The probabilistic paradigm represents the robot’s momentary belief by a probability
density function over the space of all locations.
The example of the robot determining in front of which door it stands illustrates many aspects of the
probabilistic paradigm. Stated probabilistically, the robot perception problem is a state estimation
problem, and the localization example uses an algorithm known as Bayes filter for posterior
estimation over the space of robot locations. The representation of information is a probability density
function. The update of this function represents the information gained through sensor measurements,
or the information lost through processes in the world that increase a robot’s uncertainty.
In contrast with traditional programming techniques probabilistic approaches:
- Tend to be more robust in the face of sensor limitations and model limitations. This enables
them to scale much better to complex real-world environments.
- Have weaker requirements on the accuracy of the robot’s models, thereby relieving the
programmer from the insurmountable burden to come up with accurate models.
- Have weaker requirements on the accuracy of robotic sensors than those made by many
reactive techniques, whose sole control input is the momentary sensor input.
Limitations of probabilistic algorithms;
- Computational complexity
- Need to approximate.
- Inherently less efficient then non-probabilistic counterparts because they consider entire
probability densities instead of a single guess. The need to approximate arises from the fact
that most robot worlds are continuous. Computing exact posterior distributions tends to be
computationally intractable.
Chapter 2: Recursive State Estimation
State estimation addresses the problem of estimating quantities form sensor data that are not
directly observable, but that can be inferred.
In most robotic applications, determining what to do is relatively easy if one only knew certain
quantities. Unfortunately these are often not directly measurable. Sensors carry only partial
information about those quantities, and their measurements are corrupted by noise. State estimation
seeks to recover state variables from the data.
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,Probabilistic state estimation algorithms compute belief distributions over possible world states.
Random variables can take on multiple values and they do so according to specific probabilistic
laws.
Probabilistic inference: process of calculating these laws for random variables that are derived from
other random variables and observed data.
p(X=x) or p(x): probability that the random variable X has value x.
Probabilities are always non negative.
Probability density functions (PDF) Of a normal distribution:
given by the Gaussian function: It always integrates to 1. But it is
not upper bounded by 1.
Normal distributions: N ( x :μ , σ 2) specifies the random variable, the mean and variance.
Multivariate distribution: normal distributions over vectors. They are characterized by density
functions: Here μ is the mean vector. Σ a positive
semidefinite and symmetric matrix called the covariance matrix. The superscript T marks the
transpose of a vector. The argument in the exponent in this PDF is quadratic in x, and the parameters
of this quadratic function are μ and Σ.
Joint distribution of two random variables X and Y p(x,y) = p(X=x and Y=y).
If they are independent we have p(x,y) = p(x)p(y).
Conditional probability: Suppose we already know that Y ’s value is y, and we would like to know
the probability that X’s value is x conditioned on that fact. Such a probability is: p(x | y) = p(X = x | Y =
y).
- If p(y)> 0 p(x | y) = p(x, y)/p(y)
- If X and Y are independent: p(x | y) = (p(x)p(y))/p(y) = p(x)
Theorem of total probability:
Bayes rule: relates p(x | y) to its inverse: p(y | x). the rule as stated her requires p(y)>0
Conditional impudence: p(x, y | z) = p(x | z) p(y | z) . It applies whenever a variable y carries no
information about a variable x if another variable’s value z is known.
Expectation of a random variable X is given by:
Entropy of a probability distribution is given by the following expression:
. It is the expected information that the value of x carries.
Robot environment/world: dynamical system that possesses internal state. Characterized by states.
State (x): collection of all aspects of the robot and its environment that can impact the future. States
that change are dynamic states and non-changing states are static states.
Typical state variables:
- Robot pose: location and orientation relative to the global coordinate frame.
- Configuration of the robots actuators / kinematic state
- Robot velocity and velocities of its joints: dynamic state
2
, - Location and features of surrounding objects in environment. Can be in the form of
landmarks, which are distinct, stationary features of the environment that can be recognized
reliably.
- Location and velocities of moving objects and people
Complete state: state xt if it is the best predictor of the future. completeness entails that knowledge of
past states, measurements, or controls carry no additional information that would help us predict the
future more accurately.
Hybrid state spaces: state spaces that contain both continuous and discrete variables.
Fundamental interactions of the robot with the environment:
- Environment sensor measurements: perception is process by which the robot uses its
sensors to obtain information about the state of its environment. Result of such a perceptual
interaction is a measurement (also called observation or percept). Typically there is some
delay.
- Control actions: change state of world by actively asserting forces on the robot’s environment.
Robot has access to two different data streams:
- Environment measurement data: information about a momentary state of the environment.
- Control data: information about change of state in the environment. Examples are velocity
and odometers (sensors that measure the revolution of a robot’s wheels).
Conditional independence: certain variables are independent of others if one knows the values of a
third group of variables, the conditioning variables.
Dynamics of a robot and its environment are characterized by two probabilistic laws:
1. State transition probability: p(xt | xt−1, ut), specifies how environmental state evolves over
time as a function of robot controls ut.
2. Measurement distribution probability: p(zt | xt), specifies probabilistic law according to
which measurements z are generated from the environment state x.
State transition and measurement probabilities together describe the dynamical stochastic system of
the robot and its environment.
Belief: reflects robot’s internal knowledge about the state of the environment. Other names are state
of knowledge or information state.
Belief distribution: assigns a probability (or density value) to each possible hypothesis with regards
to the true state.
Posterior: probability distribution over the state xt at time t, conditioned on all past measurements and
past controls.
Bayes filter algorithm: calculates the belief distribution (bel) from measurement and control data.
The belief bel(xt) at time t is calculated from the belief(xt-1) at time t-1.
In line 3, the algorithm processes the control ut. This step is called control update or prediction. In line
4, the measurement update is performed. The algorithm multiplies the belief by the probability that the
measurement zt may have been observed. It does so for each hypothetical posterior state xt.
Markov assumption / complete state assumption: postulates that past and future data are
independent if one knows the current state xt.
When choosing an approximation one has to trade off a range of properties:
- Computational efficiency
- Accuracy of the approximation
- Ease of implementation.
3
, Since the Bayes filter is not a practical algorithm, in that it cannot be implemented on a digital
computer, probabilistic algorithms use tractable approximations. Such approximations may be
evaluated according to different criteria, relating to their accuracy, efficiency, and ease of
implementation.
Chapter 3. Gaussian Filters
Gaussian techniques all share the basic idea that beliefs are represented by multivariate normal
distributions, which is: . This density over the
variable x is characterized by two sets of parameters: mean μ and covariance Σ.
Gaussians can be represented in two different ways:
- Moments parameterization: consists of mean (first moment) and covariance (second
moment) of Gaussian.
- Canonical/natural parameterization: consists of information matrix and an information
vector.
Bayes filters can be implemented for both parameterizations. When using the moments
parameterization, the resulting filter is called Kalman filter. The dual of the Kalman filter is the
information filter, which represents the posterior in the canonical parameterization.
For both filters to calculate the correct posterior, three assumptions have to be fulfilled:
1. Initial belief must be Gaussian.
2. State transition probability must be composed of a function that is linear in its argument with
added independent Gaussian noise.
3. Same applies to the measurement probability. It must also be linear in its argument, with
added Gaussian noise.
Systems that meet these assumptions are called linear Gaussian systems
The Kalman Filter
Kalman filter: technique for filtering and prediction in linear Gaussian systems. It implements belief
computation for continuous states. It is not applicable to discrete or hybrid state spaces. The Kalman
filter represents beliefs by the moments parameterization: At time t, the belief is represented by the
mean μt and the covariance Σt.
Kt = Kalman gain: specifies degree to which the measurement is incorporated into new state
estimate.
Innovation: difference between actual measurements zt and expected measurement Ct μt (line 5).
The Kalman filter alternates a measurement update step (lines 5-7), in which sensor data is integrated
into present belief, with a prediction step (or control update step), which modifies belief in accordance
to an action. Update step decreases and the prediction step increases uncertainty in the robot’s belief.
Extended Kalman Filter (EFK) relaxes the linearity assumption. Here the assumption is that the state
transition probability and the measurement probabilities are governed by nonlinear functions g and h,
respectively:
It calculates a Gaussian approximation to the true belief. Idea underlying EKF approximation is
linearization, which approximates nonlinear function g by a linear function that is tangent to g at the
mean of the Gaussian. Advantage is efficiency. EFK uses (first order) Taylor expansion for
linearization.
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