Probabilistic Robotics Introduction

SA-1 2 Introduction  Robotics is the science of perceiving and manipulating the physical world through computer-controlled devices.  Desirability of “intelligent” manipulating devices. Examples…  To be intelligent, robots have to accommodate the enormous uncertainty that exists in the physical world.

SA-1 3 Introduction  Where does Uncertainty come from?  robot environments are inherently unpredictable. Assembly line, high way, private homes (highly dynamic and highly unpredictable)  Sensors range and resolution,noise…  Robot actuation motor, control noise,wear-and-tear, mechanical failure….  robot software, internal models of robots are crude and approximate. Model errors are a source of uncertainty that has often been ignored in robotics, despite the fact that most robotic models used in state-of-the-art robotics systems are rather crude.  Algorithmic approximation

SA-1 4 Introduction  As robotics is now moving into the open world, the issue of uncertainty has become a major stumbling block for the design of capable robot systems.  Managing uncertainty is possibly the most important step towards robust real- world robot systems.  Hence Probabilistic Robotics

SA-1 5 Introduction- Probabilistic Robotics  A relatively new approch robotics.  The key idea is to represent uncertainty explicitly using the calculus of probability theory.  Maintaining a probability distribution instead of a single best guess.  It outperforms alternative techniques in many real-world applications.

SA-1 6 Introduction- Probabilistic Robotics Examples using Probabilistic Robotics.  Mobile Robot Localization  The problem of estimating a robot‘s coordinates relative to an external reference frame. Map of environment is given, find out where I (the robot) am. Can consulte sensor data and can move around.  See Figure 1.1

SA-1 7 Introduction- Probabilistic Robotics Examples using Probabilistic Robotics.  Robotic Planning and Control  Coastal navigation  See Figure 1.2

SA-1 8 Implication of Probabilistic Robotics  Probabilistic robotics seamlessly integrates models with sensor data, overcoming the limitations of both at the same time.  A bit of history of robotics research  Model-based paradigm  Behavior-based paradigm  Probabilistic robotics tends to bemore robust in the face of sensor limitations and model limitations. Scale much better to complex real-world environments.  Probabilistc approachs are currently the only known working solutions to hard robotic estimation problems, such as localization and mapping

SA-1 9 Implication of Probabilistic Robotics  Probabilistic robotics have weaker requirements on the accuracy of the robot‘s models, and weaker requirements on the accuracy of robotic sensors, compared with previous approaches.  Probabilistic robotics is also criticized because of  Computational complexity, and  Need to approxiamte.

SA-1 10 Probabilistic Robotics Recursive State Estimation

SA-1 11 Recursive State Estimation  Give me the sensor data, I will tell you something.  Estimate state from sensor data.  This is the core idea of probabilistic robotics.  Probabilistic state estimation algorithms comopute belief distributions over possible world states. An example, already shown, what is it?  The purpose of this part of lecture  Introduce notions and notions that will be used.  Background preparation  Introduce the algiorthm Bayes filters, the single most important algorithm that is the basis of virtually every techniques presented in the book.

12 Pr(A) denotes probability that proposition A is true. Axioms of Probability Theory

13 A Closer Look at Axiom 3 B

14 Using the Axioms

15 Discrete Random Variables X denotes a random variable. X can take on a countable number of values in {x 1, x 2, …, x n }. P(X=x i ), or P(x i ), is the probability that the random variable X takes on value x i. P( ) is called probability mass function. E.g..

16 Continuous Random Variables X takes on values in the continuum. p(X=x), or p(x), is a probability density function. E.g. x p(x)

17 Joint and Conditional Probability P(X=x and Y=y) = P(x,y) If X and Y are independent then P(x,y) = P(x) P(y) P(x | y) is the probability of x given y P(x | y) = P(x,y) / P(y) P(x,y) = P(x | y) P(y) If X and Y are independent then P(x | y) = P(x)

18 Law of Total Probability, Marginals Discrete caseContinuous case

19 Bayes Formula

20 Normalization Algorithm:

21 Conditioning Law of total probability:

22 Bayes Rule with Background Knowledge

23 Conditional Independence equivalent to and

24 Robot Environment Interaction See Figure 2.1 The robot maintains an internal belief with regards to the state of its environment. The robot also influence its environment through its actuators. Uncertainty exists, as always….

25 Robot Environment Interaction Environment are characterized by state varibales. State can be conveniently considered as the collection of all aspects of the robot and its environment that can impact the future. Dynamic state variable and static state variable? State is denoted, the state at time t is denoted

26 Robot Environment Interaction Typical state variables: Robot pose, Configuration of the robot’s actuators, Robot velocity, Location and features of surrounding objects in the environment Landmarks are distinct, stationary features of the environment that can be recognized reliably. Location and velocities of moving objects and people… Whether sensor is broken. The list can go on and on…..

27 Robot Environment Interaction Complete State Markov Chains You want to know if tomorrow is going to rain, you only need to know….. Incomplete State A complete state is hard to obtain, we get by with a small subset of complete state. In most robotics applications, the state is continuous, meaning that is defined over a continuum. For example… Sometimes, the state can be discrete. Thus hybrid state space….

28 Robot Environment Interaction Two fundamental types of interactions between a robot and its environment. 1. The robot can influence the state of its environment through its actuators. The control actions 2. The robot can gather information about the state through its sensors. The environment sensor measurements How to probabilistically model these interactions?

29 Robot Environment Interaction A robot can, at least, hypothetically, keep a record of all past sensor measurements and control actions. Such a collection is referred to as data. Two different streams of data Environment measurement data Control data corresponds to the change of state in the time interval (t- 1;t]

30 Robot Environment Interaction Environment perception provides information about the environment’s state, and it tends to increase the robot’s knowledge. Motion (control date), on the other hand, tends to induce a loss of knowledge due to noise (uncertainty). The evolution of state and measurements is governed by probabilistic laws. (Probabilistic Robotics)

31 Robot Environment Interaction For state variable If the state variable is complete This is an example of Conditional independence (CI).

32 Robot Environment Interaction For measurement data If the state variable is complete This is another example of Conditional independence (CI).

33 Robot Environment Interaction State transition probability measurement probability