Notes for 'Probabilistic robotics' Chapter 2 (Sebastian THRUN)

Probabilistic robotics (Chapter 2)

April 14, 2020

Jiayin Xie

 

Key words:

 

Related papers:

 

Key concepts: 

1. Probability theory

 

• Random variables: a function without input and randomly output some value
  • Expectation
  • Covariance
  • entropy

 

• Probability function: a function that measures the probability of random variables that take on some value.

 

• Probability density function: in continuous case, the probability at a specific value is close to zero. Thus, we define the density function for a specific value. 

 

• Joint probability: the probability that two events happen jointly.

We use multivariate density function to describe the joint probability density.

• Conditional probability: given the fact that event A happens, the probability that event B happens.

Conditions on a random variable: P(A|B=x). 

 

Independent events: p(A|B) the probability of event A given event B is same is probability of event A

 

• Theorem of total probability: given the conditional probability, we try to integrate and get the total probability of one event.

 

• Bayes rule: given the conditional probability p(x|y), try to compute the inverse. In probabilistic robotics, we are interested in inferring the state x from the observation or sensored data y:

p(x|y)=p(y|x)p(x)p(y)

Here, the p(y|x)is called the generative model, it is the density function for y assuming x. Or in other words, it can produce the data given the x. p(x) is called the prior probability distribution, and p(x|y) is called the posterior probability distribution.

 

• Conditional independence: It applies whenever a variable y carries no information about a variable x if another variable z’s value is known.

since p(x,y|z) = p(x|z)p(y|z)

thus p(x|z) =p(x|y,z)

p(y|z) = p(y|x,z)

 

2. How to describe the interaction between robots and the environment?

• We use state to describe the aspects of robots and environments that can impact the future.   xt

 

• Markov chain: no variables prior to x_t would influence the stochastic evolution of future states, unless this dependence is mediated through the state x_t.
• Measurement: zt

3. Hidden markov model

 

• We use the probabilistic generative laws to govern the evolution of state and measurements. The probabilistic generative laws for the state is given by a probability distribution:

p(xt |x0:t-1,z1:t-1,u1:t)

 

• If the state xt-1is complete then it is a sufficient summary of all that happened in previous time steps. Thus, it is sufficient to predict the state xt:

p(xt |x0:t-1,z1:t-1,u1:t) = p(xt|xt-1,ut)

                       The equations above can also be understanded as a conditional independence.

 

• Similarly, we have generative laws for the measurements:

p(zt|x0:t,z1:t-1,u1:t)=p(zt|xt) 

4. Belief distribution

Q: We have already defined the state and measurements for the robots and the environment. Why do we need a concept called belief? 

 

A: Since the true state can never be measured directly, and thus we define the concept called the belief to distinguish the true state from ite internal belief.

 

• The belief over a state variable xt is denoted by bel(xt), which is an abbreviation for the posterior 

bel(xt)=p(xt|z1:t,u1:t)

• If we calculate the posterior before incorporating zt, just after executing the control ut, we get:

bel(xt)=p(xt|z1:t-1,u1:t)

Calculating bel(xt)from bel(xt) is called correction or measurement update.

 

Bayes filter, a general rule of thumb for calculating beliefs:

• The bayes filter is a recursive algorithm and the goal is to solve the bel(xt). There are three components in this filter: 
  • The initial belief bel(xo)
  • The measurement probability p(zt|x0:t,z1:t-1,u1:t)
  • The state transition probability p(xt|x0:t-1,z1:t-1,u1:t)
• Markov assumption: If we assume the state is complete, i.e., if we know xt-1, past measurements and controls convey no information regarding the state xt.

This gives us the following conditional independence:

• p(zt|x0:t,z1:t-1,u1:t)= p(zt|xt)
• p(xt|x0:t-1,z1:t-1,u1:t)=p(xt|xt-1,ut)

• Before we get bel(xt), we need to compute the prediction bel(xt)from bel(xt-1) based on the theorem of total probability:  

 

bel(xt)=p(xt|ut)=p(xt|xt-1,ut)p(xt-1|ut)dxt-1

here , we let bel(xt-1)=p(xt-1|ut-1), (note that we omit the ut) and we get 

bel(xt)=p(xt|xt-1,ut)bel(xt-1)dxt-1

• Then, we utilize the bayes rule to compute the p(xt|zt,ut):

p(xt|zt,ut)=p(zt|xt,ut)p(xt|ut)

 

Gaussian filters 

Q: What is the multivariate density function?

 

Q: why does the covariance matrix measure the linear relationship between variables?

 

Q: Why is gaussian distribution suitable for representing the belief ?

 

Q: What are the moments of a distribution? Are they important? 

 

• Moments representation
• Natural or canonical representation

 

• Linear Gaussian Systems describes the next state probability p(xt|ut,xt-1) must be a linear function in its arguments with added gaussian noise.  Since the posterior in continuous space is represented by probability density function. Since it is a function, thus, xtis the variable and ut, xt-1is the given arguments.

xt=Atxt-1+Btut+t