(a) What is maximum likelihood estimation?\ (b)

x

is a Gaussian random vector with distribution

xN(\\\\mu ,\\\\Lambda )

. Assume a sensor is\ used to measure

x

, each time the measurement is

y_(i)=x+v_(i)

, where

v_(i)

is the\ measurement noise at the i-th measurement

y_(i)

with distribution

v_(i)N(0,\\\\Sigma )

.\ (i) Determine the joint distribution of

z=[[x],[y_(i)]]

.\ (ii) If the sensor measures

x

once and returns a measurement

/bar (y)_(1)

, what is the\ maximum likelihood estimation of

x

?\ (iii) If the sensor measures

x

twice with measurements

/bar (y)_(1)

and

/bar (y)_(2)

, what is the\ maximum likelihood estimation of

x

?\ (c) How does an extended Kalman filter deal with the nonlinear state equation and output\ equation in a robotic system?\ (5 marks)

(a) What is maximum likelihood estimation? (5 marks) (b) x is a Gaussian random vector with distribution xN(,). Assume a sensor is used to measure x, each time the measurement is yi=x+vi, where vi is the measurement noise at the i-th measurement yi with distribution viN(0,). (i) Determine the joint distribution of z=[xyi]. (5 marks) (ii) If the sensor measures x once and returns a measurement y1, what is the maximum likelihood estimation of x ? (5 marks) (iii) If the sensor measures x twice with measurements y1 and y2, what is the maximum likelihood estimation of x ? (5 marks) (c) How does an extended Kalman filter deal with the nonlinear state equation and output equation in a robotic system