Consider a robot with its centre of mass located at x = [P, Py). The robot is equipped with…

Consider a robot with its centre of mass located at x = [P, Py). The robot is equipped with…

Question:

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Consider a robot with its centre of mass located at x = [P, Py). The robot is equipped with sensors
that can measure the squared distance between $mathbf{x}$ and four landmarks located in the
following coordinates:
Pz. Pyl
Pr2 Py.2
4
4
Pr.3 Py.3
PA PyA]
The measurement models for sensor $i=1,ldots,4$, is given by:
m,= (p. – Ps.)² + (Py – Pr.)² + ei
a. Write the measurement model above in vector form for all sensors. (1 points)
b. Obtain the Jacobian related to the measurement model in the vector form in point a. (2 points)
c. If the measurements obtained from the sensors are
[49.93427503, 34.02551511, 18.03571293, 33.97350448], and the measurement noise ei, i =
1,…,4 is zero mean with variance 2.5 x 10-3, calculate two iteration of the estimate of x, using
an initial guess (0) = [0,0] and gradient descent step size = 10-5. (7 points)

Expert Answer:

Answer rating: 100% (QA)

Part a Writing the Measurement Model in Vector Form Define Variables Let x be a 2×1 vector representing the robot s position x y Let z be an mx1 vector containing the distances measured by each landma
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