A scalar Kalman filter estimates the current robot position x given GPS measurements z. The motion model predicts x_pred = x_prev + v·dt (constant velocity). The measurement z = x + noise_measurement has variance σ_z². The filter maintains state estimate x_est and error covariance P. After prediction, P increases (uncertainty grows due to model imperfection); after measurement update, P decreases. Why does the filter trade these two sources?
AThe prediction is always correct, and the measurement is always wrong; the filter ignores measurements and trusts predictions
BThe measurement is always correct, and the prediction is always wrong; the filter ignores predictions and trusts measurements
CBoth prediction and measurement have errors. The filter computes the Kalman gain K = P/(P + σ_z²), which weights the measurement update based on relative uncertainties: if measurement noise is large, K is small and the update is small; if prediction uncertainty is large, K is large and the update is large
DThe filter randomly chooses between prediction and measurement
Multi-sensor fusion is where the Kalman filter shines. By explicitly modeling each sensor's noise and update rate, you get a principled way to combine them. This is standard in autonomous vehicles (combining GPS, IMU, wheel encoders, and sometimes LiDAR and cameras).
Question 2 True / False
A particle filter is an alternative to the Kalman filter for state estimation. Unlike the Kalman filter, which assumes Gaussian distributions, the particle filter represents the belief as a set of weighted samples (particles). When would a particle filter be preferable to a Kalman filter?
TTrue
FFalse
Answer: True
Correct. Particle filters are useful when the belief distribution is non-Gaussian (multimodal, with multiple peaks) or when the system is highly nonlinear and the Gaussian assumption is very poor. However, particle filters require more computation (many particles × propagation cost) and can suffer from particle degeneracy (a few particles dominate). The Kalman filter is efficient and optimal for linear Gaussian systems. Many robotics systems use hybrid approaches: Kalman filter for primary state (linear, single mode), particle filter for discrete mode tracking (e.g., which room am I in?) or for kidnapped robot problems (where the robot's initial position is unknown and multimodal).