11.2: The Covariance Matrix

When estimating quantities with multiple variables, such as the position of a robot that consists of its x-position, its y-position and its orientation, matrix notation has been a convenient way of writing down equations. For error propagation, we have written the variances of each input variable into the diagonal of a covariance matrix. For example, when using a differential wheel robot, uncertainty in position expressed by σ_x, σ_y and σ_θ were grounded in the uncertainty of its left and right wheel. We have entered the variances of the left and right wheel into a 2x2 matrix and obtained a 3x3 matrix that had σ_x, σ_y and σ_θ in its diagonal. Here, we set all other entries of the matrix to zero and ignored entries in the resulting matrix that were not in its diagonal. The reason we could actually do this is because uncertainty in the left and right wheel are independent random processes: there is no reason that the left wheel slips, just because the right wheel slips. Thus the covariance — the measure on how much two random variables are changing together — of these is zero. This is not the case for the robot’s position: uncertainty in one wheel will affect all output random variables (σ_x, σ_y and σ_θ) at the same time, which is expressed by their non-zero covariances — the non-zero entries off the diagonal of the output covariance matrix.