We will find a way to quickly keep track of the current sum after each move of the robot.

Assume that robot moved to the right (east) from $(a,b)$ to $(a+1,b)$. Distance to any control point changed by $1$ or $-1$ after this move. To be more precise, distance increased by $1$ for every control point with x-coordinate no greater than $a$, and decreased by $1$ for the rest of them. If we denote the number of points in the first group $f(a)$, then the sum of these distances changed by $f(a)-(N-f(a))$. We can do the same for the y-coordinate.

Next, we must figure out how to calculate $f(a)$ efficiently. One of the ways to do this is to keep the control points in two arrays, one sorted by x-coordinate and the other sorted by y-coordinate. Now we can easily binary search $f(a)$ for any value $a$.

This solution has a complexity of $\mathcal{O}(N\log N+M\log N)$. Better complexity can be achieved by precomputing all the $f(a)$ values using dynamic programming.