Solution:

If he runs up 6 steps, then he needs 45.0 seconds to reach the top.
If he runs up 15 steps, then he needs 31.5 seconds to reach the top.

The 9 additional steps take 13.5 seconds.
Therefore, each step takes 1.5 seconds.

Total steps in escalator = 6 + 45.0 / 1.5 = 36 or Total steps in escalator = 15 + 31.5 / 1.5 = 36.

If Robin did not run up any steps at all,
he would reach the top of the escalator in 54.0 seconds (i.e., 36 steps × 1.5 seconds/step).

Alternative Solution through Equations:

Let the total number of steps in the escalator be x.
The escalator moves at a constant speed given by

Speed of escalator = (x − 6)/45.0 = (x − 15)/31.5

The above equation may be solved as follows.

31.5 (x − 6) = 45.0 (x − 15); or
x = (15 × 45.0 − 6 × 31.5) / (45.0 − 31.5) = 36.

Now, Speed of escalator = (36 − 6)/45.0 = (36 − 15)/31.5 = 1/1.5 steps/second.
Time to reach top = Total Steps / Speed = 54.0 seconds.

Food for thought:

Does it matter whether the escalator is moving up or down?