45m

The total angle turned through after each of the first 4 moves is $10^{\circ}$, $30^{\circ}$, $60^{\circ}$ and $100^{\circ}$. So the robot does not face due East at the end of a move in its first complete revolution. The total angle it has turned through after each of the next 5 moves is $150^{\circ}$, $210^{\circ}$, $280^{\circ}$, $360^{\circ}$ and $450^{\circ}$, so at the end of the 9th move the robot does face due East. As the robot moves 5m in each move, the distance it travels is 45m.

Note that this solution assumes that the robot is not starting the process close to the North Pole!

*This problem is taken from the UKMT Mathematical Challenges.*