How do I compute the impulse response of a robot?
Impulse response makes sense for linear systems. If your system is nonlinear forget it. If you assume that your system is linear time invariant, you may want to measure the impulse response by exerting a short duration high energy pulse to your robot. It must have sufficiently high energy so that most of the exerted energy is transferred to kinematic energy of the robot to enable motion. 

If this is not possible you can apply longer duration pulses to your robot and then measure its response (motion). You must deconvolve then the response and the input to get the impulse response of your robot. You may try this for a number of different shape pulses and if you get impulse responses similar to each other every time, you can be fairly sure that your system is linear. 

Your comment on having an infinite impulse response is not in line with correct systems modelling, so forget about it.