1. Lasers can be used to drill the hip socket for the insertion of an artificial hip joint. The use of lasers for surgery requires high accuracy for position and velocity response. Consider the system shown in the block diagram below, which uses a DC manipulator for the laser. The amplifier gain K must be adjusted so that the steady-state error for a ramp input, r(t) = At (where A = 1 mm/s), is less than or equal to 0.1 mm, while a stable response is maintained. R(s) + Amplifier K Motor & manipulator 1 S(TS + 1)(TS + 1) Break-away/break-in points (if any) Imaginary axis crossings (if any) Y(s) Figure 1. Control system for laser surgery. a) We select a motor with a field time constant T = 0.1 s and a motor-plus-load time constant T = 0.2 s. Find the closed loop transfer function T(s). Your answer should be in canonical form such that the coefficient multiplied with the highest power of s in the denominator is one. b) Use a Routh table to find the range of amplifier gain K that yields a stable system. c) Sketch the root locus (do not use MATLAB). Find and show the following points of interest on the sketch (you must provide hand calculations for credit): i. Asymptotes (if any) ii. iii. d) Find the amplifier gain required for the steady-state error ess to a ramp input to be 0.1 mm (or less). e) Setting the gain K to the value found in part d), we find that there is a dominant pair of complex poles, meaning that we can approximate this as a second order system. Find the overshoot and 2% settling time for a unit step input. You may use roots in MATLAB to factor the poles of the closed loop transfer function. f) Do you think the system response to a unit step input is desirable for laser surgery? Explain with a sentence or two.