.A Mars rover autonomous vehicle includes a robotic arm for collecting rock samples. The dynamics of the robotic arm system have been analysed, and a root locus obtained for changes

in a controller gain k varying from 0 to oo. The root locus is shown in the attached Root Locus.pdf document, and available for download from the MEC321 Blackboard course pages. a) Write down the start points and end points of the root locus diagram. b) Describe in detail how the system's transient response changes as k is increased. c) The designers wish to achieve a damping ratio of 0.423 from the system, but with the fastest possible settling time. (i) Use the magnitude condition to determine the required value for k, noting the need for the fastest possible settling time. d) Write down the fastest possible settling time for the system, and briefly explain why this is the maximum possible value.