On February 18th, 2021, NASA's Mars 2020 Preserverance Rover is planned for landing on Mars.A landing animation video can be seen here http://bit.ly/PreserveranceThe Sky Crane, which is responsible for gracefuly

landing the rover on the designated site, can be modeled as a rigid body with thruster forces being controlled by a gimbal to produce thrusta t angles 0 as shown. a. (20%) Derive the equations of motion for the system (3 directions)The thruster angles and thrust forces are all independent input variables now.We wish to design a state-space controller to help the SkyCrane navigate to a desired position in space. b. (30%) Linearize the system and put it in state-space form, then design a controller via pole placement to achieve the following transient response characteristics - Ts = 2 s -\omega_{d}=20 \mathrm{rad} / \mathrm{s} c. (25%) Apply your controller on the nonlinear system (numerical integration) d. (25%) In reality, there are limits to the inputs. The thrust can not be negative, and there is a minimum thrust once the engine is ignited. The thruster gimbal can only operate within a specific angular range. Repeat part c with the following saturation limits: - Thrusts: 100N < F < 2000N Gimbal Angle Range: -45° < 0 < 45°-