On February 18th, 2021, NASA's Mars 2020 Preserverance Rover is planned for landing on Mars.A landing animation video can be seen here 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°-

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