MECH 3140 Final Project
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1 MECH 3140 Final Project Final presentation will be held December 7-8. The presentation will be the only deliverable for the final project and should be approximately minutes with an additional 10 minutes for questions. Each member should present on some aspect of the project and be prepared to answer questions after the presentation. Members of each group will turn in a grade for each of the other members in the group at the presentation to evaluate how much each member contributed to the project. You may split up tasks however you wish, but each member of the group must understand the project as a whole! You are to bring your presentation on a thumb-drive and as well as a print out of your slides (use multiple slides/page to save paper). One slide should show your Matlab code where you simulate the control system. The final project requires you model one of the provided systems, develop a controller to control the system, and provide a Simulink control block that will be used to evaluate the controller. Note that each team will only be allowed 3 evaluation chances and the evaluation serves as 50% of the final project grade (the remaining 50% is based on the final project presentation). You may test your control during the following scheduled times at the location specified in the problem description: Monday, December 4 4:00 PM to 8:00 PM Tuesday, December 5 9:00 AM to 12:00 PM Wednesday, December 6 4:00 PM to 8:00 PM The final project is broken into several steps below to highlight various aspects of control theory. 1. System Model: Develop the model for your system (EOMs). Your team should develop a Matlab function (or Simulink block) that takes in the system input and provides the specified outputs. Your simulation should account for any real limitations such as actuator saturation. a. Simulate your open loop system (step responses) b. Sketch the roots of the system (poles and zeros) c. Sketch the Bode Diagram of the system 2. Controller Design: Your team should develop a Matlab function that takes in the specified system outputs and commanded reference and generates the input to the system. a. What type of controller was used b. Where are the closed loop roots? c. What is the bandwidth of the controller? DC Gain? d. What is the step response of the closed loop system? e. Test the tracking ability of your controller. Does it correlate with the bandwidth of your closed loop system
2 3. Controller Test (Evaluation): You must package your Matlab control function into a Simulink block of the inputs/outputs of your Matlab control function. Someone from the GAVLAB will then evaluate your controller by checking a. How well does the system track different references b. How does it react to various disturbances c. Is the controller robust to variation in parameters d. Does the controller handle sensor noise 4. Sensor Considerations: Select sensors to provide the measurements that you used to observe the state of your system (i.e. the measurements you used in your Simulink block). You must select REAL sensors. a. Model the dynamics of your sensor if necessary b. Do your sensors have any dynamics/constraints that must be taken into account? c. Simulate your system open loop (i.e actuator->system->sensor) Notes/Comments on the Final Project 1. Make sure you have one slide showing your Matlab plant and controller functions (that shows where you calculate error and your control input and the simulation of your system). 2. Make sure you don t cheat in your simulation in terms of what measurements or inputs you have. For example, if you only have a sensor that measures error, you must derive the derivative of error from your error sensor (or show that you have a second sensor). You cannot simply take velocity from your simulation. 3. You are free to divide up tasks, but realize that you will be tested on the Final Exam on modeling, analysis, and controls (not on sensors and actuators or presentation development). So if your task is not relevant to the specifics of MECH3140 it may have costs on the exam. 4. Bring your sensor specification sheets to the presentation.
3 Ball on Beam Woltosz 2524 Derive the system model for the ball on a beam. Parameter Name Units Value m Mass of Ball kg 0.25 l Length of Rod m 2.0 R Radius of Ball m 0.1 J b Inertia of beam Nm/rad/s L Motor Inductance Henry 4.64 (10-5 ) R m Motor Resistance Ohms 0.36 J m Inertia of Motor Nm/rad/s (10-6 ) K b Motor Voltage Constant V/rad K i Motor Torque Constant Nm/A b Motor Torsional Damping Nm/rad/s 5.0 Measurements: Ball position (x), Beam Angle (θ), and desired ball position (x d ) Controllable Variables: Voltage to Motor
4 Inverted Pendulum on Cart (Segway Problem) Woltosz 2530 Derive the system model for the inverted pendulum on a cart. Parameter Name Units Value m Mass of Rider kg Your Mass l Length of Pendulum Rod m 1 M Mass of Segway Base kg 47 R Wheel Radius cm 29 J w Inertia of Wheel Nm/rad/s L Motor Inductance Henry 0.82x10-3 R Motor Resistance Ohms 1.03 J m Inertia of Motor Nm/rad/s 2 6.7x10-3 K b Motor Voltage Constant V/rad/s 2.25 K i Motor Torque Constant Nm/A 2.25 b Motor Torsional Damping Nm/rad/s 5.0 Measurements: Cart position (x), pendulum angle (θ), and desired pendulum angle (θ d ) Controllable Variables: Voltage to Motor
5 Latera Vehicle Tracking (Lane Tracking or Highway Autopilot) Woltosz 2524 Derive the system model for the autopilot system. x Fyf δ a β V car Fyr b The tire force is related to the lateral velocity at the tire through the following approximation: FFFF = CC VV cccccc VV yy tttttttt Parameter Name Units Value a Front CG distance m b Rear CG distance m C f Front Tire Cornering Stiffness N/rad 112,000 C r Rear Tire Cornering Stiffness N/rad 78,600 I z Yaw Mass Moment of Inertia Kgm V car Vehicle Speed m/s 15 m Vehicle Mass kg 1695 Note that you must include centripetal acceleration when applying Newton s lateral equations: ΣFF yy = mmyy = mm VV cccccc ψψ + VV yy Measurements: y (controlled), y d (desired), x, heading (ψ) and yaw rate (rr = ψψ ) Controllable Variables: Steer angle y
6 Heavy Truck Convoy Control Problem Woltosz 2540 Heavy vehicle convoys can offer significant savings in fuel economy, reduction in traffic congestion, and lower driver fatigue. These convoy controllers only actuate the vehicle longitudinally through the use of the engine and braking systems. You are tasked with designing a controller that will control the range between vehicles only using the engine (i.e. ignore the braking scenario). The system diagram is below. The lead truck will operate under a simple cruise controller (nominally set to 55mph); however, the vehicle is fully loaded so in the presence of a disturbances such as road grade it may not be able to achieve its set speed due to actuator saturation. Be sure that your controller can handle disturbances such as road grade, changing lead vehicle speed, and measurement error. The engine can be approximated as a first order system (from desired torque to actual torque) with a time constant of 0.25 sec and a DC Gain of one. Note: this desired torque is at the engine output flywheel. Assume no slip from engine to road. System Constants: Trailer Mass: 6,493 Kg Driveline Damping: 1.12 Engine Inertia: 2.75 Kg m^2 Nms Tractor Mass: 17,546 Kg Differential Ratio: 4.4 Trans. Inertia: Kg m^2 Engine Damping: 3.12 Nms Wheel Radius: 0.51 m Rolling Resistance: 5.82 N Cd: Front Area: m^2 Wheel Inertia:0.15Kg m^2 Gear Ratios 8 th ,9 th -1,10 th Max Engine Torque 2500 Nm Measurements: Lead Vehicle Velocity (m/s)
7 Host Vehicle Velocity (m/s) Range Between Vehicles (m) Range Rate Between Vehicles (m/s) Current Gear (gear number not ratio) Road Grade (dz/dx) Controllable Variables: Engine Torque (N-m)
8 Inverted Pendulum with Shaft Compliance and Electric Motor Dynamics Woltosz 2530 Derive the system model for the inverted pendulum and include the dynamics of the Maxon motor specified below and account for compliance in the shaft between the motor and the pendulum arm. The pendulum arm is of uniform density. The motor is attached to a stationary surface relative to the Earth at an altitude close to sea level. Parameter Name Units Value m Mass of Pendulum Arm kg l Length of Pendulum Arm m b shaft Damping of Shaft N m s/rad k shaft Stiffness of Shaft N m/rad J m Inertia of Motor kg m 2 L Motor Inductance Henry R Motor Resistance Ohms b m Bearing Loss of Motor N m s/rad K b Motor Voltage Constant V s/rad K i Motor Torque Constant Nm/A Measurements: Pendulum angle (θ 2 ), motor armature angle (θ 1 ) and desired pendulum angle (θ d ) Controllable Variables: Voltage to Motor
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