16.30/31 September 24, 2010 Prof. J. P. How and Prof. E. Frazzoli Due: October 15, 2010 T.A. B. Luders /31 Lab #1

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1 16.30/31 Septeber 24, 2010 Prof. J. P. How and Prof. E. Frazzoli Due: October 15, 2010 T.A. B. Luders 16.30/31 Lab #1 1 Introduction The Quanser helicopter is a echanical device that eulates the flight of a reduced degree of freedo (DOF) helicopter. Instead of the usual six DOF of a free-flying helicopter, the Quanser only exhibits three DOF: roll, pitch, and travel, as illustrated in Figures 1-3. The Quanser syste is actuated by two rotor speeds, and the inputs to the syste are V cyc, which is an electric voltage that results in differential change in the two rotor speeds, and V coll, which is an electric voltage that controls the speed of the two propellers collectively. The outputs of the syste are three angles: roll φ, pitch θ, and travel ψ. Please note the liits of the Quanser: voltage [ 5, 5] Volts, φ [ 40, 40 ], and θ [ 25, 30 ]. TL θ ϕ TR Ψ 1 Iages by MIT OpenCourseWare.

2 2 Physical Model The Quanser odel is derived by applying Newton s second law to the rate of change of angular oentu. The nonlinear equations of otion are I xx φ = τcyc l h gl φ sin(φ) L p φ I r ω rotor (θ cos(φ) + ψ sin(φ)), (1) I yy θ = τcoll l boo cos(φ) Mgl θ sin(θ + θ rest ) Dl boo sin(γ) + I r ω rotor φ M q θ, (2) I zz ψ = τ coll l boo sin(φ) Dl boo cos(γ), (3) where D is the induced drag, τ coll is the collective thrust, and τ cyc is the cyclic thrust. These forces are odeled individually as D = K D ψ, (4) τ coll = K τ ω coll K v ψ, (5) τ cyc = K τ ω cyc K v ψ, (6) where K D, K τ and K v are constant coefficients. In addition, the otor dynaics can be odeled by ω cyc + 6ω cyc = 780V cyc, (7) ω coll + 6ω coll = 540V coll. (8) The values of other paraeters in this odel are given in Table 1. Note that ω rotor is the rotor rotation speed, and is assued to be constant; ters containing ω rotor represent the rotors contribution to angular oentu. Additionally, γ is the vehicle s flight path angle. Table 1: Physical paraeters Paraeter Value Units Description M l boo l φ l θ l h I xx I yy K τ θ rest g L p M q [0.02, 0.2] [0.1, 0.9] kg kg N N N/s degs /s 2 ass of rotor assebly ass of whole setup length fro pivot point to heli body length of pendulu for roll axis length of pendulu for pitch axis length fro pivot point to the rotor oent of inertia about x-axis oent of inertia about y-axis coefficient of thrust theta rest value acceleration due to gravity roll daping coefficient pitch daping coefficient 2

3 Linearize these equations around a steady-state hover operating condition, i.e., φ 0 = θ 0 = ψ 0 = 0, φ 0 = θ 0 = ψ 0 = 0, ω cyc,0 = ω coll,0 = 0, and γ 0 = 0. If we ignore the dynaics in travel (ψ), the pitch (θ) and roll (φ) subsystes can be represented by the block diagras in Figure 4. In the pre-lab assignent, you will be asked to find in sybolic for (i.e., not using the tabulated paraeter values) the transfer functions M coll, Q P, M cyc, and Q R. V coll ω coll θ M coll Q P Rotor Speed (collective) Collective Voltage Pitch Angle V cyc M Q R cyc ω cyc Φ Cyclic Rotor Speed Roll Voltage (differential) Angle Iage by MIT OpenCourseWare. The linear odel does not perfectly represent the dynaics of the actual Quanser for any reasons. While there are any inaccuracies, the variations in pitch and roll daping between Quansers present the largest proble. You should assue that the daping coefficients L p and M q for any given Quanser will be soewhere in the ranges listed in Table 1. 3 Schedule Fri 9/24: Lab 1 assignent posted Mon 9/27: (optional) LabVIEW software available for personal download Wed 9/29: deadline to e-ail group ebers to TA; LabVIEW tutorial in lecture Fri 10/1: Signup sheets for lab sessions available in lecture Mon 10/4: First day of lab sessions Wed 10/6: Lab 1 pre-lab assignent due Thu 10/14: Last day of lab sessions Fri 10/15: Lab 1 writeup due 3

4 4 Pre-Lab Assignent (due 10/6) 1. Derive the linearized odel as described in Section 2, and sybolically for the transfer functions M coll, Q P, M cyc, and Q R. When foring the transfer functions, you ay assue that the rotors contribution to angular oentu is negligible (i.e. ω rotor = 0). 2. Use classical design techniques to develop controllers for the roll (φ) and pitch (θ) axes. You are encouraged to use Matlab/sisotool to assist you in your design. Reeber that you are designing the controllers to work for the entire range of daping constants L p and M q. For the noinal plants (i.e., use a daping value in the iddle of the range), your controllers ust have the following properties: Less than 10% error in tracking signals with frequency content up to 1 rad/s At least 20 of phase argin Maintain stability even for the worst-case daping For these controllers, write the controller transfer function and plot the following: Bode plot of the controller Open loop Bode plot of controller + noinal plant Siulated closed-loop step response for a 20 step coand A paragraph or two describing your design strategy, predicted perforance, and pros/cons of your design (Hint: Your roll (φ) controller is going to be the inner-loop for the travel controller in the next lab, i.e. the travel controller will be coanding roll angle. Thus it is iportant that your roll controller is fast and stable but not so iportant that it has zero steady-state error. Conversely, your pitch (θ) controller should have zero steadystate error so that you are sure to clear altitude obstacles in the next lab.) The pre-lab assignent ay be subitted either in person or online ; however, you ust bring a copleted pre-lab assignent with you when attending a lab session in order to proceed with the lab. You ust also hand in the pre-lab assignent with your lab writeup, and it will count for a substantial portion of the overall lab grade. 4

5 5 Lab Procedure 1. Lab sessions will take place between October 4 and October 14. You will be working in groups of four (please work with students in your class, either or 16.31). Each eber of the group ust coplete the pre-lab assignent, and you ust subit your own lab report. 2. Specific instructions on how to build your odel and construct and test your controllers in LabVIEW will be posted in the lab, We will arrange for soe support in the lab if you get stuck - but recognizing that labs are generally tie consuing, and the Quanser lab facility is not available 24 hours a day, please plan ahead. As a student in 16.30/31, you will be given access to download LabVIEW for use on your own coputer, and will be able to run the sae software used to siulate and control the Quansers in lab. Take advantage of this opportunity to get ore cofortable with the software and save tie in lab. 3. Apply steps of appropriate aplitude and duration; record the syste output in roll and pitch. Copare with the expected response fro your odels, and redesign your controller gains if necessary. 4. Record and save the following four step responses (to be included in your lab writeup): Responses 1-2: Roll step input of Δφ = +20 or Δφ = 20, starting fro φ 0 = 0 (vehicle is level), for two different pitch angles θ at least 10 apart Response 3 : Pitch step input of Δθ = +20, with φ 0 (vehicle is level) and arbitrary starting pitch θ 0 Response 4 : Pitch step input of Δθ = +20, with φ +20 or φ 20 (vehicle is rolling) and arbitrary starting pitch θ 0 (These should be perfored for all of your group s working controllers. Your group ay choose to coe to a consensus on a working roll/pitch controller cobination.) For either of these responses, if your controller cannot aintain stability for a pitch step of this agnitude, you should instead try to perfor as large of a positive (Δθ > 0) pitch step as is feasible. You should then additionally perfor as large of a negative (Δθ < 0) pitch step as is feasible. 5. Once you are satisfied with the perforance of your controllers on your Quanser, save your step responses and record your final controllers. Make sure you either eail the data to yourself or use SecureFX to upload it into your Athena directory so that you can access it when you write your lab report. 5

6 6 Lab Writeup (due 10/15) The lab writeup need not be particularly foral. However, you ust include the following to receive full credit: 1. Both of your classical controllers (roll and pitch). 2. The recorded step responses listed in Lab Procedure with the corresponding theoretical responses (fro your odel) overlayed in the sae plot. Give a brief description of the properties of each observed response (i.e., overshoot, rise tie, etc.). 3. A few paragraphs discussing how well the theoretical responses fro the odel atched the actual responses you easured in the lab. Give reasons for any inconsistencies and explain the iportance of accounting for the plant uncertainty. Please observe standard technical writing conventions (i.e., label all charts and figures, put data in tables, etc.). The lab assignent ay be subitted either in person or online. 6

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