Applying the dynamic modelling approach

Project Phase 1
This project requires you to apply the dynamic modelling approach used in the course to obtain
the nonlinear and linearized equations of motion for the Quanser 3 degree-of-freedom (DOF)
helicopter, shown in Figure 1. You will also do some preliminary controller design (for the pitch
and roll dynamics). This is an important prerequisite to phase 2 of the project, which will require
you to develop and test controllers for the full system.
This project is based (in part) on lab 1 from MIT OCW 16.30 Lab 1. Please refer to that lab, in
particular, read the Introduction and Physical Model sections and note the angle definitions in the
figure on page 1 and the data table on page 2. The remainder of this set of project instructions
assumes you have read those two sections.
Summary of Equations and Numerical Parameters
Nonlinear equations of motion:
????????¨= ??????????h – ????????sin?? – ??????? – ????????????????(???cos?? + ???sin??) (1)
????????¨= ????????????????????cos?? – ???????? sin(?? – ??????????) – ????????????sin?? + ??????????????????? – ??????? (2)
????????¨= ????????????????????sin?? – ????????????cos?? (3)
?? = ??????? (4)
?????????? = ?????????????? – ??????? (5)
???????? = ???????????? – ??????? (6)
????????? + 6???????? = 780???????? (7)
??????????? + 6?????????? = 540?????????? (8)
The required numerical parameters are given in Table 1.
Table 1 – Physical parameters of the 3-DOF unit MIT OCW 16.30 Lab 1
Parameter Value Units Description
?? 1.15 kg mass of rotor assembly
?? 3.57 kg mass of whole setup
?????????? 0.66 m length from pivot point to heli body
???? 0.004 m length of pendulum for roll axis
???? 0.014 m length of pendulum for pitch axis
??h 0.177 m length from pivot point to the rotor
?????? 0.036 Nms2 moment of inertia about x-axis
?????? 0.93 Nms2 moment of inertia about y-axis
???? 4.25 × 10-3 Ns coefficient of thrust
?????????? -25 deg. theta rest value
?? 9.81 m/s
acceleration due to gravity
???? [0.02, 0.2] Nms roll damping coefficient
[0.1, 0.9] Nms pitch damping coefficient
Figure 1 – 3 DOF Helicopter – adapted from Quanser website
Project Assignment
Items 1-3 can be completed after Lecture State-space model features (2/2).

  1. Explain the physical source and meaning of each term in the nonlinear equations of motion
    for the Quanser 3 DOF helicopter (equations 1-3).
  2. Linearize the equations of motion about a steady hover equilibrium. At this condition ???
    0 =
    0 = ???
    0 = 0, ??0 = ??0 = ??0 = 0 (because the “reference” travel angle can be set
    arbitrarily), ?????????,0 = 0, ???????????,0 = 0, ??0 = 0 (flight path angle), and ??0 = 0 (velocity).
    Note that the linear model does not perfectly represent the dynamics of the actual Quanser
    due not only to the approximations inherent in linearization, but due to variations in pitch
    and roll damping from one Quanser unit to another. This is why ranges are given in the MIT
    OCW document for these quantities. ???? and ???? could be anywhere in the ranges given.
    – Express your results for this problem in symbolic form.
  3. Ignoring the dynamics in travel (??), obtain state space models for the pitch and roll
    – Express your results in symbolic form.
    – Use the numerical values provided in the MIT OCW document to check the
    following for the full range of ???? and ???? values:
    • stability
    • controllability
    • observability
    • stabilizability
    • detectability
    Reporting Requirements:
    Your project report MUST be submitted electronically via Blackboard as a single zip file, with
    the following naming convention: where
    YYYY is the year for the course. For example,
    • This is an individual project.
    • The style of the document should be in APA version 6 format. Make sure you follow the
    instructions provided in the resource at the end of this document and look at the sample
    APA paper to get an idea of what the guidelines are. If you have questions, always ask!
    • The report should be maximum 10 pages in length including figures. Be as concise as
    possible while clearly communicating your thought process and findings. Appendices do
    not count towards this length constraint.
    • Your report must detail your design process, the results, and explain why you think your
    design is the best one you could come up with, e.g. justify all design choices and the
    adequacy of the performance obtained.
    • Ensure all figures are of sufficient size and clarity to be easily read. Outputs from
    MATLAB/Simulink should be generated as vector files rather than using screenshots (see
    the documentation for details). Free PostScript to vector formats such as .svg, .emf, and .eps
    is available online, look for Inkscape software; if you have difficulty finding one, come to
    see me at office hours and I will help you. Screen captures from MATLAB/Simulonk are
    not acceptable in your report.
    • Observe standard technical writing conventions – label all figures, put data in tables, etc.
    • Use three significant figures for all results and comparisons between numbers.
    • Your report should (if applicable) contain:
    – Title page Following instructions for APA version 6 cover page
    – Table of contents
    – List of figures and/or tables
    – Abstract The purpose for an abstract is to present the gist of the study/work in the
    shortest way possible where problem is introduced, methods are clear, and outcome
    is communicated. Therefore, if you read an abstract section of the report you would
    know whether it is related to what you are searching for and would not have to go
    through the complete report.
    – Introduction This introduction should be designed to interest your reader in your
    topic and provide some historical context for your project. You need to identify the
    problem you are trying to solve and its significance. At the end of this section, you
    need to transition to problem statement. Remember that at this point you have
    completed the project so all the material in this section should now represent the
    work done and there should be no ambiguity.
    – Problem Definition In this section quickly provide the summary of the problem you
    are required to solve and all the available information about your system dynamics,
    required tasks, etc.
    – Theory In this section, you need to provide theory on what you are solving and
    required equations, assumptions, etc. goes here. This is the section you need to use
    your in-text citations the most and must have an appealing flow in the context. Make
    sure you adhere to the citation style required.
    – Methodology In this section, you should discuss the methods and sources you have
    used to conduct your project and solve the problem using the theory and tools. Try
    to connect your theory section to here using the equations you have introduced
    earlier with equation numbers, etc.
    – Results and Discussions This section is the place where you need to showcase the
    hard work you put in to complete the project. In this section, using the methods you
    suggested to solve the problem on the data you have used to analyze, you need to
    clearly show through, graphs, plots, tables, and other appropriate presentation means
    what you have learned, and through the process helped solve the problem you
    intended to solve. Each result needs to be accompanied by a reasonable discussion
    on the outcomes. The discussion needs to incorporate the intentions of the study as
    well as the outcomes so that one can understand clearly how the presented results
    match with the solution you are proposing for the problem at hand. It is also
    important to make sure that all plots and tables are discussed in your discussion as
    otherwise they would have been presented for no reason. This section should
    transition to your conclusion section where you conclude the study by summarizing
    your learning outcome of the project.
    – Conclusion In your conclusion, now that you have completed the project, you need
    to clarify what you did and what you learned from it. It needs to briefly reflect on
    the problem, mention the methods used to solve it, draw on the important results
    obtained through the study and provide some suggestion for the future directions of
    the study in case someone else is interested to continue your work and improve it
    further because there is always room for improvement!
    – References You need to have credible sources. Using Mendeley is strongly
    recommended as it can save you tons of time and make sure you are doing your
    citations correctly and efficiently. Citations in APA 6 format. Any in-text citations
    should also follow APA 6 guidelines.
    – Appendices This section needs to include any additional material that is needed for
    the report to be self-sustainable but cannot fit in the page limit of the paper. Your
    MATLAB/Simulink files would go here. If you have extensive derivations, lookup tables, code, supplementary equations, etc. they all go here in their respective
    subsection numbered either numerically “Appendix 1, 2, 3…” or alphabetically
    “Appendix A, B, C…”.
    Submission file:
    • Submit your project as a single ZIP file. This file should contain
    – Your reports original file (Word document, etc.)
    – All supporting plots, figures, tables, that are required for the original report
    – PDF version of your report that should run on any computer and look the same
    – All you MATLAB/Simulink files with instructions (instruction.txt) for someone to
    be able to run and get the same results. All in structured folders under (MATLAB
    folder). Use the figure below as a guideline to structure your zip file.
    Figure 1. Structure of the .zip file to be submitted
    You will be penalized for the following:
    Page overage
    For each page over the page limit you will be deducted 10% of the total grade. This is to ensure
    all are evaluated on a fair basis. Please note that your line spacing needs to adhere to the style
    requested and page limit will be checked in that format.
    Late submissions
    The penalty for late submission is 10% for the total mark deduction for each day, up to 3 days,
    after which the mark for this component of the course will be ZERO. Note that partial marks are
    not awarded for hours and minutes during late submission.
    .Zip file
    report folder
    report.PDF report.Doc(x)

Sample Solution