A portion of your grade will be assigned based on a quarter long project involving analysis of the dynamics of a multibody system using Kane's method and SymPy/PyDy. You must select a reasonably complex system (including possibly significant damping and other interesting dynamic components). This is an ideal opportunity to get started on the dynamic analysis of a thesis topic (with free consulting!). Next you must suggest a set of well-posed questions about the system which the simulation can help you to answer, derive the differential equations of motion, study the system by simulating the system dynamics, and present you results in a well formatted written report in the form of a Jupyter notebook (or set of) including whatever charts, tables, figures, and visualizations you may need to clearly communicate your results.

Many students believe that once the equations are written and the simulation is completed, the job is finished. Actually, the task has only begun. Dynamic simulations are merely a tool to learn about the possible dynamic behaviors that a system can exhibit and how these dynamics depend on other parameters. They must be used cleverly and resourcefully to elucidate the behavior of the system. This involves asking the right questions about the system and understanding which parametric dependencies are of interest. Usually this will come from some engineering job requirements and specifications or from some scientific question which one would like answered (e.g. "Design an antenna deployment system for a communication satellite which works in the near earth orbit, deploys in less that 10 sec, ..., etc." or "What are the important parameters which limit the maximum range of achievable in a throw of the discus and how should it be launched optimally?"). In this project you will be responsible for generating this project requirement before writing the simulation to help you answer the question you have asked. A good, precise statement of an interesting question is every bit as important as good answer. Indeed, poorly defined or worded questions are often impossible to answer in a satisfying and fulfilling way.

Your grade will be based on the following aspects:

  1. Sophistication, interest, and difficulty of system and the questions(s) you will ask about it
  2. Correctness of equations/analysis and interpretation
  3. The cleverness and resourcefulness with which you use the simulation to learn about the dynamics of the system and answer the questions posed
  4. Your written discussion of the system and the presentation of the results of your study/interpretation.
  5. Clarity, coherence and general organization of your report


Friday October 18th

Report to me in writing the system of your choice, including a motivation for the problem, background on how the system works heuristically, a literature search to identify previous work on this problem, and a relatively complete discussion of the way you hope to use the dynamic simulation to learn what you have chosen to learn. This should take several pages to do completely. I urge you to begin the selection of your system immediately and to discuss it with me office hours so that I can help you in this phase. This proposal should be no longer that three pages.

Proposal Rubric

  • [10] Research question
    • Defined a research question
    • Research question is present but not articulated clearly
    • Did not define a research question
  • [10] Literature review
    • Identified more than one relevant work or extensive discussion of at least one
    • Identified and discussed at least one relevant work
    • Did not look into the literature
  • [10] System description
    • System described from a dynamics perspective
    • System described but lacking in description of dynamics
    • System not described
  • [10] Methods description
    • Clear plan on how analysis and simulation may be used to answer research question
    • Plan on how to answer question is murky
    • No discussion of how analysis and simulation may be used to answer research question
  • [10] Writing
    • Clear, coherent, and well organized
    • Writing and organization needs improvement
    • Not clear, coherent, or well organized

Thursday, December 12th

The final report is due. No submissions will be accepted after this date due to the time needed to grade them.

Project Rubric

Score will be between 50-100.

  • [5] Research question
    • Research question is present and clearly defined.
    • Research question is present but not articulated clearly.
    • Did not define a research question.
  • [10] System and model description
    • Text, equations, and figures are used to clearly describe the configuration, possible motion, inertial characteristics, and important loads acting on and of the system.
    • Text, equations, or figures are used to describe the configuration, possible motion, inertial characteristics, and important loads acting on and of the system, but not all aspects are clear.
    • System and model poorly described without text, equations, and figures.
  • [10] Model Design
    • Model is designed with sufficient complexity to answer your research question and is not overly complex.
    • Model is designed with sufficient complexity to answer your research question.
    • Model is not appropriate for answering research question.
  • [10] Model Correctness
    • Equations of motion are correct and simulation shows that the model behaves as expected.
    • Equations of motion are mostly correct and simulation shows the model mostly behaves as expected.
    • Equations of motion are incorrect and model does not behave as expected.
  • [10] Analysis and Interpretation
    • Simulation or other analysis is demonstrated through text, plots, and optionally animations to address the research question. Interpretation of the results is correct and answers the research question.
    • Simulation or other analysis is demonstrated through text, plots, and optionally animations to address the research question. Interpretation of the results is mostly correct and answers the research question.
    • Simulation or other analysis not present and research question is not answered.
  • [5] Writing
    • Clear, coherent, and well organized. All variables defined, plots labeled and explained. No extraneous elements in the notebook.
    • Writing and organization needs improvement; missing clarity, organization, variable definitions, labeled and explained plots, etc.
    • Not clear, coherent, or well organized.

Thursday, December 12th

You will present a 5 minute lightning talk to the class explaining your project, methods, and the results.

Example Notebooks

To get an idea of what you can do with Jupyter notebooks, here are some examples:

Project Ideas

Benchmark Problems

Library of Computational Benchmark Problems:


There are thousands of interesting mechanisms. Here are several collections of mechanisms to get some ideas from:

Kinetic Sculptures

Google searches for "kinetic sculptures" or "kinetic art" will provide you with many interesting multibody systems. One of my favorites are the strandbeesten from Theo Jansen:

Theo Jansen's Strandbeesten

Theo Jansen developed a multi-bar linkage that translates rotational motion into linear motion that works well for making walking machines. He deploys it in his Strandbeesten "Beach Animals":

Modeling and analyzing the leg linkages or something similar would work well for a project.


Single Track and Titling Vehicles

Single track and titling vehicles are particularly interesting because they must be both balanced and directed. There are many interesting single track vehicles that would offer opportunities for multibody modeling. For example, bicycles, scooters, motorcycles, monocycles, single wheel trailers, titling vehicles, snake boards, unicycles, etc. Wikipedia gives a good starting point.

Some good papers:

  • Sharp, R. S. The Stability and Control of Motorcycles. Journal of Mechanical Engineering Science 13, 316–329 (1971).
  • Meijaard, J. P., Papadopoulos, J. M., Ruina, A. & Schwab, A. L. Linearized dynamics equations for the balance and steer of a bicycle: A benchmark and review. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 463, 1955–1982 (2007).
  • Kooijman, J. D. G., Meijaard, J. P., Papadopoulos, J. M., Ruina, A. & Schwab, A. L. A Bicycle Can Be Self-Stable Without Gyroscopic or Caster Effects. Science 332, 339–342 (2011).
  • Karnopp, D. Tilt Control for Gyro-Stabilized Two-Wheeled Vehicles. Vehicle System Dynamics 37, 145–156 (2002).

The "Bicycle and Motorcycle Dynamics" conference has proceedings about these vehicles.


Human Locomotion

There a different "simple walking models" that could be appropriate for a class project. Here are some papers:

  • Collins, S., Ruina, A., Tedrake, R. & Wisse, M. Efficient Bipedal Robots Based on Passive-Dynamic Walkers. Science 307, 1082–1085 (2005).
  • Garcia, M., Chatterjee, A., Ruina, A. & Coleman, M. The Simplest Walking Model: Stability, Complexity, and Scaling. J Biomech Eng 120, 281–288 (1998).
  • Kuo, A. D. A Simple Model of Bipedal Walking Predicts the Preferred Speed–Step Length Relationship. J Biomech Eng 123, 264–269 (2001).

The Dynamic Walking conference has the best work on these topics. Here are the video abstracts from a past conference:

Animal Motion

Animals have evolved a very large variety of ways to locomote from hopping, sliding, flying, multi-legged walking, etc. Here are some related papers:

  • Schmitt, J. & Holmes, P. Mechanical models for insect locomotion: dynamics and stability in the horizontal plane I. Theory. Biol Cybern 83, 501–515 (2000).
  • Koditschek, D. E. & Bühler, M. Analysis of a Simplified Hopping Robot. The International Journal of Robotics Research 10, 587–605 (1991).
  • Hyon, S. H. & Mita, T. Development of a biologically inspired hopping robot-"Kenken". in Proceedings 2002 IEEE International Conference on Robotics and Automation (Cat. No.02CH37292) 4, 3984–3991 vol.4 (2002).
  • Brown, B. & Zeglin, G. The bow leg hopping robot. in Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No.98CH36146) 1, 781–786 vol.1 (1998).

Sports Biomechanics

The Skateboard

The basic skateboard dynamics offering a nice non-holomonic system to model. See this paper:

Hubbard, M. Human control of the skateboard. Journal of Biomechanics 13, 745–754 (1980).

Another interesting aspects is that skateboarders are able to jump with the skateboard seemingly attached to their feet, yet it isn't. The technique is called the "ollie" and revolutionized the sport when invented. The technique is now the foundation for hundreds of similar tricks. The skateboarder uses a combination of popping the board at and angle and then lifting the board using the friction between their foot and the surface of the board to bring the board into the air. The goal of this project would be to develop a model of a skateboard that can be "ollied" and attempt to do so.


There are numerous toys that dynamicist's find interesting, for example the walking rabbit, the oloid, the rattleback, gyroscopes, snakeboards, etc. These often provide nicely scoped models for the class project.



  • Kane, T. R. & Levinson, D. A. Realistic mathematical modeling of the rattleback. International Journal of Non-Linear Mechanics 17, 175–186 (1982).
  • Garcia, A., Hubbard, M. & Bondi, H. Spin reversal of the rattleback: theory and experiment. Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 418, 165–197 (1988).

Make Luxo the Pixar Lamp Jump!

Pixar modeled a lamp, Luxo, back in 1986 to hop around like it was alive. They used multibody dynamics and space time optimization techniques. The original paper is:

Witkin, A. & Kass, M. Spacetime Constraints. 10 (1988).

Where to Find Other Ideas