Undergraduate Student Kirk Nichols
Dates of Involvement 2007- 2009
Faculty Advisor Todd Murphy
Background
Variational integration has become an increasingly important method for modeling systems without resorting to traditional ordinary and partial differential equations. This method allows one to directly calculate numerical update laws that are not inclined to the same energetic difficulties as Ordinary Differential Equations (ODEs) and Partial Differential Equations (PDEs), such as energy change or momentum change.
Procedure
Mr. Nichols will first learn how variational integrators function for simple rigid body systems, where all calculations may be done by hand [1]. He will implement a variational integrator for a pendulum and compare it to traditional ODE representations.
Mr. Nichols will then write programming code to compute the variational update laws for a generic simple mechanical system, where the variations are defined by the difference between the potential and kinetic energy. He will then animate the results to make visualization of complicated rigid bodies easier.
Time permitting, Mr. Nichols will finish the project by observing the application of variational integrators to simplify infinite dimensional systems, such as elastic bodies.
This work is to be done in conjunction with Todd Murphy of the Department of Electrical and Computer Engineering, who also instructs in the field of computer animation, a field in which Mr. Nichols is also studying (relevant course: ECEN 5438 Robot Control).
[1] L. Kharevych, Wei Wei, Y. Tong, E. Kanso, J.E. Marsden, P. Schroder, and M. Desbrun. Geometric, Variational Integrators for Computer Animation. SIGGRAPH Symposium on Computer Animation, 2006.
Results
The variational integrator was programmed in Mathematica and worked perfectly. The system under consideration was a pendulum, providing a simple example. The system can be represented with one configuration variable, the angle theta, which emerges as the pendulum is rotated from the primary axis. This system is usually modeled using an ODE, but variational integrators discretize the equation of motion into an algebraic expression, making it less prone to error compared to other traditional methods.
Below are computer animations of a variational integrator in action.
The animation above compares a variational integrator, using the Discrete Euler-Lagrange equation (shown in black), to a forwarding integrator computed by the Linearly Implicit Euler technique (shown in red). The forward integration techniques are calculated using the Euler-Lagrange equation. The graphs display the configuration variable, theta, as a function of time. This animation is of the same one-dimensional pendulum, but as time progresses, the time step used to calculate the position of the pendulum is increased.
The expected result would consist of both graphs being perfect sinusoids. It can easily be seen that initially with a time step of 0.001 seconds, the graphs are just slightly different. As the time step increases to a value of 0.15 seconds the Linearly Implicit Euler technique of forward integration becomes completely dampened, although dissipative forces exist. However, the variational integrator continues to preserve the energy in the system.
The second animation above is of the same system as the first but instead of using Linearly Implicit Euler integration, we use Explicit Euler technique for our forward integration. Note that the Explicit Euler technique becomes unstable, even with small time steps between calculations.
The third animation above shows two pendulums starting from rest. One is using the variational integrator to calculate the next figuration, while the other is calculated using Linear Implicit Euler forward integration. The time step used is 0.02 seconds. The initial position is to clearly show the Linear Implicit Euler technique lose energy.
The fourth animation above is the same two pendulum system as in the third animation but now one pendulum is calculated with variational integrator and the other pendulum calculated using Explicit Euler techniques. Note here that the initial position is shown so that we can see the Explicit Euler technique blow up clearly.
About Kirk Nichols
At the time of this research, Mr. Nichols was in his second year of college. His majors include electrical and computer engineering as well as applied mathematics. The areas of interest of which Mr. Nichols is interested in among the specific fields of electrical and computer engineering are control systems, embedded systems, and digital signal processing. His aspirations beyond two bachelors degrees include at least a masters degree, with an open possibility of pursuing a doctorate degree.