I haven’t blogged in about two weeks, and there has been a lot of progress in PyDy. For starters, PyDy now ‘automatically’ derives the equations of motion for a rolling disc and a rigid body in space. I haven’t blogged in about two weeks, and there has been a lot of progress in PyDy. For starters, PyDy now ‘automatically’ derives the equations of motion for a rolling disc and a rigid body in space. One of the more useful methods I implemented is the output_eoms() method in the NewtonianReferenceFrame class. Once the equations of motion have been derived, this method creates python functions for them (in the form that scipy.odeint wants them in) and writes them to a file that you specify. Once you have this, you can import the function and use it for numerical integration. Additionally, this method also outputs a function that calculates the generalized speeds given the coordinates and their time derivatives, which is useful if you know the initial conditions of the coordinates time derivatives but need to determine the initial conditions of the generalized speeds when you want to perform a numerical integration. Finally, this method also optionally outputs an animate function, which takes a variable number of Points and (Axis, Angle) tuples and outputs a function which given the coordinates and parameters, will compute the inertial position and orientation of Points and/or Reference Frames. This is useful when you want to animate your simulation.
Both the rolling_disc.py and rigidbody.py files in the pydy/examples folder demonstrate the usage of these new methods.
One issue that has been repeatedly rearing its ugly head is the use of Function instances instead of Symbol instances. For everything except time differentiation, PyDy expressions could work fine with Symbols instead of Functions. I have become to realize that everything just works faster and more efficiently in Sympy with Symbols, so I will be working in the next week or so to replace the use of Functions in PyDy with Symbol instances instead. This should allow lots of things to work much better without having to call a bunch of helper methods to get expressions in their simplest form, or having to repeatedly substitute and back substitute with Symbols and Functions. For time differentiation, I will keep track of a list of symbols that represent the generalized coordinates, and then any expression that needs time differentiation will be examined to see if it has any occurrences of those coordinates. For expressions involving those coordinates, the partial derivative will be taken with respect to each coordinate, and then multiplied by the symbol that will be used to represent the time derivative of that coordinate, in essence a manual (as opposed to letting Sympy take care of it by using sympy Function) application of the chain rule. The user will be required to eclare these coordinate symbols in a special way, specifically using the .setcoords() method. Finally, I will implement method that will replace the symbols with Function instances (depending on time), so that if one wants to play with an expression interactively, all that they will have to do is call this function so that things like q1 = Symbol(’q1′) get replaced with q1 = Function(’q1′)(t), and then time differentiation will work just fine.
We really have to improve sympy that functions work as good as symbols…