Peter’s Mechanics Examples

Mechanics examples created using various scientific Python tools: SymPy, PyDy, Opty, SciPy, Matplotlib, etc.

Contents:

• See the Examples

Mechanics Animations

Comparisons

John T. Betts’ book Practical Methods for Optimal Control Using Nonlinear Programming is considered to be a foundational text in the field of optimal control using nonlinear programming. In its section 10 it gives a number of example problems with solutions. The solutions were calculated by SOS, a Fortran 95 proprietary code written by Dr. Betts and collaborators, and maintained by ASTOS GmbH.

We solved a number of them using opty. They may be found at these locations:

• location 1: https://opty.readthedocs.io/stable/examples/index.html#beginner

• location 2. https://opty.readthedocs.io/stable/examples/index.html#intermediate

• location 3: https://pydy.org/pst-notebooks/

Notes:

• The number of the examples, e.g. 10.58 refers to the numbering in the a.m. book, 3rd edition, 2020.

• Dr. John Betts was always very helpful and answered any questions I had regarding the examples.

• While most of the examples we tried to solve with opty converged easily to a result close to that given in the book, a few did not converge.

• By accuracy is meant the deviation from the solution given in the book.

List of Examples Solved with opty

Number	Name	Location	Remarks
10.1	Stiff Set of DAEs	3	Differential equation designed to be very stiff. Accuracy 1.05%
10.6	Underwater Vehicle	3	Accuracy 0.095%
10.7	Hypersensitive Control	1	Accuracy 0.143%
10.43	Free Flying Robot	3	Accuracy 2.77%
10.47	Singular Arc Problem	2	Three Phase Problem. While SOS solved it simultaneously, opty had to solve it in separate stages. Overall accuracy < 0.95%
10.50	Delay Equation	1	Accuracy 0.027%
10.57	Heat Diffusion Process with Inequality	3	Accuracy 1.17%
10.58	Heat Equation	3	This deals with the discretization of a PDE. opty converged, but the accuracy was not good.
10.73 10.74	Linear Tangent Steering	3	The same physical problem, formulated in two different ways. Accuracy 0.00012%
10.90	Nonconvex delay	3	Accuracy 0.004%
10.103 10.104	Compare DAE vs. ODE Formulation	1	The same physical problem formulated in two different ways. Accuracy 2.49%
10.113	Mixed State-Control Constraints	3	Accuracy 0.03%
10.133	Two-Strain Tuberculosis Model	3	Accuracy 0.16%
10.141	Tumor Anti angiogenesis	3	Accuracy 1.45% / 0.1975%
10.144 10.145	Van der Pol Oscillator	3	Two different formulations of the same problem. Accuracy 0.017%
10.148	Zermelo’s Problem	3	Accuracy 0.023%
	Parameter Identification Betts & Huffmann 2003	1	Accuracy 0.021%
	Brachistochrone with Obstruction	3	As per section 4.16.2 of the a.m. book this can be difficult. opty solved it with no problems.