Added blog post about opty 1.3 and sympy 1.13.

content/sympy-1.13-opty-1.3-release.rst

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+=======================================================
+Biomechanical Optimization with SymPy 1.13 and Opty 1.3
+=======================================================
+
+:date: 2024-08-05 00:00:00
+:tags: sympy, czi, biomechanics, multibody dynamics, open source software,
+       muscles, optimal control, bicycle, code generation
+:category: research
+:authors: Jason K. Moore
+:summary: SymPy 1.13 and Opty 1.3 can now be used for optimal control solutions
+          of muscle-driven biomechanical simulation. This blog post shows of a
+          solution to a muscle driven cycling time trial.
+
+`SymPy 1.13`_ was released on July 8, 2024. This release includes `all of the
+work we did under the CZI Open Source Software for Science Cycle 4 grant
+<{file}/czi-sympy-wrapup.rst>`_. For example, SymPy now includes new features
+for modeling biomechanical multibody systems. The new features include model
+construction from joints via a system manager and force actuator classes
+including a musculo-tendon model. At the close of the grant period, we
+demonstrated how these features can be used in the solution of biomechanical
+trajectory optimization and parameter estimation optimal control problems, but
+only from development branches of the various intertwined software packages.
+Today, we released `Opty 1.3`_ which can now be used alongside SymPy 1.13 to solve
+biomechanical optimal control problems. Opty can now handle symbolic equations
+of motion that have many millions of mathematical operations and tens of
+thousands of constraints and optimization variables. To help grow adoption,
+we've built out a new example gallery that demonstrates how Opty can be used to
+solve a variety of problems:
+
+.. figure:: https://objects-us-east-1.dream.io/mechmotum/opty-1.3-examples.png
+   :align: center
+   :target: https://opty.readthedocs.io/stable/examples/index.html
+
+   Opty 1.3's new example gallery.
+
+.. _SymPy 1.13: https://github.com/sympy/sympy/releases/tag/1.13.0
+.. _Opty 1.3: https://github.com/csu-hmc/opty/releases/tag/v1.3.0
+
+One example in the gallery, the "`One-Legged Cycling Time Trial
+<https://opty.readthedocs.io/stable/examples/plot_one_legged_time_trial.html>`_",
+shows how to find an optimal control solution of a musculo-tendon driven model
+that makes use of SymPy 1.13's new biomechanics features. We developed this
+example for an activity in TU Delft's 2024 Sports Engineering course to show
+the students how models can be used to study how muscles play a role in optimal
+performance. The example solves a minimum time cycling race using a model of a
+single leg and simplified muscles that drive the leg motion to propel the
+bicycle. The example in the gallery is set up with a short race distance but if
+you bump the distance up to 80 meters the cyclist starts approaching terminal
+velocity. Opty can find an optimal solution for the 17,000 variables in this
+problem in about 10 minutes on an AMD Ryzen 5 2600X 12 core machine.
+
+This produces a reasonably realistic overall motion which can be seen in the
+cadence, resulting speed, and the power to accelerate and overcome rolling and
+air resistance:
+
+.. image:: https://objects-us-east-1.dream.io/mechmotum/one-legged-80m-power.png
+   :align: center
+
+There is no limit on the fatigue of the muscles so the optimizer finds a
+bang-bang style coordination of the muscle excitations to power the leg and
+push the muscles to their maximum force production:
+
+.. figure:: https://objects-us-east-1.dream.io/mechmotum/one-legged-80m-muscle.png
+   :align: center
+
+   The four muscle forces, muscle lengths, and muscle excitations versus time
+   during the 80 meter time trial with the minimum race time shown at the top.
+
+This solution results in this overall pedaling motion:
+
+.. raw:: html
+
+   <center>
+     <video controls>
+       <source src="https://objects-us-east-1.dream.io/mechmotum/one-legged-80m.mp4" type="video/mp4">
+     </video>
+   </center>
+
+Our next step is to incorporate muscles in our 3D nonlinear bicycle-rider
+models to enable deeper understanding of how bicycles are balanced and
+controlled. Opty can quickly find optimal trajectories of the joint torque
+driven model, as shown in our `BRiM paper`_:
+
+.. figure:: https://objects-us-east-1.dream.io/mechmotum/brim-paper-opty-example.png
+   :align: center
+
+   Torque driven Opty optimal control solution for the non-linear
+   Carvallo-Whipple bicycle model.
+
+.. _BRiM paper: https://doi.org/10.59490/660179a06bf1082286458109
+
+Thanks to Timo, Sam, Peter, and all the SymPy developers for the contributions
+that helped pull all this together! We hope that others can make use of these
+models and tools for their own work.