LGVI Multibody Experiments

Numerical experiments for the analysis of time integrators & discretizations for the simulation & optimal control of rigid-flexible robotic systems.

This repository contains the numerical experiments on time integration and optimal control of rigid-flexible multibody systems reported in [1]. In the experiments, various common ODE integrators for nonstiff and stiff systems are compared to a Lie-group variational integrator in terms of numerical properties and computational efficiency. The implementation is based on the MATLAB toolbox ELARA, which must be installed on the machine.

Multibody Simulation Experiments

1. Free motion of a planar four-link pendulum
2. Nonlinear oscillations of a geometrically exact cantilever beam (modeled as a Kirchhoff beam); this is the same case also treated in [2, 3]
3. Robot manipulator with a highly flexible end link under closed-loop PD control

The first two simulation cases are each analyzed both with and without dissipation.

Optimal Control Experiments

1. Rest-to-rest motion of a planar robot manipulator (after [4])
2. TCP trajectory tracking of a three-DOF robot manipulator
3. TCP trajectory tracking of a tendon-driven continuum manipulator

Running the Simulation Studies

1. Make sure the mex files in the toolbox have been built correctly (run elara.build and check with elara.setup)

2. Run startup_sim_studies.m to add all required functions to the MATLAB path and check for dependencies.

3. Run the simulation studies in the /studies folder:

   • sim_study_integrators_run_sims: Time integration simulation study
   • sim_study_OCP_disc_run_sims: Optimal control discretization study

   The scripts save the simulation results in the /results/ folders.

4. Run the evaluation scripts to generate the output plots. Adjust the system, dissipation case, and data file paths in the script settings.

Additional Scripts

Under /tests, there are two additional validation scripts that perform simulations of the four-link pendulum and the cantilever beam of the integrator study. Additionally, validation_sim_cantilever_beam.m compares the simulation results to the literature results from [3].

Requirements

• MATLAB R2025b
• ELARA Toolbox, V0.1 (installed and available on the MATLAB path)
• CasADi V3.7.2 (only for the optimal control experiments; must be installed and available on the MATLAB path)
• Coin-HSL linear solvers for IPOPT (version 2024.05.15)

Later versions of the required software may work, but have not been tested and may lead to different results.

The code is tested on Windows 11.

Additional Software Used

• This repository includes the RADAU integrator implemented by Denis Bichsel. It is originally available here.

• The MATLAB version of the scientific color maps Crameri is included as a submodule.

• The PHCosseratRods repository (corresponding to [3]) is included as submodule for validation and comparison of the cantilever beam simulation case.

Installing the HSL linear solvers for IPOPT

Download: https://licences.stfc.ac.uk/product/coin-hsl

Installation instructions: https://github.com/casadi/casadi/wiki/Obtaining-HSL

• Create an account on the HSL licensing website (academic license) and download binaries for Windows
• Add the \bin subfolder to the PATH variable

Now, the solvers should be available in CasADi/IPOPT:

  opts = struct;
  opts.ipopt.linear_solver = 'ma97';
  NLPSolver = nlpsol('solver', 'ipopt', NLprob, opts);

Additional Information

For additional details on the implementation, see the ELARA documentation, and for background on the systems and integrators, see [1].

License

The code in this repository is licensed under the MIT License; see LICENSE.

Third-party components in third-party/ are redistributed under their respective upstream terms, notices, and citation requirements.

References

[1] M. Herrmann: Geometric Modeling and Optimal Control of Rigid-Flexible Robot Manipulators. PhD Thesis, Technical University of Munich, 2026 (in preparation).

[2] M. Herrmann and P. Kotyczka. “Relative-kinematic formulation of geometrically exact beam dynamics based on Lie group variational integrators”. In: Computer Methods in Applied Mechanics and Engineering 432 (2024), p. 117367. doi: 10.1016/j.cma.2024.117367

[3] P. L. Kinon, S. R. Eugster, and P. Betsch. “Mixed formulation and structure-preserving discretization of Cosserat rod dynamics in a port-Hamiltonian framework”. In: Computer Methods in Applied Mechanics and Engineering 458 (2026), p. 118966. doi: 10.1016/j.cma.2026.118966

[4] S. Ober-Blöbaum, O. Junge, and J. E. Marsden. “Discrete mechanics and optimal control: An analysis”. In: ESAIM: Control, Optimisation and Calculus of Variations 17.2 (2011), pp. 322–352. doi: 10.1051/cocv/2010012