## Offered Bachelor and Master Theses

This is only a partial list of available topics.

If you are interested in robotics locomotion, flexible and soft robotics, nonlinear dynamics and control, and you cannot find any theses that you like in the following list, you may want to send us an email anyway, to see if there are other opportunities.

**Dynamic locomotion algorithms for soft quadrupeds**

Locomotion of legged robots is a challenging problem due to its hybrid dynamics (discrete contact sequencing and continuous whole-body motion), and the constraints on the direction and amplitude of the contact forces. Recently, the concepts of three-dimensional Divergent Component of Motion (DCM) and Virtual Repellent Point (VRP) were introduced in [1], decomposing the second-order CoM dynamics into two first-order linear dynamics, with the CoM converging to the DCM (stable dynamics), and the DCM diverging away from the VRP (unstable dynamics). Based on this formulation, continuous closed-form DCM and CoM trajectories can be generated using a piecewise interpolation of the VRP trajectory over a sequence of waypoints [2]. This highly compact motion representation is a natural way of handling the locomotion hybrid dynamics, with the discrete contact sequencing being mapped onto the VRP waypoints. This approach has been used successfully for bipedal locomotion in [1], and dynamic multi-contact motion in [3].

This thesis will be about generalizing this concept to quadrupedal locomotion, and to understand to which extent this technique can be realized with intermitted control actions. The ultimate goal of this thesis will be the implementation of dynamic locomotion with the robot Bert [4].

Suggested as (but not limited to!)

- Master´s thesis in Robotics, Cognition and Intelligence
- Master's thesis in Mechatronics and Robotics

References

- [1] J. Englsberger, C. Ott, and A. Albu-Schäffer, "Three-dimensional bipedal walking control using divergent component of motion," in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2013, pp. 2600–2607.
- [2] G. Mesesan, J. Englsberger, C. Ott, and A. Albu-Schäffer, "Convex properties of center-of-mass trajectories for locomotion based on divergent component of motion," IEEE Robotics and Automation Letters, 3(4), 2018, pp. 3449-3456.
- [3] G. Mesesan, J. Englsberger, B. Henze, and C. Ott, "Dynamic multicontact transitions for humanoid robots using divergent component of motion," in IEEE International Conference on Robotics and Automation (ICRA), 2017, pp. 4108–4115.
- [4] D. Lakatos, et al. "Dynamic locomotion gaits of a compliantly actuated quadruped with SLIP-like articulated legs embodied in the mechanical design."
*IEEE Robotics and Automation Letters*3.4 (2018): 3908-3915.

Supervisors: George Mesesan (George.Mesesan@dlr.de), Dr. Cosimo Della Santina (cosimodellasantina@gmail.com), Prof. Dr.-Ing. Alin Albu-Schaeffer

**Resonant motions in a generalization of the elastic pendulum**

Controlling motion at low energetic cost constitutes one of the major challenges in biologic and robotic locomotion. Recent results suggest that robots can be designed and controlled to move efficiently by exploiting resonance effects of an elastic body, increasing the performance compared to a rigid body design. This motivates the study of simple mechanical systems which represent complex behaviours through few degrees of freedoms (e.g. the spring leaded inverted pendulum (SLIP) model for legged locomotion). To such end, this thesis will focus on analysing a paradigm model for an elastic leg, namely a generalization of the elastic pendulum, which includes a fully coupled elastic field rather than a solely radial one. The goal of the thesis is to gain a complete understanding of the phase portrait of the system, in particular a complete characterization of periodic and quasiperiodic motions of low period. Of particular interest will be the study of bifurcations of the phase portrait when either mechanical parameters (e.g. body inertia, spring stiffness), or energy levels change.

References

- Strogatz, S. H. (2018). Nonlinear dynamics and chaos: with applications to physics, biology, chemistry, and engineering. CRC Press.
- Geyer, H., et al. (2006). Compliant leg behaviour explains basic dynamics of walking and running. Proceedings of the Royal Society B: Biological Sciences, 273(1603), 2861-2867.
- Breitenberger, E., & Mueller, R. D. (1981). The elastic pendulum: a nonlinear paradigm. Journal of Mathematical Physics, 22(6), 1196-1210.
- Cuerno, R., et al. (1992). Deterministic chaos in the elastic pendulum: A simple laboratory for nonlinear dynamics. American Journal of Physics, 60(1), 73-79.
- Lynch, P. (2002). Resonant motions of the three-dimensional elastic pendulum. International Journal of Non-Linear Mechanics, 37(2), 345-367.

Supervisors: Prof. Dr. Oliver Junge, Prof. Dr.-Ing. Alin Albu-Schaeffer, Dr. Cosimo Della Santina (cosimodellasantina@gmail.com)

**Multiple theses in: Discovering natural oscillations in highly coupled mechanical systems with machine learning and data-driven strategies, with application to soft robotics.**

For the study of highly nonlinear mechanical systems, finding special periodic solutions which are generalization of the well-known normal modes of linear systems promise to enable the execution of complex tasks (e.g. running, swimming, periodic pick and place) with an efficiency and a robustness close to the one observed in animals.

However, the study of nonlinear normal modes is rather a niche topic, treated mainly in the context of structural mechanics for systems with Euclidean metrics, i.e., for point masses connected by nonlinear springs. Nonetheless newest results emphasize that a very rich structure of periodic and low-dimensional solutions exist also within nonlinear systems such as elastic multi-body systems encountered in the biomechanics of humans and animals or of humanoid and quadruped robots, which are characterized by a non-constant metric tensor.

Evaluating an analytic mathematical description of nonlinear normal modes for these systems is, however, no simple task. The most straightforward way to do that is to approximate the solution of a set of partial differential equations describing the geometry of these structures, using Galerkin methods. Unfortunately, this method provides only local solutions, and do not scale well with the dimension of the dynamical system.

As an alternative, this thesis will investigate the use of simulation based and data driven strategies as swarm and particle optimization, Bayesian optimization, (deep) autoencoders, and other machine learning techniques.

Whenever possible, these techniques will be guided by the theoretical knowledge of the underlying mathematical structure, allowing for effective and efficient formulations of the problem.

The expected outcome of this thesis will be a semi-professional MatLab toolbox, able to discover nonlinear normal modes in generic soft robots. Therefore, either a good knowledge of this programming language, or above average programming skills are mandatory.

Suggested as (but not limited to!)

- Master’s Thesis in Informatics (all masters)
- Master's Thesis in Mathematics in Data Science
- Master'sThesis in Mathematics in Science and Engineering

References

- Albu-Schaeffer, A., et al. (2019). One-Dimensional Solution Families of Nonlinear Systems Characterized by Scalar Functions on Riemannian Manifolds. arXiv:1911.01882.
- Renson, L., et al., (2016). Numerical computation of nonlinear normal modes in mechanical engineering. Journal of Sound and Vibration, 364, 177-206.
- Silva, V. D., & Tenenbaum, J. B. (2003). Global versus local methods in nonlinear dimensionality reduction. In Advances in neural information processing systems (pp. 721-728).
- Baldi, P. (2012). Autoencoders, unsupervised learning, and deep architectures. In Proceedings of ICML workshop on unsupervised and transfer learning (pp. 37-49).

Supervisor: Prof. Dr.-Ing. Alin Albu-Schaeffer, Dr. Cosimo Della Santina (cosimodellasantina@gmail.com)

DLR: Dynamic Motion Planning Metrics and Collision Space Formulation for Sensor-Equipped Mobile Robot