Design of nonlinear springs to shape oscillations for a double pendulum

MSc assignment

Robot arms are often designed for a range of tasks: they work for a range of payloads, have a reasonably large workspace, and come with low level controllers and motors that allow them to perform a large set of motions. However, when designing robots for specific tasks, it can lead to a huge boost in efficiency (in terms of power consumption) if the eventual task is kept in mind. For example, given the need for a robot arm that performs repetitive motions, one can add parallel elastic actuators (springs) to the robot’s joints to support the motors and to reduce the required joint torques during operation. Examples also extend to locomotion systems such as quadrupeds, where PEAs can assist jumping motions [1].

On an abstract level, a PEA is simply a potential field that is applied at the joint level. At RaM, we recently developed a framework for optimizing such potential fields to support specific periodic tasks [2]. Given a desired periodic motion, our learning algorithm would try to find a potential field such that the robot follows the given periodic motion without action of the motors. This is what is referred to as hyper-efficient behavior. In particular, we tried to learn such motions for a double pendulum, which succeeded in simulation.

However, it is an open problem to enable hyper-efficient behavior in a real double pendulum. The design of specific nonlinear springs is required, e.g., by Fourier composition of sinusoidal springs [3] or single, magnet-based elastic elements [4]. This assignment aims to tackle that problem by focusing on the design, manufacturing and testing of such nonlinear springs on a double pendulum.


The main goals are:

0. Build and characterize a double pendulum

1. Design nonlinear springs for hyper-efficient motion of a double pendulum

2. Apply the nonlinear springs to the double pendulum, and show to which degree hyper-efficient motion is achieved in practice


For more questions, please contact Yannik Wotte ( or Boi Okken (


[1] Learning-based Design and Control for Quadrupedal Robots with Parallel-Elastic Actuators; 2021; F. Bjelonic, J. Lee, P. Arm, D. Sako, D. Tateo, J. Peters, M. Hutter; DOI: 10.1109/lra.2023.3234809

[2] Discovering Efficient Periodic Behaviours in Mechanical Systems via Neural Approximators; 2023; Y. Wotte, S. Dummer, N. Botteghi, C. Brune, S. Stramigioli, F. Califano; DOI: 10.1002/oca.3025

[3] Novel Adaptive Magnetic Springs for Reliable Industrial Variable Stiffness Actuation; 2023; B. Mrak, J. Willems, J. Baake and C. Ganseman; DOI: 10.3390/act12050191

[4] Progressive Series-Elastic Actuation with Magnet-based Non-linear Elastic Elements; 2022; B. Okken, S. Stramigioli, W. Roozing; DOI: 10.1109/SSRR56537.2022.10018635