Strict Modes Everywhere - Bringing Order Into Dynamics of Mechanical Systems by a Potential Compatible With the Geodesic Flow

Arne Sachtler; Alin Albu-Schäffer

cs.ROSep 4, 2026

Strict Modes Everywhere - Bringing Order Into Dynamics of Mechanical Systems by a Potential Compatible With the Geodesic Flow

Authors: Arne Sachtler, Alin Albu-Schäffer

Organizations: Technical University of Munich (TUM), Department of Informatics; Boltzmannstr. 3, 85748 Garching · German Aerospace Center (DLR), Institute of Robotics and Mechatronics; Münchener Str. 20, 82234 Weßling, Germany

Abstract

Strict nonlinear normal modes provide very regular families of oscillations within conservative mechanical systems. However, a strict normal mode will generally be an isolated curve within the configuration space of the system. In this letter, we design a potential that will densely fill the configuration space with strict normal modes such that each configuration belongs to one mode and each mode passes through a common point, the equilibrium. As the potential can be realized by (nonlinear) elastic elements it can be used to execute a variety of periodic trajectories very efficiently. Most of the required torques will come from the elastic elements in the system and not from the actuators. We also design a controller stabilizing the system to a desired target mode and a controller performing swing-up and compensating dissipated energy. Finally, we showcase the approach for a two DoF manipulator. The experiments show that the approach performed well for the example system.

Figures & tables

Panel of Figure 1 — Fig. 1: Aim of this paper: find a potential function such that the configuration is densely packed with strict modes.

Panel of Figure 2 — Fig. 2: Double Pendulum

Panel of Figure 3 — Fig. 3: Geodesics of the example system. Restriction to a subregion results in a space where the geodesics only intersect at the equilibrium.

Panel of Table 4 — TABLE I: Comparison of different target equilibria.

Panel of Figure 6 — Fig. 5: Simulation of the double pendulum with the optimized potential for different initial conditions and visualization of the potential function.

Panel of Figure 7 — Fig. 6: Mode labeling function

Panel of Figure 8 — Fig. 7: Simulation results with activated mode selection controller when switching the desired mode step-wisely.

Panel of Figure 9 — Fig. 8: Damped system including the optimized potential, the mode selection controller and two types of energy regulation.

Panel of Figure 10 — Fig. 9: Results for the sensitivity analysis. The parasitical potential is increased over the experiments. The top row shows the evolution of the system in terms of the joint angles. The bottom row shows power flow due to the individual torques sources for three different values of $\epsilon$ .

$\epsilon$	$\int\|P_{\tilde{U}}\|$	$\int\|P_{\theta}\|$	$\int\|P_{E}\|$
0	354.56	2.86	0.03
0.01	356.23	2.85	1.87
0.05	363.25	2.94	9.44
0.1	372.43	3.74	18.75
0.5	447.18	11.72	89.26
1.0	565.60	20.92	181.83

TABLE II: Comparison of total energy fluxes

Appendix figures & tables1 asset

Supplementary material from the paper’s appendix.

Appendix

	Link 1	Link 2
Mass	$m_{1}=$ 1\text{,}\mathrm{kg}$$	$m_{2}=$ 1\text{,}\mathrm{kg}$$
Length	$l_{1}=$ 2\text{,}\mathrm{m}$$	$l_{2}=$ 2\text{,}\mathrm{m}$$
Center of Mass from Origin	$l_{c1}=$ 1\text{,}\mathrm{m}$$	$l_{c2}=$ 1\text{,}\mathrm{m}$$
Moments of Inertia	$I_{1}=$ 1\text{,}\mathrm{kg}\text{,}{\mathrm{m}}^{2}$$	$I_{2}=$ 1\text{,}\mathrm{kg}\text{,}{\mathrm{m}}^{2}$$

Appendix

TABLE III: Parameters of the Double Pendulum

Dynamical Systems

Dynamical systems studies how states evolve over time, including stability, feedback, chaos, and transitions in physical, biological, and engineered systems.

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May 27, 2026J/K move · Enter open · S save

Natural Locomotion: Principle and Method

Mirado Mortel, Luc Jaulin, Lionel Lapierre, Simon Rohou

Robot Locomotion

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Robotic locomotion can become efficient when mechanisms exploit passive dynamics, compliance, and resonance rather than track prescribed trajectories. This paper formulates natural locomotion as an exchange principle for systems whose motion is mediated by environmental constraints or interactions. A motion is natural when an internal oscillator returns periodically, the body pose drifts, and the mean Propulsion--Oscillator Exchange power (POE power) vanishes over one cycle. The selected family is a Natural Locomotion Manifold (NLM). We develop the conservative realization of this principle for continuous ideal environmental constraints: the constraints do no external work, total mechanical energy is conserved, and zero mean POE power is an internal exchange with the environment-mediated propulsive channel, not external energy input. The method is a closed/open construction. The propulsive channel is first closed to reveal an effective internal oscillator, organized by scalar action-angle structure in one effective degree of freedom or by nonlinear modal sectors in several degrees of freedom. The channel is then reopened, pose is reconstructed, and accepted cycles must preserve internal recurrence and zero mean POE power. We demonstrate the principle on two ideal nonholonomic no-slip systems: a Chaplygin-sleigh / pendulum-driven car and a three-body extension. In the scalar case, POE closure is equivalent to the missing internal return condition, giving a theorem-backed computation of the NLM family. In the multi-degree case, POE closure remains necessary but must be completed by modal identity, internal return, dynamics consistency, same fixed passive architecture, and nonzero displacement. Natural locomotion becomes a design question: which passive architectures support no, one, or several certified NLM families?

Abstract

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