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US-12617075-B2 - Nonlinear dynamic system and method for designing a nonlinear dynamic system

US12617075B2US 12617075 B2US12617075 B2US 12617075B2US-12617075-B2

Abstract

The invention relates to a dynamic nonlinear system having a plurality of degrees of freedom. The system has at least one potential element, and eigenmodes of the system are produced by means of the potential element. A potential is produced by means of the at least one potential element, and the potential causes an acceleration tangential to the basic trajectories of the system, a basic trajectory being a trajectory of the potential-free system.

Inventors

  • Alin Albu-Schäffer
  • Arne Sachtler

Assignees

  • DEUTSCHES ZENTRUM FüR LUFT-UND RAUMFAHRT E.V.
  • Technische Universität München

Dates

Publication Date
20260505
Application Date
20220207
Priority Date
20210212

Claims (18)

  1. 1 . A dynamic nonlinear system with multiple degrees of freedom, the system comprising at least one potential element designed to generate eigenmodes of the system, wherein a potential is generated by means of the at least one potential element, wherein the potential causes an acceleration to act tangential to basic trajectories of the system, and wherein the basic trajectories are trajectories of the system without the potential.
  2. 2 . The system according to claim 1 , wherein at least one potential element is a mechanical, pneumatic, hydraulic or electrical energy storage.
  3. 3 . The system according to claim 1 , wherein the at least one potential element are springs or capacitors.
  4. 4 . The system according to claim 1 , wherein at least one potential element is one or a plurality of electric motors.
  5. 5 . The system according to claim 1 , wherein the at least one potential element are nonlinear potential elements.
  6. 6 . The system according to claim 1 , wherein a plurality of potential elements are provided.
  7. 7 . The system according claim 1 , wherein every point of a configuration space of the system lies on an eigenmode.
  8. 8 . The system according to claim 1 , wherein the potential has an energy minimum.
  9. 9 . The system according claim 1 , wherein a selected eigenmode θ d , which a movement follows, is selected only based on a direction of an initial speed.
  10. 10 . The system according to claim 1 , wherein the eigenmodes are basic trajectories corresponding to a mass tensor or a mass matrix of the system.
  11. 11 . The system according to claim 1 , wherein a derivation of the potential according to configuration variables is tangential to the basic trajectories with respect to a metric.
  12. 12 . The system according to claim 1 , wherein the potential U fulfills the condition: ∂ U ∂ q ⁢ M ⁡ ( γ ) - 1 ⁢ γ . ⊥ = 0 , where q is the configuration variables of the system, M the mass inertia matrix, γ is the movement along a basic trajectory parameterized by the path length, and {dot over (γ)} ⊥ is a vector perpendicular to the time derivative of γ.
  13. 13 . The system according to claim 1 , further comprising a controller, wherein the controller generates a restoring value τ θ which depends on a deviation Δ(θ) of a movement q(θ) from a selected eigenmode θ d of the movement and acts perpendicularly on the selected eigenmode θ d of the movement.
  14. 14 . The system according to claim 13 , wherein the restoring value τ θ of the controller is given by: τ θ = M ⁡ ( q ) ⁢ ∂ U ⊥ ∂ q [ - k θ ⁢ P ⁢ Δθ ⁡ ( q ) - d θ ( q ) ⁢ θ . ] , where {dot over (θ)} is a deviation rate of a deviation from a desired mode, ∂U ⊥ /∂q is a unit vector orthogonal to ∂U/∂q, k θ p is a constant and d θ (q) is a damping.
  15. 15 . The system according to claim 13 , wherein the controller is not active if:  q - q min  < α , where q min is an energy minimum of the potential and α is a predefined limit value.
  16. 16 . The system according to claim 13 , wherein the controller acts continuously on the movement of the system or acts on the movement at a reversal point of the movement or acts on the movement when the movement passes though an energy minimum of the potential.
  17. 17 . A method for designing a dynamic nonlinear system with multiple degrees of freedom, the method comprising the steps of: selecting a starting point of the system; determining a potential of the system, the potential generating an acceleration tangential to basic trajectories of the system, the basic trajectories representing trajectories of the system without the potential; and implementing the potential by means of potential elements; wherein eigenmodes of the system are adapted to a predefined movement by adapting kinematics or inertia properties or components of the system.
  18. 18 . A system with multiple degrees of freedom, wherein the system is designed according to the method according to claim 17 .

Description

CROSS-REFERENCE TO RELATED APPLICATIONS This application is the United States national phase of International Application No. PCT/EP2022/052884 filed Feb. 7, 2022, and claims priority to German Patent Application No. 10 2021 103 397.7 filed Feb. 12, 2021, the disclosures of which are hereby incorporated by reference in their entireties. BACKGROUND OF THE INVENTION Field of the Invention The present invention relates to a nonlinear dynamic system with multiple degrees of freedom. The present invention further relates to a method for designing a nonlinear dynamic system with multiple degrees of freedom. Description of Related Art Coupled, potentially nonlinear dynamic systems are used in many technical fields. In particular in robot technology, the mechanics can be described as such a system. Additional energy storages and dissipative elements in the system improve the robustness against destruction and the interaction with other systems. Moreover, these elements can improve the energy efficiency in systems. Without the use of energy storages, forces required for movement have to be provided by actuators alone. When actuators are used, energy losses occur. Potential energy storages can contribute or even completely take over forces necessary for acceleration, so that the power requirements to the actuators are reduced. Adding energy storages to dynamic systems may result in oscillatory systems. With oscillatory linear dynamic systems or systems that are locally approximated with linear differential equations, at least n oscillation modes exist for a system with n degrees of freedom. Along a mode, the linear system can be described with only one generalized system variable (coordinate) and its time derivative. Each of these solutions describes a system behavior along a modal line, in which there is a permanent exchange between potential energy and kinetic energy. The modal line is an invariant line of movement, independent of the magnitude of the oscillation energy of the system—the overall energy only affects the amplitude of the oscillation along this line. In general, the oscillation modes are isolated in the configuration space. It is known that there is a generalization of such modal oscillations also for nonlinear systems (Albu-Schäffer, A.; Lakatos, D., Stramigioli, S., “Strict Nonlinear Normal Modes of Systems Characterized by Scalar Functions on Riemannian Manifolds”, IEEE Robotics and Automation Letters, 2021 and Albu-Schäffer, A. & Della Santina, C., “A review on nonlinear modes in conservative mechanical systems” Annual Reviews in Control, 2020, 50, 49-71). This time, the oscillation no longer occurs along a straight line in the configuration space, but in general along a bent curve. If, also in this case, the curve is invariant with respect to the oscillation case, this mode is referred to as a strict nonlinear mode. The existence and the characteristics of the strict nonlinear modes depend on the system dynamics. Like a linear system, a nonlinear system of n-th order comprises at least n oscillation modes under certain conditions—this, however, includes energy-dependent modes—strict nonlinear modes are generally much rarer (ibid). For each energy level, the modes are mostly isolated in the configuration space, so that only a small part of the configuration space is reached by modes. Oscillations without exchange of energy with the outside world are possible on the isolated oscillation modes. If other points in the configuration space are to be reached, controllers and strong actuators are required, which impress the desired dynamics. Here, it may even be necessary to fill and empty the energy storages present in the system. The impression of the desired dynamics is therefore most often associated with energy losses and a reduction of the maximum available peak power. SUMMARY OF THE INVENTION It is an object of the present invention to provide a nonlinear dynamic system which is versatile and energy-efficient. The object is achieved with a nonlinear dynamic system, a method and a nonlinear dynamic system as described herein. The nonlinear dynamic system according to the invention has multiple degrees of freedom. The nonlinear dynamic system may be a mechanical system, such as a robot arm, which, by its kinematics, provides multiple degrees of freedom in movement. As an alternative thereto, the dynamic system is an electronic system, the multiple degrees of freedom being implemented by interconnected electronic components. The system has at least two or more degrees of freedom. The system according to the invention comprises at least one potential element. By using a potential element, an oscillatory system is already provided. Here, the system oscillates along the eigenmodes if the system was initialized on one of the isolated eigenmodes. In mechanical systems, the oscillation is the actual trajectory of the system or the movement of the system expressed, for example, in generalized coordinates. I