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Strict Modes Everywhere - Bringing Order Into Dynamics of Mechanical Systems by a Potential Compatible With the Geodesic Flow
Arne Sachtler, Alin Albu-Schäffer
TL;DR
The paper addresses the difficulty that strict normal modes are usually isolated and that stiff actuation can waste mechanical energy. It designs a potential compatible with geodesic flow so strict modes densely cover configuration space, and combines this with stabilization and energy regulation. Simulations of a double pendulum show periodic motions without active control and low control torques relative to elastic torques, while physical elastic implementation remains future work.
Problem
Strict normal modes are generally isolated, while stiff actuation can dissipate mechanical energy and requires actuators to generate the required trajectory forces.
Method
The paper fits an elastic potential to geodesics launched from a chosen equilibrium, then uses mode-stabilization and energy-regulation controllers for practical operation.
Results
The double-pendulum simulations exhibit infinitely many strict modes and periodic motions without active control, with control torques much smaller than elastic-element torques.
Takeaways & Limitations
The designed potential enables efficient execution of modal oscillations and nearby trajectories while mechanics provide most required torques.
Takeaways & Limitations
Direct mechanical implementation using nonlinear elastic elements was outside the paper’s scope and is left for future work.
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Robotic applications often require periodic motion, but strict normal modes are typically isolated and nonlinear systems can behave complexly. The paper therefore seeks elastic elements that make strict modes available throughout configuration space while reducing actuator effort.
- Motivation: Periodic motions arise in applications including pick-and-place manipulation and legged or non-legged locomotion.Prior work generally studies periodic orbits or controlling robots onto modes while treating mechanics and elastic elements as given.
- Motivation: Stiffly actuated robots exchange substantial energy between electrical actuation and mechanics, with some mechanical energy dissipated as heat.Elastic elements can store and release mechanical energy, potentially aligning natural oscillations with desired trajectories so actuators mainly stabilize them.
- Background: Nonlinear normal modes provide low-dimensional regular motions within complex elastic systems, but strict modes usually occur as isolated configuration-space curves.Trajectories can become complex when initialization does not lie directly on a strict mode.
- Contribution: The paper asks whether elastic elements can be designed so every configuration belongs to a strict mode passing through a common equilibrium.Such a system would produce periodic motion from arbitrary configurations when initialized with zero or compatible velocity.
II. STRICT NONLINEAR NORMAL MODES
The paper models conservative mechanics on a configuration manifold with an inertia-induced Riemannian metric. A strict normal mode is characterized geometrically as a geodesic whose potential gradient remains tangent to the curve.
- Mechanical model: The configuration space Q is modeled as a manifold for conservative mechanical dynamics with mass, Coriolis, and potential terms.The coordinate-independent formulation equips Q with a Riemannian metric generated by the inertia tensor.
- Strict normal modes: A strict normal mode is a one-dimensional submanifold whose tangent bundle is invariant under the system dynamics.The mode is represented by a parametrized curve with line-segment topology.
- Strict normal modes: A curve is a strict normal mode exactly when it is a geodesic of the inertia metric and the potential gradient is tangent to it everywhere.These are the two geometric conditions stated by the theorem.
III. APPROACH
The approach constructs a potential compatible with geodesics launched from a chosen equilibrium, then fits a parameterized elastic potential under tangency and global potential constraints. Controllers stabilize modes and regulate energy, while the elastic potential supplies most useful torques.
- Design goal: The target is a potential whose strict modes densely cover a selected configuration-space region containing the equilibrium.Every configuration in the region must lie on at least one strict mode.
- Geodesic construction: Geodesic candidates are generated by launching the potential-free system from the chosen equilibrium in arbitrary initial-velocity directions.The resulting curves depend on direction rather than velocity magnitude and are parameterized by arc length.
- Potential fitting: The unknown elastic potential is fitted so its force is tangent to every candidate geodesic, using an orthogonal-complement formulation that removes an unknown scalar.This turns the tangency requirement into an optimization objective.
- Potential constraints: The final potential is constrained to have its minimum at equilibrium and to increase along geodesics away from equilibrium.The Lie-derivative constraint enforces strict monotonic increase along geodesics.
- Parameterization: Training samples from computed geodesics are used in a constrained optimization problem for a parameterized potential, here represented by a single-output neural network.Elastic basis functions and mechanical implementations are discussed, but physical implementation is left for future work.
- Control and actuation: The resulting elastic potential supplies useful torques for natural oscillations, leaving actuators to stabilize modes and compensate energy losses.The paper proposes mode stabilization and energy-regulation control because disturbances and dissipation prevent passive maintenance of a desired mode.
IV. CASE STUDY: DOUBLE PENDULUM
The double-pendulum case study selects target equilibria, computes geodesics in a suitable configuration-space region, and trains an optimized potential compatible with them. Simulations show periodic motions from multiple zero-velocity initial configurations without active control, while damping motivates later control.
- The target equilibrium and inertia tensor determine the shape of the strict normal modes, so the equilibrium must be selected carefully.
- For the elbow-up example, geodesics are computed from the equilibrium using sampled unit initial velocities, with a smaller region selected to avoid intersections.The larger configuration-space region contains intersecting geodesics, whereas the selected subregion avoids them.
- Three target equilibria are compared through their configuration-space and Cartesian-space geodesics, with the elbow-up setting used for the remainder of the case study.
- A two-layer tanh neural network is trained first on a parabola and then with the geodesic-compatibility loss to obtain the elastic potential.The network has 200 neurons in its first layer and 100 in its second; new training data are sampled during training to reduce overfitting.
- Zero-velocity simulations from several initial configurations produce periodic trajectories without an active controller, due to the optimized potential.The potential and geodesics are visualized, and three trajectories are shown in time plots.
- The approach yields infinitely many strict modes with no preferred mode, but damping dissipates energy and may pull the system off its mode.A stabilizing controller and an energy-injecting controller are therefore introduced for practical operation.
V. CONTROLLERS
The paper derives two controllers for the example system: one stabilizes motion on a selected strict normal mode, and the other performs swing-up while maintaining the desired energy level.
- Two controllers are derived and evaluated: one stabilizes the system onto one of infinitely many strict normal modes, while the other swings it up and maintains the desired energy level.
A. Mode Selection Controller
The mode-selection controller labels strict modes by a configuration-dependent function and uses that label to select and stabilize a desired mode. Because labeling fails at the equilibrium, control is enabled only outside an ε-ball, while experiments switch modes stepwise.
- Mode labeling: A labeling function θ: Q → X assigns each configuration to a strict mode, with dim X = n−1, but cannot be defined at the common equilibrium.
- Mode labeling: For the two-degree-of-freedom system, modes are identified by the velocity direction at equilibrium, with opposite directions representing the same mode and θ mapping into S1.
- Controller design: A neural-network representation of θ produces level sets corresponding to the strict modes, providing the basis for mode selection control.
- Controller design: The controller uses proportional feedback on deviation from the desired mode and damping computed from a desired damping ratio.
- Operating boundary: Because θ is undefined or unreliable near equilibrium, the controller is enabled only outside an ε-ball centered there.
- Mode switching: The desired mode can be changed during operation by specifying θd, and experiments increase it stepwise while applying the optimized potential.With kθ = 1, ζ = 0.7, and ε = 0.1, the controller torques remain quite low relative to the potential torques during switching.
B. Energy Regulation
The paper compares two energy-regulation controllers for swing-up and energy maintenance, finding different strengths during and after swing-up. The mode-compatible controller preserves modal motion, while damping compensation depends on structural compatibility.
- Energy regulation: Energy regulation swings the system from its initial energy toward a target level and reinjects energy lost through damping or other dissipation.The simulation uses E0 = 5.0J and Ed = 20.0J.
- Controller comparison: The mode-compatible controller τ E performs better during swing-up because it accelerates along the mode without immediately pulling the system away from it.Its control action is dynamically decoupled from the mode-selection controller.
- Controller comparison: The negative-damping controller τ nD performs better after swing-up when modeled joint friction matches its damping structure.This favorable match is described as unrealistic for real-world scenarios.
- Controller comparison: The mode-compatible controller τ E cannot fully compensate isotropic system damping because its structure differs from the damping, requiring corrective mode-selection torques.The mismatch prevents complete damping compensation while operating on the mode.
- Practical implication: The practical recommendation is to use classical damping compensation with the mode-compatible controller τ E for energy regulation and swing-up on hardware.The recommendation follows the stated limitation of the negative-damping controller's friction match.
C. Sensitivity Analysis
The sensitivity analysis tests how parasitic elastic forces affect trajectories, frequency, and control effort. Moderate potential inaccuracies preserve trajectory shape approximately, but larger deviations increase disturbances, corrective action, and reduce energy efficiency.
- Trajectory sensitivity: For ϵ ≤ 0.1, parasitic potentials change the configuration-space trajectory little, although the oscillation frequency is slightly altered.Larger settings, ϵ = 0.5 and ϵ = 1.0, produce larger disturbances.
- Trajectory sensitivity: At ϵ = 1, parasitic-potential torques are on average as large as torques from the optimized potential Ue.The parasitic terms are therefore a substantial implementation error rather than a small perturbation.
- Control sensitivity: Increasing ϵ requires increased control action, especially from the energy-regulation controller, which periodically removes and adds energy.The controller computes energy using Ue even though the implemented potential is disturbed.
- Energy efficiency: The energy-flux comparison treats elastic-element exchange as recoverable for ideal springs, while actuator energy flux is generally lost because electrical drives usually do not recover energy.The actuator-related fluxes should therefore be minimized for energy-efficient operation.
- Practical boundary: The system tolerates moderate potential inaccuracies with only slight trajectory-shape changes, but deviations from the optimized potential reduce energy efficiency.Timing and oscillation frequency can still change even when the trajectory shape remains similar.
VI. CONCLUSION
The paper develops a potential that densely equips configuration space with strict normal modes by harmonizing the potential with the system’s inertial geometry. It also presents mode stabilization and energy regulation, while leaving direct mechanical implementation of the optimized potential for future work.
- Conclusion: The method creates a potential whose strict normal modes densely cover configuration space and describe natural periodic oscillations.The design aligns the potential field with the system’s natural Riemannian metric.
- Conclusion: Direct implementation of the optimized potential using nonlinear elastic elements was outside this paper’s scope and is reserved for future work.Sensitivity analysis indicates that certain discrepancies between implemented and optimal potentials can be tolerated.
- Conclusion: The proposed controllers stabilize a desired strict mode, regulate swing-up energy, and reinject energy lost through damping or interactions.Simulations and experiments showed control torques were very low compared with torques generated by the elastic elements.
APPENDIX A
Appendix A contains Table III, titled “Parameters of the Double Pendulum.”
- APPENDIX A: Table III is titled “Parameters of the Double Pendulum.”
- APPENDIX A: The table concerns parameters of a double-pendulum system.
- APPENDIX A: No parameter values are stated in the supplied table passage.
APPENDIX B MODE IDENTIFICATION COORDINATE
The appendix constructs a smooth mode-identification coordinate from geodesic samples, handles the mode’s circular representation, and uses it for damping and mode-error calculations.
- Mode identification: Strict modes are identified by the velocity direction at equilibrium, with opposite velocity vectors assigned to the same mode.The labeling treats v and −v as equivalent.
- Mode identification: System geodesics are sampled and assigned mode labels to create training tuples for learning the mode-identification function.Each geodesic contributes sampled configuration-label pairs.
- Mode identification: Because the raw mode label lies on S1 and is undefined at equilibrium, the method replaces it with a smooth function Λ(q).A neural network with tanh activations, two hidden layers, and two outputs approximates Λ(q).
- Mode error: The circular mode coordinate requires special handling when computing its difference from a desired mode.The method represents the coordinate as a rotation matrix and orthonormalizes it using singular value decomposition.
- Damping design: The damping design first transforms the manipulator’s mass matrix into the mode-coordinate representation before computing the damping coefficient.