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A Geometric Approach to Structure-Preserving Integrators for Mechanical Systems

Viyom Vivek, David Martin de Diego, Ravi N. Banavar

Abstract

We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean) spaces, we include a short interlude reviewing the differential geometric notions required in the sequel. We then introduce retraction maps as intrinsic generalizations of the Riemannian exponential, which induce discretization maps tailored to manifold-valued dynamics. Adopting the Tulczyjew unified viewpoint, mechanical systems are formulated as Lagrangian submanifolds, providing a natural and coordinate-free foundation for the construction of structure-preserving integrators for both Hamiltonian and Lagrangian systems. The framework is specialized to Lie groups, where parallelizability allows for the global trivialization of tangent and cotangent bundles and the systematic derivation of integrators for Euler-Poincare and Lie-Poisson equations. The effectiveness of the proposed approach is illustrated through the rigid body and heavy top, and is further extended to the construction of a geometric integrator for an underactuated mechanical system-a quadrotor-demonstrating the applicability of the framework beyond fully symmetric systems and toward problems relevant in robotics and control.

A Geometric Approach to Structure-Preserving Integrators for Mechanical Systems

Abstract

We develop a geometric framework for the numerical integration of mechanical systems evolving on manifolds. After briefly reviewing classical numerical methods and highlighting their limitations and shortcomings in non-flat (non-Euclidean) spaces, we include a short interlude reviewing the differential geometric notions required in the sequel. We then introduce retraction maps as intrinsic generalizations of the Riemannian exponential, which induce discretization maps tailored to manifold-valued dynamics. Adopting the Tulczyjew unified viewpoint, mechanical systems are formulated as Lagrangian submanifolds, providing a natural and coordinate-free foundation for the construction of structure-preserving integrators for both Hamiltonian and Lagrangian systems. The framework is specialized to Lie groups, where parallelizability allows for the global trivialization of tangent and cotangent bundles and the systematic derivation of integrators for Euler-Poincare and Lie-Poisson equations. The effectiveness of the proposed approach is illustrated through the rigid body and heavy top, and is further extended to the construction of a geometric integrator for an underactuated mechanical system-a quadrotor-demonstrating the applicability of the framework beyond fully symmetric systems and toward problems relevant in robotics and control.

Paper Structure

This paper contains 74 sections, 5 theorems, 153 equations, 12 figures.

Table of Contents

  1. A Brief Introduction to Numerical Integrators
  2. What are numerical integrators and why do we need them?
  3. A quick overview of classical methods and their limitations
  4. Classical Harmonic Oscillator
  5. Explicit Euler method
  6. Implicit Euler method
  7. Symplectic Euler method
  8. Linear Symplectic Structure
  9. Poisson Structure
  10. Kepler Problem
  11. Runge-Kutta methods
  12. Partitioned Runge-Kutta methods
  13. Symplectic Partitioned Runge-Kutta methods
  14. Planar Pendulum
  15. Projection-based methods
  16. ...and 59 more sections

Key Result

Proposition 3.8

A map $\mathcal{R}^L : G \ltimes \mathfrak{g} \to G$ defined as for all $(g,\xi) \in G \times \mathfrak{g}$ is a left-trivialized retraction map on $G$ where $\tau$ is as defined above.

Figures (12)

  • Figure 1: (a) Comparison of various numerical integrators for the classical harmonic oscillator with parameter values (in standard units) $k/m=1$, $h=0.1$, and initial condition $x_0=(1,0)$ for 50 iterations. The exact solution, which traces an elliptical trajectory, is shown in black. (b) Energy versus iteration for various numerical integrators applied to the classical harmonic oscillator. The energy is shown on a logarithmic scale.
  • Figure 2: (a) Comparison of Keplerian orbits computed using the second-order Runge–Kutta (RK2) method and the Störmer–Verlet (SV) scheme, with parameter values (in standard units) $G(m_1+m_2)=1, h =0.01$, and initial conditions $x_0=(1,0,0,0.5)$ over 3000 iterations. The exact orbit is shown for reference. (b) Energy versus iteration for the RK2 and SV methods. (c) Magnitude of the angular momentum versus iteration for the RK2 and SV methods.
  • Figure 3: (a) Comparison of projected and unprojected trajectories for the embedded planar pendulum dynamics integrated using the second-order Runge–Kutta (RK2) scheme, with parameter values (in standard units) $ml^2 =1, mgl =1, h= 0.1$, and initial conditions $(\theta_0,p_0)=(1,0)$, over 1000 iterations. The exact trajectory is shown for reference. (b) Hamiltonian (energy) versus iteration for the RK2 method with and without projection.
  • Figure 4: Retraction map on $M$
  • Figure 5: Retraction maps on $\mathbb{R}^n$ (top left), on a Riemannian manifold $(M,g)$ (top right) and on a Lie group $G$ (bottom)
  • ...and 7 more figures

Theorems & Definitions (42)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Definition 1.6
  • Definition 1.7
  • Definition 1.8
  • Definition 1.9
  • Definition 1.10
  • ...and 32 more