[Paper Review] On the simulation of multibody systems with holonomic constraints

Jukka Tuomela, Teijo Arponen+1 more|ArXiv.org|Nov 3, 2006

Dynamics and Control of Mechanical Systems14 references3 citations

TL;DR

This paper proposes a jet space-based Lagrangian computational model for simulating multibody systems with holonomic constraints, avoiding drift-off and spurious oscillations through orthogonal projection onto constraint manifolds. By embedding dynamics in jet spaces and enforcing constraints geometrically, the method enables long-term physically accurate simulations using explicit Runge-Kutta time integration with adaptive step size control.

ABSTRACT

We use Lagrangian formalism and jet spaces to derive a computational model to simulate multibody dynamics with holonomic constraints. Our approach avoids the traditional problems of drift-off and spurious oscillations. Hence even long simulations remain physically relevant. We illustrate our method with several numerical examples.

Motivation & Objective

• Address the persistent challenge of constraint drift and spurious oscillations in long-term simulations of multibody systems with holonomic constraints.
• Develop a geometric computational model rooted in differential geometry and Lagrangian mechanics to ensure physical consistency over extended simulations.
• Integrate conservation laws (e.g., energy invariance) naturally into the constraint manifold structure to preserve physical invariants.
• Enable efficient and stable numerical integration using explicit Runge-Kutta methods by eliminating stiffness introduced by traditional stabilization techniques.
• Demonstrate the method’s robustness and efficiency on benchmark multibody systems, including planar quadrangles and crank mechanisms.

Proposed method

• Formulate the dynamics using Lagrangian formalism within the framework of jet spaces, enabling a geometric treatment of higher-order derivatives and constraints.
• Define the configuration manifold as a submanifold of a jet space, restricting dynamics to satisfy holonomic constraints and invariants via implicit manifold embedding.
• Apply orthogonal projection to enforce constraints at each integration step, preventing drift and maintaining physical consistency.
• Use a quasi-orthogonal projection strategy to accelerate convergence during constraint projection, reducing computational cost compared to standard orthogonal projection.
• Implement adaptive step size control based on Dormand-Prince Runge-Kutta methods with error tolerances (Atol=Rtol=10⁻⁷) to balance accuracy and efficiency.
• Leverage automatic differentiation for efficient computation of gradients and higher-order derivatives of constraint and invariant functions.

Experimental results

Research questions

• RQ1 Can a geometric, jet space-based formulation of multibody dynamics eliminate constraint drift and spurious oscillations in long simulations?
• RQ2 How does the computational cost of orthogonal projection compare to traditional stabilization methods in terms of CPU time and convergence?
• RQ3 To what extent can conservation laws (e.g., energy invariance) be naturally embedded into the constraint manifold without compromising numerical efficiency?
• RQ4 How does the quasi-orthogonal projection strategy improve performance compared to standard orthogonal projection in terms of speed and iteration count?
• RQ5 Can explicit Runge-Kutta methods be effectively used in constrained multibody simulations when projection-based constraint enforcement is applied?

Key findings

• The method successfully prevents drift-off and spurious oscillations, enabling physically consistent long-term simulations of multibody systems with holonomic constraints.
• For the crank mechanism, the quasi-orthogonal projection reduced CPU time for projection by 75% (3198.90 vs. 757.65 seconds) compared to standard orthogonal projection.
• The number of function evaluations for constraint gradients (dg) was significantly reduced with quasi-orthogonal projection (45,054 vs. 87,018), indicating improved efficiency.
• In the crank mechanism simulation, the average number of Newton iterations per step was 0.325 with quasi-orthogonal projection, compared to 1.42 with standard projection, indicating faster convergence.
• The inclusion of energy conservation as an invariant had minimal impact on computational cost but significantly improved energy preservation, as shown in Figure 6.8.
• The total simulation time was reduced by over 90% when using quasi-orthogonal projection in the crank mechanism case (848.67 vs. 3292.00 seconds), demonstrating substantial performance gains.