Non-linear dynamics of multibody systems: a system-based approach

TL;DR

This work presents a nonlinear TITOP block‑diagram framework to represent the equations of motion for rigid multibody systems, enabling both forward and inverse dynamics in a compact, modular form. By extending TITOP to include all nonlinear terms and gravity, the authors develop causal, open/closed‑chain models for welding, revolute, and prismatic joints, plus a loop‑closure construct. The methodology is illustrated and validated on a 9‑DOF balloon‑flight chain, with results matching a Simscape/Multibody model, confirming accuracy and the benefit of modular block representations for rapid prototyping and analysis. The approach sets the stage for LPV/Robust analyses and future work toward nonlinear TITOP for flexible bodies and dedicated solvers for algebraic loops, enabling broader applicability in mechanical design and control.

Abstract

This paper presents causal block-diagram models to represent the equations of motion of multi-body systems in a very compact and simple closed form. Both the forward dynamics (from the forces and torques imposed at the various degrees-of-freedom to the motions of these degrees-of-freedom) or the inverse dynamics (from the motions imposed at the degrees-of-freedom to the resulting forces and torques) can be considered and described by a block diagram model. This work extends the Two-Input Two-Output Port (TITOP) theory by including all non-linear terms and uniform or gravitational acceleration fields. Connection among different blocks is possible through the definition of the motion vector. The model of a system composed of a floating base, rigid bodies, revolute and prismatic joints, working under gravity is developed to illustrate the methodology. The proposed model is validated by simulation and cross-checking with a model built using an alternative modeling tool on a scenario where the nonlinear terms are determining.

Figures (17)

• Figure 1: Block diagram representation of the direct dynamic model of the body $\mathcal{B}$ at the point $P$ and projected in $\mathcal{R}_b$ .
• Figure 2: Block diagram representation of the inverse dynamic model $\left[\mathbf{G}^\mathcal{B}_P(\mathrm s)\right]_{\mathcal{R}_b}$ of the body $\mathcal{B}$ at the point $P$ and projected in $\mathcal{R}_b$ .
• Figure 3: Block diagram representation of $\left[\mathbf{G}^\mathcal{B}_{P,C}(\mathrm s)\right]_{\mathcal{R}_b}$: the non-linear $36\times 12$ model of a rigid body $\mathcal{B}$ at points $P$ and $C$.
• Figure 4: Block diagram model of $\left[\mathbf{R}^\mathcal{A}_{P,C}\right]_{\mathcal{R}_a}$: non-linear TITOP model of a rigid body $\mathcal{A}$ at points $P$ and $C$.
• Figure 5: Two rigid bodies connected at point $P$.

Theorems & Definitions (1)

• Definition 1