# Articulated Rigid-Body Systems

• Murilo G. Coutinho

## Abstract

The dynamic simulation of rigid-body systems covered in the previous chapter can be further extended to the case of articulated rigid-body systems, where bodies are attached to each other using joints. There are several types of joints that can be used to connect bodies, and they differ from each other by the degree of freedom of the relative motion allowed. Several methods have been proposed to address the dynamics of articulated systems, and most of them fall into one of the following two categories. In the first category, the dynamic equations describing the system’s motion are formulated using a reduced set of variables. This is the so called reduced coordinate formulation. The reduced set of variables, also known as generalized coordinates, is obtained by removing all degrees of freedom constrained by the joints. The result is a set of parameterized coordinates that fully describes the motion of the entire articulated system while assuring the joint constraints. In the second category, additional constraint forces are introduced in the system to assure the joint constraints throughout motion. This method is known as the Lagrangian formulation. The idea is to formulate equations relating the constraint forces (also referred to as the Lagrangian multipliers) with the dynamic state of the articulated system. In the case of articulated rigid-body systems, the formulation consists of building and solving a linear system (often sparse) for the joint forces. Sparsity can then be used advantageously to derive O(n) algorithms, where n is the total number of articulated bodies being considered.

## Keywords

Contact Force Hierarchical Tree Spherical Joint Joint Force Joint Contact
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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## Authors and Affiliations

• Murilo G. Coutinho
• 1
1. 1.Los AngelesUSA