# On the Configuration Space of planar closed Kinematic Chains

## TL;DR

This paper introduces a new parametrization of the configuration space of planar closed kinematic chains using diagonal lengths, simplifying analysis and potential applications in motion planning and singularity detection.

## Contribution

It provides explicit formulas for joint angles guaranteeing closedness based on diagonal lengths, revealing a simple domain structure for these parameters.

## Key findings

- Diagonal lengths form a simple-structured domain.
- Explicit formulas for joint angles are derived.
- Numerical examples demonstrate practical applicability.

## Abstract

A planar kinematic chain consists of $n$ links connected by joints. In this work we investigate the space of configurations, described in terms of joint angles, that guarantee that the kinematic chain is closed. We give explicit formulas expressing the joint angles that guarantee closedness by a new set of parameters, the diagonal lengths (the distances of the joints from the origin) of the closed kinematic chain. Moreover, it turns out that these diagonals are contained in a domain that possesses a simple structure. We expect that the new insight can be applied for several issues such as motion planning for closed kinematic chains or singularity analysis of their configuration spaces. In order to demonstrate practicality of the new method we present numerical examples.