# Controllability and Accessibility Results for N-Link Horizontal Planar Manipulators with One Unactuated Joint

## TL;DR

This paper investigates the controllability and accessibility of N-link horizontal planar manipulators with one unactuated joint, providing new theoretical results and conditions for small-time local controllability based on system dynamics and actuator configurations.

## Contribution

It introduces new controllability and accessibility results for N-link manipulators with one unactuated joint, extending previous work beyond three-link systems and considering realistic models.

## Key findings

- N-link manipulators with the first joint actuated are STLC at certain equilibrium points.
- The paper derives simple forms for vector fields to identify configurations losing full rank.
- Results are based on Lie brackets and Sussmann's theorem, applicable to various actuator configurations.

## Abstract

This paper presents the accessibility and small-time local controllability (STLC) results for $N$-link horizontal planar manipulators with only one unactuated joint. STLC is important in controls, both for design considerations and because large and swinging maneuvers may be avoided for close reconfiguration if a system is STLC. Despite the fact that controllability of underactuated horizontal planar manipulators has been extensively studied, most work focused only on three-link and global controllability. This paper thus has two contributions: 1) using Lie brackets to study the accessibility and STLC for underactuated two-link manipulators with different actuator configurations, and illustrating the results from a perspective of system dynamics, 2) obtaining the accessibility and STLC results for $N$-link manipulators with one unactuated joint by considering realistic models and different actuator configurations. It is found that an $N$-link ($N\geq 3$) with the first joint actuated is STLC for a subset of equilibrium points based on Sussmann's general theorem for STLC. Moreover, with the dynamics of $N$-link considered in the controllability analysis, it gives relatively simple forms for the nontrivial vector fields, which make it easy to determine at which configurations a model loses full rank condition for accessibility.