# Constrained Attitude Control on SO(3)

$\newcommand{\bracket}[1]{\left[ #1 \right]} \newcommand{\parenth}[1]{\left( #1 \right)} \newcommand{\tr}[1]{\mathrm{tr}\negthickspace\bracket{#1}}$

This is an extension of our previous paper presented at the 2016 American Control Conference. Here the stability proof is improved and complete. Additional numerical experiments and simulations are presented which demonstrate the controller performance.

## Abstract

This paper presents a new geometric adaptive control system with state inequality constraints for the attitude dynamics of a rigid body. The control system is designed such that the desired attitude is asymptotically stabilized, while the controlled attitude trajectory avoids undesired regions defined by an inequality constraint. In addition, we develop an adaptive update law that enables attitude stabilization in the presence of unknown disturbances. The attitude dynamics and the proposed control systems are developed on the special orthogonal group such that singularities and ambiguities of other attitude parameterizations, such as Euler angles and quaternions are completely avoided. The effectiveness of the proposed control system is demonstrated through numerical simulations and experimental results.

## Configuration Error Function

We create a modified configuration error function which utlizes the concept of a barrier function. This error function quantifies the distance between the curent and desired system configuration. The error function, $$\Psi$$ is composed of an attractive term $$A$$ and a repulsive term $$B$$.

\begin{align} \Psi(R, R_d) &= A(R, R_d) B(R) \\ A(R, R_d) &= \frac{1}{2} \tr{G \left( I - R_d^T R\right)} \\ B_i(R) &= 1 - \frac{1}{\alpha_i} \ln \left( - \frac{ r^T R^T v_i - \cos \theta_i}{1 + \cos \theta_i}\right) \end{align}

We can also extend this easily to any arbitrary number of obstacles by slightly modifying our error function:

\begin{align} \Psi = A(R) \bracket{1 + \sum_i C_i(R)} \quad C_i = B(R) - 1 . \end{align}

## BibTeX citation

@Article{kulumani2017a,
author              = {Kulumani, Shankar and Lee, Taeyoung},
title               = {Constrained Geometric Attitude Control on SO(3)},
journal             = {International Journal of Control, Automation, and Systems},
year                = {2017},
volume              = {15},
number              = {6},
month               = dec,
}