SO(3) attitude controllers and the alignment of robots with non-constant 3D vector fields

TL;DR

This work develops a geometric-control framework on $ ext{SO}(3)$ to align 3D robots with non-constant vector fields by tracking a time-varying attitude target $R_a(t)$. The core method uses a logarithmic attitude error and a feedback law $\boldsymbol{\rm \Omega}(R_e)=-k_w\log(R_e)+\mathrm{Ad}_{R_e^T}(\Omega_a)$, yielding local exponential stability via a Lyapunov function $V(R_e)=\tfrac12\|\log(R_e)\|^2$. It extends to practical scenarios, addressing unknown time variation in the vector field and providing inter-robot distance bounds under exponential tracking. The approach leverages Lie-group tools, exponential/log maps, and Adjoint invariance to enable robust, model-based heading alignment in 3D and 2D unicycle cases, with numerical simulations confirming fast, exponential convergence. The results offer a principled route to precise attitude control in robotics where heading must follow spatial vector fields, with potential for broader applications in formation control and source-seeking tasks.

Abstract

This technical note aims to introduce geometric controllers to roboticists for aligning \emph{3D robots} with non-constant 3D vector fields. This alignment entails the control of the robot's 3D attitude. We derive with excessive detail all the calculations needed for the analysis and implementation of the controllers.

Figures (7)

• Figure 1: This figure illustrates a 3D unicycle robot moving at a linear velocity $v$ and rotating at an angular velocity $\omega$. These velocities are observed from the inertial frame $\mathcal{F}_E$ but represented in the non-inertial frame $\mathcal{F}_B$, which is fixed at the body of the robot. $\{x_e, y_e, z_e\}$ and $\{x, y, z\}$ are the vector bases of $\mathcal{F}_E$ and $\mathcal{F}_B$, respectively.
• Figure 2: This figure illustrates, for two different manifolds, a pair of geodesics ($\gamma$ in blue color and $\gamma_g$ in red color) that connects $q_a, q_b \in \mathbb{T}^2$ (left) and $x,y \in S^2$ (right). Note that $\gamma_g$ is the length-minimizing geodesic.
• Figure 3: The $S^1$ manifold is a unit circle (blue) in $\mathbb{C}$, which contains every complex number $x$ such that $x^*x=1$. Its Lie algebra $T_1S^1$ is the line of imaginary numbers $i\mathbb{R}$ (red), and any tangent space $T_xS^1$ is isomorphic to the line $\mathbb{R}$ (red). Tangent vectors (red segments) wrap the manifold creating $\gamma_g$. The $\exp(\cdot)$ and $\log(\cdot)$ maps (arrows) wrap and unwrap elements of $\mathbb{R}$ to/from elements of $S^1$. This figure has been adapted from the reference so3_catalanes.
• Figure 4: A unicycle robot moving at the constant speed $v = 0.5$space unit/time unit along its $x$ body axis aligns with a time-varying attitude target $R_a(t) = [x_a \; y_a(t) \; z_a(t)]$, with $x_a,y_a,z_a \in S^2$ as in \ref{['eq: R']}. The time variation of $y(t)$ and $z(t)$ are given by the body-fixed angular velocity vector $\omega^k = [\pi,0,0]$ in radians/time unit, which is known. The control law for alignment is given by \ref{['eq: omega_control']}, with $k_\omega = 1$. Note that, at $t=8$, $x_a$ instantly changes from $[-1,1,1]$ to $[1,0,0]$. The left plot shows the time evolution of the attitude error $\mu_{R_e}$. The right plot depicts the trajectory of the unicycle along with its attitude vectors $x$, $y$ and $z$ represented in red, green and blue, respectively.
• Figure 5: A unicycle robot moving at the constant speed $v = 0.5$space unit/time unit along its $x$ body axis aligns with a time-varying attitude target $R_a(t) = [x_a(t) \; y_a(t) \; z_a(t)]$, with $x_a,y_a,z_a \in S^2$ as in \ref{['eq: R']}. The time variation of $R_a(t)$ is given by the earth-fixed angular velocity vector $w(t) = R_a(t)^\top w^k + w^u$, where $\omega^k = [\pi,0,0]$ in radians/time unit and $\omega^u = [0,0,-w_d]$ with $w_d = \pi/14$radians/time unit. The known variables are $w^k$ and $\|\omega^u\| = w_d$, so the control law for alignment is given by \ref{['eq: omega_control_know']}, with $k_\omega = \sqrt{2}w_d/\mu_{R_e}^*$ and $\mu_{R_e}^* = \delta^* = 0.4$. The left plot shows the time evolution of the attitude error $\mu_{R_e}$ and $\delta$. The right plot depicts the trajectory of the unicycle along with its attitude vectors $x$, $y$ and $z$ represented in red, green and blue, respectively.

Theorems & Definitions (21)

• Remark 1
• Example 1
• Definition 1: Length-minimizing geodesic
• Lemma 1
• proof
• Example 2
• Lemma 2
• proof
• Definition 2: Local exponential stability in trajectory tracking
• Proposition 1: 3D attitude controller