⚡ electrical engineering

A Leader-Follower Approach for The Attitude Synchronization of Multiple Rigid Body Systems on $SO(3)$

This paper proposes a distributed control strategy that achieves almost global asymptotic attitude synchronization for a group of heterogeneous rigid body systems on $SO(3)$ under an undirected, connected, acyclic graph, where the leader's constant desired orientation is known only to a single agent.

Original authors: Yiliang Li, Jun-e Feng, Abdelhamid Tayebi

Published 2026-01-28

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CC BY 4.0

Original authors: Yiliang Li, Jun-e Feng, Abdelhamid Tayebi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine a flock of drones, a fleet of satellites, or a team of underwater robots. Each one is a spinning object (a "rigid body") trying to orient itself in 3D space. The goal of this paper is to get all of them to face the exact same direction at the same time, even if they are all different shapes and sizes.

Here is the breakdown of how the authors solved this problem, using simple analogies:

The Big Challenge: The "Spinning Top" Problem

In normal math (like drawing on a flat piece of paper), getting things to agree is easy. You just tell everyone to move toward a specific point. But these robots move on a sphere (like the surface of a globe), and their orientation is described by a complex mathematical shape called SO(3).

Think of it like this: If you try to draw a map of the entire Earth on a flat piece of paper without tearing or stretching it, you can't. Similarly, you can't use standard "flat" math to perfectly control these spinning robots. If you try to force them to agree using old methods, they might get stuck in a "deadlock" or spin forever without ever settling down. The paper acknowledges that the shape of the space they live in makes this much harder than it looks.

The Setup: A Leader and a Tree

The authors propose a Leader-Follower strategy.

The Leader: Imagine one specific robot (let's call it Robot #1) is the only one holding a map. It knows the "Target Direction" (the desired orientation).
The Followers: The other robots don't know the target. They only know what their immediate neighbors are doing.
The Connection: The robots are connected like a tree (in the botanical sense, not a computer tree). There are no loops. If Robot A talks to Robot B, and Robot B talks to Robot C, Robot A doesn't talk directly to C. It's a single, branching path with no circles. This is crucial because it prevents the group from getting confused by conflicting instructions.

The Solution: A "Local Whisper" Strategy

The authors designed a new control law (a set of instructions for the robots' motors). Here is how it works:

Local Whispering: Robot #1 whispers the target direction to its neighbors. Those neighbors whisper to their neighbors, and so on. No robot needs to know the whole picture; they only need to know what the person next to them is doing.
The "Rubber Band" Effect: The math acts like invisible rubber bands. If Robot #3 is facing slightly differently than Robot #2, the "rubber band" between them pulls them toward alignment.
The "Brakes": The system also includes a damping mechanism (like brakes on a car) to stop them from overshooting and spinning out of control.

The Result: "Almost" Perfect Agreement

The paper proves that this strategy works with "Almost Global Asymptotic Stability."

What does that mean? Imagine you drop a ball into a bowl. It will almost always roll to the bottom (the target).
The "Almost": There are a few very specific, weird starting positions (like balancing a pencil perfectly on its tip) where the robots might get stuck in a wrong position. However, the paper proves these "bad spots" are so rare (mathematically speaking, they have "zero measure") that if you start the robots in any random position, they will almost certainly find the correct direction and stop spinning.
The Outcome: Eventually, every single robot faces the exact same direction as the Leader, and they all stop spinning (their speed becomes zero).

The Simulation

To prove this works, the authors ran a computer simulation with 7 different robots.

They had different weights and shapes (heterogeneous).
They started facing random directions and spinning at different speeds.
Only Robot #1 knew the target.
The Result: The simulation showed that within a short time, all 7 robots aligned perfectly with Robot #1 and stopped spinning, exactly as the math predicted.

Summary

In short, this paper provides a new "recipe" for getting a group of different, spinning robots to agree on a direction. It works even if only one robot knows the goal, and it works on the complex, curved geometry of 3D space where older methods fail. The only catch is that the robots must be connected in a "tree" shape (no loops), and while it works for almost every starting position, there are a few mathematically rare cases where it might get stuck.

Technical Summary: A Leader-Follower Approach for Attitude Synchronization of Multiple Rigid Body Systems on SO(3)

Problem Formulation
This paper addresses the leader-follower attitude synchronization problem for a group of $N$ heterogeneous rigid body systems evolving on the special orthogonal group $SO(3)$. The objective is to design a decentralized control torque $\tau_i$ for each agent such that all rigid bodies synchronize their attitudes to a constant desired orientation $R_0$ and their angular velocities converge to zero. A critical constraint of the problem is that the desired attitude $R_0$ is known only to a single agent (the leader, indexed as 1), while the remaining agents (followers) must achieve synchronization using only local information exchange. The communication topology is modeled as an undirected, connected, and acyclic graph (a tree). The systems are heterogeneous, meaning they possess distinct inertia matrices $J_i$ .

Methodology
The proposed solution utilizes a distributed control law that operates directly on the rotation matrix representation of attitude, avoiding parameterizations like Euler angles, Modified Rodrigues Parameters (MRPs), or unit quaternions, which suffer from singularities or non-uniqueness.

Graph-Theoretic Setup: The authors define a virtual orientation for the edges of the undirected tree graph to facilitate stability analysis. For any edge $k$ connecting agents $i$ and $j$ , a relative attitude $\bar{R}_k = R_j^\top R_i$ is defined.
Control Law Design: The control torque $\tau_i$ for each agent $i$ is constructed using local relative attitude errors and angular velocity damping:
$\tau_i = -k_{\bar{R}_0}\psi(A_{1,0}R_0^\top R_1) - k_{\bar{R}}\sum_{j \in \mathcal{N}(i)}\psi(A_{i,j}R_j^\top R_i) - k_\omega \omega_i$
Here, $\psi(\cdot)$ is a map from $so(3)$ to $\mathbb{R}^3$ derived from the projection onto the Lie algebra, $A_{i,j}$ are positive definite matrices with distinct eigenvalues, and $k$ terms are positive gains. The term involving $R_0$ is active only for the leader (agent 1).
Stability Analysis:
- Lyapunov Function: A potential function $V$ is constructed combining the trace of the attitude error matrices and the kinetic energy of the system.
- LaSalle's Invariance Principle: The time derivative of $V$ is shown to be non-positive ( $\dot{V} = -k_\omega \|\omega\|^2 \leq 0$ ). This establishes that trajectories converge to the largest invariant set where $\dot{V}=0$ , implying $\omega = 0$ .
- Equilibrium Characterization: The analysis identifies two sets of equilibria: a desired set $\Xi_d$ where all attitudes align with $R_0$ , and an undesired set $\Xi_u$ containing saddle points where attitudes align with $R_0$ up to a rotation of $\pi$ around specific axes.
- Almost Global Stability: Using Chetaev's theorem, the undesired equilibria are proven to be unstable. Furthermore, by linearizing the system around these unstable points and demonstrating that the Jacobian matrix possesses no eigenvalues on the imaginary axis, the authors invoke the stable manifold theorem to prove that the stable manifold of the undesired equilibria has zero Lebesgue measure. Consequently, the desired equilibrium is almost globally asymptotically stable.

Key Contributions

Direct SO(3) Formulation: The paper proposes a control strategy that avoids the singularities of local parameterizations (like MRPs) and the non-uniqueness of global parameterizations (like unit quaternions) by working directly with rotation matrices on $SO(3)$.
Heterogeneous Systems: The control law is designed for heterogeneous rigid bodies with distinct inertia matrices, a condition often relaxed in prior works.
Tree Topology: The method is specifically tailored for undirected, connected, acyclic graphs (trees), leveraging the specific structure of the incidence matrix to prove stability.
Leader-Follower with Local Knowledge: The approach successfully achieves synchronization to a reference known only to a single agent without requiring global knowledge or observers for the reference attitude.

Results
Theoretical results are validated through numerical simulations involving a network of 7 heterogeneous rigid bodies.

Convergence: The simulation results demonstrate that the relative attitudes between neighboring agents converge to the identity matrix, and the relative attitudes between all agents and the leader converge to the identity.
Velocity Damping: All angular velocities converge to zero.
Performance: The system successfully synchronizes to the constant desired attitude $R_0$ despite heterogeneous inertia matrices and diverse initial conditions, confirming the almost global asymptotic stability guarantees.

Significance and Claims
The paper claims to provide a rigorous solution to the leader-follower attitude synchronization problem on $SO(3)$ with almost global asymptotic stability guarantees. The authors emphasize that existing literature often relies on local parameterizations (limiting the domain of attraction) or unit quaternions (introducing ambiguity), or provides only local convergence results. By contrast, this work offers a control strategy that respects the intrinsic geometry of $SO(3)$ and ensures convergence from almost all initial conditions.

The authors note that the current work is limited to constant desired attitudes and acyclic (tree) communication graphs. They identify the extension to time-varying desired attitudes, communication delays, and more general graph topologies as challenging open problems for future research.