💻 computer science

Rigidity-Aware Formation Tracking under Sensing Range Constraints via Single Control Barrier Function Constraint

This paper proposes a distributed control framework that integrates formation tracking and rigidity maintenance for heterogeneous multi-robot systems with nonlinear dynamics under sensing range constraints by utilizing a single Control Barrier Function constraint within a quadratic optimization program.

Original authors: S. Saharsh, Pushpak Jagtap

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: S. Saharsh, Pushpak Jagtap

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

In the world of robotics, the ability for a group of machines to move together as a single, coordinated unit is a powerful tool. Whether navigating the cluttered aisles of a warehouse, exploring the dark depths of an underwater cave, or mapping a disaster zone, a team of robots can accomplish tasks that a single machine could never handle alone. However, keeping these teams together is not as simple as telling them to stay close. If the robots lose their sense of direction relative to one another, the group can fracture, with some members drifting off course while others remain behind. This is particularly difficult in environments where global positioning systems, like GPS, do not work. In these places, robots must rely entirely on what they can see of their immediate neighbors to figure out where they are.

The challenge lies in a delicate balance between two goals. First, the robots must track a specific path and maintain a precise shape, like a triangle or a line, to do their job. Second, they must ensure that the connections between them never break. If a robot moves too far from its neighbor, the communication link snaps, and the group loses its structural integrity. Without these connections, the robots can no longer agree on their formation, and the entire mission can fail. Researchers have long known that simply telling robots to follow a path is not enough; they must also actively manage the links that hold the group together, especially when their sensors have a limited range and cannot see far into the distance.

A team of researchers at the Indian Institute of Science has developed a new way to solve this problem, allowing a diverse group of robots to move together while strictly preserving the connections that keep them safe. Their work focuses on a scenario where robots have different physical capabilities and can only "see" other robots within a specific distance. In such a setting, a robot might accidentally drift out of range of a neighbor, causing the group to lose its rigid structure. The researchers created a control system that acts like a vigilant guardian, constantly checking the distance between neighbors and the angles at which they view one another. If a robot begins to drift too far or if the group starts to flatten into a straight line—a configuration that makes the formation unstable—the system automatically adjusts the robot's speed and direction to pull it back into a safe zone.

The core of this solution is a mathematical framework that combines the goal of reaching a destination with the goal of maintaining a safe structure. Instead of treating these as separate tasks, the researchers unified them into a single decision-making process. At every moment, each robot calculates the best possible move that satisfies both requirements. It asks itself: "How can I move toward my target while ensuring I stay close enough to my neighbors to keep the group rigid?" This calculation happens in real-time, allowing the robots to react instantly to disturbances or changes in the environment. The system is designed to be distributed, meaning no single robot acts as a central commander. Instead, every robot makes its own decisions based only on the information it can gather from its immediate neighbors, making the entire group robust and scalable.

To test their idea, the researchers ran detailed computer simulations involving a team of eight robots in a two-dimensional space. The team included two leaders that followed a predetermined path and six followers that had to maintain a specific shape relative to the leaders. The robots were not all the same; some moved like simple points, while others had more complex mechanics, similar to cars with steering wheels or differential-drive robots. The simulation introduced a significant challenge: a constant disturbance that pushed the robots off course. In one scenario, without the new safety system, a robot drifted out of range, breaking the connection with its neighbor. As a result, the group lost its rigidity, and the formation collapsed, with robots failing to reach their intended positions.

However, when the researchers applied their new control framework, the outcome was different. Even under the same disturbing forces, the system detected the impending break in the connection and adjusted the robots' movements to preserve the critical links. The group remained rigid, and the formation stayed intact, allowing the robots to successfully track the desired path. The simulations showed that the robots could maintain their shape with high precision, keeping errors in their positions and angles extremely small. The researchers also demonstrated that their method works even when the robots have different types of motion, proving that the system is flexible enough to handle a heterogeneous team.

This approach offers a distinct advantage over previous methods. Older strategies often required robots to perform complex, global calculations to ensure the group stayed connected, which could be slow and prone to errors if the network changed rapidly. Other methods focused only on keeping the robots connected, without guaranteeing that the specific shape of the formation was maintained. The new framework bridges this gap by ensuring that the group remains rigid—a state where the formation is uniquely defined and cannot be distorted—while simultaneously tracking the desired trajectory. By focusing on specific, critical connections between neighbors, the system avoids the need for heavy computational overhead, making it suitable for real-world applications where processing power and energy are limited.

The results of these simulations suggest that this method could be a significant step forward for autonomous teams operating in challenging environments. While the current work is based on computer models, the principles are grounded in the physical laws of motion and the geometry of the robots' interactions. The researchers plan to extend this work to three-dimensional spaces and to test the system on actual flying robots in the near future. For now, the study provides a clear proof of concept: by treating the maintenance of connections as an active, integral part of the movement strategy, a team of robots can stay together and get the job done, even when the world around them tries to pull them apart.

Technical Summary: Rigidity-Aware Formation Tracking under Sensing Range Constraints via Single Control Barrier Function Constraint

1. Problem Statement

The paper addresses the challenge of formation tracking and rigidity maintenance in heterogeneous multi-robot systems (MRS) with nonlinear dynamics operating under limited sensing ranges.

In GPS-denied environments (e.g., indoor, underwater, deep space), agents rely on local relative measurements. While bearing-based formation control is effective for minimal sensing requirements, it typically assumes a rigid communication graph is maintained throughout the mission. However, in practical scenarios where agents have limited sensing ranges ( $\lambda$ ), communication links can break or form dynamically, potentially causing the loss of bearing rigidity. Since rigidity is a prerequisite for establishing and preserving a unique formation, the loss of rigidity leads to formation failure even if individual tracking objectives are met.

The core problem (Problem 1) is to design a distributed control strategy for heterogeneous agents that:

Achieves a unique rigid formation defined by desired leader trajectories and bearing vectors.
Preserves bearing rigidity ( $\det(B_{ff}(t)) \neq 0$ ) at all times, despite limited sensing ranges that may cause links to drop.
Operates using only local relative information from neighbors.

2. Methodology

The proposed solution integrates formation tracking and rigidity maintenance into a single Control Barrier Function (CBF) constraint within a Quadratic Programming (QP) framework.

2.1 System and Graph Model

Dynamics: Agents are modeled as planar robots with nonlinear dynamics ( $\dot{p}_i = v_i, \dot{v}_i = f_{v_i}(v_i) + h_{v_i}(v_i)u_i$ ), covering double-integrators, differential-drive, and Ackermann-drive vehicles.
Graph: The communication topology $G(t)$ is undirected and time-varying, determined by the sensing range $\lambda$ .
Framework: A leader-follower structure is used, where leaders ( $V_L$ ) have global position access, and followers ( $V_F$ ) rely on local bearings.

2.2 Rigidity Preservation Strategy

Unlike prior works that assume rigidity is maintained, this paper treats rigidity preservation as an explicit control objective.

Reference Assignment: Based on a Henneberg-type vertex-addition sequence, each follower is assigned a reference pair of neighbors ( $j_1, j_2$ ) with non-parallel desired bearings.
Criticality: The paper establishes that maintaining the non-collinearity and sensing-range constraints of these specific reference pairs is sufficient to ensure the global rigidity of the entire graph.
Dual Targets: The controller switches between two targets based on the state of the reference links:
1. Bearing-Guided Target ( $p^b_{iT}$ ): Used when rigidity is secure. This target aligns the agent with the desired formation shape.
2. Rigidity-Critical Target ( $p^r_{iT}$ ): Activated when a reference link approaches the sensing limit or when the angle between references approaches collinearity. This target pulls the agent to a position that maximizes the safety margin for the critical links, ensuring they remain within range and non-collinear.

2.3 Control Framework: Reachability CBF

The authors employ a Reachability Control Barrier Function (reach-CBF) to unify these objectives.

Safe Set & Target Set: A safe set $X$ is defined where the agent is within the sensing range and non-collinear. A target set $R$ is defined for the desired alignment.
Single Constraint: Instead of separate constraints for tracking and safety, a single reach-CBF constraint is formulated to ensure the system reaches the target set $R$ within a finite time $\tau$ while remaining in the forward-invariant safe set $X$ .
QP Formulation: The control input $u_i$ is computed by solving a QP that minimizes the deviation from a nominal controller while satisfying the reach-CBF constraint:
$u_i = \arg \min_q \frac{1}{2} \|q - u_{nom}\|^2 \quad \text{s.t.} \quad K_f + K_h^\top q \geq \frac{\mu}{\tau}$
Stability: The closed-loop system is analyzed as a switched nonlinear system. The authors prove that under specific gain conditions and a minimum dwell-time (guaranteed by sensing and angular margins), the system is uniformly ultimately bounded, and the unique formation is achieved.

3. Key Contributions

Unified Framework: A novel control framework that unifies formation tracking and rigidity maintenance into a single reach-CBF constraint, eliminating the need for complex, multi-layered constraint handling.
Local Rigidity Maintenance: The method ensures global rigidity preservation using only local relative information by identifying and maintaining "rigidity-critical" reference pairs, removing the assumption of permanent rigidity.
Heterogeneous Support: The approach is applicable to heterogeneous systems with nonlinear dynamics (e.g., double-integrators and differential-drive robots).
Distributed Implementation: The controller requires only local neighbor data, making it scalable and suitable for decentralized operations.

4. Results

The approach was validated through simulation with 8 agents (2 leaders, 6 followers) in 2D space, including double-integrator and differential-drive models. Three scenarios were tested:

Case (i) Nominal Tracking: Agents successfully tracked a moving formation with bearing errors converging to $\approx 10^{-2}$ .
Case (ii) Disturbance without Rigidity Maintenance: Under constant disturbances applied from 0–7 s, perturbations $[-0.05, 0.04]$ and $[-0.01, 0.04]$ were added to the positions of Follower 1 (Agent 3) and Follower 2 (Agent 4), respectively. Consequently, Follower 5 (Agent 7) lost connectivity with Follower 2 (Agent 4), leading to a loss of network rigidity as $\det(B_{ff}) \to 0$ (i.e., $\log_{10}(| \det(B_{ff})|) \approx -12$ ). Although the bearing error remained small, the mean formation error increased significantly due to the loss of rigidity, and Follower 7 deviated from the desired formation.
Case (iii) Disturbance with Rigidity Maintenance: Under the same disturbances, a rigidity-critical link between Follower 4 and Follower 7 was preserved. As a result, the formation remained rigid and was successfully tracked.

Comparison: The paper contrasts its method with existing strategies that rely on global spectral estimation (e.g., rigidity eigenvalues via power iteration) or simple connectivity maintenance (Fiedler eigenvalue). The proposed method avoids global computation and ensures the stronger condition of rigidity rather than just connectivity.

5. Significance and Claims

The paper claims that rigidity preservation is critical for accurate formation tracking, particularly in the presence of disturbances and limited sensing ranges. Without explicit rigidity maintenance, formation tracking controllers can fail even if local bearing errors are minimized.

The significance of this work lies in:

Bridging the Gap: It addresses the gap between theoretical formation control (which assumes rigid graphs) and practical constraints (limited sensing).
Efficiency: By using a single CBF constraint, it simplifies the control architecture compared to methods requiring multiple constraints or global estimators.
Robustness: It provides a provably stable method to maintain the structural integrity of the formation in dynamic, constrained environments.

The authors note that future work will extend this framework to 3D, incorporate obstacle avoidance and input bounds, and validate the approach experimentally on aerial vehicles.