Relative Motion-Based Control

• Relative motion-based control is a framework where control actions stem from state differences between agents, enabling decentralized and scalable coordination.
• It leverages methodologies such as formation control, velocity alignment, and impulsive control, integrating local sensing and recursive estimation for robust performance.
• This approach has been validated in applications like UAV swarming, spacecraft maneuvers, and vehicular systems, ensuring collision avoidance and optimal control outcomes.

Relative motion-based control refers to a class of algorithms and architectures in which control actions are derived from the relative positions, velocities, or other state differences between agents, bodies, or effectors, rather than absolute states. Applications span cooperative UAV navigation, spacecraft rendezvous, multi-agent formation, mobile manipulation, and distributed vehicular systems. The field encompasses both continuous and impulsive controls, deterministic and stochastic settings, and centralized, decentralized, or fully distributed architectures exploiting vision, inertial sensing, or local communication.

1. Fundamental Concepts and Model Structures

Relative motion-based control exploits measurements or estimates of the state difference—position, velocity, orientation—between agents or with respect to targets. The approach is especially effective for scenarios without access to global localization (e.g., GPS-denied environments) or where scalability and robustness to agent loss are critical.

Typical system dynamics considered are:

• Multi-agent systems in Euclidean space, with agent $i$ state equations $\dot{x}_i = v_i$, $\dot{v}_i = u_i$, and control $u_i$ constructed from functions of $\{x_i-x_j,\,v_i-v_j\}$ for $j\ne i$—enabling distributed control based on local measurements (Liu et al., 2019).
• Discrete-time kinematics for agents and targets: $p_i(k+1) = p_i(k) + T u_i(k)$, where $u_i(k)$ is driven by relative positions and velocities (Liu et al., 2023).
• LTV/LTI dynamics in aerospace (CWH, ROE, CR3BP models) for spacecraft relative motion: $\dot{x} = A(t) x + B u$ or nonlinear $f(x,t)$, with feedback or impulsive controls derived from relative state (Jr. et al., 2019, Servadio et al., 2022, Foss et al., 3 Oct 2025).

Relative measurements come from vision systems, ranging sensors, IMU, recursive estimation (RLSE/Kalman), or communication.

2. Synthesis of Relative Motion Control Laws

Distributed Flocking and Formation

Relative-motion control laws are typified by the sum of terms promoting

• Velocity alignment: consensus on velocities via terms of the form $\sum_j a_{ij}(x) (v_j - v_i)$.
• Physical separation: repelling neighbors that are too close, e.g., $$\Lambda(\mathbf v)\sum_{j\neq i} f_0(\|x_i-x_j\|^2)(x_i-x_j)$$ with $f_0(\cdot)$ singular at the minimum separation.
• Cohesion: attracting distant neighbors to maintain group integrity, e.g., $$\Lambda(\mathbf v)\sum_{j\neq i} f_1(\|x_i-x_j\|^2)(x_j-x_i)$$ (Liu et al., 2019).

A unifying structure is: $$u_i(t) = \text{alignment} + \text{separation} + \text{cohesion}$$ Parametric functions $a_{ij}$, $f_0(\cdot)$, and $f_1(\cdot)$, along with a velocity dispersion regulator $\Lambda(\mathbf v)$, provide tunable safety and performance guarantees (collision avoidance, bounded cohesion, and convergence of velocities).

Formation tasks (e.g., enclosing a moving target) can integrate estimation and control in a tightly coupled loop, as in the Kuramoto-type oscillator framework that ensures even spacing and persistent excitation for estimator convergence, together with RLSE-based recursive localization and consensus-plus-feedforward control for shape tracking (Liu et al., 2023).

Relative Localization and Estimation

Recursive Least Squares Estimation (RLSE) with forgetting factor allows each agent to estimate inter-agent or agent–target relative position via a sequence of noisy linear measurements derived from pairwise distance and displacement changes. Persistent excitation—guaranteed by time-varying, oscillatory reference motions—yields exponential convergence of the relative position estimates (Liu et al., 2023).

Optimization-Based and Impulsive Control

In impulsive or high-dimensional domains (e.g., spacecraft relative trajectories), optimal control problems are often solved using reachable-set geometry, semi-infinite programming, or sequential convex programming, all structured around relative state models (e.g., difference in position/velocity or relative orbital elements). Key tools include:

• Support function and convex hull representations for reachable sets (Chernick et al., 2020, Hunter et al., 29 Jul 2025).
• Dual optimization (SOCP/SIP) to find globally optimal $\Delta v$ allocations while satisfying state, input, and safety constraints—impulses are scheduled at relative state maxima in the dual function (Foss et al., 3 Oct 2025, Hunter et al., 29 Jul 2025).
• Sequential convexification and time-dilation for continuous constraint satisfaction along relative trajectories, enabling path guarantees for non-linear and cislunar regimes (Spada et al., 31 Jan 2025).

3. Architectures and Implementation Approaches

Sensor and Computation Deployment

Distributed, vision-based architectures as in UAV flocking/from-only relative camera and IMU information realize entirely onboard relative motion control without global positioning or communication, leveraging VINS-Mono (visual-inertial odometry), AprilTag-type markers for inter-agent localization, and real-time estimation/control at rates up to 400 Hz (Liu et al., 2019).

Decentralized networked formations implement a multi-agent oscillator/relative-estimator pipeline, requiring only local sensing and, occasionally, broadcasted global information such as target velocity (Liu et al., 2023).

For manipulation, all arm and base degrees of freedom are embedded in a unified, relative Jacobian framework, handled in a quadratic program that incorporates task-space tracking, joint/safety constraints, manipulability, and base orientation (Haviland et al., 2021).

Planning and Optimization

High-reliability planning for systems with complex constraints (e.g., spacecraft) uses graph-search across precomputed invariant sets (e.g., forced equilibria or periodically-invariant tubes) built in relative coordinate space. Chained chance-constrained admissible set networks provide formal probabilistic safety guarantees under estimation and process noise, with each node and edge representing a relative position, tube, or transfer maneuver (Jr. et al., 2019, Frey et al., 2017).

4. Theoretical Guarantees and Optimality

• Collision Avoidance and Cohesion: Lyapunov-type functionals combining kinetic energy and pairwise potential terms strictly decrease along the trajectories, ensuring global boundedness and avoidance of pairwise collisions in distributed multi-agent systems (Liu et al., 2019, Liu et al., 2023).
• Velocity Alignment: In symmetric distributed protocols, the average velocity is preserved while the velocity dispersion converges to zero under bounded relative-feedback protocols, guaranteeing flocking or aligned motion (Liu et al., 2019).
• Exponential Convergence: RLSE-based localization with persistent excitation, combined with consensus/oscillator-based control, produces exponential decay of both localization and formation-trajectory tracking errors (Liu et al., 2023).
• Probabilistic Safety: Admissible set-graph approaches enable output-feedback optimal control that satisfies nonconvex state and input constraints with prescribed probability under noise and partial observability (Jr. et al., 2019).
• Global Optimality: Dual SIP and convex hull reachable-set methods provide global optima for impulsive reconfiguration given relative boundary conditions, regardless of system dimension or perturbations, and closed-form burn schemes in ROE or LTV spaces (Chernick et al., 2020, Foss et al., 3 Oct 2025, Hunter et al., 29 Jul 2025).
• Certifiable Obstacle Avoidance: By constructing invariant tubes or chained admissible sets, all transitions in relative state-space can be verified to avoid explicit static or time-varying obstacles (Frey et al., 2017).

5. Application Domains and Experimental Validation

Relative motion-based control underpins high-performance solutions in a variety of multi-agent and single-agent domains:

Domain	Example Applications	Key Works
UAV swarming	GPS-denied distributed flocking, formation tracking	(Liu et al., 2019, Liu et al., 2023)
Spacecraft	Rendezvous, docking, station-keeping, formation	(Jr. et al., 2019, Frey et al., 2017, Hunter et al., 29 Jul 2025, Servadio et al., 2022)
Manipulation	Holistic mobile manipulation, visual servoing	(Haviland et al., 2021)
Vehicles	CAV formation switching, conflict-free lane changes	(Cai et al., 2021)
Time series	Motion retargeting via metric-aligning alignment	(Agata et al., 26 May 2025)

Experimental results include:

• Indoor/outdoor two-UAV experiments with vision-inertial distributed flocking, maintaining $\sim2$ m separation with no external infrastructure (Liu et al., 2019).
• Three-Crazyflie quadrotor experiments tracking a moving target with only one UAV measuring the target, achieving formation errors $<0.2$ m (Liu et al., 2023).
• Onboard optimization for real-time impulsive maneuvers in cislunar CR3BP dynamics, showing sub-kilometer errors and $<$10 s solve times for full maneuver plans (Hunter et al., 29 Jul 2025).
• Mobile manipulation at $>$200 Hz, tracking moving objects with continuous closed-loop controller achieving pick-and-place in unstructured environments (Haviland et al., 2021).
• CAV formation control providing $\sim$25% fuel savings and halved travel time in highway bottleneck simulations (Cai et al., 2021).

6. Extensions, Limitations, and Outlook

Limitations and active areas include:

• Extension to high-order nonlinear dynamics requires Koopman operator, Galerkin projection, or high-order polynomial expansion; accuracy may be limited for very large maneuvers or long time horizons (Servadio et al., 2022).
• Constraints on actuation (e.g., thrust, joint velocity) and strict state constraints are handled via constrained graphs, SCP, or SIP, but uncertain future motion may require robust or tube-based planning (Frey et al., 2017, Spada et al., 31 Jan 2025).
• Impulsive control methods are best suited for actuators capable of delivering large, discrete actions; continuous-thrust problems require extensions or hybridization.
• Flow-field design (e.g., oscillator networks) requires persistent excitation for localization, which may limit performance in static or degenerate configurations (Liu et al., 2023).

Future directions include adaptive basis enrichment in Koopman expansions, multi-agent coordination in high-dimensional nonlinear and time-varying environments, robustification under high noise or communication loss, extensions to continuous low-thrust and multi-body regimes, and scalable onboard implementation for large networks of agents (Servadio et al., 2022, Spada et al., 31 Jan 2025).

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References:

• "Bird Flocking Inspired Control Strategy for Multi-UAV Collective Motion" (Liu et al., 2019)
• "Formation Control for Moving Target Enclosing via Relative Localization" (Liu et al., 2023)
• "Spacecraft Relative Motion Planning Using Chained Chance-Constrained Admissible Sets" (Jr. et al., 2019)
• "Constrained Spacecraft Relative Motion Planning Exploiting Periodic Natural Motion Trajectories and Invariance" (Frey et al., 2017)
• "Optimal Impulsive Control of Cislunar Relative Motion using Reachable Set Theory" (Hunter et al., 29 Jul 2025)
• "Closed-Form Optimal Impulsive Control of Spacecraft Relative Motion Using Reachable Set Theory" (Chernick et al., 2020)
• "A Koopman-Operator Control Optimization for Relative Motion in Space" (Servadio et al., 2022)
• "Efficient Input-Constrained Impulsive Optimal Control of Linear Systems with Application to Spacecraft Relative Motion" (Foss et al., 3 Oct 2025)
• "Impulsive Relative Motion Control with Continuous-Time Constraint Satisfaction for Cislunar Space Missions" (Spada et al., 31 Jan 2025)
• "A Holistic Approach to Reactive Mobile Manipulation" (Haviland et al., 2021)
• "Formation Control for Connected and Automated Vehicles on Multi-lane Roads: Relative Motion Planning and Conflict Resolution" (Cai et al., 2021)
• "MAMM: Motion Control via Metric-Aligning Motion Matching" (Agata et al., 26 May 2025)