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FF-MPCC: High-speed Agile Formation Flight with Model Predictive Contouring Control

Aditya Dandwate, Vit Kratky, Parakh M. Gupta, Martin Saska, Robert Penicka

arXiv:2608.21056v1cs.RO

TL;DR

Agile formation flight is difficult because individual UAVs face different dynamic demands while existing approaches struggle to combine online optimization with strict, evolving formation constraints. FF-MPCC integrates formation maintenance into decentralized MPCC, improving formation accuracy by 65% in flights up to 21 m/s while maintaining comparable or better completion times.

  • Problem

    Existing methods struggle to provide decentralized, agile formation flight that preserves prescribed, dynamically evolving formations while respecting individual UAV dynamics.

  • Method

    FF-MPCC jointly optimizes path progression and formation adherence through decentralized MPCC with heading-aware synchronization and online distributed formation fitting.

  • Results

    65% improved formation accuracy was achieved versus preplanned trajectory tracking, with comparable or better flight times at velocities up to 21 m/s.

  • Takeaways & Limitations

    The results demonstrate decentralized agile formation flight that maintains formation integrity without requiring preplanned dynamically feasible trajectories.

Abstract

from arXiv · show

Flying in a prescribed formation in an agile manner remains a challenging problem in the field of UAVs, particularly when following highly-demanding trajectories that require flight at platform limits. We address this problem by proposing a novel decentralized approach to formation flight along a given path that integrates formation maintenance into the MPCC framework, allowing UAVs to adapt their progression along complex paths while respecting individual dynamic constraints and maintaining the desired formation. To this end, we introduce a novel reparametrization and synchronization method for dynamic formation geometries together with a decentralized approach to determine the desired positions for the individual UAVs. The proposed approach allows the formation to coordinate high-speed path following without compromising formation integrity. The proposed approach is validated through extensive simulation and real-world experiments involving scenarios with varying complexity of paths and changes of required formation shape on the fly. In comparison to time-parameterized trajectory tracking, we demonstrate improved formation maintenance by 65% in high-speed flight with velocities up to 21 m/s, while achieving comparable times required to reach the goal.

SUPPLEMENTARY MATERIAL · I. INTRODUCTION · II. RELATED WORKS

FF-MPCC addresses the challenge of high-speed multi-UAV formation flight by jointly optimizing path progression and formation adherence while respecting individual vehicle dynamics. The framework combines heading-aware formation synchronization, distributed reference fitting, and decentralized control, achieving 65% better formation accuracy than a conventional baseline with comparable or better flight times at speeds up to 21 m s−1.

  • I. INTRODUCTION: Formation flight is harder than single-UAV navigation because different formation members encounter different path curvatures and therefore require different velocities and accelerations.A trajectory feasible for the formation center is not necessarily feasible for every UAV; conservative constraints can reduce flight speed and efficiency.
  • II. RELATED WORKS: Existing formation methods commonly prioritize cohesion, collision avoidance, reshaping, or environmental adaptation rather than time-optimal progression along prescribed trajectories.Time-optimal methods also inherit preplanning difficulties while needing feasible trajectories for all formation members.
  • I. INTRODUCTION: FF-MPCC jointly optimizes path progression and adherence to the desired formation geometry within a decentralized control framework.Synchronization uses a heading-aware formation representation for continuously evolving geometries parameterized by path progression.
  • II. RELATED WORKS: Distributed rigid-body fitting constructs formation-consistent target positions from neighboring UAV states, enabling each vehicle to minimize formation errors while independently adapting its motion and progress.The method is designed to adapt to disturbances while preserving formation consistency.
  • I. INTRODUCTION: 65% formation-accuracy improvement over preplanned minimum-time trajectories with NMPC tracking, while maintaining comparable or better flight times at speeds up to 21 m s−1.The evaluation includes extensive simulations and real-world experiments with acceleration limits of 30 m s−2.
  • II. RELATED WORKS: MPCC avoids prescribing reference timing by optimizing online progression along a geometric path while minimizing contouring and lag errors.This complements NMPC, which directly handles nonlinear vehicle dynamics and actuator limits for aggressive flight –.
  • II. RELATED WORKS: FF-MPCC bridges agile MPCC-based flight and formation-constrained multi-UAV flight by integrating formation constraints into nonlinear MPCC and fitting targets online from current and predicted neighbor states.The resulting coordination supports aggressive motion along a shared path while respecting vehicle dynamics and formation geometry.

III. METHODOLOGY · A. Quadrotor Dynamics

The methodology introduces the quadrotor model before formulating FF-MPCC, generating reference formation paths, and coordinating formation flight. The dynamics use position, orientation, linear and angular velocity, thrust, body torques, gravity, inertia, and drag.

  • III. METHODOLOGY: The methodology proceeds from quadrotor modeling to FF-MPCC optimal-control formulation, reference formation-path generation, and formation coordination.
  • A. Quadrotor Dynamics: The quadrotor state comprises position p ∈R3, orientation q ∈SO(3), linear velocity v ∈R3, and body-frame angular velocity ω ∈R3.
  • A. Quadrotor Dynamics: The dynamics are driven by collective body-frame thrust fT and body torques τ, and are described by world-frame differential equations.
  • A. Quadrotor Dynamics: The model accounts for mass m, inertia J, gravity g, quaternion multiplication, rotation R(q), and drag force fD.
  • A. Quadrotor Dynamics: Drag is modeled linearly as fD = −kvvB using body-frame velocity vB = RT(q)v and a vector of drag coefficients kv.
  • A. Quadrotor Dynamics: Collective thrust and body torques are calculated from the individual rotor thrusts f = [f1, f2, f3, f4].
  • A. Quadrotor Dynamics: The rotor-to-force and torque relation uses the arm length l from the center of mass to each motor and the motor torque coefficient κ.

B. Formation Flight Model Predictive Contouring Control Formulation

FF-MPCC replaces time-parameterized trajectory tracking with online arc-length path progression, while integrating contour, lag, and formation errors into a decentralized predictive control formulation. It synchronizes individually optimized UAVs through fitted formation targets and balances accuracy, cohesion, progress, and smooth constrained control.

  • Path and error formulation: FF-MPCC tracks a geometric path parameterized by arc length, letting the solver determine time progression online rather than prescribing it beforehand.Progress is represented by θ, following the MPCC formulation, instead of explicitly tracking a full state trajectory parameterized in time as in classical NMPC.
  • Path and error formulation: The controller decomposes position error into lag error along the path tangent and contour error orthogonal to that tangent.For position error e(θ_k), the contour term is computed by subtracting the scalar lag component from the full error vector.
  • Decentralized formation coordination: Decentralized interlinking preserves formation geometry by fitting the desired formation shape to current and predicted positions from all UAVs to generate each formation target position.Each ego-UAV receives other UAVs’ current and predicted positions, then minimizes its formation error alongside path-tracking errors.
  • Optimization objective: The objective minimizes lag, contour, and formation errors while maximizing arc-length speed v_θ,k, with additional penalties encouraging smooth angular velocity, progress-speed, and thrust changes.The progress term advances the UAV along the path, while weighted input-change penalties discourage noisy or bang-bang control behavior.
  • Augmented dynamics and constraints: The augmented UAV model includes thrusts, path progress θ_k, and virtual speed v_θ,k, using progress and thrust accelerations as control inputs to enforce rate limits.These augmented variables support finite-horizon FF-MPCC optimization under the system’s dynamic and control constraints.

C. Reference Formation Trajectory Generation

The reference formation trajectory couples the formation center path with a synchronized, time-varying geometry parameterized by τ. Individual UAV paths are transformed, spline-approximated, and reparameterized by a shared arc-length variable θ for consistent formation evaluation.

  • Reference formation definition: The formation reference couples the center trajectory and time-varying geometry through a shared path parameter τ, with each τ identifying both a center point and formation configuration.The geometry is represented by the matrix-valued function M(τ).
  • Reference formation definition: User-defined formation key-frames are linearly interpolated in τ, enabling constant geometries or smooth transitions between distinct configurations.The evolution is illustrated in Fig. 3.
  • Individual UAV trajectories: Each matrix column specifies a UAV’s relative position in the local formation frame, which is rotated into the world frame to generate its nominal trajectory.The rotation uses the formation center heading ψ(τ).
  • Individual UAV trajectories: The resulting UAV trajectories are sampled into approximately equidistant knots and represented as continuous piecewise cubic splines after arc-length reparameterization.The spline segments span θ from 0 to θmax.
  • Arc-length synchronization: Because optimization and disturbances can place UAVs at different progression states, both paths and formation geometry are reparameterized and evaluated using the same arc-length variable θ.Strictly monotonic mappings Fi and inverse mappings F−1_i connect τ and θ for each UAV.

D. In Flight Formation Coordination

FF-MPCC coordinates formation flight decentrally: each UAV independently computes its target position from its predicted state and communicated neighbour trajectories. Rigid-body fitting and communication-delay compensation support continuous formation maintenance while adapting to disturbances and slower members.

  • Decentralized coordination: Each UAV independently solves its optimization problem and evaluates formation error using a target position computed from current and predicted positions of the other members.The target position λ_i is computed at every shooting node of the prediction horizon.
  • Decentralized coordination: UAVs periodically broadcast their current states and locally predicted trajectories, which neighbours combine to estimate the formation state at each prediction stage.Each UAV constructs a measured formation position matrix from its own predicted states and the latest received neighbour trajectories.
  • Distributed target computation: A rigid-body transformation aligns the desired formation geometry with the measured formation, after which each UAV extracts its target position for formation-error evaluation.The extracted λ_i is used in the optimization problem, as illustrated by the distributed target-position computation procedure.
  • Communication-delay compensation: Communication latency is compensated by evaluating each neighbour trajectory at the delayed prediction time and linearly interpolating the corresponding neighbour state.The delay is δ = T_0 − T_s, and the required prediction-stage time is t_k = δ + k∆T.
  • Decentralized coordination: The distributed method removes centralized coordination while allowing UAVs to maintain geometric structure despite in-flight disturbances and slower formation members.Each UAV continuously determines its desired position using local predicted states and communicated neighbour predictions.

IV. RESULTS

FF-MPCC is evaluated through extensive simulations and real-world flights using identical custom agile quadrotors. The experiments employ RTK GPS, CubePilot/PX4 flight control, and the MRS UAV system architecture running on Khadas Vim3 Pro computers.

  • FF-MPCC is evaluated in extensive simulations and real-world flights using identical custom agile quadrotors.
  • The quadrotors have 300 mm diagonal length and weigh 1.2 kg.
  • Each quadrotor uses RTK GPS, a CubePilot flight controller with PX4 firmware, and the MRS UAV system architecture running on a Khadas Vim3 Pro computer.

A. Experiment Methodology

The experiments evaluate FF-MPCC using formation accuracy and mission completion time. Formation accuracy combines relative-spacing and spatial-shape measures, while completion time is determined by the last UAV to reach its target.

  • A. Experiment Methodology: Formation accuracy is evaluated from uniformly sampled trajectory measurements using MAPDE and SFTE.MAPDE measures pairwise distance errors, while SFTE addresses formation orientation and scalability limitations for formations with N > 3.
  • A. Experiment Methodology: MAPDE computes errors between actual and desired mutual Euclidean distances for UAV pairs over recorded trajectory samples.Desired pairwise distances derive from the reference shape matrix and are computed independently by each UAV because neighbors may use slightly different progression values.
  • A. Experiment Methodology: SFTE aligns the formation orientation, anchors the rotated shape at each UAV, and evaluates Euclidean position errors for the remaining UAVs.The desired heading comes from the tangent of the reference center path at the closest projected point, separating shape distortion from global translation errors.
  • A. Experiment Methodology: Mission completion time is the maximum arrival time among all UAVs, with arrival defined as entering a 0.75 m-radius sphere around the assigned endpoint.This definition is used consistently across all experiments.

B. Simulation Results

The method is evaluated in Gazebo simulations through an ablation study of formation-error weighting and a comparison with an NMPC-based approach.

  • B. Simulation Results: The evaluation uses Gazebo simulations to assess the proposed method.
  • B. Simulation Results: An ablation study isolates the impact of the formation error weight.
  • B. Simulation Results: The method is comparatively evaluated against an NMPC-based approach.

1) Ablation Study

The ablation study tests whether the formation-error cost is necessary for three UAVs tracking a sine-wave center path while maintaining a static equilateral formation. Setting qf = 0 produces substantially worse formation accuracy than non-zero weights, which achieve better metrics with modestly longer completion times.

  • Ablation Study: The study evaluates three UAVs tracking a sine-wave center path while maintaining a static equilateral triangular formation.The ablation varies the formation-error weight qf in the objective function.
  • Ablation Study: Non-zero qf values achieve MAPDE within 0.25 m to 0.78 m and SFTE within 1.08 m to 1.3 m, with completion times of 13.21 s to 14.98 s.The results show improved formation accuracy relative to qf = 0, while completion times are modestly longer.

2) Comparative Evaluation

FF-MPCC is evaluated against the conventional PMM+NMPC planning-and-control pipeline using consistent formation-flight metrics and 20-trial averages. It achieves superior formation accuracy across all simulations and often completes scenarios faster, while suitable gain settings can match baseline completion times without sacrificing accuracy.

  • Evaluation setup: The baseline combines minimum-time PMM trajectory planning with NMPC tracking, while individual UAV trajectories are generated by offsetting the formation-center trajectory using the prescribed formation geometry.MAPDE uses reference pairwise distances from the shape matrix, and SFTE and completion time follow the evaluation procedure in Sec. IV-A.
  • Trajectory comparison: For a sine-wave path with triangle-line-triangle formation geometry, Fig. 6 compares the reference and flown trajectories of PMM+NMPC and FF-MPCC.Dashed lines denote reference trajectories, while solid lines denote trajectories flown by individual UAVs.
  • Comparative results: Over 20 trials per scenario, FF-MPCC improves MAPDE and SFTE over PMM+NMPC by more than 30% in every simulation and completes half the scenarios faster.The comparison uses identical constant gains, with results averaged over 20 trials per scenario (Table I).
  • Comparative results: For every scenario, some FF-MPCC gain settings match PMM+NMPC completion times while retaining superior formation accuracy.FF-MPCC has longer completion times in some cases, but gain selection preserves its MAPDE and SFTE advantage.

C. Real World Experiment · V. CONCLUSION

Real-world experiments validated FF-MPCC with three UAVs on a 60 m straight path using static and transitioning formation geometries. The conclusion presents FF-MPCC as a decentralized MPCC framework that jointly optimizes path progression and formation accuracy for agile, time-efficient flight without offline dynamically feasible trajectories.

  • C. Real World Experiment: Real-world flights used three UAVs to evaluate the decentralized FF-MPCC framework under field conditions.
  • C. Real World Experiment: The experiments followed a 60 m straight-line trajectory and evaluated two representative formation profiles.
  • C. Real World Experiment: Both scenarios were successfully executed: a static equilateral triangle with 10 m sides and a transition to an 8.66 m co-linear formation.
  • V. CONCLUSION: FF-MPCC is presented as a distributed MPCC framework for decentralized multi-UAV formation flight.
  • V. CONCLUSION: The framework jointly optimizes path progression and formation accuracy, enabling agile, time-efficient flight along predefined paths without offline dynamically feasible trajectories.
  • V. CONCLUSION: The real-world validation reconstructed trajectories and captured experimental scenes for both transitional and static triangular geometries along a global straight path.
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