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L1-MPPI: L1 Adaptive Model Predictive Path Integral for Agile UAV Control

Lukáš Kotek, Ondřej Procházka, Vojtěch Vonásek, Martin Saska, Robert Pěnička

arXiv:2609.38467v1cs.RO

TL;DR

Agile UAV tracking must remain accurate despite model uncertainty and external disturbances, which complicate the nominal dynamics. L1-MPPI combines MPPI with online L1 compensation and a model of low-level control dynamics; it lowers positional RMSE by 44.37% relative to plain MPPI in the presence of uncertainties.

  • Problem

    During aggressive maneuvers, nominal UAV dynamics can diverge from the modeled system, making accurate tracking necessary despite uncertain drag, payloads, and wind gusts.

    Source: §I · p. 1

  • Method

    L1-MPPI combines MPPI trajectory tracking with online L1 adaptive compensation and incorporates low-level controller, motor, and communication-delay dynamics into the model.

    Source: §I · pp. 1–2

  • Results

    44.37% lower positional RMSE than plain MPPI in the presence of uncertainties.

    Source: §I · p. 2

  • Takeaways & Limitations

    Real-world agile-flight experiments demonstrate L1-MPPI adaptability without performance degradation, at speeds up to 13.5 m s−1 and accelerations up to 2.5 g.

    Source: §V · p. 8

Abstract

from arXiv · show

This work proposes the L1 Adaptive Model Predictive Path Integral (L1-MPPI). It cascades L1 adaptive control with the Model Predictive Path Integral (MPPI) to improve tracking of high-speed UAV trajectories. Thanks to the L1augmentation, the tracking remains accurate even under model uncertainties and external disturbances, such as an additional payload or a mismatch in the modeled aerodynamic drag. In contrast to existing MPPI approaches for UAV control that do not explicitly model aerodynamic effects, varying payloads, and typically neglect the dynamics of low-level motor controllers, our L1-MPPI approach enhances the dynamic model used in the MPPI by incorporating the low-level flight controller and motor dynamics, as well as an iterative mixing scheme that reflects the approach of the low-level controller. The proposed method demonstrates improved tracking performance in both simulation and the real world, even when the UAV is subjected to an unknown payload. In flight with 35% mass increase, our approach lowers the RMSE by 58.61% with respect to plain MPPI. Compared to the same MPPI using an online mass estimator in place of the L1 augmentation, the RMSE is lower by 38.59%. During the real-world experiments the UAV reaches speeds up to 13.50 m/s and accelerations up to 2.5 g.

SUPPLEMENTARY MATERIAL

Agile UAV tracking must contend with model uncertainty and disturbances, while prior predictive and adaptive approaches leave gaps in modeling the control hierarchy or demonstrating fast flight. L1-MPPI combines online adaptive compensation with a richer MPPI model and is evaluated in simulation and real-world flight.

I. INTRODUCTION

  • Aggressive maneuvers increase mismatch between nominal and actual UAV dynamics, and uncompensated trajectory deviations can increase collision risk or cause missed waypoints or mission failure.Uncertain drag, varying payloads, and wind gusts are examples of relevant disturbances.Source: §I · p. 1
  • L1-MPPI models low-level controller and motor dynamics plus communication delays, then uses L1 adaptation to estimate and compensate for remaining uncertainties online.The paper reports fully onboard operation at 100 Hz and real-world agile-flight evaluation, distinguishing it from the cited L1-augmented MPPI approach.Source: §I · p. 2, §II · p. 3
  • 24.78% lower positional RMSE than plain MPPI under nominal conditions and 44.37% lower in the presence of uncertainties.The L1 augmentation alone reduces positional RMSE by 51.11% with an unknown payload compared with the same controller without augmentation.Source: §I · p. 2

II. RELATED WORKS

  • Prior data-driven models can overfit or be difficult to update online, while offline training and off-board computation limit adaptation to changing dynamics.Other approaches explicitly model effects such as aerodynamic drag, motor dynamics, and iterative mixing.Source: §II · p. 2
  • Existing adaptive UAV controllers have often been tested only in hovering or low-speed flight, while MPPI approaches omit low-level controller, motor, or communication-delay dynamics.Prior L1-MPPI work was demonstrated in simulated racing, whereas L1-NMPC was reported in agile real-world flights.Source: §II · pp. 2–3

III. METHODOLOGY

The methodology combines MPPI trajectory tracking with L1 adaptive compensation and models the low-level flight controller within the predictive dynamics. Its nominal UAV model describes position, velocity, attitude, body rate, inertia, mass, and linear aerodynamic drag.

III. METHODOLOGY

  • L1-MPPI uses MPPI for nominal thrust and body-rate commands and L1 adaptive augmentation for compensation against model uncertainties and external disturbances.The L1 commands are compensating thrust and body rate, added to the MPPI command before mapping to throttle and sending it to the flight controller.Source: §III · p. 3

A. UAV Model

  • The nominal UAV model represents position, velocity, body rate, and quaternion attitude, with inertia, mass, and diagonal linear aerodynamic drag in its dynamics.Its control inputs are collective thrust and body torques.Source: A. UAV Model · p. 3
  • Because low-level flight-controller behavior influences UAV behavior, the method includes its dynamics inside the MPPI controller rather than ignoring them.The high-level controller's thrust and body-rate commands are mapped to throttle and passed to the low-level controller, which controls the motors.Source: A. UAV Model · p. 3

B. Dynamics of the Low-level flight Controller

The MPPI model includes the low-level PID controller and motor response, mapping throttle and body-rate commands into UAV force and torque while accounting for actuator limits.

B. Dynamics of the Low-level flight Controller

C. PID dynamics for long prediction steps

For longer MPPI prediction steps, the controller simplifies torque computation, then converts feasible rotor thrusts back into UAV inputs.

C. PID dynamics for long prediction steps

D. Model Predictive Path Integral

MPPI predicts trajectory tracking with both the UAV and low-level flight-controller dynamics, then updates nominal controls using costs from sampled rollouts.

D. Model Predictive Path Integral

E. L1 Adaptive Control

L1 adaptive control is cascaded with MPPI: MPPI tracks the reference using nominal commands, while L1 estimates and compensates for model uncertainties and disturbances.

E. L1 Adaptive Control

  • MPPI supplies nominal thrust and body-rate dynamics, while the cascaded L1 controller compensates for model uncertainties and external disturbances.Source: E. L1 Adaptive Control · p. 5
  • The L1 architecture uses an observer, adaptation law, and control law: observed state informs uncertainty estimates that determine the adaptive command.Source: Figure 4 · p. 5
  • A piecewise adaptation law estimates uncertainties, and the control law low-pass filters the estimated matched uncertainty to form the adaptive command.Source: E. L1 Adaptive Control · p. 5
  • Matched uncertainties comprise vertical force and body-rate terms that L1 can compensate; lateral force uncertainties are unmatched and cannot be directly compensated.Source: E. L1 Adaptive Control · p. 5
  • The first-order rotational approximation may be interpreted as uncertainty by L1; this is acceptable if its error remains small relative to actual uncertainties.Source: E. L1 Adaptive Control · p. 5

F. Considering Communication Delay

A communication delay separates MPPI's high-level command from the start of rotor throttle response. The measured delay informs the predicted state used to initialize MPPI rollouts.

F. Considering Communication Delay

  • The high-level MPPI command is delayed before rotor throttle begins because a low-level controller sits between them.The delay is illustrated by the motor response to a step command.Source: F. Considering Communication Delay · p. 6
  • The delay ∆t is characterized from the measured mean motor-throttle response to a step command.The figure highlights the interval between the applied command and motor response.Source: Figure 5 · p. 6
  • The delay measurement helps determine initial state x0 for MPPI rollouts, while the preceding iteration is propagated using ulast.The predicted state x0 initializes rollout generation.Source: F. Considering Communication Delay · p. 6

G. Separation of L1 Controller and MPPI

The measured motor state already reflects L1 control, so its contribution is removed before MPPI prediction to avoid an inconsistent prediction and controller counteraction.

G. Separation of L1 Controller and MPPI

  • Because measured motor state includes L1 adaptation, its contribution must be removed before MPPI prediction to avoid assuming the L1 command persists under uncertainty.Without removal, the MPPI and L1 controllers can counteract each other.Source: G. Separation of L1 Controller and MPPI · p. 6
  • Measured motor speeds are normalized and used to compute the current single-rotor forces.The normalized values are computed from measured motor speeds using (5), and forces from (6).Source: G. Separation of L1 Controller and MPPI · p. 6
  • The MPPI force contribution is obtained by subtracting prior-iteration L1 thrust, then the resulting normalized motor speeds initialize actuator dynamics.The L1 thrust command is from the previous iteration.Source: G. Separation of L1 Controller and MPPI · p. 6

IV. RESULTS

The section evaluates trajectory tracking and adaptation, comparing the proposed controller with existing methods and testing its component contributions.

IV. RESULTS

  • The experiments assess trajectory tracking and adaptive capabilities, compare the proposed approach with existing methods, and ablate its introduced features.The ablation assesses the contribution of each feature introduced in Sec. III.Source: §IV · p. 6
  • Simulation used a laptop, while real-world experiments used a Jetson Orin NX; a PX4 low-level controller tracked collective-throttle and body-rate commands.The real-world computer has an 8-core Arm Cortex-A78AE CPU and an Ampere GPU.Source: §IV · p. 6
  • Reference trajectories were generated offline with a polynomial planner and supplied to L1-MPPI during flight.The controller outputs desired collective throttle and body-rate commands.Source: §IV · p. 6

A. Designed Experiments

L1-MPPI improves simulated tracking over plain MPPI, especially under payload and drag mismatches, while ablations show that the effects of L1 augmentation and low-level dynamics depend on the condition. The experiments use agile trajectories and compare the method with a mass-estimator variant.

A. Designed Experiments

  • The proposed L1-MPPI controller is compared with plain MPPI and an MPPI implementation augmented with a mass estimator; ablations evaluate delay compensation, translational-only augmentation, and removing L1 augmentation.Source: A. Designed Experiments · p. 6
  • The adaptive scenarios include a nominal model, an unknown payload increasing mass by 35%, and doubled drag coefficients cx, cy, and cz.The experiments evaluate low-level dynamic modeling, L1 augmentation, and adaptation to mass and aerodynamic uncertainties.Source: A. Designed Experiments · pp. 6–7

B. Simulated Results

  • L1-MPPI improved tracking performance over plain MPPI by approximately 24.78% on average under nominal conditions, 58.61% with an additional payload, and 30.12% under increased drag.Source: B. Simulated Results · p. 7
  • Removing low-level PID dynamics worsens performance by approximately 25% nominally and 16% with payload, but improves tracking by approximately 17% under drag mismatch.A proposed explanation is that L1 drag compensation interacts less favorably with the low-level controller because drag depends on attitude.Source: B. Simulated Results · p. 7
  • Under increased drag, the mass estimator has approximately 67% higher RMSE than L1-MPPI, despite comparable performance under nominal conditions and with an additional payload.This suggests L1 adaptation handles uncertainties not represented by a mass change alone.Source: B. Simulated Results · p. 7

C. Real-world Experiments

In real-world figure-8 trials, L1 augmentation compensated for an unknown payload, while its effect on position RMSE was minor under unchanged conditions. The conclusion reports agile-flight evaluations in simulation and real-world adaptability under uncertainty.

C. Real-world Experiments

  • Real-world figure-8 trials reached 13.51 m/s and 2.5 g across unchanged-model, 79% apparent-mass-mismatch, and 35% physical-payload conditions.The apparent mass mismatch was created by reducing modeled UAV mass from 1.43 kg to 0.8 kg; the physical payload also influenced inertia.Source: C. Real-world Experiments · p. 7
  • 7.49% increase in position RMSE: L1-MPPI versus Ours (No L1) in No change; with additional payload, full augmentation compensated for the unknown payload.The authors also report full L1 augmentation outperforming Ours (No L1) in nominal settings.Source: C. Real-world Experiments · p. 7
  • Figure 6 shows reference and tracked trajectories under an additional payload for the proposed L1-MPPI controller and plain MPPI.Source: Figure 6 · p. 7

V. CONCLUSION

  • The authors describe L1-MPPI as the first MPPI-based controller incorporating low-level controller and motor dynamics, and the first adaptive MPPI suitable for onboard control.They report simulation tracking up to 17.84 m/s and 3.5 g, and real-world adaptability up to 13.5 m/s and 2.5 g.Source: §V · p. 8
  • Across simulation and flight under uncertainties, L1-MPPI outperformed an existing MPPI approach on agile trajectories and an approach using mass estimation.The real-world experiments showed adaptability without performance degradation.Source: §V · p. 8
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