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RAMP-Net: A Robust Adaptive MPC for Quadrotors via Physics-informed Neural Network

Sourav Sanyal, Kaushik Roy

arXiv:2209.09025v3cs.ROcs.AIcs.SYeess.SY

TL;DR

Robust MPC can become computationally demanding when uncertain dynamics require additional constraints, while regression-based system identification may need extensive labeled data. RAMP-Net combines ideal-dynamics physics loss with data loss in a PINN for quadrotor MPC. In simulated trajectory tracking, it reports lower normalized DTW errors than KNODE-MPC and GP-MPC on circles and lemniscates.

  • Problem

    Robust MPC adds constraints to handle uncertain dynamics, increasing the optimization burden, while data-driven regression may require substantial labeled data and training overhead.

    Source: §I · p. 1

  • Method

    RAMP-Net trains a PINN with a physics loss from ideal quadrotor dynamics and a data loss for dynamics unaccounted for by the model.

    Source: §I · p. 1

  • Results

    Average normalized % DTW errors were 46.07% for RAMP-Net, 53.63% for KNODE-MPC, and 78% for GP-MPC on circles; on lemniscates they were 47.41%, 80.65%, and 92.79%, respectively.

    Source: D. Comparison with SOTA Regression… · p. 5

  • Takeaways & Limitations

    The composite loss combines a symbolic dynamics constraint with observations of uncertain disturbances for quadrotor tracking in simulation.

    Source: C. Physics-informed Neural Networks · p. 3

    Evaluation is scoped to Hummingbird quadrotor trajectory tracking in the Gazebo simulation environment.

    Source: §I · p. 1

Abstract

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Model Predictive Control (MPC) is a state-of-the-art (SOTA) control technique which requires solving hard constrained optimization problems iteratively. For uncertain dynamics, analytical model based robust MPC imposes additional constraints, increasing the hardness of the problem. The problem exacerbates in performance-critical applications, when more compute is required in lesser time. Data-driven regression methods such as Neural Networks have been proposed in the past to approximate system dynamics. However, such models rely on high volumes of labeled data, in the absence of symbolic analytical priors. This incurs non-trivial training overheads. Physics-informed Neural Networks (PINNs) have gained traction for approximating non-linear system of ordinary differential equations (ODEs), with reasonable accuracy. In this work, we propose a Robust Adaptive MPC framework via PINNs (RAMP-Net), which uses a neural network trained partly from simple ODEs and partly from data. A physics loss is used to learn simple ODEs representing ideal dynamics. Having access to analytical functions inside the loss function acts as a regularizer, enforcing robust behavior for parametric uncertainties. On the other hand, a regular data loss is used for adapting to residual disturbances (non-parametric uncertainties), unaccounted during mathematical modelling. Experiments are performed in a simulated environment for trajectory tracking of a quadrotor. We report 7.8% to 43.2% and 8.04% to 61.5% reduction in tracking errors for speeds ranging from 0.5 to 1.75 m/s compared to two SOTA regression based MPC methods.

I. INTRODUCTION

The introduction frames robust MPC as computationally demanding under uncertain dynamics and motivates combining analytical priors with data in RAMP-Net. In simulated Hummingbird quadrotor tracking, the method reports lower tracking error and faster convergence than a regression-based method.

I. INTRODUCTION

  • Robust MPC adds conservative disturbance constraints, making accurate dynamics optimization within tight time budgets a bottleneck for high-speed applications such as agile drone navigation.MPC iteratively solves a constrained optimal-control problem.Source: §I · p. 1
  • Data-driven regression can reduce inference computation, but may require extensive labeled data and incur substantial training overhead.The introduction identifies Gaussian processes and neural networks as regression examples.Source: §I · p. 1
  • RAMP-Net trains a PINN partly on ideal quadrotor ODEs and partly on data from disturbed simulations to adapt to similar disturbances during inference.Analytical symbolic functions provide training targets at collocation points and are intended to enforce desired system behavior.Source: §I · p. 1
  • The physics loss models ideal dynamics, the data loss captures dynamics omitted from the model, and their sum trains the neural network to approximate quadrotor dynamics.The introduction presents this composite loss as the basis of RAMP-Net.Source: §I · p. 1
  • RAMP-Net achieved ∼60% less tracking error and ∼11% faster convergence than a SOTA regression-based method in Gazebo Hummingbird quadrotor tracking.The introduction reports these results for trajectory tracking.Source: §I · p. 1

II. RELATED WORK

Related work contrasts reinforcement-learning, analytical robust/adaptive MPC, and regression-based system identification approaches. The paper then formulates MPC using a moving horizon in which successive intervals are linked through their states.

II. RELATED WORK

  • Reinforcement-learning control approaches, include adversarial actor-critic methods, and RL within MPC [22],, but can be sample-inefficient and lack analytical priors.The paper also notes convergence challenges in complex or rapidly changing scenarios.Source: §II · p. 2
  • Tube-MPC bounds disturbances conservatively, while adaptive MPC, addresses parametric uncertainty but can overfit reference models and suffer model drift.The paper links conservative robust MPC constraints to greater optimization hardness under tight time budgets.Source: §II · p. 2
  • Regression-based MPC system identification includes Gaussian-process regression and Neural-ODE correction; RAMP-Net instead uses PINNs to form a composite loss.The section positions the approach alongside work integrating machine learning and MPC.Source: §II · p. 2

A. Dynamical System formulation

  • The dynamical-system formulation uses 13 quadrotor states and 4 control variables, and assumes a Lipschitz-continuous state function for a unique IVP solution.The paper adopts a multiple-shooting moving horizon, feeding each interval’s final state into the next as its initial state.Source: A. Dynamical System formulation · p. 2

B. MPC problem formulation

The MPC formulation seeks to make a quadrotor state follow a reference trajectory by repeatedly solving a constrained optimization problem. Its dynamics regressor may be model-based, data-based, or both.

B. MPC problem formulation

  • The control objective is to make the quadrotor state vector follow a reference trajectory as closely as possible.The formulation defines a quadratic cost for this objective.Source: B. MPC problem formulation · p. 2
  • At each moving-horizon interval, MPC obtains optimal control signals by solving the optimization problem iteratively in real time.The cost uses weighted semi-norms with positive semi-definite Q and R matrices.Source: B. MPC problem formulation · p. 3
  • The paper uses a PINN to evaluate the dynamics regressor φ because accurate modeling can be difficult in complex environments and robust constraints challenge high-speed computation.Traditional MPC uses nonlinear programming and numerical integration schemes such as Runge-Kutta methods.Source: B. MPC problem formulation · p. 3

C. Physics-informed Neural Networks

The PINN approach combines an ODE residual physics loss with a data-fitting loss to identify quadrotor dynamics. Symbolic dynamics constrain the physics dataset, while disturbed simulation observations supply data for fitting.

C. Physics-informed Neural Networks

  • PINNs incorporate differential equations as residuals in the loss, and this work extends the inputs to include control variables and time for MPC.The paper describes the physics loss as an ODE residual and uses a neural network to learn the dynamics regressor.Source: C. Physics-informed Neural Networks · p. 3
  • Collocation points are state-control pairs evaluated against symbolic dynamics, which regularize the network to follow the modeled quadrotor dynamics.The physics dataset supplies symbolic priors rather than labeled targets at those points.Source: C. Physics-informed Neural Networks · p. 3
  • The MLP takes control input and the previous horizon’s initial state; its output is passed through ideal dynamics and autograd to compute physics loss.Recorded observations provide the dataset used to fit the data loss.Source: C. Physics-informed Neural Networks · p. 3
  • Disturbed simulation data pairs odometry state-control measurements with ground-truth state derivatives, and the physics and data losses are summed for weight updates.The paper varies the relative sizes of the physics and data datasets in later evaluation.Source: C. Physics-informed Neural Networks · p. 3

IV. SYSTEM DESIGN

RAMP-Net uses a switch to distinguish training from inference. The design describes nominal quadrotor dynamics and robustness to uncertain dynamics in the MPC context.

IV. SYSTEM DESIGN

  • The switch S_t toggles training with S_t = ON and inference with S̄_t = ON.The architecture is presented as a logical view of RAMP-Net.Source: §IV · p. 3
  • The system design describes how the nominal quadrotor dynamics support robust behavior under uncertain dynamics in MPC.The passage places this description after summarizing the nominal dynamics model.Source: §IV · p. 3

A. Quadrotor Nominal Dynamics

The nominal quadrotor model represents its 13-state configuration and uses four rotor thrusts as control inputs. It defines collective thrust, body torque, and drag-related quantities for the dynamics.

A. Quadrotor Nominal Dynamics

  • The 13-dimensional state x comprises position p, attitude q, linear velocity v, and body-frame angular velocity ωB.Position is expressed in world coordinates, and attitude is represented by a unit quaternion.Source: A. Quadrotor Nominal Dynamics · pp. 3–4
  • The model uses four rotor thrusts T_i as control inputs and introduces them as the inputs for the nominal dynamics.The four thrusts are indexed i ∈ {0, 1, 2, 3}.Source: A. Quadrotor Nominal Dynamics · p. 4
  • The thrust-to-torque expressions use arm length L and drag constant kdrag; T_B denotes collective thrust and τ_B denotes body torque.The arm length is for the quadrotor’s X configuration.Source: A. Quadrotor Nominal Dynamics · p. 4

B. Robustness through infusing Parametric Uncertainty

RAMP-Net modifies the system dynamics by adding parametric uncertainty to the nominal quadrotor model. The uncertainty term represents deviations from nominal state dynamics and enters the physics loss.

B. Robustness through infusing Parametric Uncertainty

  • The system dynamics f are modified by adding a parametric uncertainty term to the nominal dynamics.The passage introduces this modification before specifying the uncertainty term.Source: B. Robustness through infusing… · p. 4
  • The additive uncertainty represents deviations from nominal quadrotor state dynamics and can be sampled from standard distributions such as N(μ, σ).μ and σ denote the distribution mean and standard deviation.Source: B. Robustness through infusing… · p. 4
  • The additive parametric uncertainty is included in the symbolic prior and influences the physics loss Lp.Source: B. Robustness through infusing… · p. 4

C. Adaptation to Residual Non-Parametric Uncertainty

RAMP-Net uses data-driven adaptation to account for environmental effects that cannot be parameterized using quadrotor states. A dataset collected under simulated disturbances supplies the data loss, which is combined with the physics loss for inference.

C. Adaptation to Residual Non-Parametric Uncertainty

  • Winds, disturbances, and frictional effects on rotor inertia are treated as residual effects that cannot be parameterized using quadrotor states.The method adds a non-parametric term to the dynamics to represent these effects.Source: C. Adaptation to Residual… · p. 4
  • The simulated dataset includes zero-mean Gaussian rotor-thrust noise with standard deviation 1, second-order polynomial aerodynamic drag, and asymmetric motor-voltage noise.Source: C. Adaptation to Residual… · p. 4
  • The PINN is trained with physics and data losses to identify perceived dynamics for inference.The dataset’s y_k values are logged as the integrand in Eqn. (13).Source: C. Adaptation to Residual… · p. 4

A. Methodology

The methodology tests RAMP-Net in simulated quadrotor trajectory tracking and compares it with several MPC baselines. The implementation combines a PINN with nominal dynamics and evaluates it against learned residual-dynamics methods and oracle-based references.

A. Methodology

  • The PINN used a four-layer, 128-neuron MLP, trained for 2,000 epochs with early stopping and L-BFGS on 5k samples.Parametric uncertainty was sampled from a zero-mean normal distribution with unit diagonal covariance.Source: A. Methodology · p. 4
  • Closed-loop tracking tests ran in Gazebo using the AscTec Hummingbird model from RotorS.The symbolic nominal dynamics were implemented in ACADOS, which wraps CasADi.Source: A. Methodology · pp. 4–5

B. Comparitive Schemes •

  • KNODE-MPC learns residual dynamics with Neural ODEs, while GP-MPC models them as a Gaussian Process posterior given nominal dynamics.These are the two regression-based schemes compared with RAMP-Net.Source: B. Comparitive Schemes • · p. 5
  • The Ideal baseline uses nominal dynamics with an oracle residual, whereas Nominal uses no data-driven correction.Source: B. Comparitive Schemes • · p. 5
  • The PID baseline assumes an oracle of the uncertainties and a perfectly fine-tuned adaptive PID controller.Source: B. Comparitive Schemes • · p. 5

C. Impact of data on training performance

The section measures training sensitivity to data-skewness and compares tracking performance across regression-based MPC methods. It reports normalized DTW-error reductions and shows that results depend on data availability and trajectory type.

C. Impact of data on training performance

  • Training performance is evaluated by DTW trajectory similarity and total training time, with data-skewness defined as |D|/|P|.The ratio compares regular data points with physics collocation points.Source: C. Impact of data on training… · p. 5
  • At |D|/|P| = 1/16, three of four DTW errors exceed nominal; the authors attribute this to insufficient PINN expressive power with very little data.Source: C. Impact of data on training… · p. 5
  • At zero skew, RAMP-Net reports ∼61% and ∼59% lower errors for 3m and 4m circles than KNODE-MPC, with ∼11% faster convergence.Errors fell for both methods as skew decreased, with a larger reduction for RAMP-Net; subsequent evaluations used |D|/|P| = 1.Source: C. Impact of data on training… · p. 5, Figure 4 · p. 5

D. Comparison with SOTA Regression based Methods

  • RAMP-Net's average normalized DTW errors were 46.07% for circles and 47.41% for lemniscates, versus 53.63% and 80.65% for KNODE-MPC and 78% and 92.79% for GP-MPC.The methods were trained on circles of radii 3m and 4m; evaluation also included lemniscates.Source: D. Comparison with SOTA Regression… · p. 5
  • 7.8%−43.2% and 8.04%−61.5% average DTW-error reductions are reported against MPC and GP-MPC, respectively, in Fig. 5.Source: Figure 5 · p. 6

E. Robustness Analysis for higher flight speeds

RAMP-Net reports lower tracking error and faster convergence than regression-based MPC in quadrotor tracking, with the least error among compared methods as flight speed increases. Its PINN also has lower execution latency than standard numerical integrators.

E. Robustness Analysis for higher flight speeds

  • RAMP-Net had the least tracking error, within ∼10× of ideal, with sub-linear error growth as maximum radial speed increased.The comparison used a 3 m radius and 1 m height; the environment included dynamic disturbances, wind, and translational drag.Source: E. Robustness Analysis for higher… · p. 6, Table II · p. 6

F. Latency comparison with standard integrators

VI. CONCLUSION

  • ∼60% lower tracking error while training than a SOTA regression-based method, with ∼11% faster convergence, was reported for a Hummingbird quadrotor in Gazebo.The conclusion also reports reduced tracking error across speeds of 0.5–12.5 m/s compared with two regression-based MPC methods, and three standard controllers.Source: §VI · p. 6

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