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Taylor-Informed Predictive Cost Adaptive Control for Quadrotors with Online Gravity-Trim Adaptation

Tam W. Nguyen

arXiv:2609.03351v1eess.SYcs.SY

TL;DR

Payload changes can make fixed nominal gravity compensation produce steady-state error, and sparse hover-linear quadrotor PCAC loses prediction accuracy during larger excursions. The paper adds Taylor-informed predictors and online mass and gravity-trim estimation to PCAC. In simulations, higher-order models improved tracking and prediction relative to the first-order model while preserving the standard optimization framework.

  • Problem

    Payload changes alter hover thrust, making fixed nominal gravity compensation produce steady-state error, while existing quadrotor PCAC uses a sparse hover-linear predictor whose accuracy degrades during larger excursions from hover.

    Source: §I · p. 1

  • Method

    The method uses first- through third-order Taylor expansions to form sparse sampled-data predictors and estimates mass and gravity trim from identified vertical dynamics.

    Source: §I · p. 1, §V · pp. 2–3

  • Results

    Second- and third-order models improved tracking and one-step prediction relative to the first-order model, and vertical dynamics recovered mass and trim after an abrupt payload change.

    Source: B. Case 1: Abrupt Mass Change · p. 5, C. Case 2: Aggressive Helix Tracking · p. 5, §IX · p. 6

  • Takeaways & Limitations

    The results show that local higher-order structure can improve PCAC away from hover without changing its optimization framework.

    Source: §IX · p. 6

    Future work will address motion-capture experiments and globally nonsingular attitude representations.

    Source: §IX · p. 6

Abstract

from arXiv · show

This paper develops Taylor-informed predictive cost adaptive control (PCAC) for quadrotors with online gravity-trim adaptation. First-, second-, and third-order expansions of the nonlinear dynamics about nominal hover define sparse sampled-data dictionaries for row-wise recursive least-squares identification with variable-rate forgetting. At each step, the identified predictor is linearized at the current state, and its Jacobian is fixed over the prediction horizon. The identified vertical dynamics also estimate the vehicle mass and gravity-trim input, eliminating fixed nominal gravity compensation. Simulations with an abrupt payload change and aggressive helix tracking show that the higher-order predictors improve prediction and tracking while preserving the standard PCAC formulation.

I. INTRODUCTION

Quadrotor payload changes and disturbances motivate adaptive control, while existing quadrotor PCAC uses a sparse hover-linear predictor whose accuracy degrades farther from hover. The paper introduces Taylor-informed predictors and online gravity-trim adaptation within the standard PCAC optimization.

I. INTRODUCTION

  • Payload changes alter hover thrust, so fixed nominal gravity compensation causes steady-state error when actual mass differs from nominal mass.Existing quadrotor PCAC uses a sparse hover-linear predictor whose accuracy degrades during larger excursions from hover.Source: §I · p. 1
  • First-, second-, and third-order Taylor expansions about nominal hover define sparse sampled-data dictionaries for row-wise recursive least-squares identification.Identified vertical dynamics provide online mass and gravity-trim estimates.Source: §I · p. 1
  • The predictor is linearized at each control step and its Jacobian is held fixed over the horizon, yielding an affine model for standard PCAC optimization.Source: §I · p. 1
  • Simulations evaluate an abrupt payload change and aggressive helix tracking; higher-order predictors improve tracking and one-step prediction, while trim adaptation removes fixed nominal compensation.Source: §I · p. 1

II. PROBLEM SETUP

  • The setup defines inertial and body-fixed frames, state and input variables, and quadrotor dynamics parameterized by mass, inertia, and gravity.Source: §II · p. 1

III. NOMINAL HOVER COORDINATES AND SAMPLED DYNAMICS

The nominal hover input balances thrust for a nominal mass, but a mass mismatch means the nominal hover pair is not an equilibrium. Shifting the state and input exposes this mismatch as constant vertical acceleration in sampled dynamics.

III. NOMINAL HOVER COORDINATES AND SAMPLED DYNAMICS

  • With nominal mass m0, the nominal hover state and input are not an equilibrium when the actual mass differs from m0.Source: §III · p. 2
  • Shifting state and input by the nominal hover pair makes the mass mismatch appear as a constant vertical acceleration.Source: §III · p. 2
  • Under zero-order hold, the shifted full-state-feedback dynamics are represented by a sampled-data flow map.Source: §III · p. 2

IV. TAYLOR EXPANSION ABOUT THE NOMINAL HOVER POINT

The nonlinear shifted dynamics are approximated about nominal hover using Taylor polynomials of degrees one through three. Retained terms include position, Euler-kinematic, and translational and angular-acceleration contributions, with higher-degree terms capturing additional nonlinear structure.

IV. TAYLOR EXPANSION ABOUT THE NOMINAL HOVER POINT

  • The shifted input separates thrust and torque components, with nominal torque zero and the thrust expressed relative to nominal hover thrust.Source: §IV · p. 2
  • The dynamics are expanded about the nominal hover point using degree-one, degree-two, or degree-three Taylor polynomials, with omitted terms of total degree greater than D.Source: §IV · p. 2
  • The position row is exact across expansion degrees, while the other rows depend on the selected degree.Source: §IV · p. 2
  • The expansion retains Euler-kinematic and translational-acceleration terms, alongside angular-acceleration terms determined by angular velocity, torque, and inertia.Source: §IV · p. 2

V. SPARSE TAYLOR-INFORMED SAMPLED-DATA DICTIONARY

The Taylor terms define sparse row-wise sampled-data dictionaries, with each state row using regressors through the selected degree. Degree two adds coupling terms and degree three adds cubic terms to the sparse hover-linear model.

V. SPARSE TAYLOR-INFORMED SAMPLED-DATA DICTIONARY

  • The Taylor terms define a row-wise sampled-data dictionary, with Table I listing complete degree-one regressors and Table II listing additional degree-two and degree-three monomials.Source: §V · p. 2, Table I · p. 3, Table II · p. 3
  • Each row's regressor vector concatenates table entries through the chosen degree, removes duplicates while preserving first occurrence, and omits empty entries.Source: §V · p. 2
  • Degree one is the sparse hover-linear model with a vertical trim intercept; degree two adds thrust–attitude, Euler-kinematic, and gyroscopic couplings, and degree three adds cubic terms.Source: §V · p. 3

VI. TAYLOR-INFORMED RECURSIVE IDENTIFICATION

The section applies recursive least squares independently to each state row and adapts forgetting using a variable-rate rule. Coefficients in the vertical-velocity row support online mass and gravity-trim estimation.

VI. TAYLOR-INFORMED RECURSIVE IDENTIFICATION

  • RLS is applied independently to each state row for fixed Taylor degree D ∈ {1, 2, 3}.Each transition supplies an update for the identified model.Source: §VI · p. 3
  • The forgetting factor is computed from the stacked error using the variable-rate forgetting rule, while λ_k = 1 disables forgetting.Source: §VI · p. 3
  • The vertical-velocity row’s constant and input coefficients feed mass and trim-thrust estimates after the estimated thrust coefficient is projected onto an admissible positive interval.Source: §VI · p. 3

VII. PREDICTIVE COST ADAPTIVE CONTROL

At each sample, the identified map is linearized at the current operating point, and that Jacobian is held fixed across the prediction horizon. The resulting lifted model supports a trim-centered PCAC optimization with state constraints.

VII. PREDICTIVE COST ADAPTIVE CONTROL

  • The identified map is linearized at the current operating point, and its Jacobian is held fixed for every step across the prediction horizon.Source: §VII · p. 3
  • Predicted inputs are parameterized as deviations from estimated trim, which shifts the model’s constant term to d_D,k := c_D,k + B_D,k ˆu_trim,k.Source: §VII · pp. 3–4
  • Input variation is represented by successive input-deviation differences over the horizon, with initialization following the PCAC timing convention.Source: §VII · p. 4
  • The lifted model is used to solve the PCAC optimization for the shifted reference, subject to the stated state constraints and positive semidefinite cost weights.The method applies the first optimized deviation at the next step and uses the constraint notation of.Source: §VII · p. 4

A. Simulation Setup

The simulations compare Taylor degrees under common plant, tuning, constraint, and horizon settings in two cases: an abrupt payload change and aggressive three-dimensional helix tracking. The implementation uses sampled-data identification and solves PCAC after linearizing the identified map.

A. Simulation Setup

  • The exact nonlinear plant is integrated by fixed-step RK4 with 10 substeps per sampling interval and zero-order-held input; 12 scalar RLS estimators identify the sampled-data map.They start from the exact zero-order-hold discretization of the nominal hover-linear model and share the VRF factor.Source: A. Simulation Setup · p. 4
  • At each sample, the identified map is linearized and PCAC is solved by the quadprog active-set algorithm, with at most 180 iterations and constraint tolerance 10^-7.Source: A. Simulation Setup · p. 4
  • The cases use common simulation parameters, and Tables IV–V report performance metrics based on the position components of one-step prediction error.The position-tracking response after the mass change is shown in Fig. 1; the provided table captions identify the performance scopes.Source: Table III · p. 4, A. Simulation Setup · p. 4, Figure 1 · p. 5
  • Case 1 tests mass and trim estimates under an abrupt mass change, while Case 2 tests higher-order predictors during aggressive three-dimensional helix tracking at constant mass.Both cases use noise-free full-state measurements, zero applied force and torque disturbances, and common plant, tuning, constraints, and horizon settings.Source: A. Simulation Setup · p. 4

B. Case 1: Abrupt Mass Change

In the abrupt-mass-change case, both higher-order predictors reduce tracking error, with degree two best on position RMSE and degree three best on terminal mass and trim estimates.

B. Case 1: Abrupt Mass Change

  • The mass changes from 4.34 kg to 5.60 kg at t = 13 s, and the case runs through t = 28 s.The tracking response, online estimates, and VRF factor are shown for this change.Source: B. Case 1: Abrupt Mass Change · p. 5, Figure 2 · p. 5
  • Both higher-order predictors reduce tracking error; degree two gives the lowest position RMSE, while degree three gives the most accurate terminal mass and trim estimates.Source: B. Case 1: Abrupt Mass Change · p. 5

C. Case 2: Aggressive Helix Tracking

In aggressive helix tracking, degrees two and three provide similar tracking accuracy, while degree three has the lowest one-step prediction error and control variation. The conclusion reports improved tracking and prediction over the first-order model while retaining the standard PCAC optimization framework.

C. Case 2: Aggressive Helix Tracking

IX. CONCLUSION

  • Second- and third-order models improve tracking and one-step prediction over the first-order model, while vertical dynamics recover mass and trim after an abrupt payload change.These results show that local higher-order structure can improve PCAC away from hover without changing its optimization framework.Source: §IX · p. 6
  • Motion-capture experiments and globally nonsingular attitude representations are identified as future work.Source: §IX · p. 6
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