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Contact-Aware Incremental Model Predictive Control for an Underactuated Aerial Manipulator

Darwin Liu, Tamas Keviczky, Sihao Sun

arXiv:2609.11661v1cs.RO

TL;DR

The paper addresses whether an underactuated quadrotor with a simple rigid arm can achieve robust simultaneous end-effector pose and contact-force tracking without dedicated force sensing. It combines contact-aware NMPC with whole-body INDI and validates the controllers in simulation and aerial-writing experiments, demonstrating robust five-DoF pose and force tracking across contact conditions.

  • Problem

    Underactuated aerial manipulators face nonlinear coupled dynamics and frictional or aerodynamic disturbances, while existing solutions often require fully actuated platforms or dedicated force/torque sensing.

  • Method

    The framework combines contact-aware NMPC for pose and normal-force tracking with a whole-body INDI inner loop for disturbance robustness.

  • Results

    Robust simultaneous five-DoF end-effector pose and contact-force tracking is demonstrated with a quadrotor-based platform and simple one-DoF rigid arm across varied surfaces, friction, and wind conditions.

  • Takeaways & Limitations

    The results support aerial writing with a standard underactuated quadrotor without a fully actuated platform, complex arm, or dedicated force/torque sensing.

Abstract

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We present a robust contact-aware control framework for aerial writing on an underactuated platform. The framework combines nonlinear model predictive control (NMPC) for accurate end-effector position and normal-force tracking at small reference penetration depths, with consistent performance across controller tunings, with whole-body incremental nonlinear dynamic inversion (INDI) for robustness to frictional and aerodynamic disturbances during contact. The proposed controllers are validated on a quadrotor-based aerial manipulator with a rigid, single-link, one-degree-of-freedom (DoF) arm in simulation and real-world experiments. The aerial writing experiments span vertical and inclined surfaces, multiple reference forces, different friction conditions, and wind disturbances. The results demonstrate that robust simultaneous five-DoF end-effector pose and contact-force tracking is achievable on a standard underactuated quadrotor with a simple, rigid, single-link arm, without requiring a fully actuated platform, a complex arm, or dedicated force/torque sensing.

I. INTRODUCTION

The paper targets robust simultaneous pose and contact-force control for underactuated aerial manipulation, where nonlinear dynamics, disturbances, and hardware simplicity constrain performance. It combines contact-aware NMPC with whole-body INDI and validates the approach in varied aerial-writing conditions.

  • I. INTRODUCTION: Underactuated aerial manipulators simplify hardware but make accurate control difficult because of nonlinear dynamics, friction, aerodynamic effects, and base–arm coupling.NMPC handles constraints and coupled dynamics but lacks fast disturbance rejection during frictional or windy contact.
  • I. INTRODUCTION: INDI uses onboard measurements to compensate for model uncertainty and external disturbances, but prior aerial-manipulator applications lacked a whole-body formulation and contact-rich evaluation.The paper positions whole-body INDI as an extension beyond approaches that account only for UAV base dynamics.
  • I. INTRODUCTION: Dedicated force/torque sensing and fully omnidirectional bases increase hardware complexity, motivating pose-and-force tracking with a standard underactuated quadrotor and simple arm.Prior underactuated alternatives were restricted to horizontal interactions, while sensor-based approaches incur mass, cost, and calibration penalties.
  • I. INTRODUCTION: The paper combines contact-aware NMPC with a cascaded whole-body INDI controller to improve robustness against unmodeled disturbances during contact.The controllers are evaluated on a quadrotor-based platform with a one-DoF non-compliant arm.
  • I. INTRODUCTION: The proposed controllers are validated in simulation and real-world aerial writing on vertical and inclined surfaces with different friction properties and external wind disturbances.These experiments are illustrated through chalk and marker writing scenarios, including a vertical whiteboard under wind.

B. Multibody Kinematics and Dynamics

The aerial manipulator is modeled as a floating multibody system comprising a quadrotor base and a one-DoF hinged arm. Its generalized coordinates, velocities, and constraint-consistent dynamics provide the model used for control and end-effector tracking.

  • B. Multibody Kinematics and Dynamics: The model contains two rigid bodies—the quadrotor base and arm with end-effector—connected by a one-DoF hinge joint.The joint rotates about the quadrotor body yB axis.
  • B. Multibody Kinematics and Dynamics: This morphology enables five-DoF end-effector pose tracking, leaving only rotation about the arm axis uncontrolled.The remaining uncontrolled rotation follows directly from the single-link joint structure.
  • B. Multibody Kinematics and Dynamics: The generalized coordinates comprise the base position, base-orientation quaternion, and arm joint angle.The base position is represented in three dimensions and orientation by qb = [qw, qx, qy, qz]^T.
  • B. Multibody Kinematics and Dynamics: Generalized velocities combine base linear velocity, body-frame base angular velocity, and joint velocity, while quaternion propagation maps angular velocity to coordinate derivatives.The quaternion propagation matrix is denoted Λ(qb).
  • B. Multibody Kinematics and Dynamics: Constraint-consistent multibody dynamics are derived with the projected Newton–Euler method using mass, Coriolis/centrifugal, gravity, and actuation terms.The actuation includes thrust, quadrotor torque, and arm torque; the end-effector Jacobian and contact force enter the formulation.

C. Task-Space Planning

The task-space planner specifies end-effector pose and contact-force references, then supplies a smooth configuration-space trajectory and approximate velocities for NMPC tracking. The baseline NMPC optimizes multibody tracking objectives without modeling environmental interaction forces.

  • Task-space reference generation: The planner specifies end-effector pointing direction, position, and contact force, while accounting for desired-force effects on required thrust direction.It derives an approximate configuration-space trajectory and generalized velocities for tracking.
  • Baseline NMPC: The baseline NMPC extends quadrotor agile-flight NMPC to multibody dynamics with end-effector position, attitude, and joint-tracking costs.It updates the dynamic constraints for the aerial manipulator model.
  • Baseline NMPC: The MPC state contains generalized coordinates and velocities, while inputs comprise propeller speeds and commanded arm joint position.The arm uses position control rather than torque control because of hardware constraints.
  • Baseline NMPC: Propeller speeds are mapped to body forces and torques through a control-allocation matrix that accounts for each propeller’s thrust and drag.The allocated quantities are expressed in the body frame for the multibody dynamics.
  • Baseline NMPC: The receding-horizon optimization returns desired rotor-speed commands and arm joint angles at every timestep.Stage and terminal tracking errors are weighted across base, arm, end-effector, and control-effort terms.
  • Baseline NMPC: Because the baseline NMPC does not model environmental interaction forces, its effect on force tracking is evaluated separately.

B. Contact-Aware NMPC Formulation

The contact-aware NMPC incorporates a surface model, virtual penetration, and force-aware dynamics and costs to plan non-compliant contact. Reference-based activation smoothly introduces force commands as the planned trajectory approaches the surface.

  • Contact modeling: CA-NMPC models the contact surface with a point and outward normal, then penalizes deviation between predicted and desired contact force.The force cost uses a weighted error between the predicted contact force and its reference.
  • Contact modeling: Virtual penetration corrects the estimated normal force using signed surface distance and a force-per-length gain, without representing physical displacement or stiffness.The gain is tuned to balance tracking accuracy against oscillatory contact behavior.
  • Contact activation: Reference penetration depth activates the planned contact force smoothly across a 2 cm transition boundary layer as the reference trajectory reaches the surface.The activation variable rises from 0 to 1 and is planned from the reference trajectory and known surface information.
  • Contact dynamics: The force entering the MPC dynamics uses the activated reference force as a feedforward term rather than a spring model for non-compliant interaction.This predicts the desired steady-state force independently of end-effector position, while closed-loop correction addresses actual force error.

C. External Force Estimator

The external-force estimator infers interaction force from measured acceleration and estimated thrust under a quasi-static arm assumption. It then separates the estimate into normal and in-plane components.

  • Estimator assumptions: The estimator assumes quasi-static arm motion during interaction, allowing arm dynamics to be neglected.
  • Force decomposition: External force is estimated from total mass, filtered body-frame accelerometer specific force, and thrust inferred from filtered propeller speeds.The resulting inertial-frame estimate is decomposed into normal and in-plane force magnitudes.

D. Cascaded Whole-Body INDI Controller

The cascaded whole-body INDI controller robustifies the NMPC trajectory by using incremental dynamics and a two-layer position–attitude structure. Its validity depends on fast sampling and is challenged at contact onset by rapidly changing contact force.

  • Incremental inversion: Whole-body INDI tracks the optimal NMPC trajectory by linearizing aerial-manipulator dynamics around previously sampled states, inputs, and contact forces.The incremental formulation uses a first-order expansion and filtered sampled variables.
  • Robustness: The INDI law requires no model of external force, Coriolis, or gravity and tolerates inertia mismatch, so only a coarse inertia estimate is needed.Rigid-body inertial parameters are measured from hardware, while the inertia matrix depends on arm configuration.
  • Validity and contact: INDI stability relies on sampling faster than the true dynamics change, but contact onset violates this assumption when contact force rises within one sampling interval.Achievable control frequency and accelerometer noise also bound practical validity.
  • Cascaded structure: Because direct inversion would command inadmissible thrust along underactuated directions, the controller cascades position INDI into tilt-prioritized attitude INDI.Position INDI produces a commanded thrust vector; the attitude layer generates desired angular acceleration.
  • Controller architecture: The proposed controller is summarized as CA-NMPC-CINDI, combining contact-aware NMPC with the cascaded whole-body INDI structure.

1) Position INDI:

The cascaded control structure converts the NMPC translational command into thrust and uses a tilt-prioritized attitude controller before attitude INDI generates body torque. It accounts for multibody inertial effects while leaving joint torque control unavailable in the hardware implementation.

  • Position INDI: The position INDI converts the NMPC desired translational acceleration into a commanded thrust vector, then a tilt-prioritized attitude controller generates the desired angular acceleration.This cascaded structure avoids physically inadmissible thrust components along the quadrotor’s underactuated directions.
  • Filtering: The INDI control law uses filtered propeller-speed measurements, with 3 Hz filtering for the outer layer and 6 Hz filtering for the inner layer.The filtering applies separately to the position and attitude INDI layers.
  • Attitude INDI: The attitude INDI takes desired angular acceleration as a virtual input and computes the corresponding desired body torque for the inner loop.The formulation includes inertial torque contributions from quadrotor linear accelerations and arm joint accelerations.
  • Hardware interface: Joint torque commands are not sent directly because of hardware limitations; the arm instead receives desired joint positions from the MPC.Rotor speed commands are still generated from the commanded thrust and body torque.

IV. EXPERIMENTS

The experiments test contact-aware control against simulation baselines and examine force-model sensitivity, five-DoF tracking, and disturbance rejection. Evaluation covers friction, wind, surface orientation, and the contributions of the cascaded INDI layers.

  • Research questions: The experiments isolate how contact modeling affects normal-force error and sensitivity to reference penetration depth.The study also compares full cascaded INDI with an attitude-only variant under friction and lateral wind.
  • Research questions: The evaluation asks whether the proposed design enables simultaneous five-DoF end-effector pose and force tracking on surfaces with different orientations.It also compares the method with state-of-the-art contact-aware controllers for similar underactuated aerial manipulators.
  • Experimental setup: The controller variants are evaluated in simulation and real-world experiments using spring-damper contact, smoothed Coulomb friction, and a rigid simulated wall.The real-world setup runs NMPC at 100 Hz and INDI layers at 500 Hz.
  • Experimental setup: The real platform is a 0.808 kg quadrotor with a 20.5 cm, 0.02 kg arm link and tilted rotors for additional yaw authority during sliding.The contact surface pose is assumed known a priori and supplied as fixed controller parameters.

B. Simulation Results

Simulation results show that contact-aware NMPC maintains accurate force tracking at small reference penetration depths across tunings and forces, while INDI robustification improves tracking under friction.

  • Reference Penetration Depth Sensitivity: Both CA-NMPC-CINDI controllers achieve accurate force tracking at small reference penetration depths across controller tunings and 1 N or 2 N reference forces.The baselines instead have tuning-dependent optimal penetration depths, making accurate tracking sensitive to controller parameters.
  • Tracking Accuracy Comparison: With simulated friction, CA-NMPC-CINDI achieves significantly better figure-eight tracking than CA-NMPC-SOTA and HIC.CA-NMPC-SOTA crashes during sliding, while HIC tracks force relatively accurately but has substantially worse end-effector position tracking.
  • Tracking Accuracy Comparison: Without friction, all three compared controllers achieve accurate force and pose tracking in the figure-eight simulation.The comparison uses a 1 N normal-force reference on a vertical surface.
  • Force estimation: The proposed real-world validation uses onboard estimated normal and tangential contact forces rather than an onboard force sensor.The normal-force estimate is separately validated against an environmental force/torque sensor on vertical and 45° inclined surfaces.

1) Vertical Surface:

Real-world writing experiments show that INDI-augmented controllers complete vertical and inclined writing tasks, while the full cascaded controller improves position robustness under wind. Force tracking remains consistent despite noisier estimates.

  • Vertical Surface: The INDI-augmented controllers complete vertical figure-eight writing and maintain comparable tracking performance, whereas pure CA-NMPC crashes shortly after sliding begins.Both INDI variants track the desired force consistently, with CA-NMPC-CINDI showing slightly more consistent tracking error.
  • Inclined Surface: On the 45° inclined whiteboard, the one-DoF joint enables writing in arbitrary directions, while pure CA-NMPC is excluded because it cannot produce reliable, consistent writing.The exclusion is attributed to sensitivity to aerodynamic downwash on the surface.
  • Tracking comparison: Across writing experiments, CA-NMPC-ATT-INDI and CA-NMPC-CINDI show similar attitude and force tracking, but CA-NMPC-CINDI has lower and less variable in-plane position error.Table II reports force, in-plane position, and attitude RMSE and standard deviation across vertical and inclined trajectories.
  • Wind Disturbance Test: Under approximately 5 m/s lateral wind, CA-NMPC-CINDI limits peak end-effector tracking error to 11 cm versus 20 cm for CA-NMPC-ATT-INDI, a reduction of approximately 45%.Normal-force tracking remains consistent under wind, although the estimate becomes noisier.

2) Aerial Writing on a Blackboard:

Blackboard writing provides a demanding contact test because chalk friction is larger and less smooth, while the proposed controllers maintain force tracking and improve pose tracking during difficult sliding segments.

  • 2) Aerial Writing on a Blackboard:: The experiment uses chalk on a blackboard, whose larger, less smooth friction forces make writing more challenging than whiteboard writing.The difficulty arises from transitions between static and kinetic friction regimes.
  • 2) Aerial Writing on a Blackboard:: 8 cm peak end-effector tracking error for CA-NMPC-CINDI versus 13 cm for CA-NMPC-ATT-INDI during high-acceleration segments, while both maintain accurate contact-force tracking.The CINDI variant also reduces the static tracking offset caused by friction.
  • 2) Aerial Writing on a Blackboard:: The aerial manipulator demonstrates simultaneous five-DoF end-effector pose and contact-force tracking without a compliant end-effector or omnidirectional base.The validation includes simulation and real-world writing experiments on surfaces with different orientations and friction properties, including wind disturbances.
  • 2) Aerial Writing on a Blackboard:: The contact-aware extension supports accurate force tracking at small reference penetration depths and remains robust to controller tuning compared with predictive and impedance controllers.The comparison is made against existing controllers on a similarly underactuated platform.
  • 2) Aerial Writing on a Blackboard:: CA-NMPC alone became unstable during sliding, whereas adding the INDI outer position loop improved tracking under lateral wind and high-friction interaction.Friction-induced torques on the quadrotor base were not rejected by NMPC alone.
  • 2) Aerial Writing on a Blackboard:: The force estimator was sufficient for these experiments but is sensitive to unmodeled disturbances, motivating future dedicated sensing or advanced estimation.This limitation is explicitly identified as a direction for future work.
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