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CBF-Based Control Allocation for Omnidirectional Tiltrotor Aerial Robots

Seobeom Lee, Hyondong Oh

arXiv:2610.22057v1cs.RO

TL;DR

Servo transients can make a desired wrench infeasible during reconfiguration, and prior allocation approaches leave open how to preserve the request despite configuration mismatch. The paper develops a CBF allocator that constrains servo motion while a differential stage tracks the wrench. In simulation, its measured-angle implementation maintains feasibility at every stored disturbed-cartwheel sample and has lower tracking error than the compared allocators in that maneuver.

  • Problem

    Servo transients can make a desired wrench infeasible even when the target pose is kinematically admissible, leaving the question of how to preserve the request during configuration mismatch.

    Source: §I · p. 1

  • Method

    A CBF allocator tightens twelve inverse-allocation thrust margins with an ellipsoidal reserve, uses their complete derivatives in a servo-rate QP, and follows with differential thrust allocation for tracking.

    Source: §I · p. 1

  • Results

    In the disturbed cartwheel, the measured-angle proposed method was feasible at all 1751 stored samples, while GA, ADA, and NDA violation rates were 100%, 80.7%, and 60.8%, respectively.

    Source: C. Disturbed-Cartwheel Wrench… · p. 8, §IX · p. 10

  • Takeaways & Limitations

    The study reports lower tracking error than GA, ADA, and NDA in the disturbed cartwheel, while circular tracking exhibits an attitude tradeoff.

    Source: §IX · p. 10

    Direct validation of the nominal-trajectory certificate requires records of the persistent nominal trajectory and explicit configuration-mismatch reserve.

    Source: A. Setup and Scenarios · p. 7

Abstract

from arXiv · show

Actively tilting aerial robots can generate six-dimensional wrenches, but slow servo response can make a desired wrench infeasible during reconfiguration. This paper develops a control barrier function (CBF) allocator that preserves wrench feasibility under bounded servo-configuration mismatch. An exact equivalent-wrench identity connects the thrust required at nominal and actual configurations. For a nonsingular six-channel allocation map, inverse allocation reduces robust wrench containment to twelve thrust margins tightened by an ellipsoidal reserve. Their complete derivatives yield a six-variable servo-rate quadratic program, followed by differential thrust allocation for tracking. A conditional forward-invariance result establishes actual-configuration feasibility under reserve coverage, feasible initialization, and admissible CBF inputs. A complementary MuJoCo study evaluates a measured-angle implementation against geometric, augmented differential, and normalized differential allocation in circular and cartwheel maneuvers, with and without disturbances. The proposed implementation achieves lower position error in all four recorded scenarios, with an attitude tradeoff in circular tracking. During the disturbed cartwheel, the desired wrench is feasible at all 1751 stored samples of continuous rotation; baseline violation rates range from 60.8% to 100%. These comparisons evaluate sampled feasibility; validation of the nominal-trajectory certificate requires nominal-state and reserve-coverage records.

I. INTRODUCTION

Actively tilting robots must coordinate fast thrust changes with slower servo motion, since servo transients can make a desired wrench infeasible during reconfiguration. The paper constrains servo motion to preserve a requested wrench neighborhood despite configuration mismatch.

A. Actuation Geometry and Dynamic Allocation

  • Actively tilting platforms reorient thrust directions in flight, so allocation must coordinate fast thrust changes with slower servo motion.Differential allocators use both servo motion and propeller acceleration for wrench-rate control, while instantaneous feasibility concerns thrust at fixed angles.Source: §I · p. 1, A. Actuation Geometry and Dynamic… · p. 2

I. INTRODUCTION

  • The CBF allocator preserves a specified wrench request and its uncertainty neighborhood inside the instantaneous thrust-limited set despite configuration mismatch.It reduces robust containment to twelve thrust margins and constrains servo motion while a differential stage tracks the request.Source: §I · p. 1

B. Barrier Functions and Actuator Feasibility

  • Unlike readiness barriers that preserve a wrench-rate authority measure, this method certifies a desired-wrench neighborhood inside the instantaneous thrust-limited set.The square inverse map makes the containment test exact, and an equivalent-wrench identity transfers nominal margins to actual-angle feasibility.Source: B. Barrier Functions and Actuator… · p. 2

A. Rigid-Body Dynamics and Tilt Geometry

  • The allocator works with body-frame wrenches, and inertial force requests must be rotated before applying the allocation map.The force and moment dynamics include external force and moment disturbances in their respective frames.Source: A. Rigid-Body Dynamics and Tilt Geometry · p. 2

B. Instantaneous Wrench Feasibility

At fixed rotor angles, instantaneous wrench feasibility is tested through the inverse of a nonsingular square allocation map. The proposed approach distinguishes nominal servo commands from actual angles, then uses nominal margins to support actual-configuration feasibility.

B. Instantaneous Wrench Feasibility

  • For a nonsingular square map, the thrust box maps to a parallelotope with twelve facets, and the inverse-map test gives every feasibility inequality without sampling wrench directions.The exact reduction relies on the square nonsingular map; redundant fixed-configuration maps require different treatment of internal thrust freedom.Source: B. Instantaneous Wrench Feasibility · pp. 2–3

C. Nominal and Actual Servo Configurations

  • Servo lag permits actual angles and rates to differ from the nominal trajectory and its commanded rate.The allocator distinguishes nominal required thrust, actual-configuration required thrust, generated thrust, and commanded thrust for their separate roles.Source: C. Nominal and Actual Servo… · p. 3

IV. FROM GEOMETRIC TO DIFFERENTIAL ALLOCATION

  • The baseline methods distinguish actuator endpoint selection, differential tracking, and rate-limit enforcement; the proposed method adds a containment objective.Source: §IV · p. 3

A. Geometric Allocation

  • The unconstrained minimum-norm geometric allocation gives a desired endpoint when its allocation matrix has full row rank, but thrust and angle limits need separate handling.It does not constrain the wrench set encountered during servo motion.Source: A. Geometric Allocation · p. 3

B. Differential and Augmented Differential Allocation

Differential allocation maps a wrench-rate target to actuator-rate commands, while normalized differential allocation scales commands to respect actuator rate limits. The CBF stage adds instantaneous wrench-set constraints to rate-based tracking.

B. Differential and Augmented Differential Allocation

  • ADA uses a weighted differential solution whose null-space command supports secondary objectives, but individual actuator saturation can break the commanded wrench-rate equality.The particular solution has minimum W-weighted norm, and the weight balances servo rates and propeller accelerations.Source: B. Differential and Augmented… · p. 3
  • For ideal differential realization, the wrench error decays exponentially under a constant request, while feedforward request rate cancels the driving term for changing requests.A barrier depending on the desired wrench still requires its true total derivative.Source: B. Differential and Augmented… · p. 3

C. Normalized Differential Allocation

V. CBF-BASED WRENCH-FEASIBILITY ALLOCATION

The allocator separates servo motion that preserves wrench feasibility from thrust tracking. Its robust objective links nominal and actual configurations through an exact equivalent-wrench relation and an ellipsoidal reserve.

V. CBF-BASED WRENCH-FEASIBILITY ALLOCATION

  • The servo stage preserves wrench feasibility while the thrust stage tracks the requested wrench, using a sufficient nominal containment condition reduced to twelve thrust margins and affine servo-rate constraints.The benchmark uses a measured-angle prototype, while the full theoretical objective additionally requires a persistent nominal trajectory and an explicit mismatch reserve.Source: §V · pp. 3–4

A. Robust Allocation Objective

  • Equation (20) relates nominal and actual configurations exactly: the same vector produces the desired wrench through actual allocation and the perturbed wrench through nominal allocation.The relation does not linearize angle mismatch or equate nominal and actual servo velocities.Source: A. Robust Allocation Objective · p. 4
  • An ellipsoidal bound on configuration mismatch makes the nominal containment condition sufficient for actual-configuration feasibility when the equivalent perturbation lies within that reserve.The servo stage must maintain the robust containment condition as the desired wrench changes, subject to nominal rate limits.Source: A. Robust Allocation Objective · p. 4
  • The configuration-mismatch perturbation is distinct from the tracking error, which also depends on motor output and supplies feedback to the thrust stage.Source: A. Robust Allocation Objective · p. 4

B. Exact Reduction to Thrust Margins

For a nonsingular six-channel map, ellipsoidal wrench containment reduces exactly to twelve thrust margins. Their complete derivatives account for changing servo angles, reserve geometry, and requested-wrench motion, yielding affine dependence on servo rate.

B. Exact Reduction to Thrust Margins

  • Each reserve ρi is the ellipsoid's support extent projected onto thrust channel i, leaving lower and upper gaps B−i and B+i to the thrust limits.The projected reserve has thrust units and varies with inverse-map geometry, even for fixed Qw.Source: B. Exact Reduction to Thrust Margins · pp. 4–5, Figure 2 · p. 5
  • Six-dimensional containment is equivalent to twelve scalar thrust inequalities, with each channel bounding the same ellipsoid even though worst-case perturbations may differ across channels.The support extrema need not be attained simultaneously.Source: B. Exact Reduction to Thrust Margins · pp. 4–5

C. Analytical Margin Dynamics

  • The margin derivatives are affine in servo rate despite nonlinear angle-dependent containment, and the exact dynamics use no finite-angle linearization.The derivative follows from CnAn = I and the dependence of each allocation-map column on its servo angle.Source: C. Analytical Margin Dynamics · p. 5
  • The closed form ∂ci/∂αn,j = −Gijcj exposes how servo-angle changes affect the reserve geometry through the inverse allocation map.G is defined from the servo-angle derivatives of the allocation columns and the nominal inverse map.Source: C. Analytical Margin Dynamics · p. 5
  • The derivative term b must include requested-wrench motion; substituting vw for the total ˙wd can omit that motion when w = wd.The appendix derives requested-wrench rates for representative translational and attitude controllers, including frame rotation and compensation terms.Source: C. Analytical Margin Dynamics · p. 5

D. Servo-Rate Quadratic Program

The servo-rate QP jointly enforces the twelve barrier constraints and the nominal servo-rate box. Boundary conditions require required thrust to keep pace with reserve growth, while a common rate must satisfy all channel constraints.

D. Servo-Rate Quadratic Program

  • The complete servo QP enforces twelve barrier constraints that preserve tightened thrust intervals, while its rate box bounds nominal—not actual—servo rate.The nominal trajectory is retained as the certificate state; replacing it with measured angles defines different barrier dynamics.Source: D. Servo-Rate Quadratic Program · p. 5
  • The cost penalizes servo motion and nominal required-thrust rate, producing a unique minimizer when the feasible set is nonempty.The required-thrust-rate penalty regularizes reconfiguration without imposing a hard motor-acceleration limit.Source: D. Servo-Rate Quadratic Program · p. 5

E. Boundary Behavior

  • At a lower margin boundary, required thrust must rise at least as fast as the reserve; at an upper boundary, it must fall as fast as reserve growth.Holding required thrust constant is insufficient when reconfiguration increases the reserve.Source: E. Boundary Behavior · p. 6
  • Changing γ changes the permitted approach rate to a boundary, not the thrust box or uncertainty coverage; the inward-pointing condition at the boundary is γ-independent.Qw sets the reserved authority.Source: E. Boundary Behavior · p. 6
  • A common servo rate must satisfy all six channel intervals jointly; separate feasibility of each interval is insufficient.For constant wd, nonnegative margins, and a rate box containing zero, zero rate is feasible and minimizes the cost.Source: E. Boundary Behavior · p. 6

F. Differential Thrust Allocation

Differential thrust allocation uses measured or estimated actual servo motion for tracking, while the nominal servo trajectory defines the robust certificate. The conditional invariance result transfers nonnegative nominal margins to actual-configuration feasibility under its stated assumptions.

F. Differential Thrust Allocation

  • The thrust stage accounts for wrench change caused by reorientation using measured or estimated actual servo rate, while the nominal trajectory remains the state of the robust certificate.The thrust-command rate approximates thrust-rate behavior; it is not assumed to equal the physical thrust rate.Source: F. Differential Thrust Allocation · p. 6

A. Conditional Forward Invariance

  • The conditional theorem preserves the certificate set and ensures actual-angle required-thrust margins are at least their corresponding nominal barrier margins.The stated assumptions include feasible initial margins, bounded mismatch within the reserve, and an admissible command satisfying the CBF constraints.Source: A. Conditional Forward Invariance · p. 6
  • Nonnegative margins therefore ensure actual-configuration feasibility without equating nominal and actual servo rates, but the theorem does not establish recursive QP feasibility or instantaneous thrust realization.The guarantee holds along an admissible solution of the servo QP.Source: A. Conditional Forward Invariance · p. 6

B. Reserve Selection and Conditioning

The reserve must cover configuration mismatch over the claimed operating domain, and a sufficient fixed reserve can be derived from bounds on angle mismatch and scaled wrench magnitude. Reserve size and allocation-map conditioning constrain whether robust margins can remain nonnegative.

B. Reserve Selection and Conditioning

  • A sufficient fixed reserve follows from bounds on servo-angle mismatch and scaled wrench magnitude, and its ellipsoid is a ball in scaled wrench coordinates.The reserve accounts for mismatch, wrench magnitude, and actual-map conditioning.Source: B. Reserve Selection and Conditioning · p. 6
  • A larger reserve is not always admissible: nonnegative robust margins impose a nominal conditioning bound, while actual-map conditioning remains an operating-domain assumption.For a square invertible map, the stated necessary condition is sufficient for a nonempty tightened wrench set at a fixed configuration, but not for a prescribed wrench or its rate.Source: B. Reserve Selection and Conditioning · p. 7

VII. SIMULATION STUDY

The MuJoCo study compares a measured-angle CBF allocator with geometric, augmented differential, and normalized differential allocation across four scenarios. The proposed allocator improves position tracking, with an attitude tradeoff in circular tracking.

A. Setup and Scenarios

  • The study records a six-servo, twelve-propeller vehicle in MuJoCo, with each propeller pair forming one effective channel and a 0–16 N pair-thrust range.Runs include servo and thrust dynamics and a common PID outer controller; Table III summarizes its recorded settings.Source: A. Setup and Scenarios · p. 7, Table III · p. 7
  • The recorded results use the measured-angle prototype; validating the nominal-trajectory certificate requires further records.The study evaluates tracking and actual-angle wrench feasibility, not direct validation of the nominal formulation.Source: A. Setup and Scenarios · p. 7
  • Each run contains 20 s of telemetry logged at 8 ms intervals; these records support reproducible telemetry analysis, but incomplete allocator gains, null-space targets, and actuator-model settings prevent a full simulation rerun.Source: A. Setup and Scenarios · p. 7

B. Tracking Comparison

  • The proposed allocator records position RMSEs of 0.006 m in S1 and 0.026 m in S2; ADA and NDA track both circles, while GA loses tracking in S2.The proposed method has higher attitude RMSE than ADA and NDA in both circular scenarios; these runs do not isolate the causes of the position–attitude tradeoff.Source: B. Tracking Comparison · pp. 7–8, Table IV · p. 7
  • In the undisturbed cartwheel, the proposed allocator improves both position and attitude tracking over ADA and NDA; in S4 its errors remain close to S3.S4 errors are 0.028 m and 1.16°, compared with 0.025 m and 1.15° in S3; ADA and NDA attitude errors rise to 50.01° and 30.37° in S4.Source: B. Tracking Comparison · p. 8

C. Disturbed-Cartwheel Wrench Feasibility

The disturbed-cartwheel test finds the desired wrench feasible at every stored sample during continuous rotation, while the feasibility test remains distinct from tracking performance. The paper identifies reserve coverage and nominal-trajectory records as requirements for direct certificate validation.

C. Disturbed-Cartwheel Wrench Feasibility

  • All 1751 proposed-method samples from 6–20 s are feasible, while violation rates are 100% for GA, 80.7% for ADA, and 60.8% for NDA.The proposed method’s minimum full-wrench margin is 0.89 N and minimum pair-thrust margin is 0.96 N; these are sampled, not intersample, results.Source: C. Disturbed-Cartwheel Wrench… · p. 8, Figure 4 · p. 9
  • The full six-dimensional margin tests desired-wrench membership at measured servo configurations; a nonnegative margin indicates feasibility.Its magnitude depends on force–moment scaling, and negative magnitude is not distance to the attainable set.Source: C. Disturbed-Cartwheel Wrench… · p. 8, Figure 4 · p. 9

A. Geometric Feasibility and Actuator Response

  • Attainability does not guarantee accurate tracking: motor response and estimator errors can still produce transient error, and thrust redistribution cannot fix a fixed-angle wrench-set violation.The sampled test checks whether bounded thrusts can realize the request at the measured configuration.Source: A. Geometric Feasibility and Actuator… · p. 10
  • The support-based tightening is exact for a chosen ellipsoid and square nonsingular map, but a domain-wide reserve may exceed maneuver-specific mismatch.A smaller reserve frees thrust authority but still must cover the equivalent configuration error; tracking quality alone does not establish coverage.Source: A. Geometric Feasibility and Actuator… · p. 10

B. Design and Validation Priorities

  • Direct nominal-formulation validation needs integrated nominal angles, measured angles, prescribed reserve, and stored-sample checks of tightened margin and reserve coverage.The conclusion calls for subsequent hardware evaluation with identified actuator dynamics.Source: B. Design and Validation Priorities · p. 10, §IX · p. 10

APPENDIX A ANALYTICAL DESIRED-WRENCH DERIVATIVES

The appendix gives analytical desired-wrench derivatives for the allocator’s QP, including translational PID and geometric-attitude terms. It notes rotating-frame and compensation derivatives needed in the corresponding settings.

APPENDIX A ANALYTICAL DESIRED-WRENCH DERIVATIVES

  • The QP uses the total derivative of the implemented desired-wrench command; the appendix illustrates its evaluation for translational PID and geometric attitude control.The attitude-controller illustration is associated with.Source: Appendix A · p. 10

A. Translational PID Wrench in Body Coordinates

  • For the translational PID wrench, the appendix defines position and velocity errors and gives their derivatives for evaluating the commanded wrench derivative.The rotating-frame term is required during body reorientation, and the compensation derivative is included when an NRIC or another additive compensation block is used.Source: A. Translational PID Wrench in Body… · p. 10

B. Geometric Attitude Controller

  • The geometric-attitude subsection defines auxiliary quantities for a compact torque-derivative expression and notes that attitude-integral and angular-acceleration terms must be handled consistently.Uncertainty in body angular acceleration cannot be ignored in an exact-derivative claim.Source: B. Geometric Attitude Controller · p. 10

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