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arxiv: 2111.00951 · v1 · submitted 2021-11-01 · 🧮 math.OC

Flatness-based Quadcopter Trajectory Planning and Tracking with Continuous-time Safety Guarantees

Victor Freire , Xiangru Xu This is my paper

Pith reviewed 2026-05-24 12:18 UTC · model grok-4.3

classification 🧮 math.OC
keywords quadcopter trajectory planningcontrol barrier functionsdifferential flatnessB-splinesconvex optimizationsafety-critical controltrajectory trackingcontinuous-time constraints
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The pith

Quadcopters can plan and track trajectories that satisfy all state and input constraints continuously in time through convex optimization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows how to generate quadcopter trajectories that stay safe at every moment, not just at sampled points, by using B-splines to represent smooth paths and the differential flatness property to convert all constraints into a convex second-order cone program. A quadratic program then adjusts a basic controller using control barrier functions so the vehicle follows the plan without exceeding error bounds, and this adjustment program is proven always solvable. Conditions are derived to keep the adjusted commands within thrust and tilt limits. Experiments on a small drone confirm the method works in practice.

Core claim

Using the differential flatness of quadcopters and B-spline basis functions, the work formulates a second-order cone program that produces trajectories respecting safe state and input constraints in continuous time. A quadratic program based on control barrier functions filters a nominal controller to ensure bounded tracking errors continuously, with a proof that this program remains feasible at all times. The framework also supplies conditions under which the filtered controller obeys thrust, roll angle, and pitch angle limits.

What carries the argument

B-spline parameterization under differential flatness, which converts continuous-time constraints into convex constraints for a second-order cone program, together with a control barrier function quadratic program that filters controllers while preserving feasibility.

If this is right

  • Trajectories can be optimized while guaranteeing no constraint violation at any time.
  • Tracking errors remain bounded without the safety filter ever failing to find a solution.
  • Input constraints like thrust limits can be incorporated into the safe tracking controller.
  • The method applies to real hardware as shown in flight tests.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The approach may reduce conservatism in planning by directly handling continuous constraints rather than discretizing.
  • Similar techniques could apply to other vehicles with differential flatness, such as fixed-wing aircraft.
  • Real-time replanning might become feasible if the programs solve quickly enough on embedded hardware.
  • Integration with perception systems could enable online obstacle avoidance with the same guarantees.

Load-bearing premise

Quadcopters possess the differential flatness property that lets their full dynamics be recovered from position and yaw trajectories.

What would settle it

A case in which the quadratic program for tracking becomes infeasible or a continuous-time constraint is violated during closed-loop flight.

Figures

Figures reproduced from arXiv: 2111.00951 by Victor Freire, Xiangru Xu.

Figure 1
Figure 1. Figure 1: Configuration of the proposed convex optimization-based control [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Inertial (or world) coordinate frame W (black) and body coordinate frame B (blue). The intermediate coordinate frame C (red) is also shown. Crazyflie2.1 figure from [26]. u = [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Conditions (63a)-(63b) shrink the cone that bounds the Euler angles [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: The position trajectory generated in Example 1 with the parameters [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Visualization of constraints enforced for the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: Verification of the bounded-tracking property of (CBF-QP) with [PITH_FULL_IMAGE:figures/full_fig_p013_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Experimental setup of the cluttered environment and the trajectories [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 8
Figure 8. Figure 8: Visualization of the experimental data for Example 2. [PITH_FULL_IMAGE:figures/full_fig_p014_8.png] view at source ↗
read the original abstract

This work presents a convex optimization framework for the planning and tracking of quadcopter trajectories with continuous-time safety guarantees. Using B-spline basis functions and the differential flatness property of quadcopters, a second-order cone program is formulated to generate optimal trajectories that respect safe state and input constraints in the continuous-time sense. A quadratic program (QP) based on control barrier functions is proposed to guarantee bounded trajectory tracking in continuous time by filtering a nominal controller, where the QP is shown to be always feasible. Furthermore, conditions that ensure the safe tracking controller respects thrust, roll angle, and pitch angle constraints are also proposed. The effectiveness of the proposed framework is demonstrated by real-world experiments using a Crazyflie2.1 nano quadcopter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes a convex optimization framework for quadcopter trajectory planning and tracking that achieves continuous-time safety guarantees. Differential flatness maps states and inputs to position derivatives; B-splines with the convex-hull property are used to encode all continuous-time state/input constraints as convex (SOC) constraints inside a second-order cone program for planning. A CBF-based QP is then introduced to filter a nominal controller for bounded tracking, with a proof that this QP is always feasible; additional conditions are given to ensure the resulting controller respects thrust, roll, and pitch limits. Real-world experiments on a Crazyflie 2.1 are presented to illustrate performance.

Significance. If the feasibility result and continuous-time constraint satisfaction hold, the work supplies a practical, convex-optimization pipeline that converts differential-flatness properties into enforceable SOC constraints and supplies a provably feasible safety filter. This combination is useful for real-time quadcopter applications where both planning and tracking must respect actuator and state bounds without discretization artifacts.

major comments (2)
  1. [§4.2] §4.2 (QP feasibility): the claim that the CBF-QP is always feasible rests on the additional thrust/roll/pitch conditions derived in §4.3. The manuscript should explicitly state whether these conditions are checked a priori or enforced inside the QP; if they are only sufficient but not necessary, the 'always feasible' statement requires a precise qualification.
  2. [§3.3] §3.3 (B-spline SOC encoding): the mapping from B-spline control points to continuous-time thrust and attitude bounds via the convex-hull property is load-bearing for the continuous-time guarantee. The paper should verify that the chosen degree and knot spacing preserve the required inclusion for all admissible trajectories, not only for the optimized ones.
minor comments (2)
  1. [§2] Notation for the flat outputs and their derivatives is introduced inconsistently between §2 and §3; a single table of symbols would improve readability.
  2. [Figure 4] Figure 4 (tracking error plots) lacks explicit indication of the time intervals where the safety filter is active; overlaying the QP activation signal would clarify the contribution of the CBF layer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback and positive assessment of the manuscript. Below we respond point-by-point to the major comments.

read point-by-point responses

  1. Referee: [§4.2] §4.2 (QP feasibility): the claim that the CBF-QP is always feasible rests on the additional thrust/roll/pitch conditions derived in §4.3. The manuscript should explicitly state whether these conditions are checked a priori or enforced inside the QP; if they are only sufficient but not necessary, the 'always feasible' statement requires a precise qualification.

    Authors: We agree that the feasibility result in §4.2 depends on the sufficient conditions derived in §4.3. These conditions are not enforced inside the QP; they are intended to be verified a priori from the planned trajectory, system parameters, and actuator limits. The 'always feasible' claim is therefore qualified by these sufficient (but not necessary) conditions. In the revision we will add explicit qualifying language in both §4.2 and §4.3 stating that feasibility holds when the a-priori conditions are satisfied. revision: yes

  2. Referee: [§3.3] §3.3 (B-spline SOC encoding): the mapping from B-spline control points to continuous-time thrust and attitude bounds via the convex-hull property is load-bearing for the continuous-time guarantee. The paper should verify that the chosen degree and knot spacing preserve the required inclusion for all admissible trajectories, not only for the optimized ones.

    Authors: The convex-hull property of B-splines holds for any admissible set of control points once the degree and knot vector are fixed; it is a general property of the basis and does not depend on whether the control points result from optimization. Consequently, any trajectory whose control points satisfy the SOC constraints automatically satisfies the continuous-time bounds, regardless of how the control points were obtained. The chosen degree and knot spacing are selected to ensure the spline class can represent the required maneuvers, but the inclusion itself is independent of those choices. We will add a clarifying remark in §3.3 emphasizing this generality. revision: partial

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The paper's derivation relies on the standard differential flatness property of quadcopters (a known system property) combined with B-spline parameterization to encode continuous-time constraints as convex SOCP constraints. The CBF-QP for tracking is shown feasible under explicitly proposed additional conditions on thrust/attitude bounds; these steps are independent mathematical arguments and do not reduce by construction to fitted parameters, self-definitions, or load-bearing self-citations. No ansatz is smuggled via citation, and no renaming of known results occurs. The framework is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on the domain assumption of differential flatness for quadrotors and the technical premise that B-splines can encode continuous-time constraints inside a convex program. No free parameters or invented entities are visible in the abstract.

axioms (1)
  • domain assumption Quadcopters are differentially flat systems whose states and inputs can be expressed as functions of position and yaw and their derivatives.
    Invoked to reduce the trajectory planning problem to a lower-dimensional space amenable to B-spline parameterization.

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