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Quadrotor Guidance for Tasks along Elliptical Boundary

Ravi Khandelwal, Ashwini Ratnoo

arXiv:2610.32248v1eess.SYcs.SY

TL;DR

Inspection along elliptical boundaries calls for viewpoint guidance as well as circumnavigation, including time spent in selected sectors. The paper develops bifurcation-based guidance for these tasks, and reports convergence to desired viewpoints except at the unstable angular fixed point, with simulated trajectory tracking under bounded disturbances.

  • Problem

    Existing elliptical circumnavigation does not inherently accommodate hovering at specific viewpoints or switching between designated viewpoints, while constant-speed motion may be inadequate where closer observation is needed.

    Source: §I · p. 2

  • Method

    A bifurcation-based guidance framework uses decoupled elliptic radial and angle dynamics, matching a stable fixed point to a desired viewpoint and varying parameters to support multiple trajectory tasks.

    Source: §VIII · p. 28

  • Results

    The guidance formulation converges to the desired viewpoint from every initial condition except those corresponding to the unstable fixed point of the elliptic-angle dynamics.

    Source: §VIII · p. 28

  • Takeaways & Limitations

    Closed-form, algebraically simpler velocity expressions and parameterized settling times support the framework’s practical implementation across inspection trajectory modes.

    Source: §VIII · p. 32

Abstract

from arXiv · show

This paper presents a guidance method that enables a quadrotor uncrewed aerial vehicle to perform inspection tasks along an elliptical boundary enclosing a structure of interest. The proposed guidance approach leverages a bifurcation theory-based framework that supports multiple inspection tasks, including convergence to a desired viewpoint, switching between viewpoints along the boundary, constant-speed elliptical circumnavigation, and circumnavigation with sectorial dwell-time constraint. Stability analysis of the proposed guidance dynamics establishes convergence of the generated reference trajectories to the desired behavior. The control parameters in the proposed method further allow the settling time of the reference trajectory to be specified for each task. The resulting reference velocity expressions are closed-form and algebraically simple, facilitating practical implementation. A qualitative comparison with existing guidance methods highlights the flexibility of the proposed approach in achieving different inspection tasks through a unified framework, while relying on simpler guidance computations. The proposed guidance method is validated through numerical simulations using a six-degree-of-freedom quadrotor model with model uncertainties, and a cascaded sliding-mode controller that tracks the generated reference trajectories under bounded disturbances.

I. Introduction

The paper targets inspection along elliptical boundaries, where existing approaches often separate viewpoint tasks from circumnavigation. It proposes a bifurcation-based framework for multiple tasks, using a decoupled elliptic position representation.

I. Introduction

  • Existing elliptical circumnavigation methods do not inherently support hovering at specific viewpoints or switching between designated viewpoints, and constant speed may not suit sectors needing closer observation.The paper motivates sectorial dwell-time-constrained circumnavigation for those regions.Source: §I · p. 2

III. Bifurcation-Based Guidance Method

  • The proposed guidance dynamics use a saddle-node infinite-period bifurcation, with a stable fixed point matched to the desired viewpoint.The paper reports convergence from every initial condition except those corresponding to the unstable fixed point of the elliptic-angle dynamics.Source: §I · p. 3

I. Introduction

  • A unified bifurcation-based framework varies a parameter to support reaching a viewpoint, switching between viewpoints, and continuous elliptical circumnavigation.The framework also includes dwell-time-constrained circumnavigation within a specified boundary sector.Source: §I · p. 3
  • Closed-form settling-time expressions allow control parameters to be tuned for the desired convergence behavior of reference trajectories.The framework also supports dwell-time-constrained circumnavigation in a specified angular sector.Source: §I · p. 3

II. Problem Formulation

  • The guidance problem is to generate velocity components that take a quadrotor from outside the ellipse to a desired viewpoint, then support boundary switching and continuous circumnavigation.The formulation assumes the structure and inspection locations are known, and projected onto a predefined elliptical boundary; camera orientation is maintained by a gimballed camera or independent yaw control.Source: §II · pp. 4–5

A. Two-Variable Position Representation (𝛽, 𝜙)

  • A decoupled representation (β, φ) describes elliptical position, with β scaling both ellipse axes and facilitating bifurcation-based guidance design.The boundary corresponds to β=1; β<1 and β>1 denote points inside and outside it, respectively.Source: A. Two-Variable Position… · p. 5

B. Proposed Dynamics

The proposed radial dynamics have a stable fixed point set at the desired elliptical boundary. Their solution converges to that boundary, and the radial control parameter specifies the settling time.

B. Proposed Dynamics

  • The dynamics distinguish the elliptic radial parameter ρ, the radial control parameter λβ, and the angular control coefficient λφ.The passage also identifies μ as the bifurcation parameter and k as a constant.Source: B. Proposed Dynamics · p. 6

1. Elliptic Radial Dynamics

  • Setting β*=βd=1 makes the radial fixed point the desired elliptical boundary.The fixed point is stable.Source: §1 · p. 7
  • The integrated radial dynamics converge to the desired boundary as time tends to infinity, with λβ controlling settling time.The settling time is defined as the time for the state to enter and remain within an ε-neighborhood of the stable fixed point.Source: §1 · p. 7

2. Elliptic Angle Dynamics

The elliptic-angle dynamics use the bifurcation parameter to place stable fixed points at desired angles. Their fixed-point structure changes at μ=±1, while angles converge to the stable point except from the unstable point.

2. Elliptic Angle Dynamics

  • The stable angle fixed point varies across a half-circle as μ changes, while the other fixed point is unstable.For k=1, the stable point spans −π/2 to π/2; for k=−1, it spans π/2 to 3π/2.Source: §2 · pp. 8–10
  • The angle converges to the desired fixed point except when its initial value coincides with the unstable fixed point.The angular control coefficient can be expressed in terms of the desired settling time.Source: §2 · pp. 10–11
  • For |μ|<1, the angle dynamics have stable and unstable fixed points that merge into a half-stable point at μ=±1; for |μ|>1, they have no fixed points.The paper identifies this transition as a saddle-node infinite-period bifurcation.Source: §2 · p. 11
  • For |μ|>1, the angle dynamics have no fixed points and instead produce stable limit cycles.The cycle direction depends on the sign of μ.Source: §2 · p. 11

IV. Inspection Tasks

The section formulates guidance logic for four inspection tasks using analytical results from the preceding section.

IV. Inspection Tasks

  • The section uses the analytical results to formulate guidance logic for four distinct inspection tasks along the boundary.Source: §IV · p. 12

A. Reaching a Viewpoint

The viewpoint task brings the quadrotor from outside the elliptical boundary to a specified viewpoint and hover state within a chosen time. Radial and angular control parameters are set from that time, and the bifurcation parameter is selected according to the desired angle’s quadrant.

A. Reaching a Viewpoint

  • The task is to arrive and hover at a specified viewpoint on the desired elliptical boundary, starting outside it within a specified time.Source: A. Reaching a Viewpoint · p. 12
  • The radial control parameter and angular control coefficient are computed using the specified arrival time as their respective settling times.Source: A. Reaching a Viewpoint · p. 12
  • The bifurcation parameter is computed according to the quadrant containing the desired elliptic angle.Source: A. Reaching a Viewpoint · p. 12

B. Switching Between Viewpoints

The guidance sets parameters to switch between boundary viewpoints in a desired duration and extends the same dynamics to circumnavigation with a specified dwell time in a sector. For the dwell task, the required bifurcation parameter has a unique feasible solution under the stated conditions.

B. Switching Between Viewpoints

  • The switching task keeps the quadrotor on the ellipse and selects the angular control coefficient using the desired endpoint and switching duration.The desired elliptic angle is obtained from the target line-of-sight angle; the bifurcation parameter and angular coefficient are then computed using that angle and the desired time.Source: B. Switching Between Viewpoints · p. 12

C. Dwell Time Constrained Elliptical Circumnavigation

  • For sectorial dwell, the angular dynamics have a stable limit cycle when |μ| > 1, with rotation direction determined by the sign of μ.The dwell requirement is the time spent within the sector centered at the desired viewpoint.Source: C. Dwell Time Constrained Elliptical… · p. 12
  • For given λφ > 0, k > 0, Tdw > 0, and φ2 > φ1, Eq. (33) has exactly one feasible μ*, satisfying μ* > kM.The equation is nonlinear and transcendental, so μ is determined numerically; the proposition establishes existence and uniqueness.Source: C. Dwell Time Constrained Elliptical… · p. 13

D. Constant Speed Elliptical Circumnavigation

The proposed angular dynamics allow constant-speed circumnavigation when their parameters are selected as established in Proposition 5. The paper also specifies the quadrotor model and aerodynamic-drag formulation used for trajectory tracking.

D. Constant Speed Elliptical Circumnavigation

  • With k = 0, μ = 1, and β = 1, Proposition 5 gives an angular control coefficient λφ that ensures circumnavigation at desired constant speed vd.The coefficient is obtained by enforcing constant speed along the elliptical boundary.Source: D. Constant Speed Elliptical… · p. 14

A. Quadrotor Equations of Motion

  • The quadrotor equations of motion use a North-East-Up world frame and a body-fixed frame, with rotor thrust summed into u1 and roll, pitch, and yaw torques represented by u2–u4.The model includes bounded lumped uncertainty terms for unmodeled dynamics and effects such as rotor inertia and gyroscopic coupling.Source: A. Quadrotor Equations of Motion · pp. 15–16

1. Aerodynamic Drag Formulation

  • Aerodynamic drag is based on the quadrotor’s velocity relative to the surrounding air and uses quadratic drag expressed along body axes.The body-frame drag force is transformed using the rotation matrix; its coefficients and effective reference areas are specified for each body axis.Source: §1 · pp. 16–17

2. Dryden Wind Turbulence Model

The simulation models wind with a Dryden turbulence formulation and tracks generated reference trajectories using a cascaded sliding-mode controller. The controller combines position and attitude loops and filters intermediate attitude commands.

2. Dryden Wind Turbulence Model

  • The Dryden wind model combines mean world-frame wind with body-frame turbulent gust components rotated into the world frame.The gusts are generated by shaping filters driven by mutually independent, zero-mean Gaussian white-noise processes.Source: §2 · p. 17
  • For low-altitude operation defined as h ≤ 1000 ft, turbulence scale lengths and standard deviations are set from the MIL-F-8785C specification.The scale lengths are converted to metres for the simulation, and the shaping-filter transfer functions use the resulting turbulence parameters.Source: §2 · pp. 17–18

B. Cascaded Sliding-Mode Control Framework for Trajectory Tracking

  • The guidance block generates fixed-altitude reference trajectories, while the cascaded controller uses an outer-loop position controller and an inner-loop attitude controller.Reference positions are integrated from reference velocities, and desired accelerations are obtained by differentiating the velocity equations.Source: B. Cascaded Sliding-Mode Control… · pp. 18–19
  • Second-order command filters produce smooth attitude commands and their derivatives, while sliding-mode laws generate control commands for tracking under bounded disturbances.The filters avoid cumbersome direct differentiation of the intermediate roll and pitch commands; aerodynamic drag and lumped uncertainties are treated as bounded disturbances.Source: B. Cascaded Sliding-Mode Control… · pp. 19–20

VI. Qualitative Comparison With Prior Works

The comparison describes a unified guidance formulation that covers four trajectory modes with simpler analytical velocity expressions than the compared approaches. Simulations report convergence to desired viewpoints within prescribed settling times using a 6-DOF quadrotor model.

VI. Qualitative Comparison With Prior Works

  • The qualitative comparison reports simpler analytical velocity expressions and four trajectory modes from one bifurcation parameter and constant within a unified formulation.The modes include viewpoint hovering and switching, which the cited fixed-wing strategies are not suited to perform, and prescribed sectorial dwell, which the compared methods do not provide.Source: §VI · p. 20, Table 1 · p. 21

VII. Simulation Studies

  • The four boundary-task scenarios are simulated with a 6-DOF quadrotor model that includes model uncertainties and Dryden-model wind disturbances.The reported simulation setup uses aerodynamic parameters from Ref. and specifies bounded time-varying disturbance terms.Source: §VII · p. 22

A. Case 1: Rendezvous with a Viewpoint

  • The rendezvous simulation reaches the desired boundary viewpoint with negligible error within the prescribed 20 s settling time.For the case starting at (9 m, 8 m), the target line-of-sight angle is 160 deg; the reported response also closely tracks the 5 m reference altitude.Source: A. Case 1: Rendezvous with a Viewpoint · p. 22, Figure 9 · p. 23
  • Across all scenarios, the quadrotor reaches the elliptical boundary within the specified settling time, while line-of-sight and elliptic angles converge to desired viewpoints with negligible errors.Desired viewpoints, settling times, and guidance parameters are summarized in Table 2; trajectories and response plots appear in Figs. 11 and 12.Source: A. Case 1: Rendezvous with a Viewpoint · p. 23, Table 2 · p. 23

B. Case 2: Switching Between Viewpoints

Simulations demonstrate viewpoint switching, constant-speed elliptical circumnavigation, and sector-constrained dwell, while the conclusion presents these tasks as part of a unified bifurcation-based guidance framework. The method also provides closed-form velocity expressions and is evaluated with a 6-DOF quadrotor model under bounded disturbances.

B. Case 2: Switching Between Viewpoints

C. Case 3: Constant Speed Elliptical Circumnavigation

D. Case 4: Dwell Time Constrained Elliptical Circumnavigation

VIII. Conclusions

  • The framework combines viewpoint convergence, viewpoint switching, and continuous elliptical circumnavigation, with convergence except from the unstable fixed point of the elliptic-angle dynamics.The conclusion also describes sectorial dwell for a prescribed duration, achieved through stable limit cycles.Source: §VIII · pp. 28, 32
  • The method yields closed-form, algebraically simpler velocity expressions and is demonstrated in simulations tracking reference trajectories with a 6-DOF model under bounded disturbances.The disturbances include model uncertainties and wind gusts modeled using the Dryden model.Source: §VIII · p. 32

Appendix: Proof of Proposition 3

The proof divides the angular dynamics into cases according to the initial angular rate and parameter choices, then identifies the stable fixed point for each value of k.

Appendix: Proof of Proposition 3

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