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Direct Vector Field Guidance for Path Following

Amit Shivam, Nikhil Kumar Singh, Luís Tiago Paiva, Fernando A. C. C. Fontes

arXiv:2610.13576v1eess.SYcs.SY

TL;DR

Turn-constrained UAVs need path-following guidance whose transient steering commands respect vehicle maneuvering limits. The paper proposes a tanh-shaped direct vector field and derives a gain bound from the commanded turn-rate. Its analysis establishes path convergence and turn-rate feasibility under the stated conditions.

  • Problem

    Direct vector fields’ maneuvering feasibility remains largely unexplored, although aggressive convergence profiles can demand transient turn rates beyond constrained UAVs’ steering capability.

    Source: §I · p. 1

  • Method

    The paper uses a tanh-shaped course-angle command based on lateral error and analytically derives turn-rate bounds for gain selection.

    Source: A. Guidance Law · p. 2, §I · p. 2

  • Results

    For any positive shaping gain and constant positive ground speed, the desired path is globally asymptotically stable under ideal course following, and gains satisfying the derived bound ensure turn-rate feasibility.

    Source: C. Stability Properties · p. 3

  • Takeaways & Limitations

    The gain-selection rule provides a compact mechanism for enforcing turn-rate feasibility across different maneuverability regimes.

    Source: §V · p. 6

Abstract

from arXiv · show

This paper presents a path-following guidance framework for a turn-rate-constrained unmanned aerial vehicle (UAVs). The proposed guidance strategy uses commanded course angle as a tanh-hyperbolic shaping function of the lateral deviation from the desired path. Stability characteristics illustrating asymptotic convergence to the desired path are established using Lyapunov analysis. Analytical expressions for the turn-rate and corresponding maximum turn-rate bound are derived, enabling direct gain selection for a turn-rate-limited vehicle. Numerical simulations are performed for straight-line following scenarios under both ideal and non-ideal initial heading conditions. Furthermore, a comparative study against representative arctan-, arcsine-, and arccos-based vector-field guidance laws is conducted using analytically feasible gains selected from their respective turn-rate bounds. Comparative numerical results demonstrate the proposed formulation achieves reliable boundary convergence while maintaining smooth steering behavior and a compact analytical gain-design framework.

I. INTRODUCTION

The paper develops turn-rate-aware direct vector-field guidance because transient steering demands can exceed turn-constrained UAVs’ maneuvering limits. It relates the guidance shaping parameter to those limits and frames the objective as smooth, turn-feasible convergence to a prescribed path.

I. INTRODUCTION

  • Fixed-wing UAVs must follow prescribed paths while respecting maximum turn-rate and minimum turning-radius constraints.The introduction identifies geo-fencing, perimeter surveillance, infrastructure inspection, and persistent monitoring as motivating applications.Source: §I · p. 1
  • Aggressive convergence profiles can produce transient turn-rate demands that exceed the steering capability of turn-constrained UAVs.The framework targets maneuvering feasibility while preserving smooth saturation and convergence properties.Source: §I · pp. 1–2
  • The paper develops a direct vector-field law and analytically relates its shaping gain to prescribed vehicle turn-rate limits.Its contributions include deriving turn-rate profiles and bounds, comparing arctan-, arcsine-, and arccosine-based fields, and simulating ideal and non-ideal heading conditions.Source: §I · p. 2

II. PROBLEM STATEMENT

  • The path-following objective is to design the commanded course angle for smooth convergence while ensuring turn-rate feasibility.The stated constraint is |χ̇d| ≤ ωmax.Source: §II · p. 2

III. PROPOSED VECTOR FIELD GUIDANCE METHOD

The guidance law uses a tanh-shaped course-angle command based on lateral deviation from the path. Its resulting lateral motion points toward the path, and the plotted command transitions continuously across the lateral-error domain.

A. Guidance Law

  • The proposed law shapes the commanded course angle using a tanh-hyperbolic profile of lateral deviation.The positive shaping gain controls the profile, while the π/2 term aligns desired motion with the positive Y-axis on the path.Source: A. Guidance Law · p. 2
  • For either sign of lateral error, the vector field directs the UAV toward the desired path.Under ideal course following, the lateral dynamics are ẋ = −Vg tanh(k_atanh x).Source: A. Guidance Law · p. 2
  • Figure 2 plots the continuous course-angle transition and the corresponding vector fields.The course-angle profile appears in Fig. 2a and the vector fields in Fig. 2b.Source: A. Guidance Law · p. 2

B. Turn-Rate Characteristics

The section derives the commanded turn-rate profile and its maximum, then uses that bound to relate the shaping gain to the vehicle’s turn-rate limit.

B. Turn-Rate Characteristics

  • The commanded turn-rate magnitude depends on the shaping gain, ground speed, and lateral position through the tanh and sech terms.The supplied expression is |χ̇d| = k_atanh Vg |sech(k_atanh x) tanh(k_atanh x)|.Source: B. Turn-Rate Characteristics · p. 3
  • For z in [0, 1], the maximum commanded turn-rate is analyzed by equating the derivative of g2(z) to zero, giving z = 0 or z² = 1/2.Source: B. Turn-Rate Characteristics · p. 3
  • The derivation identifies the maximum commanded turn-rate and provides an explicit analytical relation between shaping gain and vehicle turn-rate limit.Figure 3 shows turn-rate profiles for different shaping gains and their effect on peak turn-rate magnitude.Source: B. Turn-Rate Characteristics · p. 3

C. Stability Properties

The proposed guidance law makes the desired path globally asymptotically stable under ideal course following, while a gain condition ensures commanded turn-rate feasibility. Increasing the shaping gain speeds convergence but raises peak turn-rate demand.

C. Stability Properties

  • For any k_atanh > 0 and constant Vg > 0, the desired path is globally asymptotically stable under ideal course following, and condition (10) ensures the commanded turn rate stays within ωmax.The proof uses the sign of x tanh(k_atanh x) to establish convergence; the turn-rate condition holds for all x.Source: C. Stability Properties · p. 3
  • Increasing k_atanh speeds convergence toward the path but raises the maximum commanded turn rate, making (10) the largest feasible shaping gain.Source: C. Stability Properties · p. 3

D. Comparative Analytical Discussion

The section introduces an analytical comparison of commonly used direct vector-field guidance laws for turn-limited path following.

D. Comparative Analytical Discussion

1) Arctan Vector Field:

The section considers the arctangent-based formulation developed in and its lateral dynamics and turn-rate profile.

1) Arctan Vector Field:

  • The arctangent-based vector-field formulation is attributed to, with its lateral dynamics and turn-rate profile analyzed.Source: 1) Arctan Vector Field · p. 4
  • The section identifies a maximum turn-rate for the arctangent formulation, but the supplied passage does not specify its value or where it occurs.Source: 1) Arctan Vector Field · p. 4

2) Arcsine Vector Field:

The arcsine formulation is analyzed through its turn-rate profile and maximum turn-rate. Compared with arctangent guidance, it requires lower peak turn rate for comparable convergence when initial heading directions are the same.

2) Arcsine Vector Field:

  • The arcsine vector-field law is attributed to, and its turn-rate profile and maximum are analytically obtained.Source: 2) Arcsine Vector Field · p. 4
  • For comparable convergence from the same initial heading directions, the arcsine formulation has lower peak turn-rate requirements than the arctangent vector field.Source: 2) Arcsine Vector Field · p. 4

3) Arccos Vector Field:

The Arccos vector field has a derived maximum commanded turn-rate and a gain constraint, and Table I compares its turn-rate characteristics with other direct vector fields. The comparison cautions against reading its numerical coefficients as convergence speeds.

3) Arccos Vector Field:

  • The maximum commanded turn-rate is derived for the Arccos vector field, with its shaping gain required to satisfy a feasibility condition.The supplied passages do not give the bound's explicit expression.Source: 3) Arccos Vector Field · p. 4
  • Table I compares the considered direct vector-field laws' turn-rate characteristics and feasible gain selection using ρ = ωmax/Vg.The numerical coefficients are not convergence speeds because shaping profiles differ, and the arcsine gain has a different dimension.Source: 3) Arccos Vector Field · p. 4
  • The comparison characterizes the proposed vector field as combining analytical tractability and smooth convergence with explicit turn-feasibility-based gain selection.Source: 3) Arccos Vector Field · p. 4

IV. SIMULATION RESULTS

The simulations examine straight-line following with an ideal initial heading and with substantial heading offsets. The proposed guidance law preserves turn-rate feasibility in the reported feasible-gain case and across the non-ideal initializations.

A. Sample scenarios with ideal initial heading

  • With k_atanh = 0.0400, the peak turn-rate is 0.40 rad s−1 within the 0.5 rad s−1 limit; k_atanh = 0.1250 peaks at 1.25 rad s−1 and exceeds it.Both gains drive the UAV toward the path with similar convergence, while the larger gain produces steeper course-angle variation.Source: A. Sample scenarios with ideal initial… · p. 4, Figure 4 · p. 5

B. Scenarios with non-ideal initial heading

  • With initial heading offsets of +90° and −60°, trajectories converge toward the desired path and turn-rate profiles remain within the prescribed envelope.Course-angle error shows a transient correction phase followed by convergence to the vector-field direction.Source: B. Scenarios with non-ideal initial… · pp. 4–5

C. Comparative study

The comparison finds that the proposed vector field uses moderate steering effort between the arctan and arccos extremes, while retaining smooth steering and an analytical gain rule. The conclusion frames it as a compromise and identifies broader operating conditions for future study.

C. Comparative study

  • Gains for the compared laws were selected under analytical feasibility conditions for ρ values 0.015, 0.025, and 0.035, corresponding to turn-rate limits of 0.3, 0.5, and 0.7 rad s−1.The study uses a constant ground speed of 20 m s−1.Source: C. Comparative study · p. 5
  • Arccos has the highest average control effort, arctan the lowest, and the proposed field lies between them; effort rises with ρ across all laws.The proposed field is smoother than the more aggressive arcsine and arccosine fields, with potentially slightly longer settling time at the same maneuverability ratio.Source: C. Comparative study · pp. 5–6, Figure 6 · p. 6

V. CONCLUSION

  • The conclusion presents the proposed law as a balance of steering smoothness, convergence characteristics, and analytical tractability, with a compact gain rule for turn-rate feasibility.Source: §V · p. 6
  • Future work is directed toward curved boundaries, closed-orbit surveillance, wind-perturbed operations, and high-fidelity six-degree-of-freedom UAV models.The paper also identifies curvature-aware and geometry-constrained guidance as future developments for the gain-design methodology.Source: §V · p. 6
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