Nonlinear Guidance for Arrival Time and Arrival Angle Control Using Trajectory Shaping

Abstract: This paper proposes a nonlinear trajectory shaping guidance strategy for arrival time and angle control of a constant-speed fixed-wing unmanned aerial vehicle. The look angle is parameterized by a fourth-order polynomial. The nonlinear guidance problem is transformed into solving two coupled nonlinear integral equations with respect to two unknown guidance parameters. Directly solving these equations is challenging due to the strong coupling between the parameters. To address this, a two-stage solution procedure is developed. In the first stage, an analytical warm start is constructed. Specifically, by applying approximations, a linear relationship between the two guidance parameters is obtained. With this relationship, a scalar quadratic equation in the first guidance parameter is derived, leading to a good initial guess for the first guidance parameter. In the second stage, this initial guess is refined by solving a one-dimensional nonlinear equation. The obtained solution is finally used as the initial guess for solving the original two-dimensional nonlinear system of equations. Numerical simulations demonstrate that for the test cases, the obtained solution is very close to the open-loop optimal solution, even in highly nonlinear scenarios where some existing methods fail to generate a feasible solution.

• The paper introduces a nonlinear guidance law that simultaneously controls UAV arrival time and angle using a fourth-order polynomial parameterization.
• It employs a two-stage solution with an analytical warm start followed by nonlinear refinement, ensuring rapid convergence and bounded acceleration.
• Numerical simulations validate the method's near-optimal performance in both standard and highly nonlinear multi-UAV formation scenarios.

Nonlinear Trajectory Shaping Guidance for Arrival Time and Angle Control

Introduction and Motivation

Precise guidance laws for constant-speed fixed-wing UAVs are fundamental to time-critical and orientation-sensitive autonomous missions, including multi-vehicle coordination and formation establishment. This work addresses the nonlinear guidance problem of simultaneously controlling arrival time and terminal angle, bypassing the inherent limitations of linearization-dependent approaches. The authors leverage trajectory shaping via polynomial parameterization of the look angle, ensuring the guidance law remains valid in strongly nonlinear engagement geometries.

Methodology

Problem Formulation

The UAV is modeled as a normalized-speed point mass in a 2D plane, with the core control variable being its look angle $o(t)$. The objective is to synthesize normal acceleration commands such that the UAV reaches the destination at a specified time and with a prescribed flight path angle, under bounded terminal acceleration. This framework dispenses with time-to-go estimation, which is unreliable for nonlinear kinematics.

Look Angle Parameterization and Guidance Law Construction

The look angle is parameterized by a fourth-order polynomial:

$o(t) = (t - 1)^2 (o_0 + K_1 t + K_2 t^2)$

This structure inherently enforces terminal look angle and look angle rate constraints, ensuring bounded acceleration at arrival. The guidance problem is formulated as two coupled nonlinear integral equations with respect to $K_1$ and $K_2$. The strong coupling and nonlinearity preclude straightforward solution.

Two-Stage Solution Procedure

To resolve the nonlinear system efficiently, a two-stage procedure is adopted:

1. Analytical Warm Start: Linear approximations (small-angle and near-linear range-depletion) lead to a linear relationship between $K_1$ and $K_2$, reducing the problem to a scalar quadratic equation for $K_1$. The candidate $K_1$ minimizing approximate control effort is selected.
2. Refinement via Nonlinear Solver: The warm start is used as an initialization for the one-dimensional nonlinear refinement, and subsequently, the full two-dimensional nonlinear system is solved.

This approach ensures good initial guesses, robust convergence, and computational tractability for real-time onboard implementation.

Numerical Evaluation

Closed-Loop Performance

The method is validated in closed-loop simulations with state feedback updated at high rates (10 ms intervals), enforcing acceleration saturation and switching to PN guidance near the terminal phase. The results confirm the UAV accurately achieves the desired arrival time and angle while remaining within command bounds, with guidance parameters converging smoothly to zero. Real-time efficiency is evidenced by rapid convergence (≤10 iterations initially, 1 iteration thereafter).

Comparative Studies

Two cases are evaluated:

• Case A: Standard scenario with long engagement. The proposed method achieves control effort ($J = 5.558 \times 10^3$ m² s⁻³) nearly identical to the open-loop optimal solution and marginally superior to geometry-based guidance.
• Case B: Highly nonlinear scenario requiring turn-back maneuvers. The proposed method and the optimal solver successfully find feasible solutions; prior trajectory-shaping approaches fail due to lack of robust initialization. The proposed method closely approximates the optimal result ($J = 1.7876 \times 10^4$ m² s⁻³ vs. $o(t) = (t - 1)^2 (o_0 + K_1 t + K_2 t^2)$0 m² s⁻³) and respects all constraints.

Multi-UAV Formation Establishment

The guidance law is extended to simultaneous coordinated arrival of multiple UAVs, establishing a prescribed triangular formation with common final heading and zero terminal acceleration. All vehicles achieve synchronous arrival and orientation convergence, demonstrating the method's applicability to multi-agent coordination tasks.

Theoretical and Practical Implications

The trajectory shaping and two-stage initialization resolve convergence failures encountered by prior nonlinear guidance laws, especially in challenging geometries. By obviating the need for time-to-go estimation and employing scalable analytical approximations, the method suits onboard implementation and allows real-time closed-loop operation. The approach's applicability to multi-UAV tasks offers a framework for distributed and cooperative guidance under strict temporal and angular constraints.

In practical deployments, this guidance law enables reliable execution of missions involving waypoint synchronization, group formation, and precise targeting under uncertain and nonlinear environments. Theoretically, the polynomial parameterization and initialization architecture can inform future guidance law designs for more complex scenarios, including input constraints, disturbance rejection, and adaptive control.

Future Directions

Potential research avenues include:

• Explicit incorporation of acceleration saturation limits and aerodynamic constraints within the analytical guidance law design.
• Extension to three-dimensional engagement geometries with additional orientation constraints.
• Integration with learning-based or hybrid optimization schemes to further increase real-time responsiveness and robustness.
• Closed-loop robustness analysis under stochastic disturbances and sensor noise.

Conclusion

This paper proposes a rigorous nonlinear trajectory-shaping strategy for simultaneous arrival time and angle control of UAVs, utilizing a fourth-order polynomial look angle parameterization and a two-stage solution procedure. Numerical results demonstrate the feasibility, efficiency, and near-optimality of the approach in both standard and highly nonlinear scenarios, as well as its applicability to coordinated multi-UAV formation establishment. The methodology provides a solid foundation for real-time guidance law synthesis in advanced autonomous flight operations.

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