Constraint-Based Alignment for UAV-Quadruped Docking

• Constraint-based alignment strategy is a method that steers UAV dynamics to dock with quadruped targets using finite-time convergence techniques.
• It rigorously enforces operational constraints such as field-of-view and minimum altitude via logarithmic barrier functions to prevent safety violations.
• Empirical validations show the approach reliably handles complex terrains, ensuring accurate tracking and collision-free outcomes.

A constraint-based alignment strategy constitutes a systematic approach to steering an autonomous system’s state toward a target configuration while rigorously adhering to one or more operational constraints. In the context of UAV-quadruped docking in complex terrains, constraint-based alignment is operationalized through the synthesis of a nonlinear finite-time controller (NFTSMC) and a logarithmic barrier function (BF) that together guarantee both rapid convergence and strict constraint satisfaction. This paradigm is crucial for ensuring reliable inter-platform docking when variable quadruped posture and visual limitations introduce nontrivial difficulties in the tracking and descent phases.

1. Formal Alignment Objective and Constraint Formulation

The core alignment objective is to steer the UAV inertial position $p_a(t) = [x(t), y(t), z(t)]^\top$ to coincide with the AprilTag target position $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$ mounted on the quadruped’s deck, such that the tracking error $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$ converges to zero in finite time. The system must adhere at all times to:

• A Field-of-View (FOV) constraint: $e_x^2 + e_y^2 \leq d_s^2$, preserving the target within the onboard camera’s view window.
• A minimum-altitude (collision-avoidance) constraint: $z_a(t) \geq z_t(t) + \delta_t$, maintaining a safety distance $\delta_t$ above the landing platform.

This constraint-aware formulation becomes a minimum-time tracking problem with feasibility dictated by bilateral safety and perception limits. All subsequent control design is strictly subordinated to these requirements.

2. State Variables and Error Dynamics

The system state comprises the actual UAV and AprilTag positions and their time derivatives. The tracking error $e(t)$ and its derivative $\dot{e}(t)$ are fundamental:

$$e(t) = p_a(t) - p_t(t), \qquad \dot{e}(t) = \dot{p}_a(t) - \dot{p}_t(t)$$

These variables are observable, with sensor and estimation granularity sufficient for control bandwidth.

The underlying UAV model is assumed to follow second-order nonlinear dynamics:

$M(p_a) \ddot{p}_a + G(p_a, \dot{p}_a) = u + d(t)$

where $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$0 is known, positive-definite, $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$1 encapsulates known terms (including gravity), $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$2 is the control input, and $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$3 is a bounded disturbance.

3. Nonsingular Fast Terminal Sliding Mode Controller (NFTSMC)

Finite-time convergence under bounded disturbances is secured through the terminal sliding surface:

$p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$4

Here, $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$5 and $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$6 are positive diagonal matrices, $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$7 are odd integers with $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$8, and all operations applied element-wise. This NFTSMC sliding surface dynamics provide:

• Nonsingular convergence.
• Superlinear vanishing rate (e.g., $p_t(t) = [x_t(t), y_t(t), z_t(t)]^\top$9 near $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$0).
• Compatibility with underactuated or nonlinear $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$1 under state feedback.

4. Constraint Enforcement via Logarithmic Barrier Functions

To guarantee $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$2 at every instant, a logarithmic barrier function is introduced:

$e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$3

Its gradient is calculated as:

$e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$4

This term acts as a repulsive force that grows unbounded as the constraint boundary is approached, thereby proactively rejecting states that risk violating the FOV constraint.

5. Composite NFTSMC–BF Control Law Design

The constraint-based composite control input is defined as:

$e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$5

Here, $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$6, $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$7, and $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$8 are positive definite diagonal gain matrices. Interpretation:

• The equivalent control term $e(t) = p_a(t) - p_t(t) = [e_x, e_y, e_z]^\top$9 cancels nominal plant dynamics and enforces the desired reference acceleration.
• The reaching law $e_x^2 + e_y^2 \leq d_s^2$0 robustly drives the sliding surface to zero, even under bounded uncertainties.
• The barrier term $e_x^2 + e_y^2 \leq d_s^2$1 ensures strict constraint satisfaction.

This modular structure allows simultaneous alignment and constraint enforcement with guaranteed Lyapunov stability (see next section).

6. Lyapunov Analysis and Finite-Time Guarantees

With the Lyapunov candidate $e_x^2 + e_y^2 \leq d_s^2$2, its time derivative under the joint dynamics and control law yields

$e_x^2 + e_y^2 \leq d_s^2$3

where $e_x^2 + e_y^2 \leq d_s^2$4 bounds matched disturbances. Classical terminal sliding mode theory implies finite-time convergence $e_x^2 + e_y^2 \leq d_s^2$5 in a time $e_x^2 + e_y^2 \leq d_s^2$6 dependent on controller gains and disturbance bounds. The barrier term does not admit negative definiteness, ensuring that constraint violation is structurally precluded. Once $e_x^2 + e_y^2 \leq d_s^2$7, by (3) one has $e_x^2 + e_y^2 \leq d_s^2$8 in finite time.

The guaranteed separation property is critically dependent on correct gain tuning: $e_x^2 + e_y^2 \leq d_s^2$9 must be chosen to dominate all possible disturbance contributions; $z_a(t) \geq z_t(t) + \delta_t$0 sufficient to repel the error trajectory from $z_a(t) \geq z_t(t) + \delta_t$1.

7. Collision Avoidance via Altitude Constraint

The controller sets the altitude reference by fixing $z_a(t) \geq z_t(t) + \delta_t$2 so that $z_a(t) \geq z_t(t) + \delta_t$3 is maintained throughout the tracking process. This ensures physical safety regardless of error convergence rate in the horizontal plane.

8. Operational Assumptions and Practical Regimes

The method’s reliability depends on several operational assumptions:

• $z_a(t) \geq z_t(t) + \delta_t$4 and $z_a(t) \geq z_t(t) + \delta_t$5 are known and well-behaved (positive-definite, bounded).
• Disturbances $z_a(t) \geq z_t(t) + \delta_t$6 are norm-bounded: $z_a(t) \geq z_t(t) + \delta_t$7.
• Sensor noise is bounded and slower than control bandwidth.
• Actuator saturation does not occur within the desired envelope.

Under these assumptions, the scheme guarantees finite-time convergence, strict constraint satisfaction, and operational safety.

9. Empirical Validation and Real-World Performance

The constraint-based alignment framework achieves verified UAV–quadruped docking on surfaces with the following metrics:

• Staircases with height $z_a(t) \geq z_t(t) + \delta_t$817 cm.
• Slopes $z_a(t) \geq z_t(t) + \delta_t$930 degrees.
• All outdoor runs in GPS-denied regimes. The UAV never loses tracking (AprilTag always visible), and collision events are excluded by design. All constraints are active throughout the control horizon; descent only proceeds after terminal stabilization confirmed by the Safety Period (SP) mechanism.

10. Summary and Applicability

Constraint-based alignment, as realized through NFTSMC combined with barrier functions, is a high-reliability method for UAV-quadruped docking under FOV and altitude constraints. It exhibits:

• Provable finite-time convergence of the tracking error.
• Strict enforcement of operational boundaries with singular barrier terms.
• Full design transparency: key equations (1–6) and proofs are reconstructible from the system model.

This framework is adaptable to other autonomous multi-robot interactions where safety, perception, or physical domain constraints must be enforced rigorously under nontrivial dynamic coupling and demonstrates empirical robustness to terrain complexity and posture variability (Xu et al., 25 Sep 2025).