Spline-Based Trajectory Parameterization

• Spline-based trajectory parameterization is a method that represents continuous motion using B-splines or Bézier curves, ensuring smoothness and feasibility.
• It leverages the convex hull property to translate continuous trajectory constraints, such as collision avoidance, into a finite set of control point requirements.
• The approach integrates differential flatness and mixed-integer optimization to efficiently meet dynamic constraints in real-world applications like UAVs and robotics.

Spline-based trajectory parameterization is a mathematical framework for expressing continuous trajectories through a finite set of basis functions—most notably B-splines or Bézier curves—such that the trajectory and its derivatives (e.g., velocity, acceleration) can be controlled and optimized with respect to waypoints, system dynamics, and environmental constraints. This approach is foundational in control, robotics, aerospace, and computer graphics, enabling trajectories that are smooth, dynamically feasible, and guaranteed to satisfy constraints at all times. The following sections synthesize the methodology, advantages, mathematical formulations, and applications of spline-based trajectory parameterization, emphasizing B-spline and Bézier representations in constrained motion-planning scenarios, with extensive reference to differential flatness and obstacle avoidance formulations for aerial vehicles (Stoican et al., 2016).

1. Mathematical Formulation and Differential Flatness

A nonlinear system with state $x(t) \in \mathbb{R}^n$ and input $u(t) \in \mathbb{R}^m$ can be expressed as

$\dot{x}(t) = f(x(t), u(t)).$

If the system is differentially flat, there exists a “flat output” $z(t) \in \mathbb{R}^p$ such that $z(t)$ and its finite derivatives parameterize the system’s entire state and input: $$z(t) = \gamma(x(t), u(t), \dot{u}(t), \ldots), \quad x(t) = \Theta(z(t), \dot{z}(t), \ldots), \quad u(t) = \Phi(z(t), \dot{z}(t), \ldots).$$ This property allows the trajectory generation and constraint enforcement problem to be posed directly in terms of $z(t)$, reducing otherwise complex nonlinear constraints to ones on $z(t)$ and its derivatives.

Parameterizing $z(t)$ as a B-spline results in

$z(t) = \sum_{i=0}^n B_{i,d}(t) \, p_i,$

where $p_i$ are the control points, $B_{i,d}(t)$ are recursively defined B-spline basis functions of degree $d$, and $\{\tau_i\}$ is the knot vector. The recursive definition is

$$\begin{align*} B_{i,1}(t) &= \begin{cases} 1 & \tau_i \leq t < \tau_{i+1} \ 0 & \text{otherwise} \end{cases} \ B_{i,d}(t) &= \frac{t - \tau_i}{\tau_{i+d-1} - \tau_i} B_{i,d-1}(t) + \frac{\tau_{i+d} - t}{\tau_{i+d} - \tau_{i+1}} B_{i+1,d-1}(t). \end{align*}$$

2. Convex Hull Property and Constraint Satisfaction

A key attribute of B-splines is the convex hull property: for $t \in (\tau_i,\tau_{i+1})$, $z(t)$ is contained in the convex hull of $\{p_{i-d+1}, \ldots, p_i\}$. This enables translation of continuous-time trajectory constraints—such as state, control, or collision avoidance constraints—into a small finite set of constraints on control points.

For collision avoidance, let $O_\ell$ be an obstacle in the workspace. The requirement $z(t) \notin O_\ell$ for all $t$ is guaranteed if, for each control-point group $\{p_{i-d+1}, ..., p_i\}$, the convex hull is separated from $O_\ell$. The separating hyperplane theorem is used: there exists $c_{il}$ such that

$$\max_{j \in \{i-d+1, ..., i\}} c_{il}^T p_j \leq \min_{x \in \Theta^{-1}(O_\ell)} c_{il}^T x,$$

which, if satisfied for each group, ensures that the entire spline-parameterized curve avoids $O_\ell$ (Stoican et al., 2016).

In practice, because searching for arbitrary separating hyperplanes creates a nonconvex constraint (bilinear in control points and normals), a suboptimal—but efficient—formulation selects hyperplanes from a predefined pool (e.g., obstacle support hyperplanes) and uses binary variables for mixed-integer optimization of the half-space assignment per convex hull.

3. Optimization Structure: Waypoint and Cost Embedding

Typical spline-parameterized trajectory optimization minimizes an integral cost (for instance, trajectory length, energy, or smoothness penalty) while enforcing waypoint passage and constraints: $$\min_{P} \int_{t_0}^{t_n} \Vert z'(t) \Vert_{Q} \, dt$$ subject to

$$\Theta( [p_0, ..., p_n] \, B_d(t_s), \ldots ) = w_s, \quad s = 0, ..., N$$

and convex hull–based safety and feasibility constraints on $(p_i)$. This finite-dimensional optimization inherits excellent numerical properties from the locality and uniformity of B-spline bases.

For incorporation of dynamic or input constraints (velocity, acceleration bounds), note that derivatives of a B-spline are themselves lower-degree B-splines whose coefficients are explicit linear combinations of the original control points.

4. Smoothness, Continuity, and Vehicle Dynamics

A spline of degree $d$ guarantees $C^{d-1}$ continuity: the $(d-1)$-th derivative is continuous everywhere. For practice in UAVs and other underactuated vehicles, ensuring $C^2$ or $C^3$ continuity is essential for physically plausible headings, roll commands, and to avoid chattering or hardware-damaging abrupt changes.

In the context of differentially flat systems (e.g., fixed-wing UAVs), the flat outputs correspond to position; higher derivatives (velocity, acceleration) are computed for feedforward control via

$$\Psi(t) = \arctan\left(\frac{\dot{z}_2(t)}{\dot{z}_1(t)}\right), \qquad V_a(t) = \sqrt{\dot{z}_1(t)^2 + \dot{z}_2(t)^2}, \qquad \Phi(t) = \arctan\left(\frac{\ddot{z}_2(t) \dot{z}_1(t) - \dot{z}_2(t) \ddot{z}_1(t)}{g (\dot{z}_1(t)^2 + \dot{z}_2(t)^2)^{1/2}}\right).$$

This explicit computation from the spline's higher derivatives ensures all control inputs are naturally compatible with vehicle dynamics if the spline degree is chosen high enough.

5. Collision Avoidance: Exact and Sub-optimal Formulations

Ensuring the entire trajectory segment remains outside an obstacle relies on the convex hull property and separating hyperplanes:

• Exact formulation involves searching for a separating hyperplane for each convex hull.
• Sub-optimal, mixed-integer formulation selects among a set of candidate hyperplanes (obstacle support hyperplanes), introducing binary variables $\alpha_{im}^k$ to select the half-space, with constraints such as

$$- \sigma(m) h_m^T p_j^k \leq -\sigma(m) k_m + T \alpha_{im}^k, \quad \sum_m \alpha_{im}^k \leq M-1, \quad j = i-d+1, ..., i.$$

Here, $T$ is a large relaxation constant and $\sigma(m)$ the selection sign. For inter-agent collision avoidance, similar pairwise constraints are imposed on respective convex hulls.

These constructs transform the continuous, infinite-dimensional collision avoidance problem into a finite (often mixed-integer) optimization problem with guarantees of constraint satisfaction over the full continuous trajectory.

6. Implementation Considerations and Scaling

B-spline parameterizations translate continuous-time trajectory generation into an optimization over a finite set of control points, drastically reducing the computational burden. This enables:

• Tractability: The number of variables grows linearly with trajectory complexity, not with the time discretization horizon.
• Constraint enforcement: Safety and feasibility can be validated or imposed on a small set of points per segment.
• Flexible integration with sampling-based planners or real-time optimization: Spline parameterizations post-process kinodynamic search paths into smooth, feasible, and collision-free trajectories, usable in real-time systems operating in dynamic, complex environments.

Key computational primitives include calculation of the B-spline basis and its derivatives, convex hull intersection and hyperplane separation tests, and efficient sparse matrix computations for optimization.

7. Real-World Applications and Extensions

Spline-based trajectory parameterization underpins trajectory generation for multi-agent aerial vehicles, automated mobile robotics, and manipulation (Stoican et al., 2016). Its advantages—physical constraint satisfaction at all times, high smoothness, and finite-dimensional optimization—make it tractable for embedded and real-time systems. The framework has also been pivotal in receding horizon planning, distributed multi-agent coordination, and dynamic obstacle avoidance for flat and non-flat systems through extension of the underlying principles.

Extensions include:

• Adaptive or Variable-Knot Splines: To improve representation efficiency where trajectory complexity varies.
• Mixed-Integer Nonlinear Optimization (MINLP): For more nuanced safety guarantees, risk assessment, and priority assignment in constrained environments.
• Stochastic Extensions: Incorporating uncertainty into the parameterization and constraint enforcement for robust planning.

Spline-based trajectory parameterization represents a mathematically principled, algorithmically efficient, and practically validated approach to trajectory generation and planning, with significant theoretical and application-driven support in contemporary robotics and control literature.