Bézier Polynomial Speed Planning

• Bézier polynomial speed planning is a method that employs Bernstein polynomials to parameterize trajectories under kinematic, dynamic, and safety constraints.
• It transforms infinite-dimensional optimal control challenges into finite-dimensional NLP or QP problems using convex-hull properties and derivative formulations.
• The approach enables efficient enforcement of speed, acceleration, and obstacle avoidance constraints, supporting real-time autonomous vehicle applications.

Bezier polynomial speed planning is the study and application of Bézier (or, more generally, Bernstein) polynomial parameterizations to velocity and trajectory optimization for autonomous systems constrained by kinematic, dynamic, and environmental limits. Key features include the reduction of infinite-dimensional optimal control or speed-constrained feasibility problems to tractable nonlinear or quadratic programming formulations, while leveraging the convex-hull and closed-form derivative properties intrinsic to the Bernstein basis. This enables rigorous enforcement of speed, acceleration, and safety constraints along geometric paths, supports efficient computation, and is especially relevant for real-time dynamic environments encountered in autonomous vehicle planning.

1. Mathematical Foundations of Bézier Polynomial Speed Profiles

A Bézier polynomial of degree $n$ is defined on the interval $[0,1]$ as

$$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$

A trajectory or scalar position profile (e.g., arc-length or station) over a generic interval $[T_k, T_{k+1}]$ can be parameterized as

$$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$

with $h_k = T_{k+1} - T_k$ and $c_i^k$ the control points.

Instantaneous velocity, acceleration, and jerk are also Bézier polynomials of order $n-1$, $n-2$, $n-3$ respectively, with control points constructed via the hodograph property: $[0,1]$0 and similarly for higher derivatives. This algebraic structure is exploited to impose dynamic constraints directly on the polynomial coefficients, transforming state and control integral constraints into finite sets of linear or norm inequalities on the associated control points (Kielas-Jensen et al., 2020, Li et al., 2021).

2. Constraint Enforcement via Bernstein Convex-Hull Property

The Bernstein convex-hull property asserts that a Bézier curve lies entirely within the convex hull of its control points. For a velocity profile $[0,1]$1 represented as a Bézier polynomial, imposing

$[0,1]$2

(where $[0,1]$3 are the Bernstein derivative control points and $[0,1]$4 is a constant or position-dependent speed cap) suffices to ensure speed constraints at all continuous times, since $[0,1]$5 is always a convex combination of $[0,1]$6. Further, when physical constraints require position, velocity, acceleration, or jerk limits, each can be translated to simple norm or component-wise inequalities on the corresponding Bézier coefficients: $[0,1]$7 for $[0,1]$8 (derivative order) and $[0,1]$9 (Li et al., 2021). This finite test replaces the need for dense or adaptive sampling, offering a necessary and sufficient condition for global feasibility within the domain.

3. From Infinite-Dimensional Optimal Control to Finite-Dimensional NLP

Consider the canonical minimum-time speed planning problem along a prescribed $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$0 path $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$1 parametrized by arc-length $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$2, with speed law $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$3. The target is minimization of the traversal time,

$$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$4

subject to bounds: $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$5 where $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$6 is curvature, and $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$7 tangential and normal acceleration bounds.

By substitution $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$8, the problem becomes a convex optimization over absolutely continuous profiles $$B_{i,n}(\tau) = \binom{n}{i} \tau^i (1 - \tau)^{n - i}, \quad i = 0, \ldots, n.$$9, further simplifying to algebraic constraints when paths are parameterized as Bézier curves. The infinite-dimensional problem is approximated by the finite expansion

$[T_k, T_{k+1}]$0

with all state, dynamic, and endpoint constraints enforced by tightening the feasible set of control points (Kielas-Jensen et al., 2020). The resulting NLP solves for $[T_k, T_{k+1}]$1 over control points, with cost integrals replaced by weighted sums (via Bernstein’s quadrature property).

4. Lattice-Theoretic Minimum-Time Speed Planning along Bézier Paths

For fixed paths, Consolini et al. (Consolini et al., 2018) demonstrate a complete lattice structure in the set of feasible speed profiles satisfying kinematic and dynamic bounds. Let $[T_k, T_{k+1}]$2 and define the “meet” operator $[T_k, T_{k+1}]$3 as the pointwise minimum of two ODE-integrated envelopes:

• Forward envelope $[T_k, T_{k+1}]$4, integrating from $[T_k, T_{k+1}]$5 under maximal acceleration $[T_k, T_{k+1}]$6.
• Backward envelope $[T_k, T_{k+1}]$7, integrating from $[T_k, T_{k+1}]$8 under maximal deceleration $[T_k, T_{k+1}]$9.

The optimal feasible solution is $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$0, with feasibility condition $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$1 everywhere. For Bézier paths,

$$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$2

arc-length $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$3 and curvature $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$4 can be precomputed and substituted directly. The resulting speed profile exhibits the characteristic “bathtub” shape: it accelerates at regions of low curvature, decelerates as tight bends approach, and re-accelerates after (Consolini et al., 2018).

5. Piecewise Bézier/Trapezoidal Corridor Methods for Obstacle Avoidance

In dynamic environments, safety envelopes in $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$5 space are non-convex due to moving obstacles. The dynamic programming (DP) based method (Li et al., 2021) discretizes time and station, building a finite S-T graph and searching for cost-minimizing waypoint sequences. Each feasible segment is then parameterized as a Bézier polynomial, with control points constrained to lie within convex “trapezoidal corridors”: $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$6 for all $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$7, guaranteeing that $$s_k(t) = h_k \sum_{i=0}^{n} c_i^{k} B_{i,n}\left(\frac{t-T_k}{h_k}\right), \quad t \in [T_k, T_{k+1}],$$8 lies within prescribed affine bounds. Physical feasibility (kinematics, jerk) is enforced analogously. Objective functions weight safety, tracking, velocity, and comfort via a convex quadratic cost, yielding a QP. This approach has demonstrated both computational efficiency and robust feasibility in simulation and real-vehicle driving tests when compared to alternative real-time planners.

6. Geometric, Computational, and Theoretical Guarantees

The convex-hull and endpoint interpolation properties uniquely enable Bézier-based methods to provide certifiable satisfaction of trajectory constraints independently of path complexity or polynomial degree. The decomposition of derivative bounds into simple algebraic inequalities on control points ensures efficient, real-time solvability even in the presence of complex dynamic obstacles and comfort requirements. In the lattice-theoretic approach, the solution structure (forward/backward sweep, supremum of feasible profiles) further admits algorithms of linear complexity in the discretization, supporting applications in embedded planning systems (Consolini et al., 2018, Li et al., 2021, Kielas-Jensen et al., 2020).

Reference	Key Contribution	Implementation Approach
Consolini et al. (Consolini et al., 2018)	Minimum-time speed planning as lattice optimization; ODE sweeps and feasibility test	Discretize arc-length, forward/backward ODEs, pointwise min/max
Kielas-Jensen & Cichella (Kielas-Jensen et al., 2020)	Bernstein transcription of infinite-dimensional OC to NLP; convex-hull enforcement	Algebraic constraints on control points, matrix algebra
Yu et al. (Li et al., 2021)	Piecewise Bézier with DP-guided trapezoidal corridors for safety and real-time QP	DP for corridor selection, QP for profile optimization

7. Practical Applications and Performance in Autonomous Systems

Real-world deployments focus on autonomous driving, aerial vehicle guidance, and multi-agent coordination, where continuous feasibility with respect to state, actuator, and safety constraints is essential. BeBOT and related Bernstein-based solvers are configured for rapid computation and universal constraint satisfaction, independent of path complexity, for tasks ranging from single-agent obstacle avoidance to coordinated, multi-vehicle trajectory planning with provable safety (Kielas-Jensen et al., 2020, Li et al., 2021). In practical scenarios (e.g., on C++/OSQP), average solve times of 5–8 ms per trajectory have been reported, and smoothness, safety with time-varying constraints, and solution optimality observed across extensive simulated and hardware tests.

A plausible implication is that the inherent compatibility of Bézier polynomial speed planning with both convex optimization and continuous-time constraint satisfaction will maintain its central role in real-time trajectory generation for autonomous vehicles and higher-level robotic path planners.