Bezier Reachable Polytopes

• BRPs are sets of Bezier polynomial trajectories parameterized by control point vectors, providing a finite representation of infinite constraint verifications for safe motion planning.
• They convert state and input constraints into finite linear inequalities by leveraging the convex-hull and affine properties of the Bernstein–Bezier basis.
• BRPs are integral to layered motion planning systems, enabling real-time path planning and robust control certification in dynamic, cluttered environments.

A Bezier Reachable Polytope (BRP) is the set of Bezier polynomial trajectories—parameterized by control-point vectors—such that any reference curve in this set can be tracked by a controlled dynamical system while satisfying state and input constraints over a given time interval. BRPs leverage the convex-hull and affine properties of the Bernstein–Bezier basis to encode infinite-time continuous reachability and constraint satisfaction as finite collections of linear inequalities. This enables efficient, exact certificates for robust motion planning, especially in layered control architectures where tractable compositional reasoning over trajectory segments is required. Specialized applications have also exploited BRP-based graph structures for real-time path planning in cluttered and dynamic environments.

1. Mathematical Foundations

Let $x_d: [0,T] \to \mathbb{R}^d$ denote a degree-$n$ Bezier curve with control points $p_0, \dots, p_n \in \mathbb{R}^d$, expanded in the Bernstein basis $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$ as

$$x_d(t) = \sum_{i=0}^n B_{i,n}\left(\frac{t}{T}\right) p_i.$$

Key properties supporting the utility of BRPs include:

• Convex-Hull Property: $x_d(t) \in \operatorname{conv}\{p_0, \dots, p_n\}$ for all $t \in [0, T]$.
• Affine Input Representation: When system dynamics are linear (or feedback-linearizable), constraints on $x_d$ and the input $u_d$ can be translated to affine inequalities involving the control points.

Given state constraints ($Cx \le d$), input constraints ($n$0), and boundary conditions ($n$1), the set of admissible control-point vectors $n$2 forms a (possibly high-dimensional) convex polytope:

$n$3

where $n$4 and $n$5 accumulate all constraints and $n$6 enforces boundary consistency (Csomay-Shanklin et al., 2024).

2. Construction and Reachability Certificates

The explicit construction of a BRP proceeds as follows:

1. Boundary Encoding: Start with prescribed start and end conditions for the trajectory segment.
2. State Constraints: By the convex-hull property, require $n$7 for all $n$8.
3. Input Constraints: For feedback-linearizable or linear systems, use tracking and Lipschitz error bounds to enforce input limits as affine inequalities on the control points. This is possible since affine functions of the Bezier curve (and its derivatives) evaluate to affine functions of control points.
4. Polytope Assembly: Aggregate all linear constraints to yield the half-space representation $n$9 of the BRP.

Theorems in the literature confirm that the BRP is convex, and inclusion of $p_0, \dots, p_n \in \mathbb{R}^d$0 in this set certifies that the corresponding trajectory can be safely tracked under all modeled constraints (Csomay-Shanklin et al., 2024). This reduces the infinite constraint-checking problem over $p_0, \dots, p_n \in \mathbb{R}^d$1 to finitely many checks at the control points.

3. Role in Motion Planning Architectures

BRPs form the mathematical backbone of layered planner–tracker architectures. A typical motion planning stack proceeds as follows:

• High-level Planner: Samples or searches over a discretized set of boundary points (waypoints) in state space.
• Trajectory Segment Certification: For each candidate segment connecting waypoints $p_0, \dots, p_n \in \mathbb{R}^d$2, the planner solves for $p_0, \dots, p_n \in \mathbb{R}^d$3.
• Feasibility Graph: The set of feasible waypoints and corresponding admissible segments defines a directed graph, where edges correspond to BRP-certified Bezier curves.
• Global Pathfinding: Efficient search algorithms (e.g., Dijkstra, A*) identify admissible long-horizon paths by chaining BRP-certified segments.
• Low-Level Control: Each segment is supplied as a reference to a tracking controller. By BRP construction, the controller is guaranteed to remain within state/input bounds if the constraints used in the BRP are accurate (Csomay-Shanklin et al., 2024, Csomay-Shanklin et al., 2024).

4. Algorithmic Implementation and Real-Time Execution

BRPs enable efficient real-time planning due to their polytopic structure:

• Edge Generation: For each candidate segment, efficient offline precomputation of the matrices $p_0, \dots, p_n \in \mathbb{R}^d$4 accelerates the online feasibility check (linear algebra per trajectory).
• Obstacle Avoidance: Convex-hull property enables collision-checking for Bezier segments by verifying that the convex hull of the control points avoids all obstacles. A staged edge-cut heuristic accelerates this process: (1) first, reject if any control point enters an obstacle; (2) then, attempt separation via linear classifiers; (3) finally, resort to a quadratic program if required (Csomay-Shanklin et al., 2024).
• Parallelization: High-degree parallelism, e.g., using GPU kernels, allows for batched evaluation of edge feasibility and obstacle separation, enabling large-scale graphs (e.g., 50,000 edges) to be pruned, searched, and replanned in real time (graph-cut and replanning at ~10 Hz, tube-MPC refinement at 200 Hz for a high-dimensional hopping robot) (Csomay-Shanklin et al., 2024).

5. Geometric Structure: Connections to Toric Degenerations

The control-mesh limits of rational Bezier patches (especially in higher dimensions or for surface patches) are characterized by the structure of regular control surfaces induced by toric degenerations (Garcia-Puente et al., 2010):

• Toric Degenerations: By scaling the weights of a rational Bezier patch via a one-parameter family $p_0, \dots, p_n \in \mathbb{R}^d$5 (with lifting function $p_0, \dots, p_n \in \mathbb{R}^d$6), the patch degenerates to a piecewise union of lower-degree patches corresponding to a regular decomposition (S_\lambda) of the index set $p_0, \dots, p_n \in \mathbb{R}^d$7.
• Limit Surfaces: These limits are always regular control surfaces—$p_0, \dots, p_n \in \mathbb{R}^d$8 splines determined by the combinatorics of the regular decomposition—forming precisely the set of possible “control-mesh limits.”
• Classification: Every BRP arising as such a degeneration is associated with a regular decomposition encoded by the lifting function λ. Conversely, only regular decompositions correspond to BRP limits.

This geometric theory ensures that only “regular” control-mesh structures (not arbitrary ones) can arise as limiting objects of rational Bezier patches (Garcia-Puente et al., 2010).

6. Example: Construction in Practice

Consider a cubic (3rd-order) Bezier in $p_0, \dots, p_n \in \mathbb{R}^d$9 with endpoints $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$0 and $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$1, states constrained to a box $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$2, and acceleration bounded by $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$3. Constraints translate to:

• $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$4, $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$5 for all $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$6.
• Input constraints on the second Bezier differences, e.g., $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$7 and $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$8.

All feasible $B_{i,n}(s) = \binom{n}{i} s^i (1-s)^{n-i}$9 pairs (with $$x_d(t) = \sum_{i=0}^n B_{i,n}\left(\frac{t}{T}\right) p_i.$$0 fixed) form a convex polygon in control-point space, directly characterizing those Bezier curves that are safe to execute (Csomay-Shanklin et al., 2024).

7. Applications, Limitations, and Extensions

BRPs support computationally tractable, certified safe trajectory design in both simulation and hardware deployments:

• Motion Planning in Cluttered Environments: Layered planners using BRPs robustly navigate complex domains with hundreds of obstacles at high update rates (Csomay-Shanklin et al., 2024).
• Certificates for Layered Architectures: By making attainable trajectory sets explicit and finite-dimensional, BRPs facilitate provable control-system guarantees spanning high-level planning and low-level tracking (Csomay-Shanklin et al., 2024).

Typical limitations and extension areas include:

• Conservatism: Use of global Lipschitz constants and worst-case tubes can make BRPs conservative; this can be reduced using time subdivision (k-refinement) or tighter local estimates (Csomay-Shanklin et al., 2024).
• Scalability: Polytope dimension grows as $$x_d(t) = \sum_{i=0}^n B_{i,n}\left(\frac{t}{T}\right) p_i.$$1; practical systems balance trajectory degree with constraint complexity.
• Obstacle Incorporation: Obstacles can be incorporated as additional linear constraints on control points.
• Generalizations: Possible extensions include BRPs using B-splines, handling polyhedral input limits, and integration with sampling-based or mixed-integer planners.

Table: Summary of BRP Approaches Across Key Papers

Aspect	(Csomay-Shanklin et al., 2024)	(Csomay-Shanklin et al., 2024)	(Garcia-Puente et al., 2010)
Core Focus	Layered motion planning certificates	Real-time planning in cluttered env.	Geometric control mesh limits
Polytope Construction	State/input constraints via control points	Kinodynamic edge filtering and cutting	Toric degeneration structure
Main Output	Convex polytope in control-point space	Kinodynamic graph for MPC/planning	Regular control surfaces

This framework provides an efficient, mathematically grounded connection between trajectory parameterization, control-theoretic reachability, and practical, robust motion generation in diverse domains.