Piecewise Linear Trajectory Approach

• The piecewise linear trajectory approach is a modeling technique that decomposes complex system trajectories into concatenated linear segments for tractable optimization.
• This method is widely applied in motion planning, control, and filtering, with real-world use cases in robotics, vehicle navigation, and high-dimensional data search.
• It integrates nonconvex constraints into convex frameworks using SDP and MILP, ensuring scalability, error bounds, and feasibility for real-time applications.

A trajectory piecewise linear approach refers collectively to methodologies that represent, approximate, or plan system trajectories as concatenations of linear (or affine) segments over time, space, or feature domains. These approaches provide computational tractability for motion planning, optimal control, filtering, system identification, or search in high-dimensional spaces. The frameworks admit rigorous encoding of system constraints, obstacle avoidance, and optimality conditions, often nonlinear or nonconvex in origin, into optimization or algorithmic forms based on piecewise-linear (PL) parameterizations. Rapid advances span motion planning, predictive control, stochastic filtering, scheduling, learning, and information retrieval.

1. Piecewise Linear Trajectory Parameterization

The foundational construct is the parameterization of a trajectory as a sequence of breakpoints or waypoints, generating a piecewise-linear curve in the state, input, or feature space.

A continuous-time or discrete-time path $x:[0,T]\to\mathbb{R}^n$ is represented as: $$x(t) = x_{i-1} + \tau_i(t) \cdot (x_i - x_{i-1}), \quad t \in [t_{i-1}, t_i],\quad i=1,\dots,s$$ where $T$ is the time horizon, $\{t_i\}_{i=0}^s$ is a (possibly nonuniform) grid, and $\{x_i\}$ are the breakpoints (Khadir et al., 2020, Shafa et al., 3 Oct 2025, Le et al., 2024). For planning under waypoints, $w_i = (t_i, p_i)$, the piecewise-linear signal is: $$\xi(t) = p_i + \frac{p_{i+1} - p_i}{t_{i+1} - t_i} (t - t_i) \quad \text{for}~ t \in [t_i, t_{i+1}]$$ (Le et al., 2024).

PL frameworks can be extended from Euclidean domains (robotics, vehicle trajectories (Plessen et al., 2017)) to feature-space data (audio or video descriptors (0710.4180)) or in the context of system identification with sampled data (Wang et al., 2024, Wang et al., 2023). For hybrid systems or those with switching regimes, the PL approaches become piecewise-affine (PWA), where each segment is a linear or affine map possibly defined on polytopic regions (Block et al., 2023, Han et al., 2019).

2. Algorithmic Construction and Optimization

Motion Planning via Global Polynomial and SDP Optimization

In complex environments with static, moving, or morphing obstacles, PL motions enable global optimality frameworks. Obstacle-free path segments are parameterized by the breakpoints, and the path length is minimized: $L(X) = \sum_{i=1}^s \|x_i - x_{i-1}\|_2$ Continuous-time, nonconvex collision constraints are enforced polynomially along each linear segment: $$g_k(t, x_{i-1} + \tau(x_i - x_{i-1})) \ge 0,\quad \forall\,\tau\in[0,1],\,\forall\,k$$ The resulting problem is a polynomial optimization, which admits a hierarchy of semidefinite programming (SDP) relaxations (Lasserre moment-SOS hierarchy) for global lower bounds. Practical planning is achieved with the Moment Motion Planner (MMP), which alternates pseudo-moment solutions and rank-one penalizations to efficiently extract feasible near-optimal PL trajectories (Khadir et al., 2020).

Mixed-Integer Linear Programming for Temporal Logic and Scheduling

For planning under temporal logic specifications (e.g., STL), PWL trajectories are synthesized so that their segments satisfy constraints in both space and time. The quantitative STL semantics are encoded recursively over segments; constraints are compiled as MILPs with time-robustness metrics, employing big-M encodings and auxiliary indicator variables to ensure satisfactory satisfaction of logic (Le et al., 2024).

PWL value functions are also critical in bilevel and mixed-integer nonlinear programming (MINLP) formulations, as in nonlinear dynamic scheduling of VTOLs. The lower-level optimal control is reduced to a value function, further piecewise-linearized for computational tractability within a single MINLP (Nikitina et al., 2023).

Lattice Piecewise Linear Approximation

Successive linearization along a reference path or nominal trajectory underpins lattice PL methods. Local Taylor expansions around sampled states yield affine models, which are then assembled into a global lattice PWL (max-min) structure: $$f_\text{LTPWL}(x) = \max_{i} \min_{j \in I_{\ge,i}} l_j^f(x)$$ Enables batch estimation, lookup, and region-free evaluation—applied both to explicit MPC law approximation and nonlinear Kalman filtering with attention mechanisms (Wang et al., 2024, Wang et al., 2023).

3. Theoretical Guarantees and Error Analysis

PL approaches are backed by (i) convergence of SDP relaxations (Khadir et al., 2020), (ii) soundness theorems for STL satisfaction (Le et al., 2024), and (iii) explicit error bounds for function approximation (Wang et al., 2023). For PL surrogates of nonlinear dynamics, the approximation error is locally $O(\|x-x_i\|^2)$ (Taylor remainder), and can be made arbitrarily small via node densification (Wang et al., 2024, Wang et al., 2023).

Safety, convergence, and reachability for unknown dynamics are provable via local Lipschitz-constrained proxy systems and reachability set under-approximations (Shafa et al., 3 Oct 2025). In PWA learning, sublinear simulation and prediction regret can be achieved under mild stochastic smoothing (Block et al., 2023).

For search in high-dimensional time-series, segment-based KL embeddings with projection-distance features provide tight distance preservation, ensuring the same search result as the unreduced sequence within bounded error (0710.4180).

4. Applications Across Domains

Motion Planning and Robotics

• Optimal motion in constrained environments: Efficiently solves global planning problems with complex, time-varying obstacles, outperforming sampling-based and direct optimization in success rate, path length, and smoothness (Khadir et al., 2020).
• Automated vehicle trajectory planning: SLP-based spatial planners leveraging PL trajectories can handle tight road and vehicle-dimension constraints, yielding smoother and higher-speed paths than clothoid chain approaches (Plessen et al., 2017).
• Temporal logic and mission planning: Synthesis of PWL reference trajectories compatible with STL or LTL specifications, with time-robustness margins (Le et al., 2024).
• Hybrid multi-contact feedback: Local PWA funnel feedback and online LP control enable robust real-time tracking and disturbance rejection in non-smooth robotic manipulation (Han et al., 2019).

Control and Estimation

• Nonlinear MPC: Lattice PWL surrogates rapidly approximate both system dynamics and optimal control law, reducing both offline and online computational load by orders of magnitude without sacrificing closed-loop accuracy (Wang et al., 2023).
• Nonlinear filtering: Self-attention Kalman architectures exploit LTPWL approximations for robust, parallelizable batch estimation in nonlinear systems (Wang et al., 2024).

Machine Learning and Online Adaptation

• PWA online learning: Epoch-based oracle-efficient algorithms achieve polynomial-regret prediction and simulation in PWA systems, capitalizing on smoothing, region labeling, and label alignment strategies (Block et al., 2023).
• Adaptive trajectory tracking under uncertainty: Reachable predictive control guarantees convergence to PL waypoints solely via local data-driven linearization and proxy reachability set construction (Shafa et al., 3 Oct 2025).

Information Retrieval

• Fast search in high-dimensional time series: Dynamic segmentation and PL feature representation in audio or video enables large-scale, equivalence-preserving search with order-of-magnitude speedup and no accuracy loss (0710.4180).

5. Computational and Algorithmic Considerations

Piecewise linearization reduces the inherent nonconvexity and computational complexity of planning, control, or estimation over nonlinear and hybrid systems:

• SDP and MILP scalability: PWL abstractions allow convex or mixed-integer optimization with tighter variable counts, especially beneficial for high-dimension or long-horizon scenarios (Khadir et al., 2020, Le et al., 2024, Nikitina et al., 2023).
• Max-min representations: Lattice PWL forms eliminate explicit region-testing, supporting efficient evaluation and batch estimation (Wang et al., 2024, Wang et al., 2023).
• Online feasibility: Sequential LPs, small online QPs/LPs for funnel tracking, and table-lookup for control law retrieval enable real-time deployment (Plessen et al., 2017, Han et al., 2019, Wang et al., 2023).
• Error and adaptivity: Tuning node density, region clustering, or projection-rank permits fine-grained control of the accuracy-speed tradeoff (Wang et al., 2024, Wang et al., 2023, 0710.4180).

6. Generalizations, Extensions, and Impact

PL frameworks generalize beyond classical state or input trajectory planning to analytic modeling of chaotic attractors (Morosetti, 2021), generic data streams, and hybrid or partially unknown systems (Shafa et al., 3 Oct 2025). They are functionally agnostic as long as local continuity and segment-wise (locally) low-rank structure are present (0710.4180).

Modern trends assimilate PWL/lattice approximations into data-driven architectures (e.g., neural networks for Kalman gain adaptation (Wang et al., 2024)) or event-triggered, model-free optimal control (Shafa et al., 3 Oct 2025).

The piecewise linear trajectory approach constitutes a central modeling, synthesis, and computational tool unifying optimization, control, learning, and information retrieval, enabling high-performance solutions in domains where full global models are intractable or unavailable.