Trajectory Based Optimization Design

• TBOD is a trajectory-focused design methodology that embeds system-specific dynamics and feasibility constraints directly into the optimization loop.
• It employs diverse formulations such as bi-level optimization, linear programming, and consensus-based polynomial methods to handle nonconvex constraints efficiently.
• TBOD enables rapid, reliable trajectory generation across fields—from aerospace to robotics—by integrating advanced constraint encoding and decomposition strategies.

Trajectory Based Optimization Design (TBOD) is used in recent arXiv literature to denote a trajectory-centric optimization methodology in which trajectories are the primary design variables and modeling, sensing, feasibility, and implementation choices are embedded directly into the optimization loop. In different instantiations, TBOD appears as bi-level trajectory optimization with optimization-based dynamics, minimum-time planning from input-output data, parallel piecewise-polynomial trajectory optimization, observer tuning from prerecorded trajectories, probabilistic path design for experimental design, and airline-centric four-dimensional trajectory orchestration in aviation (Howell et al., 2021, Li et al., 2023, Yu et al., 14 Jul 2025, Oliva et al., 1 Oct 2025, Attia, 16 Jan 2026, Costas et al., 2024).

1. Definition and conceptual scope

The cited works use TBOD as a design philosophy or methodology rather than as a single algorithm. In its most direct form, TBOD means designing a trajectory by solving an optimization problem over states, controls, path parameters, or trajectory policies, while placing system-specific structure inside the mathematical program instead of treating it as an external post-processing step. That structure may be dynamics returned by an embedded optimizer, path predictions produced by Hankel data equations, consensus-enforced continuity constraints on piecewise polynomials, observer update laws tuned from recorded runs, or probabilistic trajectory policies over a navigation mesh (Howell et al., 2021, Li et al., 2023, Yu et al., 14 Jul 2025, Oliva et al., 1 Oct 2025, Attia, 16 Jan 2026).

A central feature of TBOD is that feasibility is not reduced to a generic “trajectory smoothing” stage. Instead, feasibility is encoded in whatever representation is appropriate for the task: complementarity and cone constraints for contact and friction, polyhedral or ellipsoidal separating and containing constraints for collision avoidance, holonomic loop constraints in humanoids, wake-turbulence separation in terminal air traffic control, or FF-ICE-valid procedural constraints in airline operations (Howell et al., 2021, Fan et al., 2023, Zhang et al., 2024, Pang et al., 20 Apr 2026, Costas et al., 2024).

TBOD is also used outside motion generation in a narrow robotics sense. “Trajectory-based” may refer to optimization over control histories, over path probabilities, over aircraft 4D operational plans, or over observer parameters evaluated on measurement trajectories. This suggests that TBOD is best understood as a trajectory-centered design pattern whose precise semantics depend on the surrounding domain (Oliva et al., 1 Oct 2025, Attia, 16 Jan 2026, Costas et al., 2024).

2. Canonical mathematical structures

A canonical TBOD formulation appears in bi-level trajectory optimization with optimization-based dynamics. Over a discrete horizon $t=0,\ldots,T$, the upper level optimizes the trajectory

$$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$

where the transition map $d(\cdot)$ is itself returned by solving a lower-level constrained optimization at each time step. Auxiliary variables such as impulses, friction forces, cone variables, and slacks are optimized in the lower level and then fed into the state update $x_{t+1}=F(x_t,u_t,y_t^\ast)$ (Howell et al., 2021).

Other TBOD formulations keep the trajectory as the design object but change the model class. In minimum-time data-based planning, the trajectory is represented by stacked input-output segments constrained by Hankel matrices, and the free-final-time objective is converted into an exponentially weighted slack cost

$$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$

yielding a linear program that recovers a minimum-time trajectory for sufficiently large $\theta$ (Li et al., 2023). In probabilistic path experimental design, the discrete path $\tau$ is treated as a random variable generated by a Markov policy, and the design problem becomes

$$\theta^\star \in \arg\max_{\theta}\,\mathbb{E}_{\tau\sim\pi_\theta}[U(\tau)],$$

so that the optimized object is a trajectory distribution rather than a single deterministic path (Attia, 16 Jan 2026).

In polynomial and spline-based TBOD, the trajectory is parameterized explicitly. TOP represents each segment by a polynomial

$$\sigma_i(t)=\mathbf{c}_i^\top \beta(t-t_{i-1}), \qquad \beta(\tau)=(1,\tau,\dots,\tau^d)^\top,$$

and optimizes coefficients subject to continuity, convex corridor, and sampled dynamic-feasibility constraints (Yu et al., 14 Jul 2025). RAPTOR uses degree-$V$ Bézier polynomials for actuated humanoid coordinates and optimizes the coefficients together with reset quantities across single-support phases (Zhang et al., 2024). In SE(3)-based ISAC, the trajectory design variable is the control twist $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$0 entering the rigid-body update

$$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$1

and the objective is derived from the conditional Posterior Cramer-Rao bound (CPCRB) (Xu et al., 20 Jan 2025).

3. Constraints, uncertainty, and differentiability

A defining strength of TBOD is the direct incorporation of difficult constraints into the optimization model. In optimization-based dynamics, inequality and cone constraints are handled by interior-point formulations, with residuals defined on stationarity, equality, and complementarity blocks, and sensitivities computed by implicit differentiation: $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$2 This allows classical upper-level methods such as iLQR to receive smooth gradients even when the modeled physics includes joint limits, Coulomb friction, impacts, and thrust cones (Howell et al., 2021).

Uncertainty is handled in several distinct TBOD styles. In robust aerospace trajectory design, polynomial chaos expansion (PCE) converts stochastic path constraints into deterministic moment constraints such as

$$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$3

and the resulting deterministic problem is optimized by a constrained NSGA-II pipeline over a trajectory ensemble (Takubo et al., 2022). In observability-aware trajectory optimization, Lie-derivative-based observability matrices and interval-filtering information matrices are used as the objective, so trajectory design is explicitly driven by estimation quality rather than only control effort (Grebe et al., 2021). In SE(3)-based UAV-enabled ISAC, the trajectory is optimized against the CPCRB,

$$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$4

to improve range, Doppler, and angle estimation under rigid-body and radar constraints (Xu et al., 20 Jan 2025).

For collision-rich or interaction-rich settings, TBOD uses geometry-aware constraint constructions. A unified collision-avoidance framework formulates differentiable separating constraints through the hyperplane separation theorem and smooth containing constraints through the S-procedure and geometric tests for polytopes and ellipsoids (Fan et al., 2023). In controllable multi-vehicle interaction design, local relative topology is encoded by the differentiable metric

$$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$5

evaluated at the closest-approach time and used as a sign-constrained penalty to select clockwise or counterclockwise passing patterns (Ma et al., 7 Mar 2025).

4. Solver architectures and computational patterns

Across the literature, TBOD spans a wide solver spectrum. The common pattern is that solver choice is driven by the trajectory representation and by where feasibility is imposed.

Formulation family	Core representation	Representative solver pattern
Optimization-based dynamics	Bi-level state-control trajectory	Interior-point + implicit differentiation + iLQR
Data-based minimum-time planning	Hankel trajectory segments	Linear programming
Parallel piecewise-polynomial planning	Consensus over segment coefficients	CADMM on CPU/GPU
Learned warm-start planning	Decision-vector trajectory samples	Diffusion model + SQP
Koopman mixed-boundary planning	Lifted linear lower level	Upper-level nonlinear search + lower-level convex QP
Sparse-roadmap refinement	Multi-query path seed	Roadmap query + TrajOpt SQP

The computational consequences are substantial. In the data-based LP approach, after eliminating the DeePC coefficients, the spacecraft case study solved in $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$6 ms on a desktop, about $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$7 faster than a comparable mixed-integer program (Li et al., 2023). TOP reports constant-time per-iteration wall-clock complexity with respect to the number of segments under sufficient parallel resources, more than a tenfold speedup for one hundred segments, and $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$8 s for a $$\min_{\{x_t,u_t\}_{t=0}^{T-1}} \; J=\sum_{t=0}^{T-1}\ell(x_t,u_t)+\ell_T(x_T) \quad \text{s.t.} \quad x_{t+1}=d(x_t,u_t;\theta),$$9-segment trajectory on an NVIDIA GeForce RTX 4060 Laptop GPU (Yu et al., 14 Jul 2025). DiffuSolve+ reports $d(\cdot)$0 s versus Uniform $d(\cdot)$1 s on the tabletop task, corresponding to $d(\cdot)$2 faster 25%-quantile sampling-plus-solving time (Li et al., 2024). RAPTOR computes a six-step periodic Digit gait in $d(\cdot)$3 seconds wall-clock time (Zhang et al., 2024). The roadmap-plus-TrajOpt framework achieves combined runtimes of $d(\cdot)$4 s in “tabletop with a pole” and $d(\cdot)$5 s in “shelf with boxes” while maintaining high success rates (Dai et al., 2018).

A recurring architectural theme is decomposition. TOP decomposes by segment and consensus variable (Yu et al., 14 Jul 2025). Koopman mixed-boundary trajectory optimization decomposes the problem into a low-dimensional upper level over $d(\cdot)$6 and a convex lifted lower level (Abou-Taleb et al., 2024). DiffuSolve separates generative multi-modal initialization from deterministic local refinement (Li et al., 2024). The TMA arrival framework separates landing-order selection by MILP from per-aircraft trajectory realization by NLP (Pang et al., 20 Apr 2026). This suggests that TBOD frequently achieves tractability not by simplifying the trajectory objective itself, but by relocating nonconvexity to a lower-dimensional or better-structured subproblem.

5. Representative application domains

The breadth of TBOD is visible in the domains where the same trajectory-centered logic is instantiated.

Domain	TBOD instantiation	Reported outcome
Spacecraft relative motion	Data-based minimum-time LP	$d(\cdot)$7 steps, LP $d(\cdot)$8 ms, $d(\cdot)$9 faster than MIP
Long-horizon motion planning	TOP with CADMM	$x_{t+1}=F(x_t,u_t,y_t^\ast)$0 s for $x_{t+1}=F(x_t,u_t,y_t^\ast)$1 segments on GPU
Humanoid locomotion	RAPTOR whole-body gait optimization	Six-step periodic gait in $x_{t+1}=F(x_t,u_t,y_t^\ast)$2 s
Sensor fusion	Offline observer tuning from trajectories	Yaw MAE $x_{t+1}=F(x_t,u_t,y_t^\ast)$3 vs EKF $x_{t+1}=F(x_t,u_t,y_t^\ast)$4
Airline operations	FOC-centric 4D trajectory management	$x_{t+1}=F(x_t,u_t,y_t^\ast)$5 CONCUR, $x_{t+1}=F(x_t,u_t,y_t^\ast)$6 NEGOTIATE, latencies $x_{t+1}=F(x_t,u_t,y_t^\ast)$7–$x_{t+1}=F(x_t,u_t,y_t^\ast)$8 s
TMA arrival management	MILP+NLP sequencing and trajectory realization	FEFS/FOFFS/CPS$x_{t+1}=F(x_t,u_t,y_t^\ast)$9/CPS$$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$0 under $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$1 s per entry

In robotics, TBOD covers collision-aware and contact-rich planning, humanoid gait synthesis, and interaction design. RAPTOR directly optimizes continuous whole-body trajectories under rigid-body dynamics, closed-loop kinematic constraints, stance wrench bounds, and hybrid reset conditions (Zhang et al., 2024). Topology-driven multi-vehicle planning uses a differentiable local homotopy invariant to produce controllable interaction patterns from identical initial values (Ma et al., 7 Mar 2025). Observability-aware optimization and SE(3)-based CPCRB-guided design extend the same logic to estimation-limited flight, where the trajectory is chosen to improve calibration or tracking quality rather than merely to reach a destination (Grebe et al., 2021, Xu et al., 20 Jan 2025).

In data-driven and stochastic settings, TBOD includes LP-based minimum-time planning directly from input-output data (Li et al., 2023), robust trajectory optimization under wind uncertainty through MOEA+PCE (Takubo et al., 2022), diffusion-based warm-start generation for nonconvex trajectory NLPs (Li et al., 2024), and policy-based path optimization for experimental design on a navigation mesh (Attia, 16 Jan 2026). The common thread is that the trajectory remains the design object even when the model is nonparametric, learned, or probabilistic.

In aviation, TBOD has two notably different meanings. In airline operations, it extends Trajectory-Based Operations (TBO) by placing the Flight Operations Center at the center of trajectory generation, validation, negotiation, execution, and monitoring through FF-ICE, SWIM, and EFB/FMS integration (Costas et al., 2024). In terminal air traffic control, it denotes a runway-sequencing and scheduling framework that computes per-aircraft speed profiles and path extensions rather than abstract node delays, and reports near real-time per-entry optimization (Pang et al., 20 Apr 2026).

Sensor fusion and estimation provide another extension. “Trajectory Based Observer Design” uses prerecorded measurement trajectories to tune observer parameters offline, producing a lightweight online observer; on a rover localization task it matches EKF position accuracy while improving orientation substantially, with TBOD yaw MAE $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$2 versus EKF $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$3 (Oliva et al., 1 Oct 2025).

6. Trade-offs, misconceptions, and outlook

A common misconception is that TBOD denotes a single solver family. The cited literature does not support that interpretation. TBOD encompasses interior-point bi-level optimal control, LP reformulations, ADMM consensus schemes, stochastic evolutionary search, diffusion-model warm starts, observer tuning, and operational air-traffic trajectory orchestration. Another misconception is to treat TBOD as interchangeable with trajectory-based operations in air traffic management; one cited aviation work explicitly presents TBOD as an airline-centric instantiation that extends TBO rather than as a synonym for it (Costas et al., 2024).

The dominant trade-off is between model expressiveness and computational burden. Optimization-based dynamics provide strong constraint satisfaction and smooth sensitivities, but “the interior-point solve dominates” and the overall rollout cost is $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$4 in a generic dense count (Howell et al., 2021). Data-based LP formulations are convex and fast, but their guarantees rely on controllability, persistency of excitation of order $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$5, controlled invariance of $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$6, and exact or noise-free DeePC equalities in the base formulation (Li et al., 2023). Parallel CADMM achieves near-constant per-iteration wall-clock behavior with respect to segment count only under sufficient parallel resources and convex prox-friendly constraints (Yu et al., 14 Jul 2025). Diffusion warm-starting reduces runtime and improves diversity, but it depends on a large offline corpus of locally optimal trajectories and, in the constrained variant, on differentiable constraint functions (Li et al., 2024).

Task-specific assumptions remain central. RAPTOR assumes single support is fully actuated with unique inverse kinematics and uses a fixed contact schedule (Zhang et al., 2024). Koopman mixed-boundary optimization cannot completely convexify general problems with unknown terminal time and mixed boundary constraints; instead it pushes nonconvexity into a low-dimensional upper level (Abou-Taleb et al., 2024). In TMA sequencing, CPS beyond $$J(\epsilon)=\sum_{t=T_0}^{T_1}\theta^{t-T_0}\|\epsilon_t\|_1,$$7 yields diminishing returns with rapidly increasing solver cost (Pang et al., 20 Apr 2026). In FOC-centric aviation TBOD, operational gains depend on continued FF-ICE/SWIM deployment, multi-stakeholder collaboration, and acceptable message latency distributions (Costas et al., 2024).

The overall research direction points toward richer decomposition, tighter integration between estimation and control objectives, and broader hardware awareness. This suggests that TBOD is evolving less as a unified theory than as a reusable design grammar: choose a trajectory representation, encode feasibility where it is structurally natural, select an optimization architecture matched to that representation, and expose the resulting trajectory object directly to planning, control, sensing, or operational decision-making.