Initialization from Existing Trajectories

Updated 4 April 2026

Initialization from Existing Trajectories is a method that uses previously recorded or optimized paths as informative seeds to enhance convergence in non-convex optimization and control tasks.
It encompasses warm-starting, analytical adaptation, learning-based, in-context, and template anchoring approaches, each balancing speed, accuracy, and resource demands.
This technique is pivotal in robotics and autonomous systems, achieving up to 160× speed-ups and significant error reductions in trajectory planning and system identification.

Initialization from existing trajectories refers to the use of solution trajectories—recorded, synthesized, or optimized under similar but not identical conditions—as informative seeds or priors for learning, inference, or optimization in tasks involving motion, planning, prediction, model identification, or control. This approach is pivotal in fields such as robotics, autonomous driving, system identification, and trajectory prediction, where optimization landscapes are highly non-convex, local methods are sensitive to initialization, or computation must be performed under severe time or resource constraints.

1. Principles and Categories of Initialization from Trajectories

Multiple paradigms exist for leveraging existing trajectories:

Warm-starting optimization: Existing solutions are reused to seed an iterative trajectory optimizer, reducing iterations and accelerating convergence, especially for similar tasks or under small perturbations (Srikanth et al., 2020, Shehadeh et al., 7 Mar 2026, Melon et al., 2021).
Analytical deformation or adaptation: Given a previous solution at one set of task parameters, local sensitivity analysis (e.g., differentiation through the optimal solution) provides a first-order update direction to morph the trajectory to new conditions (Srikanth et al., 2020).
Learning-based initialization: Neural models or policies are trained using collections of expert or optimized trajectories, enabling them to generate high-quality initial guesses for new instances (Shehadeh et al., 7 Mar 2026, Yoon et al., 3 Feb 2026, Kim et al., 2021, Melon et al., 2021).
Contextual or in-context initialization: For prediction tasks, the model conditions directly on a set of retrieved exemplar trajectories (“in-context examples”) most similar to the current situation, without gradient-based learning (Fujii et al., 1 Jun 2025).
Template anchoring: A library of primitive or meta-action–conditioned trajectories is used to select and anchor stochastic generative models such as diffusion planners (Ding et al., 11 Mar 2026).

These paradigms are not mutually exclusive and often interleave: learning-based approaches may produce initializations for analytic optimizers, and in-context conditioning often exploits a database of prior motion exemplars.

2. Analytical Adaptation of Existing Trajectories through Sensitivity Analysis

A canonical strategy for rapid trajectory adaptation is formalized in "Fast Adaptation of Manipulator Trajectories to Task Perturbation By Differentiating through the Optimal Solution" (Srikanth et al., 2020). Here, the trajectory optimization is expressed as a parametric nonlinear program

$\xi^*(p) = \text{argmin}_\xi\,f(\xi,p) \quad \text{s.t.}\quad g_i(\xi,p)\leq 0,\;h_j(\xi,p)=0$

with cost terms for smoothness, boundary/via-point tracking, and orientation error, and box constraints on joint trajectories. When task parameters $p$ (boundary conditions, goal waypoints, etc.) are perturbed to $p' = p + \Delta p$ , instead of re-solving the full problem, the Karush–Kuhn–Tucker (KKT) optimality system is differentiated with respect to $p$ , yielding the argmin-Jacobian: $\frac{\partial\xi^*}{\partial p} = -\left[\nabla_\xi^2 f \right]^{-1}\nabla^2_{\xi,p} f$ This Jacobian provides a first-order deformation direction. An iterative line-search algorithm then finds an admissible step size along this direction to guarantee a decrease in the cost function. Empirically, this approach produces near-optimal adapted trajectories with worst-case $160\times$ speed-up compared to warm-started full optimizations. It applies to boundary, via-point, or goal perturbations in manipulator tasks, with sub-0.1 rad orientation errors and smoothness within 35% of the average cost over a wide perturbation range.

This paradigm demonstrates that when local differentiability is available, analytical trajectory sensitivity can effect real-time adaptation under small-to-moderate task changes, obviating the need for repeated global optimization (Srikanth et al., 2020).

3. Learning-Based Initialization using Demonstration or Synthesized Trajectories

Several works exploit large sets of existing trajectories to train initialization models:

3.1 Imitation-Driven and Reinforcement Learning Initialization

In "Learning-based Initialization of Trajectory Optimization for Path-following Problems of Redundant Manipulators" (Yoon et al., 3 Feb 2026), a policy is trained via reinforcement learning, imitating expert trajectories synthesized by an offline trajectory optimization solver (TORM). To preserve path-following while avoiding transfer of sub-optimal null-space motions, the imitation reward uses a null-space projector: $P_{\text{null}}(q_i) = I_d - J(q_i)^\dagger J(q_i)$ and the RL objective encourages matching only the null-space components of the expert demonstration. The policy is deployed to generate $\xi_{init}$ , which is supplied as a warm-start to the trajectory optimizer. This results in substantially higher convergence rates and final optimality than linear or greedy inverse-kinematic initializations, both in simulation and on 7-DOF hardware.

3.2 Data-driven Network Priors in Motion and Planning

In "Efficient Trajectory Optimization for Autonomous Racing via Formula-1 Data-Driven Initialization" (Shehadeh et al., 7 Mar 2026), Formula 1 telemetry is aligned and denoised into a geometric dataset, then a neural network learns to map local track geometry to raceline offsets. This network then seeds a continuous minimum-time optimal control solver. Compared to centerline or minimum-curvature initializations, the learned prior halves total optimization time, reduces average solver iterations (e.g., 434.5 vs. 521.6), and achieves expert-level lap times on 17 tracks and in hardware (7.090s→6.640s, $p<0.01$ ).

In legged locomotion, a Conditional Variational Autoencoder is trained on segmented expert trajectories (flat ground or terrain) (Melon et al., 2021). Initialization is performed by nearest-neighbor retrieval in task/condition space, followed by decoding the latent-mode trajectory into the optimizer’s decision variables. This strategy reduces IPOPT iterations by more than $2\times$ and yields higher-quality gaits relative to heuristic or interpolated guesses.

3.3 Guided Policy Search and Sequential Convex Programming

"Guided Policy Search using Sequential Convex Programming for Initialization of Trajectory Optimization Algorithms" (Kim et al., 2021) alternates between (i) single-step sequential convexification of the trajectory, with local perturbations generated by LQR feedback; and (ii) supervised learning (policy update). The resulting policy generalizes across initial conditions and, when used to generate initial trajectories for nonlinear guidance (e.g., 6-DOF descent), reduces required nonlinear optimizer iterations from $p$ 0 to $p$ 1 on average, with high success rates.

4. In-Context Learning and Retrieval-based Initialization

Initialization can be performed nonparametrically by retrieval and contextual conditioning on a set of existing trajectories.

In the TrajICL framework for pedestrian trajectory prediction (Fujii et al., 1 Jun 2025), a transformer-based predictor receives the observed trajectory and an in-context pool of $p$ 2 demonstration trajectories $p$ 3 selected for spatio-temporal similarity (STES) to the observed motion. Optionally, a prediction-guided example selection (PG-ES) reranks candidates by full future similarity, further refining the context set. The model then predicts the future trajectory in a purely forward-pass scheme, leveraging the structure of retrieved exemplars. This in-context approach achieves minimum FDE reductions of $p$ 4 versus baseline and surpasses even fine-tuning for both in-domain and out-of-domain pedestrian benchmarks.

Qualitatively, this inference-time selection of examples from an existing database collapses adaptation latency to a lookup and forward computation, trading off expressive capacity with retrieval diversity and the quality of underlying exemplars.

5. Initialization in System Identification: Model Learning from Multiple Trajectories

When system identification is conducted from data, initialization from diverse existing trajectories enables both consistent parameter learning and quantifiable error rates.

In "Learning the Dynamics of Autonomous Linear Systems From Multiple Trajectories" (Xin et al., 2022), $p$ 5 independent short-length trajectories with known (possibly nonzero mean) initial states are used to estimate the system matrix $p$ 6 of $p$ 7. For zero-mean $p$ 8, a one-step regression achieves $p$ 9 error, uniformly in stability properties. For $p' = p + \Delta p$ 0, taking trajectory lengths $p' = p + \Delta p$ 1 (i.e., initializing and running for $p' = p + \Delta p$ 2 steps) achieves

$p' = p + \Delta p$ 3

if $p' = p + \Delta p$ 4 is strictly stable, and $p' = p + \Delta p$ 5 if $p' = p + \Delta p$ 6 is marginally stable.

In the nonlinear setting, "Learning Linearized Models from Nonlinear Systems under Initialization Constraints with Finite Data" (Xin et al., 8 May 2025) provides deterministic experiment design: the system is initialized from a cover of directions within the feasible set, with $p' = p + \Delta p$ 7 one-step experiments centered about $p' = p + \Delta p$ 8 and radius $p' = p + \Delta p$ 9. Regularized least-squares recovers the Jacobians of the linearization, and the finite-sample error is a trade-off between variance ( $p$ 0), Taylor bias ( $p$ 1), and regularization bias. Critically, this approach ensures excitation and minimizes extrapolation bias by constraining the initialization region.

6. Generative Inverse Problems and Trajectory Recovery

In simulation-driven environments, neural networks may be trained to invert a simulated database of nonlinear trajectories, predicting initial conditions from observed or desired outcomes (Ntakouris, 2021). A uniform grid of simulation runs is created, and a deep regressor learns to solve the inverse mapping given a target, e.g., predicting the initial velocity orientation for a projectile. In this setup, the trajectory database itself forms the initialization basis; the approach achieves sub-meter endpoint accuracy for 2km-range ballistic arcs.

7. Template Anchoring and Diffusion-Based Planning

In integrated cognitive-planning systems such as KnowDiffuser (Ding et al., 11 Mar 2026), initialization from existing trajectory templates is a central pillar: a meta-action LLM selects from a library of time-averaged, clustered driving behaviors (e.g., "turn left and cruise"), and the corresponding averaged trajectory is used as a prior/anchor for a diffusion-based generative planner. During inference, only two noise-steps are injected to corrupt the template, followed by a single denoising (reverse) pass. This guarantees both semantic coherence and physical feasibility in the output, achieving $p$ 2 ADE and reputable closed-loop success on nuPlan. The template anchor mechanism allows the stochastic planner to retain structural features of human-like driving.

8. Practical Significance and Empirical Impact

Initialization from existing trajectories consistently yields:

Domain	Initialization Approach	Speed-up / Gain	Reference
Manipulator Task Adaptation	Analytical solution sensitivity, line-search	$p$ 3 speed, $p$ 4 orientation	(Srikanth et al., 2020)
Autonomous Racing	Neural raceline prior	Halves solver runtime, expert-level lap times	(Shehadeh et al., 7 Mar 2026)
Redundant Manipulation	Example-guided RL policy	$p$ 5– $p$ 6 convergence gain vs. baselines	(Yoon et al., 3 Feb 2026)
Legged Locomotion	CVAE latent-mode regression, NN lookup	Doubled convergence speed	(Melon et al., 2021)
Powered Descent Guidance	GPS with SCP/LQR feedback	$p$ 7 fewer optimizer calls	(Kim et al., 2021)
Pedestrian Prediction	In-context retrieval, STES & PG-ES	$p$ 8– $p$ 9 lower FDE vs. baselines/fine-tuning	(Fujii et al., 1 Jun 2025)
System Identification (LTI/Nonlinear)	Multiple short initializations with design constraints	Sharp finite-sample error rates	(Xin et al., 2022 Xin et al., 8 May 2025)
Diffusion-based Planning	LM-driven template prior anchoring	SOTA AD metrics, semantic-physical alignment	(Ding et al., 11 Mar 2026)

These diverse approaches share the fundamental premise that informative initialization enables efficient, robust, and high-quality downstream optimization or inference, especially in non-convex, high-dimensional, or data-limited regimes. Empirical results consistently show strong gains in speed (typically $\frac{\partial\xi^*}{\partial p} = -\left[\nabla_\xi^2 f \right]^{-1}\nabla^2_{\xi,p} f$ 0– $\frac{\partial\xi^*}{\partial p} = -\left[\nabla_\xi^2 f \right]^{-1}\nabla^2_{\xi,p} f$ 1), solution optimality, convergence reliability, and transferability (simulation-to-real or cross-domain adaptation).

9. Limitations and Open Challenges

Limitations include:

Sensitivity to the coverage, diversity, and quality of existing trajectory libraries or demonstration datasets; poor or non-representative initializations can lead to suboptimal or non-convergent solutions (Yoon et al., 3 Feb 2026, Melon et al., 2021).
For highly nonlinear or discontinuous domains, local first-order adaptation (e.g., gradient-based) may not suffice for large task perturbations; active-set changes may invalidate analytical models (Srikanth et al., 2020).
Retrieval-based and in-context approaches are bottlenecked by pool diversity and retrieval efficacy—random or poorly indexed pools saturate performance (Fujii et al., 1 Jun 2025).
In sample-based system identification, initialization constraints limit the region over which true linearization is possible; balancing signal-to-noise and Taylor remainder bias is nontrivial in strongly nonlinear dynamics (Xin et al., 8 May 2025).
Template-anchored generative models may be restricted in expressivity by the granularity of the meta-action or template library (Ding et al., 11 Mar 2026).

Open research continues into adaptive library augmentation, joint optimization of libraries and retrieval mechanisms, robust aggregation of warm-started solutions, and scalable strategies for high-dimensional, multi-modal, or open-world trajectory domains.

Collectively, initialization from existing trajectories constitutes a foundational cross-domain methodology, grounding adaptive, robust, and sample-efficient planning, learning, and prediction in both theory and practice.