Tool-Use Trajectory Generation

Updated 27 November 2025

Tool-use trajectory generation is the process of synthesizing, representing, and optimizing control sequences that enable agents to manipulate tools effectively.
It integrates continuous motion planning, symbolic tool-call sequencing, and multimodal observation to address challenges in robotics and intelligent agent systems.
It employs methods like gradient-based optimization, reinforcement learning, and constraint-driven planning to improve performance, efficiency, and adaptability in real-world tasks.

Tool-use trajectory generation refers to the synthesis, representation, and optimization of control sequences that enable an agent—typically a robot or an intelligent software agent—to manipulate or coordinate tools to achieve specified goals. This field integrates action-parameter optimization, state-transition modeling, environment interaction, reward assignment, and—depending on the domain—poses, symbolic tool-calling, or multimodal observation streams. The following sections detail the core methodologies, representational paradigms, learning and optimization approaches, and current evaluation metrics as established in recent research.

1. Formal Representations of Tool-Use Trajectories

Tool-use trajectories are formalized according to the nature of the manipulation or reasoning domain:

a) Continuous-Control (Robotics)

A tool-use trajectory is typically expressed as a sequence of joint configurations, end-effector poses, or tool-centric actions. For example, trajectories may be represented as

$x_t = [\theta_t^{\rm start}; \theta_t^{\rm end}]$

where $x_t$ denotes concatenated joint angle vectors, or as splines/waypoint sequences in $\mathbb{R}^n$ joint or task space (Kawaharazuka et al., 16 Jul 2024).

b) Symbolic Tool-Call Sequences (Reasoning Agents)

In agentic LLM or VLM settings, a trajectory is an ordered or unordered sequence of tool calls and their arguments:

$\tau = [(t_1, a_1), ..., (t_n, a_n)]$

where $t_i$ is the tool identity and $a_i$ is its argument vector. For dependent (sequential) use, a dependency graph $D$ specifies bindings between outputs and subsequent inputs (He et al., 6 Oct 2025).

c) Multimodal and Point-Cloud Trajectories

Some frameworks represent the trajectory by discrete scene states, e.g., a sequence of 3D point clouds encoding the tool pose at each timestep, accommodating arbitrary tool geometry and scene configuration (Qi et al., 2023).

d) Observation-Action Streams

In reinforcement learning with external tools, a trajectory alternates between observation and action tokens:

$\tau = (o_0, a_0, o_1, a_1, ..., o_n, a_n)$

with each action block potentially invoking a tool and receiving structured, often multimodal, outputs (Jiang et al., 1 Sep 2025, Ashraf et al., 9 Oct 2025).

2. Trajectory Generation: Optimization and Learning Techniques

Methods for trajectory generation span a spectrum from analytical optimization to imitation and RL paradigms.

a) Gradient-Based Trajectory Backpropagation

A learned forward model $f(s,\phi,u)$ predicts task-state transitions given current state $s$ , tool $\phi$ , and trajectory parameter $u$ . Optimization minimizes a loss $L(\phi, \{u\}) = \text{MSE}(f(s_{\text{current}}, t, u), s_{\text{target}}')$ via backpropagation over $u$ . The gradient $\partial L/\partial u$ is used to iteratively update $u$ , typically via normalized step (Kawaharazuka et al., 16 Jul 2024).

b) Trajectory Optimization for Robotics

Direct methods represent trajectories as sequences of waypoints and optimize a cost functional (time, effort):

$\min_{\{\mathbf{q}_i\}, \{\Delta t_i\}} \sum_k \|\mathbf{s}(\tau_{k+1})-\mathbf{s}(\tau_k)\|^2 + w_{\text{time}}\sum_i \Delta t_i$

subject to kinematic, dynamic, collision, and swept-volume constraints. Spline interpolation, time-variable keypoints, and nonlinear programming (SQP, STOMP, CasADi solvers) are used to find feasible and efficient trajectories (Yang et al., 2020, Yang et al., 2020).

c) RL and Stepwise Feedback in Symbolic Settings

Agentic tool-use with LLMs and VLMs employs methods such as PPO or GRPO, taking advantage of per-step (sub-trajectory) reward signals. PORTool explores trajectory trees, computing both trajectory- and fork-relative advantages to reinforce optimal tool-call prefixes (Wu et al., 29 Oct 2025). SWiRL constructs synthetic trajectories, decomposes them into steps, and uses a stepwise reward model to provide RL signal (Goldie et al., 7 Apr 2025). VerlTool generalizes RLVR to ARLT, masking out observation tokens and structuring optimization to focus on action tokens (Jiang et al., 1 Sep 2025, He et al., 6 Oct 2025).

d) Model-Based and Constraint-Driven Planning

Physics-based and model-driven planners, as in FEM + symbolic regression pipelines, identify key physical variables influencing the effect of tool use (e.g., force, angle, contact area), and set optimal-control objectives to enact desired outcomes while minimizing cost functionals over the joint robot+tool state (Zhang et al., 2022).

3. Task Constraints, Environment Models, and Reward Specifications

Trajectory generation is fundamentally constrained by the interplay between tool, environment, and task goals:

Geometric and Kinematic Constraints: Joint limits, workspace boundaries, and exclusion (collision) zones (Yang et al., 2020, Yang et al., 2020).
Dynamic and Physical Constraints: Actuator torque, velocity, and acceleration limits; physics-informed approximations or learned contact models (e.g., solid mechanics, soil–tool interaction) (Yang et al., 2020, Trupin et al., 2 May 2025).
Objective Criteria: Minimization of energy, time, torque, or maximization of effect (e.g., excised material, separation count) as learned or encoded in the optimization loss (Kawaharazuka et al., 16 Jul 2024, Zhang et al., 2022).
Reward Design: Stepwise reward assignment for tool-use reasoning, including process-oriented (formatting, successful calls) and outcome-oriented (correct answer) signals (Goldie et al., 7 Apr 2025, Wu et al., 29 Oct 2025).
Sequential and Parallel Dependencies: Directed acyclic graphs encode inter-call data dependencies, with parallel or chain-planning supported in both physical and symbolic tool-use environments (He et al., 6 Oct 2025).

4. Architecture and Pipeline Realizations

Tool-use trajectory pipelines are increasingly multi-component and modular:

Domain	Main Representation	Core Method
Robotic Manipulation	$\mathbb{R}^n$ , SE(3),	Joint-space splines, physics-based optimization
	point clouds	or DNN models + gradient descent/backprop
Agentic Reasoning	Symbolic (API call seqs)	RL (PPO, DPO, GRPO), supervised SFT, stepwise RL
Multimodal/Hybrid	Action-observation pairs	VLM/LLM policies with pipeline for tool selection

Pipelines include state-context encoders (CNNs, PointNets, or transformers), trajectory decoders, and, where relevant, preference or reward models. Architecture modularity enables rapid substitution or extension to new tools/environments (Ashraf et al., 9 Oct 2025, Jiang et al., 1 Sep 2025).

5. Evaluation Metrics and Benchmarking

Performance of tool-use trajectory generation is assessed via:

Physical Tasks: Chamfer distance (block positional error), joint-effort or path-length metrics, fill factors, or task success rates in simulation or hardware (Kawaharazuka et al., 16 Jul 2024, Yang et al., 2020).
Symbolic Agentic Tasks: Trajectory exact-match (order-sensitive/insensitive), inclusion (partial tool coverage), argument correctness, dependency/order satisfaction, and final answer accuracy (He et al., 6 Oct 2025, Ashraf et al., 9 Oct 2025).
Generalization: Transfer to previously unseen tools, environments, or task types, often measured via normalized task success rates across other domains (Qi et al., 2023, Goldie et al., 7 Apr 2025).
Scalability/Robustness: Performance scaling with trajectory length, tool diversity, and query complexity; revealed bottlenecks at mid-length trajectory planning (He et al., 6 Oct 2025).

6. Notable Systems, Benchmarks, and Empirical Findings

Recent research has established the following methodological and empirical findings:

DNN-based forward models support joint optimization of tool shape and trajectory through gradient-based updates in representation space, yielding rapid convergence and 50–80% error reduction over random selection (Kawaharazuka et al., 16 Jul 2024).
Time-variable keypoint optimization in excavator trajectory planning enables reduced solution dimensionality, adaptive phase timing, and robust tracking under varying soil conditions (Yang et al., 2020).
TRAJECT-Bench formalizes trajectory-level evaluation metrics, exposing a “short-to-mid length” bottleneck in agentic tool planning; parallel planning is generally more tractable for LLM-based agents (He et al., 6 Oct 2025).
Tree-structured RL (PORTool), stepwise RL (SWiRL), and multimodal preference RL (MATRIX) yield superior task completion by credit assignment both at trajectory and action-fork levels, surpassing prior SFT or non-stepwise RL (Ashraf et al., 9 Oct 2025, Goldie et al., 7 Apr 2025, Wu et al., 29 Oct 2025).
Multimodal and physical task pipelines (ToolGen, iTUP, Recon-Act) explicitly ground trajectories in scene geometry, contact semantics, and/or web interaction context, enabling robust zero-shot transfer and improved success rates (Qi et al., 2023, Trupin et al., 2 May 2025, He et al., 25 Sep 2025).

Tool-use trajectory generation thus occupies a central position in both physical and cognitive tool-using agents, representing a confluence of task-specific constraint optimization, learned state-transition modeling, hierarchical action composition, and feedback-driven refinement. The field continues to evolve with cross-pollination between continuous control, symbolic and hybrid multimodal planning, and data-driven or physics-informed objective functions.