Trajectory-conditioned Diffusion Policy (TDP)

• TDP is a conditional generative control paradigm that uses diffusion processes to produce full-horizon motion trajectories based on task context.
• It integrates diverse conditioning signals—such as geometry, robot state, and language—to enable long-horizon planning and manipulation.
• Empirical studies report significant gains in success rate, smoothness, and efficiency compared to conventional robotic control methods.

Searching arXiv for the cited TDP-related papers and closely related work. Trajectory-conditioned Diffusion Policy (TDP) is a conditional generative control paradigm in which a diffusion model samples entire motion trajectories while being explicitly conditioned on task-relevant context. In robotics, the central idea is to replace hand-tuned primitives, purely local heuristics, or observation-only denoisers with a learned reverse-diffusion process that outputs temporally coherent action sequences or trajectory segments adapted to geometry, language, robot state, prior motion, task parameters, or explicit plan-like signals (Chen et al., 3 Oct 2025). Under this umbrella, recent work includes direct long-horizon trajectory generation for industrial coverage path planning, voxel-trajectory-conditioned manipulation, diffusion-generated 2D trajectory guidance for language-conditioned imitation, temporally reused diffusion planning, SE(3)-equivariant trajectory diffusion, execution-trace-conditioned action denoising, warm-start policies for trajectory optimization, and projected diffusion predictive control with explicit constraints (Huang et al., 6 Jul 2025).

1. Definition, scope, and nomenclature

In its most explicit formulation, TDP treats the “policy” as a map from context $c$ to a distribution over complete future trajectories, rather than a map from the current observation to a single action. In 3D-CovDiffusion, conditioning variables can include geometry, robot state, prior trajectory prefixes, and task parameters, and the sampled object is a variable-length sequence $\tau = [a_1,\dots,a_H]$ with $a_t \in \mathbb R^6$ denoting end-effector pose (Chen et al., 3 Oct 2025). In VLM-TDP, the diffusion model is conditioned on both observations $o_t$ and a voxelized trajectory $\tau$ for the current sub-task, where the trajectory serves as an explicit spatial plan for long-horizon manipulation (Huang et al., 6 Jul 2025).

The nomenclature is not fully standardized. The paper “Diffusion Trajectory-guided Policy for Long-horizon Robot Manipulation” states that, in the requested context, “Trajectory-conditioned Diffusion Policy (TDP)” is equivalent to DTP, its own “Diffusion Trajectory-guided Policy,” and uses a two-stage formulation in which a diffusion model first generates a 2D end-effector trajectory and a downstream policy is conditioned on that trajectory (Fan et al., 14 Feb 2025). By contrast, “Efficient Diffusion Planning with Temporal Diffusion” uses the abbreviation TDP to mean “Temporal Diffusion Planner,” a planning-oriented method that distributes denoising across the time dimension and refines an existing plan instead of regenerating a new one at every step (Guo et al., 26 Nov 2025). This terminological overlap is substantive rather than merely cosmetic: some works use diffusion to generate the control trajectory itself, while others use diffusion-generated trajectories as intermediate guidance for another policy.

A common misconception is that TDP denotes a single fixed architecture. The surveyed works instead show a family resemblance: a diffusion process is conditioned on trajectory-scale context or produces a trajectory-scale object, and that representation then structures downstream control. This suggests that TDP is better understood as a design pattern than as one canonical algorithm.

2. Probabilistic formulation

Most Euclidean TDP variants adopt a DDPM- or DDIM-style noising process over trajectories or action chunks. In 3D-CovDiffusion, for a clean trajectory $x_0 \in \mathbb R^{H \times d}$ with $d=6$, the forward process is written as

$$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$

with closed form

$$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$

and training-time perturbation

$$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$

The reverse process is conditioned on context $\tau = [a_1,\dots,a_H]$0 through $\tau = [a_1,\dots,a_H]$1 and optimized with the standard $\tau = [a_1,\dots,a_H]$2-prediction objective, including a masking variant for variable-length sequences (Chen et al., 3 Oct 2025).

VLM-TDP uses the same general conditional denoising structure, but the conditioned variable is the action sequence for a manipulation sub-task, with the denoiser written as $\tau = [a_1,\dots,a_H]$3 and trained with

$\tau = [a_1,\dots,a_H]$4

Its inference procedure is DDPM-style iterative refinement of an action sample initialized from Gaussian noise, while the conditioning includes both observation features and a voxel-based trajectory representation (Huang et al., 6 Jul 2025).

Other variants alter what the denoiser predicts or how the trajectory variable is parameterized. The Sobolev-trained warm-start method uses DDPM forward noising on a trajectory chunk $\tau = [a_1,\dots,a_H]$5 but trains the denoiser with an $\tau = [a_1,\dots,a_H]$6-prediction objective,

$\tau = [a_1,\dots,a_H]$7

and augments it with first-order supervision derived from trajectory-optimization feedback gains (Hellard et al., 21 Apr 2026). ET-SEED departs further by defining diffusion on the SE(3) manifold, using an interpolation-plus-perturbation formula

$\tau = [a_1,\dots,a_H]$8

and training a relative-pose predictor under a geodesic loss on SE(3) (Tie et al., 2024).

These formulations share the same statistical core—iterative denoising of a structured future object—but differ in whether the object is an action chunk, a full trajectory, a state-action plan, a 2D particle trajectory, or an SE(3) trajectory. The practical consequences are significant: output space, conditioning pathway, and execution regime all change with that choice.

3. Conditioning variables and trajectory representations

The most salient axis of variation in TDP is the conditioning signal. Representative instantiations are summarized below.

System	Generated object	Conditioning signal
3D-CovDiffusion	Variable-length 6-DoF pose sequence	Partial point cloud $\tau = [a_1,\dots,a_H]$9 and 4-pose trajectory prefix $a_t \in \mathbb R^6$0
VLM-TDP	Action sequence for a sub-task	Image history, robot state, voxel trajectory $a_t \in \mathbb R^6$1
DTP	2D image-plane end-effector trajectory, then actions	Language, image observations, robot states
ET-SEED	Full SE(3) end-effector trajectory	Colored point cloud $a_t \in \mathbb R^6$2
TF-DP	Short action chunk	Multi-view images, end-effector position, projected execution trace
Sobolev-trained TDP	Trajectory chunk for warm-start	Observation history, task parameters, inpainted past actions
Temporal Diffusion Planner	State-action trajectory plan	Current state, prior plan, value guidance
DPCC	Feasible horizon trajectory	Current state and goal; optionally desired trajectory

In 3D-CovDiffusion, the conditioning vector is explicitly constructed by concatenating a 64-dimensional geometry embedding and a 64-dimensional trajectory-prefix embedding to obtain $a_t \in \mathbb R^6$3. Geometry is encoded by a PointNet-style MLP with per-point layers $a_t \in \mathbb R^6$4, global max pooling, and a projection to 64 dimensions; the trajectory prefix is encoded by a 2-layer MLP mapping flattened recent poses from $a_t \in \mathbb R^6$5. The fused vector is injected into each residual block via FiLM,

$a_t \in \mathbb R^6$6

making the denoiser jointly geometry-aware and trajectory-aware (Chen et al., 3 Oct 2025).

VLM-TDP uses a different factorization. Here the key conditioning object is a voxel-based spatial trajectory for each sub-task: a tensor $a_t \in \mathbb R^6$7 with $a_t \in \mathbb R^6$8 in the experiments, where visited voxels are labeled by visitation order and non-trajectory voxels are zero. A 3-layer 3D CNN encodes this tensor, and the resulting feature is concatenated with historical image features and flattened robot state. No cross-attention or FiLM is reported; the conditioning enters by concatenation and remains constant within a sub-task (Huang et al., 6 Jul 2025).

DTP uses diffusion to generate a 2D particle trajectory in the image plane, $a_t \in \mathbb R^6$9, from language instruction and observation. That trajectory is then tokenized, reduced by a diffusion trajectory resampler, and consumed by a GR-1-style causal transformer policy together with frozen CLIP language features, frozen MAE visual features, and robot-state tokens (Fan et al., 14 Feb 2025). TF-DP uses execution history more directly: the end-effector trace is projected into image space, converted into a trace image and a Gaussian trace-focused field, and used to construct an execution-aware observation tuple $o_t$0 (Hu et al., 7 Feb 2026).

A second misconception is that “trajectory conditioning” always means conditioning on an externally planned trajectory. The surveyed papers show several distinct meanings: conditioning on recent trajectory prefixes, on diffusion-generated trajectories, on VLM-generated sub-task paths, on projected execution traces, or on a previous partially denoised plan. The common element is not the source of the trajectory information, but its explicit role in structuring the denoiser’s future prediction.

4. Architectural and control regimes

One major design split concerns whether diffusion produces the final executable trajectory directly or only an intermediate representation. 3D-CovDiffusion is an end-to-end trajectory generator for coverage path planning: it predicts long-horizon 6-DoF end-effector trajectories, handles variable horizon by masking, and synthesizes multiple segments of variable length that are stitched into a continuous program by aligning segment endpoints. No explicit timing, velocity, or acceleration profiles are modeled; smoothness is treated as an emergent property of denoising and is evaluated post hoc via jerk-based metrics (Chen et al., 3 Oct 2025).

VLM-TDP and DTP are more explicitly hierarchical. VLM-TDP decomposes long-horizon tasks into sub-tasks separated by gripper events, uses GPT-4o to generate voxel trajectories per sub-task via mask-based visual prompting, and conditions a diffusion action denoiser on those trajectories. TDP is trained on demonstration-derived trajectories and validated with VLM-generated trajectories, so the conditioning channel is deliberately subjected to distribution shift at test time (Huang et al., 6 Jul 2025). DTP also uses two stages, but with a different split: a generative vision-language diffusion trajectory model first predicts a 2D end-effector trajectory, and a GR-1-style policy then predicts actions conditioned on that trajectory, with a joint training objective $o_t$1 (Fan et al., 14 Feb 2025).

Temporal Diffusion Planner changes the execution regime rather than the representation. Its key idea is “progressive vagueness”: near-term steps in the plan are fully denoised while far-future steps are intentionally kept less denoised, implemented by assigning each trajectory element an effective diffusion index $o_t$2. At each environment step, only a small number of denoising steps are applied to refine the remaining plan, and an automated replanning criterion decides when full regeneration is necessary (Guo et al., 26 Nov 2025).

Other papers embed TDP in planning or control loops with stronger structure. DPCC inserts a projection onto the feasible set $o_t$3 at every reverse-diffusion step and then executes only the first action in an MPC loop, thereby transforming a trajectory diffusion model into a constrained predictive controller (Römer et al., 2024). The Sobolev-trained warm-start method uses the diffusion policy to generate a trajectory chunk that initializes a gradient-based trajectory optimizer; the optimizer is then run from both the nominal initial guess and the diffusion-produced warm start, and the better converged solution is retained (Hellard et al., 21 Apr 2026). ET-SEED, by contrast, is an open-loop full-trajectory generator on SE(3) with one final equivariant denoising step and $o_t$4 invariant steps, allowing trajectory-level SE(3) equivariance without requiring per-step equivariant reverse kernels (Tie et al., 2024).

These regimes imply distinct notions of “policy.” In some TDP systems the policy is the denoiser itself; in others it is a downstream action model guided by a denoised trajectory; in still others it is a receding-horizon planner that repeatedly samples and repairs a future plan.

5. Empirical behavior across domains

In industrial coverage path planning, 3D-CovDiffusion reports average improvements of approximately $o_t$5 in Point-wise Chamfer Distance, approximately $o_t$6 in smoothness, and approximately $o_t$7 in coverage relative to prior methods. Its reported per-category values include, for Windows, PCD $o_t$8 versus PaintNet $o_t$9, coverage $\tau$0 versus PaintNet $\tau$1, and smoothness $\tau$2 versus Point-Wise $\tau$3; for Containers, the category with only 88 samples, performance drops to PCD $\tau$4, coverage $\tau$5, and smoothness $\tau$6, which the authors attribute to deep cavities, rims, high curvature, self-occlusions, and data scarcity (Chen et al., 3 Oct 2025).

In long-horizon manipulation, VLM-TDP reports an average $\tau$7 increase in success rate over classical diffusion policies across benchmarks, more than $\tau$8 improvement on long-horizon tasks, and approximately $\tau$9 reduction in performance degradation under noisy images or altered environments. On the seven-task single-view RLBench benchmark, average success rates are $x_0 \in \mathbb R^{H \times d}$0 for Diffusion Policy, $x_0 \in \mathbb R^{H \times d}$1 for TDP, $x_0 \in \mathbb R^{H \times d}$2 for VLM-TDP, $x_0 \in \mathbb R^{H \times d}$3 for 3D Diffusion Policy, and $x_0 \in \mathbb R^{H \times d}$4 for Lang-o3dp. In real-world experiments, “Pick Three Bananas” improves from $x_0 \in \mathbb R^{H \times d}$5 with Diffusion Policy to $x_0 \in \mathbb R^{H \times d}$6 with TDP (Huang et al., 6 Jul 2025).

DTP shows analogous gains on CALVIN. In the known-scene D$x_0 \in \mathbb R^{H \times d}$7D setting, Task-5 success rises from $x_0 \in \mathbb R^{H \times d}$8 for the best baseline HULC++ to $x_0 \in \mathbb R^{H \times d}$9 for DTP/TDP, with average chain length increasing from $d=6$0 to $d=6$1. In the unseen-scene ABC$d=6$2D setting, Task-5 success is $d=6$3 for DTP/TDP versus $d=6$4 for the best baseline 3D Diffuser Actor. In the data-efficient 10% ABCD$d=6$5D setting, Task-5 success improves from $d=6$6 for GR-1 to $d=6$7 for DTP/TDP. Real-world success rates are also markedly higher on several tasks, including OpenTrash and OpenSideDrawer (Fan et al., 14 Feb 2025).

Efficiency-focused planning variants report a different class of gains. Temporal Diffusion Planner improves decision-making frequency by $d=6$8 to $d=6$9 times relative to diffusion planners that fully replan at every time step, with runtime per decision of approximately $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$0 to $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$1 s versus approximately $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$2 to $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$3 s for prior methods, while maintaining higher or comparable normalized scores on D4RL benchmarks (Guo et al., 26 Nov 2025). The Sobolev-trained warm-start policy reduces solve times by $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$4 to $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$5 for gradient-based trajectory optimization, with the quadrotor task with 9 obstacles dropping from $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$6 s without warm-start to $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$7 s with Sob+Diff, and the inverted double pendulum going from about $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$8 s without warm-start to about $$q(x_t \mid x_{t-1}) = \mathcal N(x_t; \sqrt{1-\beta_t}\,x_{t-1}, \beta_t I),$$9 s total time in one reported setting (Hellard et al., 21 Apr 2026).

History-aware conditioning also yields large empirical effects. TF-DP raises average success under multi-modal action ambiguity from $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$0 for vanilla Diffusion Policy to $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$1, and under visual disturbances from $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$2 to $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$3, while adding only $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$4 s per action chunk, or approximately $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$5 runtime overhead (Hu et al., 7 Feb 2026). ET-SEED, evaluated on manipulation tasks with rigid, articulated, and deformable objects, reports substantially higher New Pose success rates than DP3, DP3+Aug, and EquiBot with only 25 or 50 demonstrations; for example, on Open Door with 50 demonstrations it reaches $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$6 versus $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$7, $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$8, and $$q(x_t \mid x_0) = \mathcal N(x_t; \sqrt{\bar\alpha_t}\,x_0, (1-\bar\alpha_t)I),$$9 for those baselines respectively, and on real-robot Open Bottle Cap with 20 demonstrations it reaches $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$0 success versus $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$1, $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$2, and $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$3 (Tie et al., 2024).

Constraint-aware diffusion changes the operating point again. In the 2D manipulator “Avoiding” task, DPCC with tightening achieves approximately $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$4 goal success and approximately $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$5 constraint-plus-goal success for the DPCC-C selection rule, with approximately $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$6 violation steps and approximately $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$7 time steps to goal, at roughly $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$8 ms per control action (Römer et al., 2024).

6. Limitations, misconceptions, and emerging directions

The empirical breadth of TDP should not be confused with universal feasibility guarantees. Several representative systems explicitly do not impose hard kinematic or collision constraints during generation. 3D-CovDiffusion states that collision or kinematic feasibility constraints are not hard-coded, and that smoothness and coverage are not optimized directly in training; the total training loss is only $$x_t = \sqrt{\bar\alpha_t}\,x_0 + \sqrt{1-\bar\alpha_t}\,\epsilon,\quad \epsilon \sim \mathcal N(0,I).$$9, with smoothness and coverage treated as evaluation metrics (Chen et al., 3 Oct 2025). VLM-TDP likewise does not add explicit feasibility checks or optimization-based trajectory refinement, and its voxel trajectories can become ambiguous at coarse $\tau = [a_1,\dots,a_H]$00 resolution when objects are close (Huang et al., 6 Jul 2025). DTP depends on accurate diffusion-generated trajectories; the paper explicitly notes that if the DTM produces poor trajectories, policy performance can degrade, and that 2D image-plane trajectories do not use depth (Fan et al., 14 Feb 2025).

Several methods inherit modality-specific failure modes. TF-DP depends on calibrated projection of the end-effector trace and can be misled by calibration errors, odometry noise, or history variables not captured by the trace (Hu et al., 7 Feb 2026). ET-SEED improves substantially on deformable tasks but also notes that those tasks are not perfectly SE(3)-equivariant and that open-loop execution can drift without feedback correction; real-world performance also depends on robust point-cloud construction and segmentation (Tie et al., 2024). The Sobolev-trained warm-start method assumes access to a gradient-based trajectory optimizer that exposes feedback gains $\tau = [a_1,\dots,a_H]$01 and differentiable dynamics, and it does not address contact-rich tasks (Hellard et al., 21 Apr 2026). Temporal Diffusion Planner requires tuning of the replanning threshold and can drift if partial refinements are carried too long, especially on lower-quality datasets (Guo et al., 26 Nov 2025).

A further misconception is that all TDPs solve long-horizon control in the same way. The surveyed papers show at least four distinct strategies: direct denoising of executable long-horizon trajectories, generation of intermediate trajectory guidance for a downstream policy, partial reuse of previously denoised plans, and insertion of optimization or projection inside the reverse process. DPCC demonstrates that explicit online constraints can be layered onto a trajectory diffusion model without retraining, but at the cost of solving a projection problem inside every reverse step and managing model mismatch via constraint tightening (Römer et al., 2024).

The emerging directions stated across the papers are correspondingly diverse. They include adding collision or jerk constraints through post-processing or cost-guided diffusion, improving robustness through more balanced data, using 3D-conditioned trajectories to handle occlusion and spatial reasoning, incorporating richer multi-modal traces, exploring better deviation metrics and hierarchical planning, extending to contact-rich settings with differentiable simulators, and integrating replanning or visual servoing into open-loop trajectory generators (Chen et al., 3 Oct 2025). A plausible implication is that future TDP work will be shaped less by the diffusion backbone itself than by how trajectory-scale context is represented, injected, repaired, and coupled to downstream execution.