---
title: SE(2) Goal-Conditioned Trajectories
url: https://www.emergentmind.com/topics/goal-conditioned-se-2-trajectories
type: topic
---

# SE(2) Goal-Conditioned Trajectories

A goal-conditioned SE(2) trajectory is a time-parameterized path in the configuration space of planar rigid body transformations (SE(2): translation in ℝ² plus rotation θ), where the trajectory is explicitly specified or optimized with respect to a target end pose or sequence of poses. Such trajectories are foundational in robotics, autonomous navigation, and legged locomotion, forming the basis for both geometric planners and learning-based controllers tasked with efficiently reaching specified positions and orientations in the plane.

## 1. Mathematical Formulation and Representations

Goal-conditioned SE(2) trajectory generation formalizes the problem as learning or planning a sequence of poses $X = [(x_1, y_1, \theta_1), \ldots, (x_H, y_H, \theta_H)] \in \mathbb{R}^{H \times 3}$ that starts at an initial state $s_0$ and terminates at a designated goal $g = (x_g, y_g, \theta_g) \in \mathrm{SE}(2)$. The problem is typically posed either as an MDP with goal conditioning, a supervised trajectory prediction task, or a generative modeling framework.

A standard goal-conditioned MDP is defined by:
- **State space** $\mathcal{S}$: including proprioceptive and global kinematic state variables.
- **Goal space** $\mathcal{G} = \mathrm{SE}(2)$: specifying target poses.
- **Action space** $\mathcal{A}$: e.g., joint commands or velocity inputs.
- **Transition** and **reward** depend explicitly or implicitly on the goal $g$ [2508.14098].

Alternative representations include *sub-goal trees*, where the trajectory is recursively partitioned around midpoints conditioned on start and goal, enabling O(log T) parallel inference [1906.05329], as well as latent generative models (e.g., diffusion, VAEs) that sample from $p_\theta(X|C)$, with $C$ explicitly encoding the goal and context [2509.03658].

## 2. Core Methodologies for Goal-Conditioned SE(2) Trajectories

Several methodology classes for synthesizing SE(2) goal-conditioned trajectories have emerged:

**A. Reinforcement Learning with Direct Goal Conditioning**  
Modern approaches such as GoTo optimize time, energy, and precision for SE(2) reaching by structuring the reward to couple translation and rotation errors. GoTo leverages a *constellation-based* reward:  
- $N$ landmark points are anchored to the robot's local frame and mapped to the goal pose, and the per-timestep reward is a negative exponential of their mean squared distance after transforming to the intended goal pose.
- This induces simultaneous reduction of position and heading errors, yielding fluid, integrated motion patterns, as opposed to rigid, marching-style velocity-tracking gaits [2508.14098].

**B. Sub-Goal Trees and Hierarchical Planning**  
The sub-goal tree (SGT) framework recursively decomposes trajectory prediction into midpoint-prediction tasks conditioned on the endpoints, enabling parallel sampling and divide-and-conquer style dynamic programming. The key propagation rule is:
$$ V_k(s, s') = \min_{s_m} [ V_{k-1}(s, s_m) + V_{k-1}(s_m, s') ] $$
with appropriate SE(2) geodesic midpoints and cost metrics [1906.05329].

**C. Generative Latent Models with Goal Conditioning**  
Recent goal-conditioned diffusion models, such as Efficient Virtuoso, encode an entire trajectory into a compact latent (via PCA) and apply learned denoising in this space. Goal input is fused via a transformer-based encoder:
- Endpoint or sparse route goals are embedded and provided as context for the noise-prediction network during denoising.
- Ablations show that a route (multi-step goal) input substantially outperforms a single endpoint in trajectory fidelity (e.g., minADE=0.2541 m with route vs. 0.4510 m with endpoint on Waymo) [2509.03658].

**D. Projective Quasimetric Planning**  
ProQ introduces learnable, asymmetric latent-space distance functions over SE(2), then performs coverage with sparse latent keypoints (landmarks) that serve as dynamic sub-goals for long-horizon planning. At inference, a graph search over latent landmarks and shortest-hop control generates robust multi-step plans [2506.18847].

**E. Equivariant Trajectory Networks**  
SE(2)-equivariant GNNs (PEP) preserve the geometric structure under arbitrary roto-translation, supporting goal-conditioned planning via a route-attraction module that softly guides the trajectory along a high-level path while enforcing equivariance [2403.11304].

## 3. Reward Functions, Cost Metrics, and Conditioning Schemes

Reward and cost design in SE(2) goal conditioning directly impacts trajectory naturalness, efficiency, and convergence:

- **Constellation-based reward**: Incorporates both translation of the robot's xy-position and rotation (heading), penalizing weighted position and orientation errors in a unified geometric term:  
  $d_\text{con} = \|c - c^*\|^2 + I_c (\theta - \theta^*)^2$ with reward $r_\text{con} = \exp(-w_c d_\text{con})$ [2508.14098].

- **Sub-goal interpolation metrics**: Employ manifold midpoints computed via SE(2) log/exp maps or by separately interpolating xy and using shortest-arc averaging for θ. Edge costs typically take the form  
  $c(s_a, s_b) = w_t \|t_b-t_a\|_2 + w_\theta |\mathrm{wrap}(\theta_b-\theta_a)|$ [1906.05329].

- **Latent geometry and quasimetric**: ProQ designs a projective quasimetric $D(z_i, z_j)$ mixing max/mean distances in learned embedding space, produces a sparse set of latent keypoints, and uses Floyd–Warshall for planning via sub-goal graphs [2506.18847].

- **Transformer goal encoding**: Efficient Virtuoso and PEP highlight the benefits of richer goal representations (multi-step vs. endpoint), confirming precise route information is essential for high-fidelity trajectory synthesis [2509.03658, 2403.11304].

## 4. Architectures, Algorithms, and Sample Efficiency

Summary of key architectural design choices and their roles:

| Method        | Input Modality                                   | Architecture                      | Notable Properties                  |
|---------------|--------------------------------------------------|------------------------------------|-------------------------------------|
| GoTo          | qₜ (joint), pᵣ, Δx, Δy, Δθ                      | 2-layer LSTM, PD setpoints         | No kinematic planner, direct RL     |
| SGT           | (s₀, s_T), interpolated midpoints                | MDN, multi-level recursion         | O(log T) concurrent inference       |
| Virtuoso      | (x,y)[,θ], context, multi-step or endpoint goal  | MLP denoiser, Transformer encoder  | Latent diffusion, 2-stage norm      |
| ProQ          | (x,y,sinθ,cosθ), learned latent sub-goals        | MLP for φ, graph search            | Compositional, OOD-constrained      |
| PEP           | Past (multi-agent), high-level route             | EqMotion GNN + route attraction    | SE(2)-equivariance, sample efficient|

GoTo’s RL approach (on Agility Digit-V3) achieves superior time, energy, and step metrics compared to marching-style baselines; constellation-based reward ensures fluid turning and trajectory integration [2508.14098]. SGT’s divide-and-conquer yields exponential speedup in trajectory prediction and competitive error rates [1906.05329]. Efficient Virtuoso’s Transformer-based latent diffusion attains state-of-the-art minADE with explicit goal route conditioning [2509.03658]. ProQ demonstrates robust global planning and value-estimation stability on long-horizon navigation [2506.18847]. SE(2)-equivariant GNNs match or exceed dataset SOTA in planning L2 and collision metrics with minimal sample requirements [2403.11304].

## 5. Benchmarking, Evaluation Metrics, and Empirical Insights

Goal-conditioned SE(2) trajectory research employs performance metrics tailored to precision, efficiency, and control stability:

- **Final Position and Orientation Error**: $||(x_f, y_f) - (x_g, y_g)||_2$ and $|\theta_f - \theta_g|$.
- **Time-to-Target**: Time until trajectory satisfies tight spatial and rotational accuracy.
- **Footstep Count / Energy per Meter**: Key for legged/humanoid systems to ensure natural, efficient locomotion [2508.14098].
- **minADE, minFDE, MissRate@2m**: Minimum average/final displacement error over sampled trajectories; standard in autonomous driving [2509.03658].
- **Sample Efficiency**: Measured by test set coverage with small training sets (e.g., PEP’s use of only ≈607 nuScenes scenes; maintains equivariance) [2403.11304].
- **Equivariance Consistency**: Output trajectory invariance under input roto-translation, critical for physical plausibility [2403.11304].

GoTo achieves a 0.63× energy, 0.56× time, and 0.67× step reduction vs. normalizing baseline in short-range tasks; in real-world trials, 90.23% success under domain randomization [2508.14098]. Efficient Virtuoso obtains minADE=0.2541 m with sparse route goal vs. 0.4510 m for endpoint, demonstrating the necessity of structurally rich goals for planning [2509.03658]. SGT reports a ∼15x inference speedup over sequential methods in 2D planning [1906.05329].

## 6. Key Insights, Extensions, and Open Directions

- *Unified translation-rotation reward (e.g., constellation reward) enables coupled error reduction for seamless turn-and-go motion, critical for humanoids and any agent with body orientation* [2508.14098].
- *Hierarchical, divide-and-conquer decomposition (SGT, ProQ) unlocks compositionality, low-depth inference, and improved generalization: particularly valuable in long-horizon or multi-modal planning* [1906.05329, 2506.18847].
- *Rich, route-based goal conditioning is generally necessary for high-fidelity SE(2) trajectory prediction—even powerful models (diffusion, GNNs) see significant performance drops when reduced to endpoint-only goals* [2509.03658, 2403.11304].
- *SE(2)-equivariance in the planning model architecture leads to output stability, robust generalization, and reduced sample complexity—enabling reliable planning under arbitrary scene frame selection* [2403.11304].
- Future work directions include scaling these techniques to SE(3) (full 3D pose), integrating manipulation and end-effector objectives, and leveraging auxiliary geometric and social interaction priors in goal-conditioned settings [2508.14098, 2403.11304].

## 7. Notable Applications and Comparative Summary

Goal-conditioned SE(2) trajectories provide the foundation for:
- Short-range agile movements in legged robots and humanoids, directly synthesizing joint-level actions to reach arbitrary poses [2508.14098].
- Autonomous driving, with route-following and multi-agent interaction in dense urban scenes [2509.03658, 2403.11304].
- Sample-efficient planning across modalities and hardware, accommodating sim-to-real transfer [2508.14098, 2403.11304].
- Long-horizon navigation and global planning by decomposing complex goals into robust, reachable sub-goals [1906.05329, 2506.18847].

Direct pose-reaching and latent-geometry-based approaches are consistently superior to indirect velocity-tracking or rigid hierarchical planners, notably in settings with complex kinematics, multi-step interactions, or severe sample constraints.

**Key references:**  
- "No More Marching: Learning Humanoid Locomotion for Short-Range SE(2) Targets" [2508.14098]  
- "Sub-Goal Trees -- a Framework for Goal-Directed Trajectory Prediction and Optimization" [1906.05329]  
- "Efficient Virtuoso: A Latent Diffusion Transformer Model for Goal-Conditioned Trajectory Planning" [2509.03658]  
- "Offline Goal-Conditioned Reinforcement Learning with Projective Quasimetric Planning" [2506.18847]  
- "Pioneering SE(2)-Equivariant Trajectory Planning for Automated Driving" [2403.11304]

Source: https://www.emergentmind.com/topics/goal-conditioned-se-2-trajectories