1. Bibliographic Information

1.1. Title

Riemannian Flow Matching Policy for Robot Motion Learning

1.2. Authors

The authors are Max Braun, Noémie Jaquier, Leonel Rozo, and Tamim Asfour.

• Max Braun, Noémie Jaquier, and Tamim Asfour are affiliated with the Institute for Anthropomatics and Robotics at the Karlsruhe Institute of Technology (KIT), Germany. This group is well-known for its research in humanoid robotics, machine learning for robotics, and motion synthesis.
• Leonel Rozo is affiliated with the Bosch Center for Artificial Intelligence (BCAI). His research focuses on robot learning, probabilistic methods, and human-robot interaction. The authors' collective expertise lies at the intersection of robotics, machine learning, and geometry, which is central to this paper's topic.

1.3. Journal/Conference

The paper was submitted to arXiv, a preprint server for academic articles. This means it has not yet undergone formal peer review for publication in a conference or journal, but it represents the latest research from the authors' group.

1.4. Publication Year

The paper was published on arXiv on March 15, 2024.

1.5. Abstract

The paper introduces Riemannian Flow Matching Policies (RFMP), a new model for learning robot visuomotor policies. RFMP is built upon flow matching, a technique known for its efficient training and fast inference. The model is designed to capture the high-dimensional, multimodal action distributions common in robotics tasks. A key feature of RFMP is its inherent geometric awareness, as it operates on Riemannian manifolds, which naturally represent robot states (like orientation). The authors conduct proof-of-concept experiments on both state-based and vision-based tasks, comparing RFMP to Diffusion Policies. The results show that while both methods can learn the tasks, RFMP produces smoother action trajectories and has significantly lower inference times.

1.6. Original Source Link

• Original Source Link: https://arxiv.org/abs/2403.10672

• PDF Link: https://arxiv.org/pdf/2403.10672v2.pdf

• Publication Status: This is a preprint, not yet officially published in a peer-reviewed venue.

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2. Executive Summary

2.1. Background & Motivation

The core problem this paper addresses is the challenge of learning complex and realistic robot motion policies. Modern robots operating in unstructured environments need to generate movements that are not just a single, fixed trajectory but can adapt to different situations, often requiring the policy to represent a multimodal distribution of possible actions (e.g., multiple valid ways to pick up an object).

Deep generative models, particularly diffusion models, have shown great promise in this area. They can learn highly complex distributions from data. However, they have two major drawbacks for robotics:

1. High Inference Cost: Generating an action with a diffusion model requires solving a stochastic differential equation (SDE), which involves many iterative denoising steps. This is computationally expensive and can be too slow for real-time, reactive robot control.

2. Geometric Ignorance: Robot states, especially those involving orientation (like the pose of a robot's hand), do not live in a flat Euclidean space but on curved spaces called Riemannian manifolds. Standard diffusion models are designed for Euclidean space and applying them naively to manifold data can lead to invalid or inaccurate results. While Riemannian diffusion models exist, they add even more computational complexity.

   The paper's innovative idea is to use a more recent generative modeling technique, Flow Matching (FM), as the foundation for a robot policy. FM offers the same power to model complex distributions as diffusion models but with a much simpler and faster inference process, as it only requires solving a deterministic ordinary differential equation (ODE). By building on a version of FM designed for Riemannian manifolds, the authors propose a solution that is both computationally efficient and geometrically sound.

2.2. Main Contributions / Findings

The paper makes two primary contributions:

1. Proposal of Riemannian Flow Matching Policies (RFMP): This is a novel framework for learning robot sensorimotor policies. It is the first, to the authors' knowledge, to apply flow matching methods to this problem. RFMP is designed to be conditioned on sensory inputs (like state or vision) and to respect the underlying geometry of the robot's state space.

2. Empirical Validation and Comparison: The authors demonstrate RFMP's effectiveness on the standard LASA benchmark dataset. They compare it directly against a state-of-the-art baseline, Diffusion Policies (DP).

   The key findings from their experiments are:

• Superior Smoothness: RFMP generates significantly smoother motion trajectories compared to DP, which is critical for safe and predictable robot behavior.

• Faster Inference: RFMP achieves a substantial reduction in inference time (30-45% faster in their experiments) compared to DP, making it more suitable for real-time applications.

• Robustness: RFMP's performance is more stable when the prediction horizon is shortened, whereas DP's performance degrades, producing jerkier motions.

• Geometric Fidelity: Unlike standard DP, RFMP is inherently designed for Riemannian manifolds, ensuring that generated actions (e.g., orientations on a sphere) are always valid and respect the data's intrinsic geometry.

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3. Prerequisite Knowledge & Related Work

3.1. Foundational Concepts

3.1.1. Deep Generative Models

Deep generative models are a class of machine learning models that learn to capture the underlying probability distribution of a training dataset. Instead of just classifying data, their goal is to be able to generate new data samples that are statistically similar to the training data. In robotics, this is powerful for learning a policy, which can be seen as a distribution of possible actions conditioned on the robot's state or sensory input. This allows the robot to generate diverse and context-appropriate behaviors rather than executing a single, pre-programmed motion.

3.1.2. Riemannian Manifolds

Imagine trying to draw a straight line on the surface of a globe. The "straightest" possible path is a "great circle," which is curved from a 3D perspective. A Riemannian manifold is a mathematical concept for such curved spaces.

• Manifold ($\mathcal{M}$): A space that, on a small enough scale, looks like a flat Euclidean space ($\mathbb{R}^d$). The Earth's surface is a 2D manifold.
• Tangent Space ($\mathcal{T}_{\mathbf{x}}\mathcal{M}$): At any point $x$ on the manifold, we can imagine a flat plane that just touches the surface at that point. This plane is the tangent space. It's a standard vector space where we can do calculus, representing instantaneous velocities or directions.
• Exponential Map ($\exp_{\mathbf{x}}$): This map takes a vector in the tangent space at $x$ and "projects" it onto the manifold by following a straight path (a geodesic) for a certain distance. It's how you move from the flat tangent space back to the curved manifold.
• Logarithmic Map ($\mathrm{Log}_{\mathbf{x}}$): This is the inverse of the exponential map. It takes two points on the manifold, $x$ and $y$, and tells you the tangent vector at $x$ that points directly towards $y$ along the shortest path.
• Robot Poses: A robot's end-effector pose consists of position (in $\mathbb{R}^3$) and orientation. Orientation can be represented as a quaternion (on the sphere manifold $S^3$) or a rotation matrix (on the special orthogonal group manifold SO(3)). Treating these as simple vectors in Euclidean space ignores their geometric constraints and can lead to problems (e.g., generating an invalid rotation).

3.1.3. Flow Matching (FM)

Flow Matching is a recent generative modeling technique that provides an alternative to diffusion models. The core idea is to learn a vector field that defines a "flow" to transform samples from a simple base distribution (e.g., a standard Gaussian, $p_0$) into samples from a complex target data distribution ($p_1$).

• Intuition: Imagine a calm pool of water (the base distribution). We want to create a complex wave pattern (the target distribution). Flow matching learns the water currents (the vector field) at every point and every moment in time that will shape the calm water into the desired waves.
• Mechanism: This vector field, $u_t(\mathbf{x})$, is used in an ordinary differential equation (ODE): $\frac{d\phi_t(\mathbf{x})}{dt} = u_t(\phi_t(\mathbf{x}))$. Solving this ODE from $t=0$ to $t=1$ pushes a starting point $\mathbf{x}_0 \sim p_0$ to a final point $\mathbf{x}_1 \sim p_1$.
• Conditional Flow Matching (CFM): Directly learning the vector field for the entire distribution is hard. CFM makes this tractable by defining a simpler conditional probability path $p_t(\mathbf{x} | \mathbf{z})$ between two points and learning a conditional vector field $u_t(\mathbf{x} | \mathbf{z})$. By choosing the conditioning variable $\mathbf{z}$ to be a sample from the target distribution ($\mathbf{x}_1$), the model can learn to "flow" any point towards that specific target sample. This simplifies the training objective significantly.

3.2. Previous Works

3.2.1. Normalizing Flows

These were among the first flow-based generative models applied to robotics. They learn a diffeomorphism, which is a complex, invertible function that directly maps the base distribution to the target distribution.

• Advantage: They provide an exact likelihood for any data point.
• Disadvantage: They have restrictive architectural constraints to ensure invertibility, and training can be slow and unstable because it often involves integrating an ODE over time.

3.2.2. Diffusion Models

Diffusion models have become extremely popular in robotics. They consist of two processes:

1. Forward Process: Gradually add noise to a data sample over many steps until it becomes pure noise from a known distribution (e.g., Gaussian).
2. Reverse Process: Train a neural network (often called a score function or denoiser) to reverse this process, step by step. To generate a new sample, one starts with pure noise and iteratively applies the learned denoiser to reconstruct a clean data sample.

• Diffusion Policies (DP) [4]: This is a key baseline in the paper. It uses a diffusion model to represent the policy $\pi(\mathbf{a} | \mathbf{o})$. Given an observation $\mathbf{o}$, it generates an action sequence $\mathbf{a}$ by starting with random noise and conditioning the denoising process on $\mathbf{o}$.
• Disadvantage: The iterative denoising process is equivalent to solving a stochastic differential equation (SDE), which requires many function evaluations (e.g., 100 in the paper's experiments), making inference slow.

3.2.3. Riemannian Flow Matching

The paper builds directly on "Flow Matching on General Geometries" [19], which extended FM to Riemannian manifolds. The key innovations were:

• Geodesic Flow: Instead of linear interpolation in Euclidean space, it defines the flow between two points on a manifold using the geodesic (the shortest path between them).
• Riemannian Loss: The loss function for training the vector field is computed using the manifold's intrinsic metric, ensuring the learned flow respects the geometry.

3.3. Technological Evolution

The field of generative robot policies has evolved in search of models that are expressive, stable to train, and efficient at inference.

1. Normalizing Flows: Offered a direct mapping but were hard to train and architecturally constrained.
2. Diffusion Models: Became dominant due to their stable training and ability to model extremely complex distributions. However, their slow, iterative inference is a major bottleneck for robotics.
3. Flow Matching (This Paper): Emerges as a solution that aims to combine the best of both worlds. It is simulation-free and stable to train like diffusion models but offers much faster inference by solving a deterministic ODE instead of a stochastic SDE. This paper is the first to bring this latest evolution to robot policy learning, and further, to the Riemannian setting.

3.4. Differentiation Analysis

Compared to the main related works, RFMP's approach is different in two key ways:

• vs. Diffusion Policies (DP):
  • Inference: RFMP uses ODE integration, which is deterministic and much faster than the SDE integration required by DP.
  • Geometry: RFMP is built on Riemannian Flow Matching, allowing it to natively and correctly handle manifold-valued data like robot poses. Standard DP is designed for Euclidean space and either ignores geometry or requires complex and computationally expensive Riemannian diffusion extensions.
• vs. Normalizing Flows:

  • Training: RFMP training is "simulation-free" and more stable. It avoids the need to backpropagate through an ODE solver, which is a major bottleneck for normalizing flows. It directly regresses the vector field in a simple supervised learning manner.

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4. Methodology

4.1. Principles

The core principle of a Riemannian Flow Matching Policy (RFMP) is to represent a robot's policy, $\pi_{\pmb{\theta}}(\mathbf{a}|\mathbf{o})$, as a conditional vector field $v_t(\mathbf{a} | \mathbf{o}; \pmb{\theta})$ defined on a Riemannian manifold. This vector field learns to "flow" a random action $\mathbf{a}_0$ (sampled from a simple base distribution) towards an expert action $\mathbf{a}_1$ (from a demonstration), conditioned on an observation $\mathbf{o}$. The training process involves minimizing the difference between the learned vector field and a target "ground truth" vector field derived from geodesic paths on the manifold. Once trained, generating an action is a fast, one-shot process of solving an ordinary differential equation (ODE) guided by the learned vector field.

4.2. Core Methodology In-depth (Layer by Layer)

The methodology can be broken down into the training and inference phases.

4.2.1. RFMP Training

The goal of training is to learn the parameters $\pmb{\theta}$ of a neural network that approximates the ideal vector field.

1. Data Structuring (Receding Horizon)

To ensure temporal consistency, the model doesn't predict a single action at a time. Instead, it predicts a future horizon of actions.

• Action Horizon: The action vector $\mathbf{a}$ consists of a sequence of future actions: $\mathbf{a} = [\mathbf{a}_\tau, \mathbf{a}_{\tau+1}, \dots, \mathbf{a}_{\tau+T_a}]$, where $T_a$ is the prediction horizon length.
• Observation Vector: To provide more context about the motion, the observation vector $\mathbf{o}$ is constructed from multiple points in the past: $\mathbf{o} = [\mathbf{o}_{\tau-1}, \mathbf{o}_c, \tau-c]$. It includes:
  • $\mathbf{o}_{\tau-1}$: The most recent (reference) observation.
  • $\mathbf{o}_c$: A context observation from a past timestep $c$, where $c$ is sampled randomly from previous steps.
  • $\tau-c$: The time difference between the current prediction and the context observation. This explicitly gives the model information about the temporal distance.

2. The RFMP Loss Function

The model is trained by minimizing the Riemannian Conditional Flow Matching (RCFM) loss, adapted for policies. The paper presents this as the RFMP loss:

$ \mathcal{L}{\mathrm{RFMP}}(\pmb{\theta}) = \mathbb{E}{t, q(\mathbf{a}_1), p_t(\mathbf{a}|\mathbf{a}1)} | v_t(\mathbf{a}|\mathbf{o}; \pmb{\theta}) - u_t(\mathbf{a}|\mathbf{a}1) |^2{g{\mathbf{a}}} $

This formula looks complex, but it's a straightforward supervised learning objective. Let's break it down:

• $\mathbb{E}_{...}$: This means we are taking the average loss over many different samples.
• $t \sim \mathcal{U}[0, 1]$: We sample a random time $t$ between 0 and 1.
• $\mathbf{a}_1 \sim q(\mathbf{a}_1)$: We sample a "target" action sequence $\mathbf{a}_1$ from the expert demonstration dataset. $q(\mathbf{a}_1)$ is the distribution of expert actions.
• $\mathbf{a} \sim p_t(\mathbf{a}|\mathbf{a}_1)$: We generate an "intermediate" action $\mathbf{a}$ by moving partway from a random noise sample $\mathbf{a}_0$ towards our target $\mathbf{a}_1$. The paper uses a Gaussian CFM formulation, where $\mathbf{a}$ is sampled from a distribution centered along the path from a base sample to $\mathbf{a}_1$.
• $v_t(\mathbf{a}|\mathbf{o}; \pmb{\theta})$: This is the output of our neural network. It's our predicted vector (direction and magnitude) at the point $\mathbf{a}$ and time $t$, given the observation $\mathbf{o}$.
• $u_t(\mathbf{a}|\mathbf{a}_1)$: This is the "ground truth" or target vector that our network should predict. It's the ideal vector that pushes $\mathbf{a}$ towards $\mathbf{a}_1$. It is derived from the geodesic flow.
• $\| \cdot \|^2_{g_{\mathbf{a}}}$: This is the squared norm (or length) of the difference vector, calculated using the Riemannian metric $g$ at point $\mathbf{a}$. This ensures that distances and directions are measured correctly on the curved manifold, making the process geometry-aware.

3. Defining the Target Vector Field

The "ground truth" vector field $u_t$ is defined by the velocity along the geodesic (shortest path) connecting a base sample $\mathbf{a}_0$ and the target sample $\mathbf{a}_1$. The paper uses the geodesic flow equation:

$ \mathbf{x}t = \mathrm{Exp}{\mathbf{x}0}(t \cdot \mathrm{Log}{\mathbf{x}_0}(\mathbf{x}_1)), \quad t \in [0, 1] $

• $\mathrm{Log}_{\mathbf{x}_0}(\mathbf{x}_1)$: Computes the tangent vector at point $\mathbf{x}_0$ that points towards $\mathbf{x}_1$ along the shortest path on the manifold.

• $t \cdot \mathrm{Log}_{\mathbf{x}_0}(\mathbf{x}_1)$: Scales this vector by time $t$. At $t=0$, it's a zero vector (we are at the start). At $t=1$, it's the full vector to reach $\mathbf{x}_1$.

• $\mathrm{Exp}_{\mathbf{x}_0}(\dots)$: Maps this scaled tangent vector back onto the manifold, giving the point $\mathbf{x}_t$ that is $t$-fraction of the way along the geodesic from $\mathbf{x}_0$ to $\mathbf{x}_1$.

  The target vector field $u_t$ is the time derivative of this path, $d\mathbf{x}_t/dt$. The paper uses a specific form of conditional flow matching from Lipman et al. [8] and Tong et al. [22], where the target vector field is given by:

$ u_t(\mathbf{x} | \mathbf{x}_1) = \frac{\mathbf{x}_1 - (1-\sigma)\mathbf{x}}{1 - (1-\sigma)t} $

(Note: This is the Euclidean version from the background section. The Riemannian equivalent involves parallel transport and derivatives of the geodesic equation, which the paper abstracts for simplicity but implements based on [19]).

4. Training Algorithm Summary

The training loop is described in Algorithm 1:

1. while not converged do:

2.  Sample a random time step $t \in [0, 1]$.
   

3.  Sample a target action sequence $\mathbf{a}_1$ from the expert demonstration data.
   

4.  Sample a noise action sequence $\mathbf{a}_0$ from the base distribution (e.g., a wrapped Gaussian on the manifold).
   

5.  Sample a corresponding observation vector $\mathbf{o}$.
   

6.  Compute the intermediate action $\mathbf{a}_t$ and the target vector field $u_t(\mathbf{a}_t | \mathbf{a}_1)$ based on the geodesic flow.
   

7.  Pass $\mathbf{a}_t$, $t$, and $\mathbf{o}$ to the neural network to get the predicted vector field $v_t(\mathbf{a}_t|\mathbf{o}; \pmb{\theta})$.
   

8.  Calculate the loss $\mathcal{L}_{\mathrm{RFMP}}$ by comparing $v_t$ and $u_t$.
   

9.  Update the network parameters $\pmb{\theta}$ using gradient descent.
   

10. end while

4.2.2. RFMP Inference

Once the vector field $v_t(\mathbf{a} | \mathbf{o}; \pmb{\theta})$ is trained, generating an action sequence is fast and deterministic.

1. At the current time step $\tau$, construct the observation vector $\mathbf{o} = [\mathbf{o}_{\tau-1}, \mathbf{o}_c, \tau-c]$.

2. Draw a single random sample $\mathbf{a}_0$ from the base distribution $p_0$. This serves as the starting point for the flow.

3. Solve the ordinary differential equation (ODE) from $t=0$ to $t=1$: $ \frac{d\mathbf{a}(t)}{dt} = v_t(\mathbf{a}(t) | \mathbf{o}; \pmb{\theta}) \quad \text{with initial condition} \quad \mathbf{a}(0) = \mathbf{a}_0 $ This is done using a standard off-the-shelf ODE solver (like DOPRI for Euclidean space or a Riemannian Euler method). The result is the final action sequence $\mathbf{a}(1) = [\mathbf{a}_\tau, \dots, \mathbf{a}_{\tau+T_a}]$.

4. Execute the first $T_e$ actions from this predicted sequence, where $T_e < T_a$.

5. Move to the next time step and repeat the process. This is known as receding horizon control, which helps make the executed trajectory smoother and more robust to errors.

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5. Experimental Setup

5.1. Datasets

5.1.1. LASA Dataset

The primary dataset used is the LASA Handwriting Dataset [12].

• Source and Scale: It contains 2D trajectories of 26 handwritten shapes, recorded from human demonstrations. The paper uses 7 demonstrations for each shape, with each trajectory containing 200 timesteps.
• Characteristics and Domain: This is a classic benchmark for imitation learning and learning dynamical systems. The shapes (like 'S', 'W', 'L') have varying curvatures and complexities, making them a good testbed for motion generation.
• Usage in the Paper: The authors use the dataset in two settings to test both the Euclidean and Riemannian capabilities of RFMP:
  1. Euclidean ($\mathbb{R}^2$): The original 2D position data.
  2. Riemannian ($S^2$): The 2D data is projected onto the surface of a 2D sphere to create a task on a Riemannian manifold.
• Multimodal Task: A multimodal dataset was created by mirroring the 'L' shape trajectories on the $S^2$ sphere, forcing the model to learn a distribution with two distinct modes.

5.1.2. Visuomotor Adaptation

For the visuomotor experiments, the authors generated visual observations corresponding to the LASA trajectories.

• Data Example: Raw $48 \times 48$ grayscale images were created to depict the progress of the drawing task over time. An example from the paper (Figure 4) shows the 'S' shape being partially drawn on a 2D plane and on a 3D sphere.

  该图像是插图，展示了两种不同的空间表示。左侧为平面空间 $\mathbb{R}^2$ 中的曲线示例，右侧为球面空间 $S^2$ 中的曲线示例。两者体现了不同几何结构下的曲线形状，强调了在机器人运动学习中考虑几何信息的重要性。

• Rationale: Using these images as input tests the model's ability to learn a visuomotor policy, mapping visual inputs to motor actions, which is a more realistic scenario for many robotic applications.

5.2. Evaluation Metrics

5.2.1. Dynamic Time Warping Distance (DTWD)

• Conceptual Definition: DTWD measures the similarity between two time series that may vary in speed or timing. It finds the optimal alignment between the two sequences by "warping" the time axis of one to match the other. A lower DTWD value indicates that the generated trajectory is closer in shape to the expert demonstration, signifying higher reproduction accuracy.
• Mathematical Formula: Given two sequences $X = (x_1, x_2, \dots, x_n)$ and $Y = (y_1, y_2, \dots, y_m)$, the DTW cost is computed recursively. Let D(i, j) be the DTW distance between the prefixes X[1:i] and Y[1:j]. $ D(i, j) = \text{cost}(x_i, y_j) + \min \begin{cases} D(i-1, j) \ D(i, j-1) \ D(i-1, j-1) \end{cases} $
• Symbol Explanation:
  • $\text{cost}(x_i, y_j)$: The distance (e.g., Euclidean) between point $x_i$ and point $y_j$.
  • D(i, j): The cumulative cost to align the sequences up to points $i$ and $j$. The final DTWD is the value D(n, m) in the top-right corner of the cost matrix.

5.2.2. Jerkiness

• Conceptual Definition: Jerk is the third derivative of position with respect to time ($j(t) = \dddot{p}(t)$). It quantifies how rapidly acceleration changes. A trajectory with high jerk feels "jerky" or unsmooth. In robotics, minimizing jerk is crucial for reducing mechanical stress, saving energy, and ensuring safe, predictable movements. Lower jerkiness values indicate a smoother trajectory.
• Mathematical Formula: The total jerkiness of a trajectory is typically calculated as the integrated squared jerk over its duration $T$: $ \text{Jerkiness} = \int_{0}^{T} \left| \frac{d^3\mathbf{p}(t)}{dt^3} \right|^2 dt $
• Symbol Explanation:
  • $\mathbf{p}(t)$: The position vector of the trajectory at time $t$.
  • $\frac{d^3\mathbf{p}(t)}{dt^3}$: The third time derivative of the position (jerk vector).
  • $\|\cdot\|^2$: The squared Euclidean norm.
  • In discrete time, this is approximated using finite differences.

5.3. Baselines

The primary baseline model for comparison is Diffusion Policies (DP), as proposed in [4].

• Description: DP is a state-of-the-art method that uses a diffusion model to represent the robot policy. It has demonstrated strong performance on complex visuomotor tasks.

• Implementation Details: The authors use the official DP implementation, which employs a large CNN-based architecture with 256 million parameters. For inference, it uses the iDDPM algorithm with 100 denoising steps.

• Representativeness: Choosing DP as a baseline is appropriate because it represents the current leading paradigm (diffusion models) for learning expressive robot policies. A favorable comparison against such a strong baseline robustly highlights the advantages of the proposed RFMP method.

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6. Results & Analysis

6.1. Core Results Analysis

6.1.1. Trajectory-based Policies

This experiment tests the ability to generate trajectories based on past states. The results are shown in Figure 2 and Table I.

The following figure (Figure 2 from the original paper) shows the trajectories learned by RFMP (left) and DP (right) for various LASA shapes. Blue curves start from the same initial conditions as the demonstrations, while orange curves start from randomly perturbed initial conditions.

该图像是图表，展示了Riemannian Flow Matching Policies (RFMP) 和Diffusion Policies (DP)的学习轨迹对比。左上和左下显示了RFMP的轨迹，右上和右下则为DP的轨迹，颜色分别代表不同的策略。图中轨迹的变化展示了两种策略在多模态动作生成中的表现。

The following are the results from Table I of the original paper, showing DTWD (accuracy) and Jerkiness (smoothness) for the reproductions in Figure 2.

Analysis:

• Smoothness: RFMP consistently produces trajectories with much lower jerkiness than DP across most tasks (e.g., 2120 vs. 8172 for 'S' in $\mathbb{R}^2$). This is a key advantage, as smoother motions are physically more desirable for robots.
• Accuracy (DTWD): DP achieves a slightly lower (better) DTWD on the unimodal tasks. The authors suggest this is because DP tends to "memorize" the training data and aggressively steers trajectories back to the demonstration paths. In contrast, RFMP exhibits more variance and generalization when starting from new points (see orange curves in Fig. 2), leading to a slightly higher average distance.
• Geometric Fidelity: A critical observation is made for the $S^2$ task. DP, being a Euclidean model, produces trajectories that sometimes leave the manifold surface (i.e., go inside the sphere). RFMP, being intrinsically Riemannian, guarantees that all generated trajectories remain on the manifold.
• Multimodality: In the multimodal 'L' task, RFMP performs better than DP in both accuracy (DTWD 6.14 vs. 7.06) and smoothness. Both models successfully learn the bimodal distribution.

6.1.2. Ablation Studies: Effect of Prediction Horizon

The authors investigate how performance changes with shorter prediction horizons ($T_a$). Shorter horizons are computationally cheaper but provide the model with less future information. The results are shown in Figure 3 and Table II.

The following figure (Figure 3 from the original paper) shows trajectories for RFMP (top) and DP (bottom) with varying prediction horizons $T_a = \{2, 4, 8\}$.

该图像是示意图，展示了在不同空间（$ext{R}^2$ 和 $ext{S}^2$）上，RFMP 和 DP 方法生成的轨迹对比。图中包含四个子图，分别表示 RFMP 在 $ext{R}^2$ (a)、RFMP 在 $ext{S}^2$ (b)、DP 在 $ext{R}^2$ (c) 和 DP 在 $ext{S}^2$ (d) 的轨迹情况。

The following are the results from Table II of the original paper, quantifying performance for different prediction horizons ($T_a$).

Analysis:

• RFMP Robustness: RFMP maintains relatively smooth trajectories even with a very short prediction horizon of $T_a=2$. Although jerkiness increases slightly, the trajectories remain visually smooth.
• DP Degradation: DP's performance degrades significantly with shorter horizons. Jerkiness becomes very high (e.g., 11905 for $T_a=2$), and the orange trajectories in Figure 3 show erratic, jerky behavior. The authors hypothesize this is due to the inherent stochasticity of the diffusion inference process, which is more sensitive to a lack of future context. This highlights a major practical advantage of RFMP.

6.1.3. Visuomotor Policies

This experiment tests the models when conditioned on visual inputs. Results are in Figure 5 and Table IV.

The following figure (Figure 5 from the original paper) shows trajectories from the learned visuomotor policies for RFMP and DP.

该图像是示意图，展示了RFMP的动作轨迹。上方展示了在欧几里得空间中对应的S和J形状轨迹，下方展示了这些轨迹在Riemannian流形上的对应，可视化了RFMP在不同任务中的表现。

The following are the results from Table IV of the original paper, comparing the visuomotor policies.

Analysis:

• Competitive Performance: RFMP performs competitively with DP in terms of accuracy (DTWD), even though RFMP uses a much simpler MLP network (32K parameters) compared to DP's large CNN (256M parameters).
• Smoothness Advantage: RFMP continues to show an advantage in smoothness on the Riemannian manifold tasks (e.g., jerkiness of 3590 vs. 5903 for 'S' on $S^2$).

6.1.4. Inference Time

This is one of the most important results, presented in Table III.

The following are the results from Table III of the original paper, showing inference time per prediction step in milliseconds.

Analysis:

• Euclidean Speed-up: In Euclidean space ($\mathbb{R}^2$), RFMP is significantly faster than DP. For trajectory-based policies, it's ~30% faster (803ms vs. 1142ms). For visuomotor policies, it's ~45% faster (1355ms vs. 2462ms). This is a direct result of using an efficient ODE solver instead of a costly SDE solver.

• Riemannian Comparison: On the sphere ($S^2$), RFMP appears slower for the trajectory-based case (1539ms vs. 1147ms). However, the authors correctly argue this is an unfair comparison. RFMP is using a computationally more expensive but geometrically correct Riemannian ODE solver. DP is using its standard, cheaper Euclidean SDE solver and simply ignoring the manifold geometry. A fair comparison would require a Riemannian Diffusion Policy, which would likely be even slower than the standard DP. In the more complex visuomotor case, RFMP is already faster even with the Riemannian solver (2351ms vs. 2662ms).

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7. Conclusion & Reflections

7.1. Conclusion Summary

The paper successfully introduces Riemannian Flow Matching Policies (RFMP), a novel framework for learning robot motion policies. RFMP leverages the efficient training and fast inference of flow matching models and extends them to operate on Riemannian manifolds, which are the natural domain for robot states involving orientation. Through experiments on the LASA benchmark, the authors demonstrate that RFMP achieves task performance comparable to state-of-the-art Diffusion Policies (DP) while offering three significant advantages:

1. Faster Inference: RFMP is significantly quicker at generating actions, making it more viable for real-time robotics.

2. Smoother Trajectories: It produces motions with lower jerk, which is crucial for robot hardware longevity and safety.

3. Inherent Geometric Awareness: It correctly handles data on curved manifolds, a critical feature that standard DP lacks.

   These attributes position RFMP as a compelling and promising alternative to diffusion models for learning complex sensorimotor skills in robotics.

7.2. Limitations & Future Work

The authors acknowledge the following limitations and future directions:

• Proof-of-Concept: The current experiments are conducted on a 2D benchmark dataset. The next logical step is to evaluate RFMP on real-world robotic platforms with high-dimensional state spaces (e.g., a 7-DOF arm operating in SE(3) space).
• Model Representation: The vector field in the experiments was parameterized by a simple MLP. Future work could explore more powerful network architectures (like transformers or graph neural networks) to potentially capture more complex dynamics.
• Prior Information: The base distribution used was a simple Gaussian. More informative priors could be used to inject domain knowledge or structure into the learning process.

7.3. Personal Insights & Critique

This paper presents a strong case for Flow Matching as a next-generation tool for robot policy learning.

• Key Insight: The main takeaway is that the "best" generative model is not just about expressive power, but about the trade-off between power, training stability, and inference efficiency. For robotics, where real-time performance is often a hard constraint, the speed of RFMP is a killer feature.
• Elegance of the Approach: The integration of Riemannian geometry is not an afterthought but a core part of the framework. This is a much more principled approach than treating all data as Euclidean and hoping for the best, which is a common shortcut in applied ML. It directly addresses the "single tangent space fallacy" mentioned in one of their other works [32].
• Potential for Extension: The framework is general. It could be applied to a wide range of robotics problems beyond motion generation, such as task and motion planning, or even in a reinforcement learning context where a fast, expressive policy is needed.
• Critique and Open Questions:
  • The difference in generalization behavior between RFMP (more variance) and DP (more memorization) is fascinating. Is one universally better than the other? This likely depends on the task. For tasks requiring creativity or exploration, RFMP's behavior might be preferable. For tasks requiring strict adherence to a style or safety constraint, DP's tendency to stay close to demonstrations might be an advantage. This trade-off needs further study.
  • While the experiments are clean and the comparisons are fair, their simplicity (2D trajectories) leaves open the question of how well RFMP scales to the complexity of full-body humanoid motion or dexterous manipulation in 3D, where the dimensionality of the manifold is much higher. This is noted as future work and will be the ultimate test of the method's practical utility.