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Probabilistic Movement Primitives

Probabilistic Movement Primitives (ProMPs)

Probabilistic extension of DMPs that captures movement variability and generates movement distributions from multiple demonstrations.

Family: Dynamic Movement Primitives Status: 📋 Planned

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Overview

Probabilistic Movement Primitives (ProMPs) extend the basic DMP framework to handle movement variability and uncertainty. Unlike standard DMPs that learn from single demonstrations, ProMPs learn from multiple demonstrations to capture the natural variability in human movements.

The key innovation of ProMPs is the probabilistic formulation that allows for: - Learning movement distributions from multiple demonstrations - Generating new movements that respect the learned variability - Handling correlations between different joints or dimensions - Conditioning on via-points or partial observations - Modulating movement characteristics through conditioning

ProMPs are particularly valuable in robotics applications where movements need to be both reproducible and adaptable to different contexts while maintaining natural variability.

Mathematical Formulation

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Problem Definition

Given:

  • Multiple demonstrations: {y_demo^(i)(t)} for i = 1, ..., N
  • Basis functions: ψ(t) = [ψ_1(t), ..., ψ_K(t)]^T
  • Weight vectors: w^(i) ∈ ℝ^K for each demonstration

Learn a probabilistic model: p(w) = N(w | μ_w, Σ_w)

Where: - μ_w = (1/N) Σ_{i=1}^N w^(i) - Σ_w = (1/N) Σ_{i=1}^N (w^(i) - μ_w)(w^(i) - μ_w)^T

The trajectory is generated as: y(t) = Ψ(t)^T w + ε(t)

Where ε(t) ~ N(0, Σ_y) is observation noise.

Key Properties

Probabilistic Weights

p(w) = N(w | μ_w, Σ_w)

Weights follow a multivariate Gaussian distribution

Movement Distribution

p(y(t)) = N(y(t) | Ψ(t)^T μ_w, Ψ(t)^T Σ_w Ψ(t) + Σ_y)

Trajectory follows a time-varying Gaussian distribution

Via-point Conditioning

p(w | y_obs) = N(w | μ_w|obs, Σ_w|obs)

Can condition on observed via-points using Gaussian conditioning

Key Properties

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  • Movement Variability

    Captures natural variability in human movements

  • Multi-demonstration Learning

    Learns from multiple demonstrations to build robust models

  • Probabilistic Generation

    Generates movements with appropriate variability

  • Via-point Conditioning

    Can condition on partial observations or via-points

Implementation Approaches

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Standard ProMP implementation with Gaussian weight distribution

Complexity:

  • Time: O(N × T × K + K^3)
  • Space: O(K^2 + N × K)

Advantages

  • Captures movement variability naturally

  • Learns from multiple demonstrations

  • Probabilistic trajectory generation

  • Via-point conditioning capabilities

Disadvantages

  • Computational cost scales with basis functions

  • Requires multiple demonstrations

  • Gaussian assumption may be limiting

ProMPs that capture correlations between different dimensions

Complexity:

  • Time: O(N × T × K + K^3 × d^2)
  • Space: O(K^2 × d^2 + N × K × d)

Advantages

  • Captures cross-dimensional correlations

  • More realistic movement modeling

  • Joint learning across dimensions

  • Better generalization

Disadvantages

  • Higher computational cost

  • More complex parameter estimation

  • Requires more demonstrations

Complete Implementation

The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:

Complexity Analysis

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Time & Space Complexity Comparison

Approach Time Complexity Space Complexity Notes
Basic ProMP Learning O(N × T × K + K^3) O(K^2 + N × K) Learning time scales with demonstrations, trajectory length, and basis functions

Use Cases & Applications

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Application Categories

Human-Robot Interaction

  • Imitation Learning: Learning from human demonstrations with natural variability

  • Collaborative Tasks: Adapting to human partner's movement style

  • Assistive Robotics: Learning assistive movements with appropriate variability

  • Social Robotics: Generating natural, human-like movements

Robotic Manipulation

  • Grasping: Learning grasping strategies with variability

  • Assembly: Learning assembly movements with natural variation

  • Tool Use: Learning tool manipulation with appropriate variability

  • Packaging: Learning packaging movements with natural variation

Locomotion and Navigation

  • Walking: Learning walking patterns with natural gait variation

  • Running: Learning running patterns with stride variability

  • Navigation: Learning navigation behaviors with path variation

  • Dancing: Learning dance movements with artistic variation

Medical and Rehabilitation

  • Physical Therapy: Learning therapeutic movements with patient-specific variation

  • Surgery: Learning surgical movements with appropriate precision variation

  • Rehabilitation: Learning recovery movements with natural variation

  • Prosthetics: Learning prosthetic control with user-specific variation

Sports and Entertainment

  • Sports Training: Learning athletic movements with natural variation

  • Dance: Learning dance movements with artistic variation

  • Music: Learning musical instrument playing with expressive variation

  • Gaming: Learning game movements with natural variation

Educational Value

  • Probabilistic Modeling: Understanding how to model uncertainty in movements

  • Multi-demonstration Learning: Learning from multiple examples to build robust models

  • Gaussian Processes: Understanding probabilistic function approximation

  • Conditioning: Understanding how to condition probabilistic models on observations

References & Further Reading

:material-library: Core Papers

:material-book:
Probabilistic movement primitives
2013 • Advances in Neural Information Processing Systems • Original ProMP paper
:material-book:
Using probabilistic movement primitives in robotics
2018 • Autonomous Robots • Comprehensive ProMP review and applications

:material-library: ProMP Extensions

:material-book:
Learning interaction for collaborative tasks with probabilistic movement primitives
2016 • IEEE-RAS International Conference on Humanoid Robots • ProMPs for human-robot interaction
:material-book:
Learning motor skills from partially observed movements executed at different speeds
2015 • IEEE/RSJ International Conference on Intelligent Robots and Systems • ProMPs with partial observations

:material-web: Online Resources

:material-link:
Probabilistic Movement Primitives
Wikipedia article on ProMPs
:material-link:
ProMP Tutorial
ResearchGate tutorial on ProMPs
:material-link:
ProMP Implementation
Python implementation of ProMPs

:material-code-tags: Implementation & Practice

:material-link:
PyDMPs
Python library for DMPs and ProMPs
:material-link:
ProMPy
Python implementation of ProMPs
:material-link:
ROS ProMP Package
ROS integration for ProMPs

Interactive Learning

Try implementing the different approaches yourself! This progression will give you deep insight into the algorithm's principles and applications.

Pro Tip: Start with the simplest implementation and gradually work your way up to more complex variants.

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Related Algorithms in Dynamic Movement Primitives:

  • DMPs with Obstacle Avoidance - DMPs enhanced with real-time obstacle avoidance capabilities using repulsive forces and safe navigation in cluttered environments.

  • Spatially Coupled Bimanual DMPs - DMPs for coordinated dual-arm movements with spatial coupling between arms for synchronized manipulation tasks and hand-eye coordination.

  • Constrained Dynamic Movement Primitives (CDMPs) - DMPs with safety constraints and operational requirements that ensure movements comply with safety limits and operational constraints.

  • DMPs for Human-Robot Interaction - DMPs specialized for human-robot interaction including imitation learning, collaborative tasks, and social robot behaviors.

  • Multi-task DMP Learning - DMPs that learn from multiple demonstrations across different tasks, enabling task generalization and cross-task knowledge transfer.

  • Geometry-aware Dynamic Movement Primitives - DMPs that operate with symmetric positive definite matrices to handle stiffness and damping matrices for impedance control applications.

  • Online DMP Adaptation - DMPs with real-time parameter updates, continuous learning from feedback, and adaptive behavior modification during execution.

  • Temporal Dynamic Movement Primitives - DMPs that generate time-based movements with rhythmic pattern learning, beat and tempo adaptation for temporal movement generation.

  • DMPs for Manipulation - DMPs specialized for robotic manipulation tasks including grasping movements, assembly tasks, and tool use behaviors.

  • Basic Dynamic Movement Primitives (DMPs) - Fundamental DMP framework for learning and reproducing point-to-point and rhythmic movements with temporal and spatial scaling.

  • Hierarchical Dynamic Movement Primitives - DMPs organized in hierarchical structures for multi-level movement decomposition, complex behavior composition, and task hierarchy learning.

  • DMPs for Locomotion - DMPs specialized for walking pattern generation, gait adaptation, and terrain-aware movement in legged robots and humanoid systems.

  • Reinforcement Learning DMPs - DMPs enhanced with reinforcement learning for parameter optimization, reward-driven learning, and policy gradient methods for movement refinement.