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Hierarchical DMPs

Hierarchical Dynamic Movement Primitives

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

Family: Dynamic Movement Primitives Status: 📋 Planned

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Overview

Hierarchical Dynamic Movement Primitives extend the basic DMP framework to handle complex behaviors through multi-level decomposition and composition. This approach enables the learning and execution of sophisticated tasks by breaking them down into simpler sub-tasks and organizing them in a hierarchical structure.

The key innovation of hierarchical DMPs is the integration of: - Multi-level movement decomposition from high-level tasks to low-level primitives - Complex behavior composition through hierarchical organization - Task hierarchy learning from demonstrations - Adaptive behavior selection based on context and goals - Robust execution through hierarchical error handling and recovery

These DMPs are particularly valuable in applications requiring complex, multi-step behaviors, such as assembly tasks, cooking, household chores, and any task that can be naturally decomposed into a hierarchy of simpler movements.

Mathematical Formulation

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Problem Definition

Given:

  • Hierarchical task structure: T = {T_1, T_2, ..., T_N} with levels L = {L_1, L_2, ..., L_M}
  • High-level DMP: τẏ_H = α_y(β_y(g_H - y_H) - ẏ_H) + f_H(x_H)
  • Low-level DMPs: τẏ_L_i = α_y(β_y(g_L_i - y_L_i) - ẏ_L_i) + f_L_i(x_L_i)
  • Hierarchy weights: W = {w_ij} for connections between levels
  • Context variables: C = {c_1, c_2, ..., c_K}

The hierarchical DMP system is: τẏ_H = α_y(β_y(g_H - y_H) - ẏ_H) + f_H(x_H) + Σ_j w_Hj * y_L_j τẏ_L_i = α_y(β_y(g_L_i - y_L_i) - ẏ_L_i) + f_L_i(x_L_i) + Σ_k w_ik * y_L_k + w_iH * y_H

Where: - y_H is the high-level state - y_L_i are the low-level states - w_ij are the hierarchy weights - C influences the hierarchy structure

Key Properties

Hierarchical Decomposition

T = T_1 ∪ T_2 ∪ ... ∪ T_N with T_i ∩ T_j = ∅ for i ≠ j

Tasks are decomposed into non-overlapping sub-tasks

Multi-level Composition

y_H = f_compose(y_L_1, y_L_2, ..., y_L_N)

High-level behaviors are composed from low-level primitives

Adaptive Hierarchy

W(t) = W_0 + ΔW(C(t))

Hierarchy weights adapt based on context

Key Properties

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  • Multi-level Decomposition

    Decomposes complex tasks into simpler sub-tasks

  • Hierarchical Composition

    Composes complex behaviors from simpler primitives

  • Task Hierarchy Learning

    Learns hierarchical task structures from demonstrations

  • Adaptive Behavior Selection

    Selects appropriate behaviors based on context

Implementation Approaches

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Basic hierarchical DMPs with two-level hierarchy

Complexity:

  • Time: O(T × K × L)
  • Space: O(K × L)

Advantages

  • Multi-level task decomposition

  • Complex behavior composition

  • Hierarchical error handling

  • Adaptive behavior selection

Disadvantages

  • Higher computational cost

  • Complex parameter tuning

  • Requires careful hierarchy design

Hierarchical DMPs with adaptive hierarchy structure

Complexity:

  • Time: O(T × K × L + T × C)
  • Space: O(K × L + C)

Advantages

  • Adaptive hierarchy structure

  • Performance-based adaptation

  • Context-aware behavior selection

  • Continuous learning and improvement

Disadvantages

  • Higher computational cost

  • Complex adaptation mechanisms

  • Requires performance metrics

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 Hierarchical DMP O(T × K × L) O(K × L) Time complexity scales with trajectory length, basis functions, and hierarchy levels

Use Cases & Applications

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Application Categories

Complex Assembly Tasks

  • Multi-step Assembly: Assembling complex products with multiple steps

  • Precision Assembly: Precise assembly tasks with hierarchical precision

  • Quality Control: Quality control tasks with hierarchical inspection

  • Packaging: Packaging tasks with hierarchical organization

Household Tasks

  • Cooking: Cooking tasks with hierarchical recipe execution

  • Cleaning: Cleaning tasks with hierarchical room organization

  • Laundry: Laundry tasks with hierarchical sorting and washing

  • Gardening: Gardening tasks with hierarchical plant care

Industrial Automation

  • Manufacturing: Manufacturing tasks with hierarchical production

  • Quality Control: Quality control tasks with hierarchical inspection

  • Maintenance: Maintenance tasks with hierarchical service

  • Logistics: Logistics tasks with hierarchical organization

Service Robotics

  • Healthcare: Healthcare tasks with hierarchical patient care

  • Education: Education tasks with hierarchical learning

  • Entertainment: Entertainment tasks with hierarchical performance

  • Security: Security tasks with hierarchical monitoring

Human-Robot Interaction

  • Collaborative Tasks: Collaborative tasks with hierarchical coordination

  • Assistive Tasks: Assistive tasks with hierarchical assistance

  • Social Interaction: Social interaction tasks with hierarchical behavior

  • Learning Tasks: Learning tasks with hierarchical skill acquisition

Educational Value

  • Hierarchical Systems: Understanding hierarchical system design

  • Task Decomposition: Understanding task decomposition techniques

  • Behavior Composition: Understanding behavior composition methods

  • Adaptive Systems: Understanding adaptive system mechanisms

References & Further Reading

:material-library: Core Papers

:material-book:
Coupling movement primitives: Interaction with the environment and bimanual tasks
2014 • IEEE Transactions on Robotics • Original work on coupled DMPs and hierarchical organization
:material-book:
Learning from demonstration with movement primitives
2013 • IEEE International Conference on Robotics and Automation • DMPs with hierarchical task decomposition

:material-library: Hierarchical Systems

:material-book:
Hierarchically organized behavior and its neural foundations: A reinforcement learning perspective
2009 • Cognition • Hierarchical behavior organization in biological systems
:material-book:
Hierarchical reinforcement learning with the MAXQ value function decomposition
2000 • Journal of Artificial Intelligence Research • Hierarchical reinforcement learning

:material-web: Online Resources

:material-link:
Hierarchical Systems
Wikipedia article on hierarchical systems
:material-link:
Task Decomposition
Wikipedia article on task decomposition
:material-link:
Behavior Composition
Wikipedia article on behavior composition

:material-code-tags: Implementation & Practice

:material-link:
ROS Behavior Trees
Behavior tree library for hierarchical behavior
:material-link:
ROS State Machine
ROS state machine for hierarchical control
:material-link:
ROS Task Planning
ROS task planning framework

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.

  • Probabilistic Movement Primitives (ProMPs) - Probabilistic extension of DMPs that captures movement variability and generates movement distributions from multiple demonstrations.

  • 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.