Temporal DMPs

Temporal Dynamic Movement Primitives

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

Family: Dynamic Movement Primitives Status: 📋 Planned

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Overview

Temporal Dynamic Movement Primitives extend the basic DMP framework to handle time-based movement generation with emphasis on rhythmic patterns, beat adaptation, and tempo control. These DMPs are particularly valuable for applications requiring precise temporal coordination, such as musical performance, dance, rhythmic locomotion, and any task that requires synchronization with external timing cues.

The key innovation of temporal DMPs is the integration of: - Time-based canonical systems that can adapt to external timing cues - Rhythmic pattern learning and generation - Beat and tempo adaptation mechanisms - Synchronization with external rhythms and metronomes - Temporal scaling and modulation capabilities

These DMPs are particularly valuable in applications requiring precise temporal coordination, such as musical instrument playing, dance performance, rhythmic locomotion, and synchronized multi-agent systems.

Mathematical Formulation

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Problem Definition

Given:

• Time-based canonical system: τφ̇ = ω(t) where ω(t) is the time-varying frequency
• Rhythmic DMP: τẏ = α_y(β_y(g - y) - ẏ) + f(φ)
• Beat adaptation: ω(t) = ω_0 + k_beat * (t_beat - t_expected)
• Tempo adaptation: τ(t) = τ_0 + k_tempo * (tempo_target - tempo_current)
• Phase synchronization: φ_sync = φ + k_sync * (φ_external - φ)

The temporal DMP becomes: τ(t)ẏ = α_y(β_y(g - y) - ẏ) + f(φ_sync) τ(t)φ̇ = ω(t)

Where: - ω(t) is the adaptive frequency - τ(t) is the adaptive time constant - φ_sync is the synchronized phase

Key Properties

Adaptive Frequency

ω(t) = ω_0 + k_beat * (t_beat - t_expected)

Frequency adapts to external beat cues

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Tempo Adaptation

τ(t) = τ_0 + k_tempo * (tempo_target - tempo_current)

Time constant adapts to target tempo

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Phase Synchronization

φ_sync = φ + k_sync * (φ_external - φ)

Phase synchronizes with external rhythm

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Key Properties

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• Rhythmic Generation

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  Generates rhythmic movements with precise timing

• Beat Adaptation

  ---

  Adapts to external beat cues and timing

• Tempo Control

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  Controls movement tempo and speed

• Phase Synchronization

  ---

  Synchronizes with external rhythms and metronomes

Implementation Approaches

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Rhythmic Temporal DMPsMetronome Synchronized DMPs

Temporal DMPs for rhythmic movement generation with beat adaptation

Complexity:

• Time: O(T × K)
• Space: O(K)

Advantages

• Rhythmic movement generation

• Beat and tempo adaptation

• Phase synchronization

• Temporal scaling capabilities

Disadvantages

• Requires external timing cues

• Complex parameter tuning

• May not handle all temporal patterns

Temporal DMPs that synchronize with external metronomes

Complexity:

• Time: O(T × K + T × M)
• Space: O(K + M)

Advantages

• Metronome synchronization

• External timing cue integration

• Robust temporal coordination

• Musical performance capabilities

Disadvantages

• Requires external metronome

• Sensitive to metronome accuracy

• Higher computational cost

Complete Implementation

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

• Main implementation with rhythmic and metronome-synchronized DMPs: src/algokit/dynamic_movement_primitives/temporal_dmps.py

• Comprehensive test suite including temporal coordination tests: tests/unit/dynamic_movement_primitives/test_temporal_dmps.py

Complexity Analysis

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Time & Space Complexity Comparison

Approach	Time Complexity	Space Complexity	Notes
Rhythmic Temporal DMP	O(T × K)	O(K)	Time complexity scales with trajectory length and basis functions

Use Cases & Applications

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Application Categories

Musical Performance

• Instrument Playing: Playing musical instruments with precise timing

• Conducting: Conducting orchestras with precise beat patterns

• Dancing: Dancing with precise rhythm and timing

• Singing: Singing with precise tempo and rhythm

Rhythmic Locomotion

• Walking: Walking with precise gait timing

• Running: Running with precise stride timing

• Swimming: Swimming with precise stroke timing

• Cycling: Cycling with precise pedal timing

Synchronized Systems

• Multi-Robot Coordination: Coordinating multiple robots with precise timing

• Human-Robot Interaction: Interacting with humans with precise timing

• Synchronized Manipulation: Manipulating objects with precise timing

• Synchronized Assembly: Assembling parts with precise timing

Entertainment and Arts

• Dance: Dancing with precise rhythm and timing

• Theater: Performing theatrical movements with precise timing

• Sports: Performing sports movements with precise timing

• Gaming: Performing game movements with precise timing

Rehabilitation and Therapy

• Physical Therapy: Performing therapeutic exercises with precise timing

• Speech Therapy: Performing speech exercises with precise timing

• Occupational Therapy: Performing occupational tasks with precise timing

• Music Therapy: Performing music therapy with precise timing

Educational Value

• Temporal Coordination: Understanding temporal coordination in robotics

• Rhythmic Patterns: Understanding rhythmic pattern generation

• Beat Adaptation: Understanding beat adaptation mechanisms

• Phase Synchronization: Understanding phase synchronization techniques

References & Further Reading

:material-library: Core Papers

:material-book:

On-line learning and modulation of periodic movements with nonlinear dynamical systems

2009 • Autonomous Robots • Original work on rhythmic DMPs and temporal adaptation

:material-book:

Dynamical movement primitives: Learning attractor landscapes for motor skills

2013 • Biological Cybernetics • Comprehensive review of DMPs including temporal aspects

:material-library: Temporal Coordination

:material-book:

Sensorimotor synchronization: A review of the tapping literature

2005 • Psychonomic Bulletin & Review • Review of sensorimotor synchronization in humans

:material-book:

The dynamics of attending: How people track time-varying events

1999 • Psychological Review • Dynamics of temporal attention and synchronization

:material-web: Online Resources

:material-link:

Rhythmic Movement

Wikipedia article on rhythmic movement

:material-link:

Temporal Coordination

Wikipedia article on temporal coordination

:material-link:

Beat Synchronization

Wikipedia article on beat synchronization

:material-code-tags: Implementation & Practice

:material-link:

MIDI

MIDI protocol for musical instrument communication

:material-link:

Audio Processing

Audio signal processing for rhythm detection

:material-link:

Tempo Detection

Tempo detection algorithms

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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Navigation

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.

• 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.

• 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.