Basic DMPs

Basic Dynamic Movement Primitives (DMPs)

Fundamental DMP framework for learning and reproducing point-to-point and rhythmic movements with temporal and spatial scaling.

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Overview

Basic Dynamic Movement Primitives (DMPs) are a fundamental framework for learning and reproducing complex movements in robotics. DMPs provide a way to encode movements as dynamical systems that can be learned from demonstrations and then generalized to new situations through temporal and spatial scaling.

The core idea of DMPs is to decompose complex movements into a set of simple dynamical systems that can be combined to create sophisticated behaviors. Each DMP consists of a canonical system that provides temporal structure and a transformation system that generates the actual movement trajectory.

DMPs are particularly powerful because they can handle both discrete (point-to-point) and rhythmic (periodic) movements, making them suitable for a wide range of robotic applications from manipulation to locomotion.

Mathematical Formulation

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Problem Definition

Given:

• Demonstrated trajectory: y_demo(t) ∈ ℝ^d
• Desired start position: y_0 ∈ ℝ^d
• Desired goal position: g ∈ ℝ^d
• Temporal scaling: τ > 0
• Spatial scaling: s ∈ ℝ^d

Learn a dynamical system that reproduces the movement:

For discrete DMPs: τẏ = α_y(β_y(g - y) - ẏ) + f(x) τẋ = -α_x x

For rhythmic DMPs: τẏ = α_y(β_y(g - y) - ẏ) + f(φ) τφ̇ = 1

Where f(x) or f(φ) is the forcing function learned from demonstrations.

Key Properties

Temporal Scaling

τẏ = α_y(β_y(g - y) - ẏ) + f(x)

Allows movement speed adjustment through τ parameter

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Spatial Scaling

s = (g_new - y_0_new) / (g_demo - y_0_demo)

Enables movement amplitude scaling to new start/goal positions

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Stability

ẏ → 0, y → g as t → ∞

Guaranteed convergence to goal position

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Key Properties

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• Movement Learning

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  Learns complex movements from demonstrations

• Temporal Scaling

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  Can adjust movement speed while preserving shape

• Spatial Scaling

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  Can adapt movements to new start/goal positions

• Stability

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  Guaranteed convergence to goal position

Implementation Approaches

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Discrete DMPsRhythmic DMPsMulti-dimensional DMPs

Point-to-point movements with temporal and spatial scaling

Complexity:

• Time: O(T × N_basis)
• Space: O(N_basis × d)

Advantages

• Simple and intuitive framework

• Guaranteed stability and convergence

• Temporal and spatial scaling capabilities

• Smooth trajectory generation

Disadvantages

• Limited to single demonstrations

• No obstacle avoidance

• Fixed basis function placement

Periodic movements with phase-based canonical system

Complexity:

• Time: O(T × N_basis)
• Space: O(N_basis × d)

Advantages

• Handles periodic movements naturally

• Phase-based canonical system

• Can generate multiple periods

• Smooth periodic trajectories

Disadvantages

• Limited to periodic movements

• No temporal scaling of period

• Fixed basis function placement

DMPs for high-dimensional spaces like joint or Cartesian coordinates

Complexity:

• Time: O(T × N_basis × d)
• Space: O(N_basis × d)

Advantages

• Handles high-dimensional spaces

• Dimension-specific scaling

• Independent learning per dimension

• Suitable for joint and Cartesian spaces

Disadvantages

• No cross-dimensional coupling

• Computational cost scales with dimensions

• May not capture coordinated movements

Complete Implementation

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

• Main implementation with discrete, rhythmic, and multi-dimensional DMPs: src/algokit/dynamic_movement_primitives/basic_dmps.py

• Comprehensive test suite including learning and generation tests: tests/unit/dynamic_movement_primitives/test_basic_dmps.py

Complexity Analysis

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Time & Space Complexity Comparison

Approach	Time Complexity	Space Complexity	Notes
Discrete DMP Learning	O(T × N_basis × d)	O(N_basis × d)	Time complexity for learning scales with trajectory length, basis functions, and dimensions

Use Cases & Applications

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Application Categories

Robotic Manipulation

• Pick and Place: Learning manipulation trajectories

• Assembly Tasks: Learning complex assembly movements

• Tool Use: Learning to use tools with proper trajectories

• Grasping: Learning grasping movements with approach trajectories

Humanoid Robotics

• Walking: Learning walking patterns and gaits

• Reaching: Learning arm reaching movements

• Balancing: Learning balance recovery movements

• Gesture: Learning human-like gestures and expressions

Industrial Robotics

• Welding: Learning welding trajectories

• Painting: Learning painting patterns

• Packaging: Learning packaging movements

• Quality Control: Learning inspection movements

Service Robotics

• Cleaning: Learning cleaning patterns

• Cooking: Learning cooking movements

• Caregiving: Learning assistive movements

• Entertainment: Learning dance and performance movements

Research Applications

• Movement Analysis: Studying human movement patterns

• Rehabilitation: Learning therapeutic movements

• Sports: Learning athletic movements

• Art: Learning artistic and creative movements

Educational Value

• Dynamical Systems: Understanding how movements can be encoded as dynamical systems

• Function Approximation: Learning to approximate complex functions with basis functions

• Temporal Scaling: Understanding how to scale movements in time

• Spatial Scaling: Understanding how to adapt movements to new spatial contexts

References & Further Reading

:material-library: Core Papers

:material-book:

Movement imitation with nonlinear dynamical systems in humanoid robots

2002 • IEEE International Conference on Robotics and Automation • Original DMP paper

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Dynamical movement primitives: Learning attractor landscapes for motor skills

2013 • Biological Cybernetics • Comprehensive DMP review and extensions

:material-library: DMP Extensions

:material-book:

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

2009 • Autonomous Robots • Rhythmic DMPs and online learning

:material-book:

Learning and generalization of motor skills by learning from demonstration

2009 • IEEE International Conference on Robotics and Automation • Multi-dimensional DMPs and generalization

:material-web: Online Resources

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Dynamic Movement Primitives

Wikipedia article on DMPs

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DMP Tutorial

ResearchGate tutorial on DMPs

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DMP Implementation

Python implementation of DMPs

:material-code-tags: Implementation & Practice

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PyDMPs

Python library for DMPs

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DMPy

Another Python DMP implementation

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ROS DMP Package

ROS integration for DMPs

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.

• 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.

• 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.