Constrained DMPs
Constrained Dynamic Movement Primitives (CDMPs)
DMPs with safety constraints and operational requirements that ensure movements comply with safety limits and operational constraints.
Family: Dynamic Movement Primitives Status: 📋 Planned
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Overview
Constrained Dynamic Movement Primitives (CDMPs) extend the basic DMP framework to incorporate safety constraints and operational requirements. These constraints ensure that generated movements comply with safety limits, operational boundaries, and other critical requirements that are essential for safe robot operation.
The key innovation of CDMPs is the integration of constraint handling mechanisms that can: - Enforce joint limits and workspace boundaries - Ensure collision-free movements - Maintain operational safety requirements - Prevent dangerous movements that could harm humans or equipment - Adapt movements to comply with changing constraints
CDMPs are particularly valuable in safety-critical applications where movements must be both effective and safe, such as human-robot collaboration, medical robotics, and industrial automation.
Mathematical Formulation¶
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Problem Definition
Given:
- Basic DMP: τẏ = α_y(β_y(g - y) - ẏ) + f(x)
- Safety constraints: g_safety(y, ẏ, ÿ) ≤ 0
- Operational constraints: g_ops(y, ẏ, ÿ) ≤ 0
- Joint limits: y_min ≤ y ≤ y_max
- Velocity limits: ẏ_min ≤ ẏ ≤ ẏ_max
- Acceleration limits: ÿ_min ≤ ÿ ≤ ÿ_max
Find constrained DMP that satisfies: min ||f(x) - f_original(x)||² subject to: - g_safety(y, ẏ, ÿ) ≤ 0 - g_ops(y, ẏ, ÿ) ≤ 0 - y_min ≤ y ≤ y_max - ẏ_min ≤ ẏ ≤ ẏ_max - ÿ_min ≤ ÿ ≤ ÿ_max
Key Properties
Constraint Satisfaction
g_safety(y, ẏ, ÿ) ≤ 0, g_ops(y, ẏ, ÿ) ≤ 0
All safety and operational constraints must be satisfied
Minimal Deviation
min ||f(x) - f_original(x)||²
Minimize deviation from original DMP while satisfying constraints
Real-time Adaptation
f_constrained(x) = f_original(x) + Δf(x)
Constraints can be adapted in real-time based on changing conditions
Key Properties¶
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Safety Guarantees
Ensures movements comply with safety requirements
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Constraint Handling
Handles multiple types of constraints simultaneously
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Real-time Adaptation
Can adapt constraints based on changing conditions
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Minimal Deviation
Minimizes deviation from original movement while satisfying constraints
Implementation Approaches¶
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CDMPs using penalty methods for constraint handling
Complexity:
- Time: O(T × K × C)
- Space: O(K + C)
Advantages
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Guarantees constraint satisfaction
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Flexible constraint handling
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Real-time constraint adaptation
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Minimal deviation from original movement
Disadvantages
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Computational overhead for constraint optimization
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May not find feasible solutions
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Requires careful constraint formulation
CDMPs using barrier methods for constraint handling
Complexity:
- Time: O(T × K × C × log(1/ε))
- Space: O(K + C)
Advantages
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Strong constraint satisfaction guarantees
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Smooth convergence to feasible region
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Theoretically well-founded
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Handles inequality constraints naturally
Disadvantages
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Higher computational cost
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Sensitive to barrier weight tuning
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May converge slowly
Complete Implementation
The full implementation with error handling, comprehensive testing, and additional variants is available in the source code:
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Main implementation with penalty and barrier methods:
src/algokit/dynamic_movement_primitives/constrained_dmps.py -
Comprehensive test suite including constraint satisfaction tests:
tests/unit/dynamic_movement_primitives/test_constrained_dmps.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Penalty Method CDMP | O(T × K × C) | O(K + C) | Time complexity scales with trajectory length, basis functions, and constraints |
Use Cases & Applications¶
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Application Categories
Human-Robot Collaboration
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Safe Handover: Ensuring safe object handover between human and robot
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Collaborative Assembly: Safe assembly tasks with human partners
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Assistive Robotics: Safe assistance for elderly or disabled users
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Social Robotics: Safe interaction in social environments
Medical Robotics
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Surgical Robotics: Ensuring safe surgical movements within patient anatomy
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Rehabilitation: Safe therapeutic movements for patient recovery
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Prosthetics: Safe prosthetic control with user safety limits
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Diagnostic: Safe diagnostic movements in medical imaging
Industrial Automation
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Manufacturing: Safe manufacturing operations with equipment protection
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Quality Control: Safe inspection movements in production lines
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Packaging: Safe packaging operations with product protection
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Maintenance: Safe maintenance operations in industrial environments
Autonomous Vehicles
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Path Planning: Safe path planning with obstacle avoidance
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Traffic: Safe traffic navigation with vehicle constraints
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Parking: Safe parking maneuvers with space constraints
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Emergency: Safe emergency maneuvers with vehicle limits
Space and Aerospace
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Satellite Operations: Safe satellite maneuvers with orbital constraints
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Spacecraft: Safe spacecraft operations with fuel and power limits
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Aircraft: Safe aircraft operations with flight envelope constraints
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Rockets: Safe rocket operations with structural and fuel limits
Educational Value
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Constraint Optimization: Understanding how to handle constraints in optimization
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Safety in Robotics: Learning to ensure safety in robotic systems
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Penalty Methods: Understanding penalty-based constraint handling
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Barrier Methods: Understanding barrier-based constraint handling
References & Further Reading¶
:material-library: Core Papers
:material-library: Constraint Handling
:material-web: Online Resources
:material-code-tags: Implementation & Practice
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:
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DMPs with Obstacle Avoidance - DMPs enhanced with real-time obstacle avoidance capabilities using repulsive forces and safe navigation in cluttered environments.
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Spatially Coupled Bimanual DMPs - DMPs for coordinated dual-arm movements with spatial coupling between arms for synchronized manipulation tasks and hand-eye coordination.
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DMPs for Human-Robot Interaction - DMPs specialized for human-robot interaction including imitation learning, collaborative tasks, and social robot behaviors.
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Multi-task DMP Learning - DMPs that learn from multiple demonstrations across different tasks, enabling task generalization and cross-task knowledge transfer.
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Geometry-aware Dynamic Movement Primitives - DMPs that operate with symmetric positive definite matrices to handle stiffness and damping matrices for impedance control applications.
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Online DMP Adaptation - DMPs with real-time parameter updates, continuous learning from feedback, and adaptive behavior modification during execution.
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Temporal Dynamic Movement Primitives - DMPs that generate time-based movements with rhythmic pattern learning, beat and tempo adaptation for temporal movement generation.
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DMPs for Manipulation - DMPs specialized for robotic manipulation tasks including grasping movements, assembly tasks, and tool use behaviors.
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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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Probabilistic Movement Primitives (ProMPs) - Probabilistic extension of DMPs that captures movement variability and generates movement distributions from multiple demonstrations.
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Hierarchical Dynamic Movement Primitives - DMPs organized in hierarchical structures for multi-level movement decomposition, complex behavior composition, and task hierarchy learning.
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DMPs for Locomotion - DMPs specialized for walking pattern generation, gait adaptation, and terrain-aware movement in legged robots and humanoid systems.
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Reinforcement Learning DMPs - DMPs enhanced with reinforcement learning for parameter optimization, reward-driven learning, and policy gradient methods for movement refinement.