Geometry-Aware DMPs
Geometry-aware Dynamic Movement Primitives
DMPs that operate with symmetric positive definite matrices to handle stiffness and damping matrices for impedance control applications.
Family: Dynamic Movement Primitives Status: 📋 Planned
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Overview
Geometry-aware Dynamic Movement Primitives extend the basic DMP framework to operate with symmetric positive definite (SPD) matrices, enabling the handling of stiffness and damping matrices that are fundamental to impedance control applications. This approach is particularly valuable in robotics where the robot's interaction with the environment requires careful control of forces and torques.
The key innovation of geometry-aware DMPs is the use of Riemannian geometry on the manifold of SPD matrices, which allows for: - Natural interpolation between stiffness and damping matrices - Proper handling of matrix-valued parameters - Impedance control with learned stiffness and damping profiles - Smooth transitions between different impedance characteristics - Preservation of matrix properties during learning and execution
These DMPs are particularly valuable in applications requiring precise force control, such as assembly tasks, manipulation in uncertain environments, and human-robot interaction where the robot must adapt its impedance to the task requirements.
Mathematical Formulation¶
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Problem Definition
Given:
- Stiffness matrices: K(t) ∈ SPD(n) for t ∈ [0, T]
- Damping matrices: D(t) ∈ SPD(n) for t ∈ [0, T]
- SPD manifold: M = {A ∈ ℝ^(n×n) : A = A^T, A ≻ 0}
- Riemannian metric: g_A(X, Y) = tr(A^(-1) X A^(-1) Y)
Learn DMPs that operate on the SPD manifold: K(t) = K_0 * exp(Σ_{i=1}^K w_i^K ψ_i(t)) D(t) = D_0 * exp(Σ_{i=1}^K w_i^D ψ_i(t))
Where: - K_0, D_0 are initial SPD matrices - w_i^K, w_i^D are learned weights - ψ_i(t) are basis functions - exp is the matrix exponential
Key Properties
SPD Manifold Operations
K(t) = K_0 * exp(Σ_{i=1}^K w_i^K ψ_i(t))
Operations preserve the SPD property of matrices
Riemannian Interpolation
K(t) = K_0^(1-t) * K_1^t
Natural interpolation between SPD matrices
Impedance Control
τ = K(t)(q_d - q) + D(t)(q̇_d - q̇)
Generated matrices can be used directly for impedance control
Key Properties¶
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SPD Matrix Handling
Properly handles symmetric positive definite matrices
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Riemannian Geometry
Uses Riemannian geometry for natural matrix operations
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Impedance Control
Generates stiffness and damping matrices for impedance control
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Matrix Interpolation
Natural interpolation between SPD matrices
Implementation Approaches¶
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DMPs that operate on the manifold of SPD matrices
Complexity:
- Time: O(T × K × n^3)
- Space: O(K × n^2)
Advantages
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Proper handling of SPD matrices
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Natural interpolation between matrices
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Direct application to impedance control
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Preserves matrix properties
Disadvantages
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High computational cost for matrix operations
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Complex learning process
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Requires matrix exponential computations
DMPs using log-Euclidean geometry for SPD matrices
Complexity:
- Time: O(T × K × n^3)
- Space: O(K × n^2)
Advantages
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Simpler geometry than full Riemannian
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Easier to implement and understand
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Still preserves SPD property
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Lower computational cost
Disadvantages
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Less geometrically principled
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May not capture all matrix relationships
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Still requires matrix exponential
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 SPD and log-Euclidean DMPs:
src/algokit/dynamic_movement_primitives/geometry_aware_dmps.py -
Comprehensive test suite including matrix property tests:
tests/unit/dynamic_movement_primitives/test_geometry_aware_dmps.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| SPD Matrix DMP Learning | O(T × K × n^3) | O(K × n^2) | Learning time scales with trajectory length, basis functions, and matrix size |
Use Cases & Applications¶
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Application Categories
Impedance Control
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Assembly Tasks: Learning appropriate stiffness and damping for assembly
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Force Control: Learning force control strategies with variable impedance
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Compliance Control: Learning compliant behaviors for safe interaction
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Hybrid Control: Learning position/force control with impedance modulation
Human-Robot Interaction
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Physical Interaction: Learning appropriate impedance for human interaction
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Collaborative Tasks: Learning impedance for collaborative manipulation
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Assistive Robotics: Learning impedance for assistive tasks
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Social Robotics: Learning impedance for social interaction
Manipulation
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Grasping: Learning impedance for grasping different objects
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Tool Use: Learning impedance for tool manipulation
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Assembly: Learning impedance for assembly tasks
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Packaging: Learning impedance for packaging operations
Locomotion
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Walking: Learning impedance for walking on different terrains
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Running: Learning impedance for running with different speeds
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Jumping: Learning impedance for jumping and landing
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Balancing: Learning impedance for balance control
Medical Robotics
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Surgery: Learning impedance for surgical procedures
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Rehabilitation: Learning impedance for therapeutic exercises
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Prosthetics: Learning impedance for prosthetic control
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Diagnostic: Learning impedance for diagnostic procedures
Educational Value
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Riemannian Geometry: Understanding geometry on manifolds of matrices
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SPD Matrices: Understanding symmetric positive definite matrices
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Impedance Control: Understanding impedance control in robotics
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Matrix Operations: Understanding matrix exponential and logarithm
References & Further Reading¶
:material-library: Core Papers
:material-library: DMP Extensions
: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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Constrained Dynamic Movement Primitives (CDMPs) - DMPs with safety constraints and operational requirements that ensure movements comply with safety limits and operational constraints.
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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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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.