DMPs Locomotion
DMPs for Locomotion
DMPs specialized for walking pattern generation, gait adaptation, and terrain-aware movement in legged robots and humanoid systems.
Family: Dynamic Movement Primitives Status: 📋 Planned
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Overview
DMPs for Locomotion extend the basic DMP framework to handle walking pattern generation, gait adaptation, and terrain-aware movement in legged robots and humanoid systems. This approach enables robots to learn and adapt their walking patterns to different terrains, speeds, and environmental conditions.
The key innovation of locomotion DMPs is the integration of: - Walking pattern generation with proper gait cycles - Gait adaptation mechanisms for different speeds and terrains - Terrain-aware movement with ground contact modeling - Balance and stability considerations - Rhythmic movement coordination for multiple limbs
These DMPs are particularly valuable in applications requiring legged locomotion, such as humanoid robots, quadruped robots, and any system that needs to walk or run on different terrains.
Mathematical Formulation¶
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Problem Definition
Given:
- Leg joint angles: θ(t) = [θ_1(t), θ_2(t), ..., θ_n(t)]
- Gait phase: φ(t) ∈ [0, 2π] where φ = 0 is heel strike
- Terrain height: h(x, y) at position (x, y)
- Desired speed: v_desired
- Balance constraints: C_balance(θ, θ̇, θ̈) ≤ 0
The locomotion DMP becomes: τθ̈ = α_y(β_y(θ_ref - θ) - θ̇) + f(φ) + f_terrain(θ, h) + f_balance(θ, θ̇, θ̈) τφ̇ = ω_gait
Where: - f(φ) is the rhythmic forcing function for gait - f_terrain(θ, h) adapts to terrain height - f_balance(θ, θ̇, θ̈) maintains balance - ω_gait is the gait frequency
Key Properties
Gait Cycle
φ(t) = ω_gait * t mod 2π
Gait phase cycles through walking phases
Terrain Adaptation
f_terrain(θ, h) = k_terrain * (h_desired - h_actual)
Adapts to terrain height variations
Balance Control
f_balance(θ, θ̇, θ̈) = k_balance * (COM_desired - COM_actual)
Maintains center of mass balance
Key Properties¶
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Walking Pattern Generation
Generates natural walking patterns with proper gait cycles
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Gait Adaptation
Adapts gait to different speeds and terrains
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Terrain Awareness
Adapts to terrain variations and obstacles
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Balance Maintenance
Maintains balance and stability during walking
Implementation Approaches¶
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DMPs for generating walking patterns with gait cycles
Complexity:
- Time: O(T × K × L)
- Space: O(K × L)
Advantages
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Natural walking pattern generation
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Gait cycle coordination
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Multi-leg coordination
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Rhythmic movement patterns
Disadvantages
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Requires gait parameter tuning
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May not handle all terrain types
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Complex multi-leg coordination
DMPs that adapt to different terrains and obstacles
Complexity:
- Time: O(T × K × L + T × T_terrain)
- Space: O(K × L + T_terrain)
Advantages
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Terrain adaptation
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Obstacle avoidance
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Ground contact modeling
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Robust locomotion
Disadvantages
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Requires terrain information
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Complex terrain modeling
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Higher computational cost
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 walking pattern and terrain-adaptive DMPs:
src/algokit/dynamic_movement_primitives/locomotion_dmps.py -
Comprehensive test suite including locomotion tests:
tests/unit/dynamic_movement_primitives/test_locomotion_dmps.py
Complexity Analysis¶
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Time & Space Complexity Comparison
| Approach | Time Complexity | Space Complexity | Notes |
|---|---|---|---|
| Walking Pattern DMP | O(T × K × L) | O(K × L) | Time complexity scales with trajectory length, basis functions, and number of legs |
Use Cases & Applications¶
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Application Categories
Humanoid Robotics
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Walking: Learning to walk on different terrains
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Running: Learning to run with different speeds
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Stairs: Learning to climb stairs
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Balance: Learning to maintain balance during walking
Quadruped Robotics
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Walking: Learning quadruped walking gaits
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Running: Learning quadruped running gaits
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Terrain: Learning to walk on rough terrain
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Obstacles: Learning to avoid obstacles
Hexapod Robotics
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Walking: Learning hexapod walking gaits
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Climbing: Learning to climb over obstacles
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Rough Terrain: Learning to walk on rough terrain
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Speed: Learning to walk at different speeds
Bipedal Robotics
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Walking: Learning bipedal walking
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Running: Learning bipedal running
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Balance: Learning to maintain balance
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Turning: Learning to turn while walking
Research Applications
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Gait Analysis: Studying human and animal gaits
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Rehabilitation: Developing rehabilitation techniques
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Sports: Analyzing sports movements
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Biomechanics: Studying biomechanical principles
Educational Value
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Locomotion: Understanding locomotion principles and gaits
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Gait Analysis: Understanding gait analysis and generation
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Terrain Adaptation: Understanding terrain adaptation mechanisms
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Multi-leg Coordination: Understanding coordination between multiple legs
References & Further Reading¶
:material-library: Core Papers
:material-library: Locomotion
: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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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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Reinforcement Learning DMPs - DMPs enhanced with reinforcement learning for parameter optimization, reward-driven learning, and policy gradient methods for movement refinement.