The spring-mass model for running and hopping

1989/01/01 by R. Blickhan, Reinhard Blickhan · 1,669 citations
Engineering · Mathematics · Medicine · #Classical mechanics #Contact force #Displacement (psychology) #Energy (signal processing) #Engineering #Geometry #Kinematics #Lower Extremity Biomechanics and Pathologies #Mathematics #Mechanics #Optics #Parameter space #Particle displacement #Physics #Range (aeronautics) #Sports Dynamics and Biomechanics #Sports Performance and Training #Spring (device)

paper · doi:10.1016/0021-9290(89)90224-8

published in Journal of Biomechanics 22(11-12), 1217-1227 (Elsevier BV)

openalex publication_date 1989/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05

Cited by

• Muscle contributions to propulsion and support during running
• Leg stiffness measures depend on computational method
• Leg stiffness and joint stiffness while running to and jumping over an obstacle
• Non-extension movements inducing over half the mechanical energy directly contributing to jumping height in human running single-leg jump
• Flexible mechanisms: the diverse roles of biological springs in vertebrate movement
• Running stability is enhanced by a proximo-distal gradient in joint neuromechanical control
• Unsteady locomotion: integrating muscle function with whole body dynamics and neuromuscular control
• The landing–take-off asymmetry in human running
• Spring-mass running: simple approximate solution and application to gait stability
• Optimal Control of a Differentially Flat 2D Spring-Loaded Inverted Pendulum Model
• Humanoid Robot Running Through Random Stepping Stones and Jumping Over Obstacles: Step Adaptation Using Spring-Mass Trajectories
• Variable-Impedance Muscle Coordination under Slow-Rate Control Frequencies and Limited Observation Conditions Evaluated through Legged Locomotion
• Self-Scalable Tanh (Stan): Multi-Scale Solutions for Physics-Informed Neural Networks
• Stable and Robust SLIP Model Control via Energy Conservation-Based Feedback Cancellation for Quadrupedal Applications
• Individualization of Footwear for Optimizing Running Economy: A Theoretical Framework
• Frequency and leg stiffness adaptation in human vertical hopping before, during and after added load
• A Simple Method for Measuring Stiffness during Running
• Optimal Reduced-order Modeling of Bipedal Locomotion
• Incorporating Human-Inspired Ankle Characteristics in a Forced-Oscillation-Based Reduced-Order Model for Walking
• Reduced-Order Model-Guided Reinforcement Learning for Demonstration-Free Humanoid Locomotion
• The Surprising Effectiveness of Linear Models for Whole-Body Model-Predictive Control
• Hierarchical Reduced-Order Model Predictive Control for Robust Locomotion on Humanoid Robots
• A Layered Control Perspective on Legged Locomotion: Embedding Reduced Order Models via Hybrid Zero Dynamics
• Different External Cues Elicit Specific Kinetic Strategies During a Drop Jump in Well-Trained Adolescent Soccer Players
• VisionLaw: Inferring Interpretable Intrinsic Dynamics from Visual Observations via Bilevel Optimization
• Efficient Anytime CLF Reactive Planning System for a Bipedal Robot on Undulating Terrain
• Symbolic Learning of Interpretable Reduced-Order Models for Jumping Quadruped Robots
• A Factor-Graph Approach for Optimization Problems with Dynamics Constraints
• Zero Dynamics, Pendulum Models, and Angular Momentum in Feedback Control of Bipedal Locomotion
• Angular Momentum about the Contact Point for Control of Bipedal Locomotion: Validation in a LIP-based Controller
• Replication Study of the Effect of Different Loading Conditions on Running Mechanics at Different Velocities
• Biologically inspired jumping robots: A comprehensive review
• Dynamic Bipedal MPC with Foot-level Obstacle Avoidance and Adjustable Step Timing
• A Novel Method to Categorize Stretch-Shortening Cycle Performance Across Maturity in Youth Soccer Players
• Impaired Stretch-Shortening Cycle Function Persists Despite Improvements in Reactive Strength After Anterior Cruciate Ligament Reconstruction
• Towards dynamic trot gait locomotion: Design, control, and experiments with Cheetah-cub, a compliant quadruped robot
• Learning plastic matching of robot dynamics in closed-loop central pattern generators
• Oncilla robot: a versatile open-source quadruped research robot with compliant pantograph legs
• Cognitive Developmental Robotics: A Survey
• Trunk orientation causes asymmetries in leg function in small bird terrestrial locomotion
• Running Springs: Speed and Animal Size
• How Animals Move: An Integrative View
• Robustness: a new SLIP model based criterion for gait transitions in bipedal locomotion
• A review on locomotion robophysics: the study of movement at the intersection of robotics, soft matter and dynamical systems
• Versatile, Robust, and Explosive Locomotion with Rigid and Articulated Compliant Quadrupeds
• Compliant leg behaviour explains basic dynamics of walking and running. [europepmc]
• Running stability is enhanced by a proximo-distal gradient in joint neuromechanical control. [europepmc]
• Unsteady locomotion: integrating muscle function with whole body dynamics and neuromuscular control. [europepmc]
• Dimensional reduction in sensorimotor systems: a framework for understanding muscle coordination of posture. [europepmc]
• Old men running: mechanical work and elastic bounce. [europepmc]
• Motions of the running horse and cheetah revisited: fundamental mechanics of the transverse and rotary gallop. [europepmc]
• Rate-dependent control strategies stabilize limb forces during human locomotion. [europepmc]
• Fifteen observations on the structure of energy-minimizing gaits in many simple biped models. [europepmc]
• Muscle contributions to propulsion and support during running. [europepmc]
• A collisional perspective on quadrupedal gait dynamics. [europepmc]
• The anatomical arrangement of muscle and tendon enhances limb versatility and locomotor performance. [europepmc]
• Leg stiffness of sprinters using running-specific prostheses. [europepmc]
• Review and perspective: neuromechanical considerations for predicting muscle activation patterns for movement. [europepmc]
• Trunk orientation causes asymmetries in leg function in small bird terrestrial locomotion. [europepmc]
• Optimizing Locomotion Controllers Using Biologically-Based Actuators and Objectives. [europepmc]
• Contribution of elastic tissues to the mechanics and energetics of muscle function during movement. [europepmc]
• Simulating Ideal Assistive Devices to Reduce the Metabolic Cost of Running. [europepmc]
• Fundamentals of soft robot locomotion. [europepmc]
• Running in highly cushioned shoes increases leg stiffness and amplifies impact loading. [europepmc]
• Neuromusculoskeletal model that walks and runs across a speed range with a few motor control parameter changes based on the muscle synergy hypothesis. [europepmc]
• Validity of the Stryd Power Meter in Measuring Running Parameters at Submaximal Speeds. [europepmc]
• Humans Optimize Ground Contact Time and Leg Stiffness to Minimize the Metabolic Cost of Running. [europepmc]
• Application of Leg, Vertical, and Joint Stiffness in Running Performance: A Literature Overview. [europepmc]
• Predicting continuous ground reaction forces from accelerometers during uphill and downhill running: a recurrent neural network solution. [europepmc]
• The biomechanics of skipping gaits: a third locomotion paradigm? [europepmc]