Dynamic Stability

The Zero-Moment Point (ZMP) is a foundational dynamic stability criterion for legged locomotion. It is defined as the point on the ground where the net moment of the inertial forces (from acceleration) and gravitational forces has no horizontal component. If the ZMP remains within the support polygon (the convex hull of ground contact points), the robot is dynamically stable. This principle underpins the stability of many statically stable walking gaits for humanoid robots.

Reduced-Order Models are simplified dynamic representations that capture the essential physics of locomotion for control design. Key models include:

• Linear Inverted Pendulum Model (LIPM): Models walking by treating the robot as a point mass atop a massless leg, assuming constant center-of-mass height. It enables efficient calculation of the ZMP and Capture Point.
• Spring-Loaded Inverted Pendulum (SLIP): Models running and hopping by representing a leg as a massless spring. It captures the passive, dynamic energy exchange between kinetic and potential energy that characterizes agile, dynamic gaits.

Derived from the LIPM, these concepts are central to reactive balance control.

• Divergent Component of Motion (DCM): A state variable that captures the unstable part of the center of mass dynamics. It naturally diverges from the desired motion.
• Capture Point (CP): The point on the ground where the robot can place its foot so that the DCM converges to a stop, allowing the robot to come to a complete halt in one step. Controllers use the CP for push recovery and reactive locomotion, calculating optimal foot placements in real-time to arrest falling motion.

This framework coordinates the entire robot to achieve dynamic tasks.

• Centroidal Dynamics: Describes the relationship between the net external Ground Reaction Forces (GRFs) and the motion of the robot's center of mass and its centroidal angular momentum. It is the high-level dynamic model used for planning.
• Whole-Body Control (WBC): A hierarchical controller that uses Quadratic Program (QP) formulations to compute joint torques. It simultaneously executes multiple tasks (e.g., tracking desired centroidal dynamics, foot trajectories) while strictly respecting physical constraints like joint limits and friction cones.

Model Predictive Control (MPC) is a dominant advanced control method for dynamic legged robots. At each control cycle (e.g., 1 kHz), the MPC solver:

1. Uses an internal dynamic model (often centroidal) to predict future system states over a short time horizon.
2. Solves an optimization to find the optimal sequence of future Ground Reaction Forces and footstep locations.
3. Executes only the first step of the plan, then re-plans with new sensor data. This allows the robot to proactively adjust to disturbances and complex terrain, rather than just reacting.

Evaluating dynamic stability involves balancing performance metrics:

• Stability Margins: Quantitative measures like the distance of the ZMP from the edge of the support polygon, or the size of the viable Capture Point region.
• Cost of Transport (CoT): The primary efficiency metric, calculated as energy used per unit weight per unit distance traveled. Highly stable, conservative gaits often have higher CoT.
• Robustness: Measured by the maximum external push impulse or terrain height variation the controller can reject without falling. The goal of modern control is to achieve high robustness with a low CoT, mimicking the efficiency of biological systems.

The Zero-Moment Point (ZMP) is a foundational dynamic stability criterion for legged locomotion. It is defined as the point on the ground plane where the net moment of the inertial and gravitational forces acting on the robot has zero horizontal component. If the ZMP remains within the support polygon (the convex hull of the foot contacts), the robot is dynamically stable. This principle is central to the control of many modern humanoid robots, where controllers actively regulate foot placement and center of mass motion to keep the ZMP bounded.

The Capture Point is a predictive stability metric derived from the Linear Inverted Pendulum Model (LIPM). It defines the specific point on the ground where a robot must place its foot to bring its center of mass to a complete stop within a single step, given its current velocity. The capture point is calculated by combining the current CoM position with its velocity, scaled by the system's natural frequency. It is a crucial concept for push recovery and reactive locomotion, providing a direct target for stabilizing foot placement in response to disturbances.

The Linear Inverted Pendulum Model (LIPM) is a canonical simplified model for analyzing bipedal walking dynamics. It makes two key assumptions:

• The robot's total mass is concentrated at a single point (the Center of Mass).
• The CoM height remains constant during motion. These assumptions linearize the dynamics, making the equations of motion analytically tractable. The LIPM is the basis for calculating the Divergent Component of Motion (DCM) and Capture Point, providing the theoretical foundation for many real-time walking controllers used in humanoid robotics.

The Divergent Component of Motion (DCM) is a state variable that decomposes the Center of Mass dynamics from the LIPM into stable and unstable parts. It is mathematically defined as the sum of the CoM position and its velocity scaled by the system's natural frequency. The DCM dynamics are inherently unstable and must be controlled through appropriate foot placement. Controlling the DCM to a desired location (often within the support polygon) is equivalent to controlling the robot's future capturability and is a central method in model predictive control strategies for walking.

A Reduced-Order Model (ROM) is a simplified dynamic representation that captures the essential locomotion dynamics while ignoring the full complexity of the multi-body system. Examples include the LIPM for walking and the Spring-Loaded Inverted Pendulum (SLIP) for running. These models are vital for real-time planning and control because they are computationally efficient and provide intuitive insights. Controllers often use a ROM for high-level step planning and a full whole-body controller to execute the plan on the actual robot, respecting all joint constraints.

Push recovery encompasses the suite of strategies a legged robot employs to maintain balance after an unexpected external disturbance, such as a shove or an uneven terrain impact. These strategies operate at different time scales and complexities:

• Ankle Strategy: Using ground reaction forces to generate a correcting moment.
• Hip Strategy: Using angular momentum of the torso to counteract the push.
• Step Strategy: Taking one or more rapid steps to reposition the support polygon and capture point.
• Lunge Strategy: A large, dynamic motion involving multiple joints. Advanced systems hierarchically combine these strategies based on the magnitude of the disturbance.