Support Polygon

The support polygon is defined as the convex hull connecting all points where the robot's feet (or other contact surfaces) touch the ground. This means you draw the smallest convex shape that encloses every contact point. For a biped with two feet flat on the ground, the polygon is a line segment. For a quadruped in a static stance with four feet, it is typically a quadrilateral. Key implications:

• The robot's Center of Mass (CoM) vertical projection must lie within this polygon for static stability.
• The shape and size change discretely with each footstep or contact break/formation.

The primary function of the support polygon is to provide a static stability criterion. A robot is statically stable if the vertical projection of its Center of Mass (CoM) lies inside the polygon. This is a necessary condition for the robot to remain balanced without requiring corrective motion.

• Stability Margin: The shortest distance from the CoM projection to the polygon's boundary. A larger margin indicates greater robustness to disturbances.
• This criterion is foundational for quasi-static walking, where the robot moves slowly enough that dynamic forces are negligible.

The Center of Pressure (CoP) is the point on the ground where the total Ground Reaction Force (GRF) vector acts. For a robot to be in static equilibrium (not tipping), the CoP must also lie within the support polygon.

• In static cases on flat, rigid ground, the CoP coincides with the CoM projection.
• Under dynamic conditions or on compliant terrain, the CoP can move within the polygon. Controlling the CoP location is a core objective of balance controllers.
• The CoP cannot reach the polygon's edge unless a contact is about to break (initiating a tip-over).

The support polygon is not static during locomotion; it evolves dynamically with the gait sequence. As feet lift and place, the polygon's shape and area change abruptly.

• Gait Planning: Algorithms must ensure the CoM trajectory is such that its projection remains within the sequence of successive polygons.
• In dynamic walking/running (e.g., using the Zero-Moment Point criterion), the CoM projection can briefly leave the polygon, relying on momentum.
• For multi-legged robots, gaits like a trot have a small, moving polygon, while a crawl maintains a large, stable polygon.

For robots making contact on non-horizontal surfaces (e.g., climbing robots or on steep slopes), the 2D polygon concept extends to a 3D support volume or convex hull of contact points in 3D space.

• Stability is then evaluated by checking if the Center of Mass lies within the vertical projection (a prism) of this volume.
• The friction cone at each contact point becomes critical, as forces must be applied within these cones to prevent slipping, further constraining feasible CoM positions.
• This is essential for multi-contact planning in climbing and manipulation.

The support polygon is a purely geometric and static concept. It must be contrasted with dynamic stability metrics used for moving robots:

• Zero-Moment Point (ZMP): The dynamic extension of the CoP. For stability, the ZMP must remain within the support polygon. This is the core principle of many bipedal walking controllers.
• Capture Point: A point on the ground where the robot can step to come to a complete stop. It accounts for momentum, which the static polygon ignores.
• Divergent Component of Motion (DCM): Encodes the unstable part of the dynamics. Controlling the DCM relative to the support polygon enables dynamic balance.
• The static polygon provides the feasible region within which these dynamic points must be managed.

The Zero-Moment Point (ZMP) is a dynamic stability criterion. It is defined as the point on the ground plane where the net moment of the inertial and gravitational forces has no horizontal component. For a robot to be dynamically stable, the ZMP must lie within the support polygon. It is the primary metric used for stability control in many bipedal robots.

• Key Insight: A ZMP at the polygon's edge indicates marginal stability; outside indicates a fall.
• Application: The Honda ASIMO and Boston Dynamics' Atlas robots use ZMP-based controllers.

The Center of Pressure (CoP) is the point on the contact surface where the total Ground Reaction Force (GRF) vector is applied. It is a measurable quantity from force-torque sensors. For a single, flat foot contact, the CoP and ZMP are equivalent. The CoP's location relative to the support polygon's edges is monitored in real-time for balance control.

• Measurement: Directly obtained from foot-mounted six-axis force/torque sensors.
• Static vs. Dynamic: In purely static scenarios, the CoP must be inside the support polygon to prevent tipping.

Ground Reaction Force (GRF) is the force vector exerted by the ground on a robot's foot. It is the physical manifestation of Newton's third law and is fundamental to all locomotion. The GRF has:

• A normal component supporting the robot's weight.
• Frictional components that propel, brake, and prevent slipping.

The vector sum of all GRFs across contacting feet must equal the robot's net weight and inertial forces. Optimizing GRF distribution is a core task in whole-body control.

The Linear Inverted Pendulum Model (LIPM) is a simplified, reduced-order model for bipedal walking. It treats the robot as a point mass (the Center of Mass) atop a massless leg, with a constant height constraint. This linearization makes dynamics tractable for real-time planning.

• Core Assumption: Constant Center of Mass height.
• Utility: Generates analytically solvable trajectories for the Center of Mass, which are then used to plan stable foot placements that keep the ZMP within the evolving support polygon.

The Capture Point is a point on the ground where a legged robot can place its foot to come to a complete stop in one step. It is derived from the Linear Inverted Pendulum Model (LIPM) and the robot's current Center of Mass state (position and velocity).

• Function: A predictive stability metric. Stepping to the Capture Point is a guaranteed recovery strategy.
• Calculation: Capture Point = Center of Mass position + (Center of Mass velocity / √(g/h)), where g is gravity and h is constant CoM height.

Whole-Body Control (WBC) is a hierarchical control framework that coordinates all of a robot's joints to execute multiple tasks simultaneously. For legged robots, high-priority tasks always include maintaining balance by enforcing ZMP/CoP constraints within the support polygon.

• Method: Typically formulated as a Quadratic Program (QP) that solves for joint torques or accelerations.
• Task Examples: 1) Balance (strict), 2) Foot trajectory tracking, 3) Arm manipulation, 4) Posture regulation.