Capture Point

The Capture Point is a point on the ground where a legged robot can place its supporting foot to bring its Center of Mass (CoM) to a complete stop in a single step, assuming the Linear Inverted Pendulum Model (LIPM). It is calculated from the robot's current CoM position and velocity. The formula is: ξ = x + (ẋ / ω₀), where x is the CoM position, ẋ is the CoM velocity, and ω₀ = √(g / z₀) is the natural frequency of the pendulum (g is gravity, z₀ is constant CoM height). This point represents the unstable manifold of the LIPM dynamics; stepping directly onto it 'captures' the divergent motion.

The Capture Point is intrinsically linked to the Divergent Component of Motion (DCM). In the 3D LIPM, the DCM is a 3D point that captures the unstable part of the dynamics. The projection of the DCM onto the ground plane is the Capture Point. This means the DCM extends the concept into the vertical direction, but for foot placement planning on flat ground, the 2D Capture Point is the critical target. Controlling the DCM's trajectory is equivalent to controlling the future location of the Capture Point.

The Capture Point and Zero-Moment Point (ZMP) are complementary but distinct stability concepts.

• ZMP: A measure of dynamic balance. It is the point where the net moment of inertial/gravitational forces is zero. The ZMP must remain inside the support polygon for the robot not to rotate and fall.
• Capture Point: A prescriptive target for future action. It answers "Where should I step next to regain balance?" A robot can have its ZMP within the support polygon (momentarily balanced) but still have a Capture Point far outside it, indicating it is diverging and will fall unless a step is taken.

Capture Point theory is the foundation for reactive locomotion and push recovery controllers. When a robot is pushed, its CoM velocity changes instantly, causing the Capture Point to jump. The control law is simple: step to the instantaneous Capture Point. This generates a reflex-like balancing behavior. Advanced implementations use the Capture Point dynamics to plan multiple steps ahead, calculating not just the next foot placement but a trajectory of future Capture Points to achieve a desired stopping location or velocity.

For real robots with finite foot size and torque limits, stepping exactly to a point is not always feasible. This leads to the concept of the Capture Region (or viable region). It is the area on the ground where, if a foot is placed, the robot can eventually come to a stop without taking another step. Its size depends on:

• Maximum leg swing speed
• Foot size and friction
• Joint torque limits
• Terrain geometry Controllers check if the computed Capture Point lies within the dynamically feasible Capture Region. If not, they command the best possible step inside the region.

The Capture Point's predictive power relies on the assumptions of the Linear Inverted Pendulum Model:

• Constant Center of Mass height
• Point mass representation
• No angular momentum about the CoM
• Massless legs In practice, robots have Centroidal Angular Momentum, variable CoM height, and swing leg dynamics. These factors introduce error. Therefore, Capture Point is often used as a high-level guide for a Model Predictive Control (MPC) or Whole-Body Control (WBC) layer that accounts for full-body dynamics and actuator constraints to execute the step.

This is the most direct application. When a legged robot is subjected to an unexpected external push, its center of mass (CoM) velocity changes instantly. The controller can compute the new instantaneous Capture Point in real-time and command the swing leg to step to that location. By placing the foot at the Capture Point, the robot can absorb the disturbance and come to a balanced stop in a single step, preventing a fall. This is far more efficient than complex whole-body adjustments.

For sustained walking or running, planners use the Capture Point as a forward-looking stability constraint. Instead of stepping to the current Capture Point to stop, they plan a sequence of future foot placements that keep the Divergent Component of Motion (DCM)—the forward-propagated Capture Point—within manageable bounds. This ensures that at any moment during the planned motion, there exists a viable emergency stopping step. It transforms the Capture Point from a reactive tool into a proactive stability governor for cyclic gaits.

On uneven or constrained terrain, not every geometrically feasible footstep is dynamically viable. The Capture Point framework allows a robot to evaluate candidate footstep locations. The controller projects the robot's state forward to the expected time of foot contact, calculates the Capture Point at that future time, and then checks if the candidate step is close enough to that point to ensure stability. This enables the robot to choose steps that are both safe from obstacles and dynamically sound, essential for traversing rubble, stairs, or slopes.

High-performance locomotion controllers often embed Capture Point dynamics within a Model Predictive Control (MPC) optimization. The MPC's internal model is frequently a Linear Inverted Pendulum Model (LIPM), which directly provides the Capture Point equation. The optimizer solves for optimal foot placements and Center of Mass (CoM) accelerations over a future horizon, with constraints that the DCM/Capture Point trajectory remains controllable. This merges the long-horizon foresight of MPC with the stability guarantees of the Capture Point.

A critical maneuver for any mobile robot is a controlled, stable stop from a walking or running gait. Using the Capture Point, a planner can generate an optimal stopping sequence: it computes the final Capture Point where the robot wants to stand (often directly under the CoM), then works backward to plan the last one or two steps that will guide the DCM to that point. This results in a smooth, decelerating transition from dynamic motion to a static standing pose, without overshoot or balance loss.

While derived from the simplified LIPM, the Capture Point concept informs control in full-dimensional robots. In Whole-Body Control (WBC) hierarchies, a high-level task might be to drive the robot's actual DCM (computed from its full centroidal dynamics) to a desired Capture Point. The lower-level WBC optimizer then solves for joint torques that achieve this while satisfying contact constraints and other tasks. This allows the intuitive high-level reasoning of the Capture Point to be executed on complex, nonlinear hardware like humanoid robots.

The Linear Inverted Pendulum Model (LIPM) is the foundational simplified dynamic model upon which the Capture Point is derived. It represents a walking robot as a point mass (the Center of Mass) atop a massless telescoping leg, with two key assumptions:

• Constant height of the Center of Mass.
• Motion confined to a horizontal plane. This linearization transforms the inherently non-linear dynamics into a set of decoupled, linear equations, allowing for closed-form solutions for the robot's motion and the explicit calculation of the Capture Point. It is the workhorse model for real-time balance and footstep planning in bipedal robotics.

The Divergent Component of Motion (DCM) is a state variable intrinsically linked to the Capture Point. For the 3D Linear Inverted Pendulum Model, the DCM (ξ) is defined as: ξ = CoM + (CoM_velocity / ω), where ω is the pendulum's natural frequency.

• Key Relationship: When projected onto the ground plane, the DCM is equivalent to the Capture Point. It represents the unstable part of the Center of Mass dynamics.
• Use in Control: Controllers often regulate the DCM's trajectory to ensure stability. Placing a footstep at the instantaneous Capture Point drives the DCM to that location, bringing the system to a stop.

The Zero-Moment Point (ZMP) is a stability criterion distinct from the Capture Point's stepping target. It is defined as the point on the ground where the net moment of the inertial and gravitational forces has no horizontal component.

• Static vs. Dynamic: The ZMP must remain within the support polygon for the robot to not rotate about its edge and fall. The Capture Point can be far outside the support polygon, indicating the need for a step.
• Practical Difference: ZMP is used for balance maintenance within a given stance (e.g., in preview control). The Capture Point is used for step recovery and gait timing, answering where and when to step next.

The Center of Pressure (CoP) is the physical instantiation of the Zero-Moment Point. It is the point on the sole of the foot where the resultant Ground Reaction Force (GRF) vector acts.

• In Ideal Rigid Contact: On a flat, rigid surface with sufficient friction, the measured CoP and the calculated ZMP are equivalent.
• Critical Distinction from Capture Point: The CoP/ZMP is a measure of current force distribution. The Capture Point is a predicted future stepping target based on the current state. A controller manipulates the CoP (e.g., by applying ankle torques) to influence the Capture Point's evolution.

A Reduced-Order Model (ROM) is a simplified representation of a complex system that captures only the dynamics essential for a specific task. The Linear Inverted Pendulum Model is the quintessential ROM for walking.

• Purpose: Full-body robot dynamics are high-dimensional and computationally expensive. ROMs like the LIPM enable real-time prediction and control.
• Capture Point's Role: The Capture Point is a powerful output of this model reduction. It provides a simple, low-dimensional target (a 2D point) that can be tracked by the high-dimensional whole-body controller to achieve stable locomotion, effectively bridging the gap between simple planning and complex execution.

Model Predictive Control (MPC) is an advanced control methodology that leverages the predictive power of models like the LIPM to optimize future actions. It is the primary framework for implementing Capture Point-based control in practice.

• How it Works: At each control cycle, the MPC solver uses the current robot state and the LIPM to predict future Capture Point evolution over a horizon. It then solves an optimization to find the optimal sequence of future footstep locations (Capture Points) that minimizes a cost (e.g., deviation from a path, energy) while satisfying constraints (e.g., step length, kinematic limits).
• Outcome: Only the first step of the optimized plan is executed, and the process repeats, enabling robust, reactive locomotion over uneven terrain.