Floating Base Dynamics

The system's configuration is described by floating base coordinates, typically the 6-DOF pose (position and orientation) of a base link (e.g., the pelvis or torso) relative to a world frame, plus the joint angles. The key property is that the six base coordinates are unactuated—no direct motor torque acts on them. Their motion is governed entirely through dynamic coupling with the actuated limbs via the equations of motion. This necessitates control strategies that use limb motion to indirectly control the base.

The dynamics are derived using Lagrangian or Newton-Euler methods, resulting in a matrix equation:

M(q) * v̇ + C(q, v) * v + G(q) = Sᵀ * τ + J_c(q)ᵀ * f

• M(q): The system's mass matrix, which is symmetric, positive-definite, and configuration-dependent.
• C(q, v): The Coriolis and centrifugal forces matrix.
• G(q): The gravitational forces vector.
• S: A selection matrix that maps actuated joint torques (τ) into the generalized force space.
• J_c(q): The contact Jacobian mapping external contact forces (f) (e.g., Ground Reaction Forces) into generalized forces.

This structure explicitly separates actuated inputs (τ) from external contact forces (f).

Intermittent contact with the environment introduces kinematic constraints. When a foot is firmly planted (sticking contact), its velocity is constrained to zero: J_c(q) * v = 0. Differentiating this yields the acceleration-level constraint: J_c(q) * v̇ + J̇_c(q) * v = 0. These constraints are holonomic for position and non-holonomic for velocity. The contact forces f become Lagrange multipliers that enforce these constraints. The dynamics are therefore hybrid, switching between different constraint regimes as feet make and break contact.

A critical property is the decomposition between the floating base and the centroidal dynamics. The net external wrench (force and moment) on the robot equals the rate of change of its centroidal momentum (linear momentum of the center of mass and angular momentum about the CoM). This is described by the Newton-Euler equations:

Σ f_ext = m * a_com Σ (r_i × f_i + μ_i) = L̇_com

where L_com is the centroidal angular momentum. These equations provide a high-level, force-based description that is directly coupled to the detailed floating-base model via the Centroidal Momentum Matrix.

Because the base is unactuated, the system is underactuated when not in full static contact. The dynamic consistency property states that any feasible motion of the center of mass and centroidal angular momentum must be achievable through some combination of joint accelerations and contact forces. This is formalized by projecting the full dynamics into the null-space of the contact constraints and the range-space of the actuation. Control algorithms like Whole-Body Control exploit this by prioritizing tasks (e.g., CoM motion) that satisfy the underactuated dynamics.

For simulation and model-based control (e.g., MPC), the equations must be solved efficiently. Featherstone's Articulated Body Algorithm (ABA) computes forward dynamics (v̇ given τ and f) in O(n) time. Its inverse, the Recursive Newton-Euler Algorithm (RNEA), computes inverse dynamics (τ given v̇ and f). For floating bases, these algorithms are initialized with a fictitious 6-DOF joint connecting the base to the world. The Composite Rigid Body Algorithm (CRBA) is used to compute the mass matrix M(q). These algorithms are foundational for real-time control.

Centroidal dynamics describes the relationship between the net external forces and moments acting on a robot and the motion of its center of mass and its centroidal angular momentum. It is a crucial simplification for whole-body control, as it aggregates the complex multi-body dynamics into the motion of a single rigid body (the centroidal frame) located at the CoM. This allows planners to reason about balance and angular momentum without solving the full floating base equations at every iteration.

• Key Application: Used in Whole-Body Control (WBC) to generate dynamically consistent motions.
• Core Equation: Relates the time derivative of the centroidal angular momentum to the net external moment about the CoM.

Whole-Body Control (WBC) is a hierarchical control framework that coordinates all degrees of freedom of a legged robot to execute multiple tasks simultaneously—like maintaining balance, tracking foot trajectories, and avoiding joint limits—while strictly respecting physical constraints like contact forces and torque limits. It directly utilizes the floating base dynamics model to solve for joint torques and contact forces in real-time, often formulated as a Quadratic Program (QP).

• Primary Input: A set of prioritized tasks (e.g., CoM position is higher priority than arm posture).
• Core Mechanism: Solves an optimization problem that includes the full rigid-body dynamics equations, treating the floating base explicitly.

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. The most common ROMs, like the Linear Inverted Pendulum (LIPM) and Spring-Loaded Inverted Pendulum (SLIP), are derived from or are compatible with the floating base framework. They treat the robot as a point mass or a simple mechanical system, enabling fast planning for footstep placement and center of mass trajectory generation.

• Examples: LIPM for walking, SLIP for running/hopping.
• Purpose: Enables real-time model predictive control by drastically reducing computational complexity.

A Quadratic Program (QP) formulation is the standard mathematical optimization structure used to solve real-time control problems in legged robotics, such as inverse dynamics and Whole-Body Control. The floating base dynamics equations provide the linear equality constraints that ensure physical consistency. The QP minimizes a quadratic cost (e.g., tracking error, energy use) subject to these dynamics constraints and other linear constraints for contact forces, friction cones, and torque limits.

• Standard Form: Minimize a quadratic cost function subject to linear equality and inequality constraints.
• Solver Requirement: Must solve in milliseconds (e.g., 1-5 ms) for real-time control, using libraries like OSQP or qpOASES.

Inverse dynamics is the computation of the joint torques or forces required to produce a desired acceleration for a robot, given its kinematic structure, mass properties, and current state of motion. For a floating base system, this is an underdetermined problem because the base is unactuated. The solution requires solving for both the joint torques and the contact wrenches simultaneously, which is typically done using a QP formulation that incorporates the floating base dynamics as a constraint.

• Core Challenge: The six unactuated base degrees of freedom mean there is no unique solution without additional optimization criteria.
• Standard Approach: Solved as a QP that minimizes torque or contact force magnitudes while satisfying dynamics.

The Ground Reaction Force (GRF) is the force vector exerted by the ground on a robot's foot during contact. The Center of Pressure (CoP) is the point on the contact surface where the total GRF is considered to act. These are not inputs but critical outputs of the floating base dynamics. When solving for control, the GRFs at each foot must be computed to satisfy the system's momentum dynamics. The location of the aggregate CoP relative to the support polygon is a direct indicator of stability.

• Dynamic Role: GRFs are the primary external forces that appear on the right-hand side of the floating base dynamics equation.
• Stability Rule: For static stability, the CoP must remain within the support polygon. For dynamic stability, metrics like the Zero-Moment Point (ZMP) are used.