Model Predictive Control (MPC)

This is the defining mechanism of MPC. At each control time step, the algorithm solves a finite-horizon optimal control problem. It uses an internal dynamic model to predict future system states over a prediction horizon (N steps). It computes a sequence of optimal control inputs, but only the first control action from this sequence is applied to the system. At the next time step, the horizon "recedes" forward, and the optimization is repeated with new state measurements, providing continuous feedback and disturbance rejection.

• Core Mechanism: Plan, execute first step, re-plan.
• Benefit: Inherently accounts for new information and model errors.

MPC formulations directly incorporate hard and soft constraints into the online optimization problem. This is a major advantage over methods that handle constraints via ad-hoc methods or saturation. Common constraints in legged robotics include:

• Actuator Limits: Joint torque, velocity, and position bounds.
• Kinematic Limits: Self-collision avoidance, joint range of motion.
• Dynamic Stability: Keeping the Center of Pressure (CoP) within the support polygon.
• Friction Cone Constraints: Ensuring ground reaction forces remain within limits to prevent slipping.

By optimizing within these boundaries, MPC generates control actions that are feasible and safe for the physical hardware.

MPC naturally handles Multiple-Input, Multiple-Output (MIMO) systems where control variables are highly coupled. In a legged robot, moving one leg affects the forces on all others and the robot's overall balance. A single MPC formulation can simultaneously optimize:

• Center of Mass (CoM) trajectory.
• Swing foot trajectories.
• Ground reaction forces at each foot.
• Body orientation (pitch, roll).

The optimizer finds the coordinated set of control inputs (e.g., joint torques) that best achieves all these objectives while respecting constraints, avoiding the need for decoupled, single-loop controllers that can conflict.

MPC generates a feedforward control signal based on the predicted future evolution of the system from its internal model. This proactive action is crucial for managing the inherent delays and dynamics of a physical robot. When a disturbance (e.g., a push or uneven terrain) causes the measured state to deviate from the prediction, the receding horizon mechanism automatically corrects for it in the next optimization cycle.

• Feedforward: Anticipates required forces for planned motions.
• Feedback: Corrects for model inaccuracies and external pushes. This combination provides robust performance even with imperfect models.

The desired robot behavior is encoded in the MPC's cost function (or objective function). The optimizer minimizes this cost over the prediction horizon. Engineers specify high-level tasks by designing appropriate costs. Examples for locomotion include:

• Tracking Cost: Penalize deviation from a desired CoM or foot trajectory.
• Effort Cost: Minimize joint torques or power consumption to improve efficiency.
• Smoothness Cost: Penalize jerk in motions for stable, fluid movement.
• Terminal Cost: Encourage the robot to reach a specific state (e.g., balanced stance) by the end of the horizon. This flexibility allows a single MPC framework to be re-tasked for walking, running, or manipulation by changing the cost function.

In practical robotic systems, MPC often operates as a mid-level controller within a hierarchy. A higher-level motion planner or gait generator (e.g., using Linear Inverted Pendulum Model (LIPM)) might provide reference footstep locations and CoM trajectories over a longer time span. The MPC layer then translates these higher-level plans into dynamically feasible, constraint-satisfying whole-body motions and torques over its shorter horizon. This decoupling allows for complex long-horizon task planning while MPC handles the intricate, short-timescale dynamics and stabilization.

Whole-Body Control (WBC) is a hierarchical control framework that coordinates all of a robot's degrees of freedom to execute multiple tasks simultaneously, such as maintaining balance, tracking foot trajectories, and avoiding joint limits. While MPC often plans the high-level motion of the center of mass or feet, WBC translates these plans into precise joint torque commands for the entire robot in real-time. It typically solves a Quadratic Program (QP) at a high frequency (e.g., 1 kHz) to distribute forces and compute optimal torques that respect physical constraints like friction cones and torque limits.

A Quadratic Program (QP) is a mathematical optimization problem with a quadratic cost function and linear constraints. It is the computational workhorse for both MPC and Whole-Body Control. In legged robot MPC:

• The cost function typically penalizes deviations from a desired trajectory (e.g., for center of mass) and control effort.
• Linear constraints encode the robot's linearized dynamics (the 'model' in MPC), friction cone conditions for stable foot contact, and actuator limits.
• Solving this QP at each control step (e.g., every 20-50 ms) yields the optimal sequence of control inputs (e.g., ground reaction forces). Efficient, real-time QP solvers are critical for deployment.

Centroidal dynamics describes the relationship between the net external wrenches (forces and moments) acting on a robot and the motion of its center of mass (CoM) and its centroidal angular momentum. For legged locomotion MPC, this model is often used because it dramatically simplifies the full floating base dynamics. The planner assumes the robot's total mass is concentrated at its CoM and controls it by optimizing the Ground Reaction Forces (GRFs) at the feet. This abstraction allows for fast, real-time optimization while capturing the essential dynamics for balance and movement.

A Reduced-Order Model (ROM) is a simplified dynamic representation used to make complex planning and control problems tractable. MPC for legged robots almost always relies on a ROM rather than the full nonlinear dynamics. Key examples include:

• Linear Inverted Pendulum Model (LIPM): Assumes constant CoM height, leading to linear dynamics.
• Spring-Loaded Inverted Pendulum (SLIP): Models the leg as a spring, useful for running. These models capture the core locomotion dynamics (CoM motion, balance) while ignoring complex joint-level details, enabling the fast optimization required for real-time MPC. The gap between the ROM and the full robot is then bridged by a whole-body controller.

Contact-implicit planning is an advanced trajectory optimization method that does not pre-specify contact sequences (e.g., which foot touches down and when). Instead, it allows the optimizer to discover the optimal contact modes (stick, slip, break) as part of the solution. This contrasts with many traditional MPC implementations that use a pre-defined gait schedule. Contact-implicit planning uses complementarity constraints or smooth approximations to model the discontinuous nature of contact, making it highly versatile for complex terrains where contact timing is uncertain. It represents the frontier of MPC for highly dynamic and adaptive locomotion.

The Linear Inverted Pendulum Model (LIPM) is the most prevalent Reduced-Order Model for bipedal walking MPC. It makes two key assumptions:

1. The robot's total mass is concentrated at a single point (the Center of Mass).
2. The CoM height remains constant. These assumptions linearize the dynamics, transforming the nonlinear equations of motion into a simple, linear system. This linearity is what makes real-time MPC feasible, as the optimization problem becomes a Quadratic Program. The LIPM, when combined with the Zero-Moment Point (ZMP) stability criterion, forms the basis for many widely deployed walking controllers in humanoid robotics.