Quadratic Program (QP) Formulation

The objective function that the QP minimizes. It is mathematically expressed as (1/2) x^T Q x + c^T x, where x is the vector of optimization variables (e.g., joint accelerations, contact forces).

• Purpose: Encodes the primary control task, such as tracking a desired center-of-mass trajectory or minimizing joint torques.
• Matrix Q: A positive semi-definite matrix that weights the importance of different terms in the cost.
• Real Example: For a humanoid robot, the primary cost might be to minimize the error between the current and desired centroidal momentum, heavily weighted by Q.

Hard mathematical rules that the solution must satisfy exactly. Represented as A_eq x = b_eq.

• Core Use: Enforce the fundamental physics of the system, primarily the floating-base dynamics equations (F = ma). This ensures any solution is dynamically consistent.
• Other Uses: Enforcing contact constraints (e.g., a foot in contact must have zero velocity) or specific task constraints (e.g., keeping the torso upright).
• Critical Role: These constraints are non-negotiable; a QP solver will fail if it cannot find a solution that satisfies them.

Boundaries that define the feasible region of solutions, expressed as A_ineq x ≤ b_ineq.

• Actuator Limits: Enforce maximum joint torques, velocities, and accelerations based on motor specifications.
• Friction Cones: Ensure contact forces lie within a pyramid defined by the coefficient of friction, preventing foot slippage. This is a linear approximation of a non-linear constraint.
• Stability Margins: Constrain the Center of Pressure (CoP) to remain within the support polygon for balance.
• Self-Collision Avoidance: Often formulated as linear constraints on joint positions to keep links apart.

The vector of unknowns the QP solves for at each control time step (typically every 1-10 ms).

• Typical Variables:
  • Joint accelerations (or torques via inverse dynamics).
  • Contact forces at each foot.
  • Centroidal spatial forces (net force and moment on the robot).
• Dimensionality: For a complex humanoid with 30 joints and 2 feet in contact, x may have 50+ dimensions. The QP efficiently finds the optimal combination of these variables to satisfy all tasks and constraints.

A method to manage competing objectives when all cannot be fully satisfied. It is implemented via a sequence of QPs or a single weighted QP.

• Strict Priority (Sequential QP): Solve a QP for the highest-priority task (e.g., dynamic consistency), then solve a subsequent QP for the next task (e.g., foot tracking) in the null space of the first.
• Weighted Priority (Single QP): Assign different weights in the Q matrix of the cost function. Higher-priority tasks get larger weights, but all tasks are solved simultaneously, potentially with trade-offs.
• Example: Balance is always highest priority, followed by swing foot trajectory, then torso orientation.

Whole-Body Control uses QP to coordinate all robot joints to execute multiple, often competing, tasks simultaneously. The QP's quadratic cost function prioritizes tasks (e.g., foot tracking is more important than torso orientation), while linear equality and inequality constraints enforce physical laws like dynamics and contact stability. This allows a humanoid to walk while moving its arms, all while respecting actuator limits.

Standard inverse dynamics computes torques for a desired motion but can violate physical limits. A QP formulation solves this by finding the torques that best achieve the desired joint accelerations while strictly enforcing:

• Torque limits of each actuator.
• Friction cone constraints to prevent foot slipping (|f_tangential| ≤ μ * f_normal).
• Unilateral force constraints (feet can only push, not pull, on the ground). This yields physically feasible torques for stable locomotion on real hardware.

Controlling the robot's aggregate linear and angular momentum (centroidal momentum) is key for balance. A QP solves for the optimal ground reaction forces (GRFs) at the feet to regulate this momentum. The cost minimizes deviation from a desired momentum rate, while constraints ensure:

• The net GRF equals the required centroidal wrench.
• Each foot's force lies within its friction pyramid.
• The Center of Pressure (CoP) remains inside the support polygon. This is the core of many dynamic balancing algorithms.

For manipulation with legs or compliant interaction, QPs distribute forces across multiple contacts. Given a desired net wrench to apply to an object or the environment, the QP solves for the individual contact forces at each foot or hand. Constraints ensure:

• The sum of forces equals the desired net force/moment.
• Each contact force satisfies friction and unilateral constraints.
• Forces stay within the actuator force-capacity limits. This is essential for robots performing cooperative lifting or pushing against surfaces.

When tasks have strict priorities (e.g., balance over arm motion), a single QP with weighted costs is insufficient. Hierarchical QP (or lexicographic optimization) solves a sequence of QPs. The first QP optimizes the highest-priority task (e.g., maintaining ZMP stability). The second QP optimizes the next task (e.g., swing foot trajectory) subject to the constraint that the solution does not degrade the first task's optimal value. This guarantees critical constraints are never violated for less important objectives.

For legged locomotion, QPs must be solved at control rates (500-1000 Hz). This requires specialized real-time QP solvers like OSQP, qpOASES, or HPIPM. These solvers exploit the problem's structure:

• The Hessian matrix (from the quadratic cost) is often constant, allowing for pre-computation.
• Warm-starting uses the previous solution to accelerate convergence.
• Active-set or interior-point methods are tailored for the dense, small-to-medium-sized QPs typical in locomotion. Integration with the robot's real-time operating system is critical for deterministic loop timing.

A hierarchical control framework for legged robots that coordinates all degrees of freedom to execute multiple tasks simultaneously. QP formulations are the standard numerical method for solving WBC problems in real-time.

• Primary Tasks: Maintaining balance (via Center of Pressure), tracking desired foot trajectories.
• Secondary Tasks: Posture control, minimizing joint torques, gaze direction.
• QP Role: The framework casts task priorities and physical constraints (like torque limits and friction cones) into a single Quadratic Program, which is solved at each control cycle (typically 1 kHz).

The dynamics describing 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. It is a critical abstraction for whole-body motion planning.

• Key Equation: The rate of change of the centroidal angular momentum equals the net external moment about the CoM.
• QP Integration: In locomotion QPs, centroidal dynamics provides the equations of motion that relate desired CoM accelerations and angular momentum rates to the required Ground Reaction Forces (GRFs) at the feet.

An advanced control method that uses an internal dynamic model to predict future system behavior over a finite time horizon and solves an optimization problem at each time step to determine optimal control inputs.

• Connection to QP: For linear or linearized dynamics with quadratic cost functions (common in legged locomotion), the MPC problem reduces to solving a Sequential Quadratic Program (SQP) or a series of QPs.
• Application: Used for high-level planning of CoM trajectories and footstep placements, while a faster WBC QP handles instantaneous torque control.

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.

• Traditional vs. QP-based: Classical inverse dynamics computes a unique solution for a fully actuated system. For underactuated or constrained systems (like floating-base robots), it becomes an optimization problem.
• QP Formulation: The desired motion is expressed as a cost function (e.g., track desired joint accelerations), while physical constraints (torque limits, contact forces) form linear inequalities. The solver finds the optimal torques that satisfy all constraints.

The Ground Reaction Force (GRF) is the force vector exerted by the ground on a robot's foot. The friction cone is a linear constraint that ensures the foot does not slip.

• Friction Cone Constraint: Enforces that the tangential components of the GRF are bounded by the normal component: sqrt(fx² + fy²) ≤ μ * fz, where μ is the coefficient of friction. This is a second-order cone constraint.
• QP Linearization: For real-time QPs, the nonlinear friction cone is often approximated as a polyhedral cone (a pyramid) using multiple linear inequalities, turning it into a set of linear constraints compatible with standard QP solvers.

A simplified dynamic representation, such as the Linear Inverted Pendulum Model (LIPM), that captures the essential dynamics for locomotion planning while ignoring the full robot's complexity.

• Purpose: Provides a tractable model for high-level MPC. The LIPM, for example, assumes constant CoM height and linearized dynamics.
• QP Bridge: The optimal CoM trajectory planned using the ROM becomes a reference for the whole-body QP. The full-body QP then computes the joint-level commands that best achieve the ROM's planned motion while satisfying all the physical constraints the ROM ignored.