Model Predictive Control (MPC)

The defining feature of MPC is its receding horizon approach. At each control timestep, the controller:

• Solves a finite-horizon optimization problem over a future prediction window.
• Applies only the first control input from the optimized sequence.
• Shifts the horizon forward by one timestep and repeats the process with new sensor feedback. This continuous re-planning provides inherent feedback and robustness to disturbances and model inaccuracies, as the system is constantly correcting its course based on the latest state.

MPC directly incorporates hard and soft constraints into the optimization problem itself, a capability lacking in classical controllers like PID. This allows for safe and optimal operation at the boundaries of performance. Common constraints include:

• State Constraints: e.g., joint position/velocity limits, safe temperature ranges.
• Input Constraints: e.g., actuator torque/speed limits, voltage bounds.
• Output Constraints: e.g., end-effector force limits, payload capacity. The optimizer finds the best control sequence that satisfies all these constraints, preventing unsafe commands from ever being sent to the physical system.

MPC naturally handles Multiple-Input, Multiple-Output (MIMO) systems where variables are tightly coupled. A single optimization coordinates all actuators to achieve the desired state across all outputs. Furthermore, Nonlinear MPC (NMPC) uses a nonlinear dynamic model in its predictions, enabling control of highly complex systems like agile drones, walking robots, or chemical processes where linear approximations fail. The trade-off is significantly higher computational cost for solving the nonlinear optimization problem in real-time.

Instead of using a fixed control law, MPC selects actions by solving a numerical optimization problem at every step. This problem is defined by:

• A dynamic model of the system to predict future states.
• A cost function (objective function) that quantifies performance (e.g., minimize error, energy, or time).
• The aforementioned system constraints. The solver finds the control sequence that minimizes the cost over the prediction horizon. This allows MPC to 'look ahead' and make globally better decisions, such as slowing down early to navigate a tight corner efficiently.

MPC seamlessly combines feedforward and feedback control. The feedforward component comes from the optimization's use of the planned future trajectory and the system model to anticipate required control actions. The feedback component is inherent in the receding horizon mechanism, where the initial state for each new optimization is the latest, measured (or estimated) state of the real system. This corrects for model errors and unmeasured disturbances, closing the loop and ensuring stability and accuracy.

The primary limitation of MPC is its computational intensity. Solving an optimization problem at control rates (often 100-1000 Hz) is demanding. This has led to specialized sub-fields:

• Explicit MPC: Pre-solves the optimization offline for all possible states, storing the solution in a lookup table for fast online evaluation.
• Fast MPC Solvers: Use specialized algorithms (e.g., ADMM, gradient-based methods) and code generation for specific problem structures.
• Approximate MPC: Employs a neural network to approximate the optimal MPC policy, trading some optimality for drastic speed increases. The feasibility of MPC is dictated by the sampling time versus the solver computation time.

Trajectory optimization is the mathematical process of computing a sequence of robot states (positions, velocities) and control inputs that minimizes a cost function (e.g., energy, time, jerk) while satisfying dynamic constraints and task goals. It is the core computational engine inside an MPC loop.

• Offline vs. Online: Can be solved once (offline) for a known task or repeatedly (online) as in MPC.
• Shooting vs. Collocation: Direct shooting methods optimize over control inputs, while direct collocation also parameterizes the state trajectory, often improving stability.
• Applications: Used for robot locomotion, manipulator motion planning, and autonomous vehicle path planning.

An inverse kinematics (IK) solver is an algorithm that computes the set of joint angles required for a robotic manipulator to achieve a desired position and orientation (pose) of its end-effector. In an MPC pipeline for manipulation, the optimized end-effector trajectory must be converted into joint commands via IK.

• Analytical vs. Numerical: Analytical IK provides closed-form solutions for simple arms, while numerical IK (e.g., using the Jacobian) iteratively solves for complex chains.
• Role in MPC: Often used to transform the high-level Cartesian-space predictions from the MPC's model into feasible low-level joint references for the robot's actuators.

A Dynamic Movement Primitive is a mathematical framework for representing and generating smooth, goal-directed robot trajectories. It encodes a baseline motion as a nonlinear dynamical system, which can be easily adapted to new start/goal positions or modulated in real-time.

• Core Components: A canonical system provides timing, and a transformation system shapes the trajectory using learned weights.
• Relation to MPC: DMPs provide a compact, robust representation of motion primitives that can serve as a warm start or a reference trajectory for a more detailed MPC optimization, reducing computational load.
• Use Case: Frequently used in learning from demonstration (LfD) to encode and generalize skilled human motions.

In robotics, the Jacobian matrix is a fundamental mathematical construct that relates the joint velocities of a manipulator to the linear and angular velocity (the twist) of its end-effector in Cartesian space. It is the first-order derivative of the forward kinematics.

• Mathematical Definition: ( \mathbf{v} = \mathbf{J}(\mathbf{q}) \dot{\mathbf{q}} ), where ( \mathbf{v} ) is the end-effector twist, ( \mathbf{J} ) is the Jacobian, and ( \dot{\mathbf{q}} ) are joint velocities.
• Critical for Control: Essential for inverse kinematics solvers, force transformation (from Cartesian forces to joint torques), and analyzing singular configurations where the robot loses mobility.
• MPC Integration: The Jacobian is often part of the linearized dynamic model used within the MPC's prediction horizon for fast optimization.

Contact-implicit trajectory optimization is an advanced planning method that optimizes robot motions without pre-specifying the sequence or timing of contacts with the environment. The solver discovers when, where, and how contacts should occur to achieve a dynamic task.

• Key Innovation: Uses complementarity constraints or penalty methods to model the discontinuous nature of contact (making/breaking, friction).
• Contrast with MPC: While MPC typically uses a pre-defined contact model, contact-implicit optimization solves for the contact model as part of the plan. This is crucial for highly dynamic tasks like legged locomotion across uneven terrain or non-prehensile manipulation (pushing, sliding).
• Computational Challenge: Significantly more complex than trajectory optimization with fixed contacts, often requiring specialized solvers.

In the context of robot control, a policy network is a neural network (often a deep neural network) that maps observations from the environment directly to low-level control actions. It represents a learned control policy, typically trained via reinforcement learning or imitation learning.

• Comparison with MPC: A policy network is a reactive, feedforward controller: fast (forward pass only) but may lack explicit long-horizon reasoning. MPC is a predictive, optimization-based controller: slower (solves an optimization each step) but explicitly reasons about future consequences.
• Hybrid Approaches: Modern systems often combine them, using a policy network to provide a warm-start or initial guess to the MPC solver, or using MPC to generate expert data for training a policy network via imitation learning.