Model Predictive Control (MPC)

This is the fundamental execution loop of MPC. At each control step, the controller:

• Solves a finite-horizon optimal control problem using the current system state.
• Applies only the first control action from the optimized sequence.
• Shifts the prediction horizon forward by one step and repeats the process. This rolling window approach provides continuous feedback correction, making the system robust to model inaccuracies and disturbances. For a fleet robot, this means recalculating its optimal path every fraction of a second based on its latest sensor data.

MPC directly incorporates hard and soft constraints into the optimization problem itself. This is a primary advantage over methods like PID control. Common constraints in fleet orchestration include:

• Kinematic/Dynamic Limits: Maximum velocity, acceleration, and turning radius.
• Safety Constraints: Minimum distance to obstacles and other agents.
• Operational Limits: Battery charge levels, payload capacity, and zone access rules. The optimizer finds the best possible control sequence that satisfies all these constraints simultaneously, ensuring plans are not just optimal but also executable and safe.

MPC's performance is fundamentally tied to the quality of its internal dynamic model. This model predicts how the system state (e.g., a robot's position, velocity, battery level) will evolve over the horizon in response to control inputs. Models can range from simple linear approximations to complex non-linear or hybrid systems. For a heterogeneous fleet, the MPC platform must maintain distinct models for different agent types (e.g., differential-drive AMR vs. omnidirectional robot vs. forklift) to accurately predict their behavior.

The optimizer minimizes a cost function that encodes competing goals. This allows the system to make nuanced trade-offs in real-time. A typical cost function for a mobile robot includes terms for:

• Trajectory Tracking: Deviation from a desired path or goal.
• Control Effort: Minimizing energy use or actuator wear.
• Comfort/Smoothness: Penalizing jerky movements.
• Progress: Encouraging forward motion toward the goal. By weighting these terms, engineers can tune the fleet's behavior, prioritizing speed, efficiency, or smoothness as needed.

Because MPC optimizes over a future horizon, it computes feedforward control actions—commands that anticipate future states and disturbances. This is combined with feedback from the current state. For example, an MPC-controlled robot approaching a sharp turn will begin decelerating before the turn is reached, as its internal model predicts the need to slow down to stay within kinematic constraints. This anticipatory behavior leads to smoother, more efficient, and safer trajectories compared to purely reactive controllers.

MPC naturally accounts for actuator delays and sensor latency by incorporating them into the prediction model. The controller 'knows' that a commanded velocity change will not take effect instantly. It optimizes the control sequence to issue commands early enough so that the desired future state is achieved at the correct time. This is critical for coordinating large, heterogeneous fleets where communication delays and mechanical response times vary significantly between agent types.

Receding Horizon Control is the implementation strategy of Model Predictive Control. At each control step, the MPC optimization is solved over a finite future window (the horizon). Only the first control action from the optimized sequence is executed. The horizon is then shifted forward in time, and the optimization is repeated with new state measurements. This creates a closed-loop, feedback-driven control policy that continuously adapts to the real world.

• Core Mechanism: Solve, execute first step, shift, repeat.
• Key Benefit: Provides robustness to model inaccuracies and disturbances by frequently re-evaluating based on fresh sensor data.
• Industry Use: The standard implementation for MPC in robotics and autonomous vehicle trajectory tracking.

Optimal Reciprocal Collision Avoidance is a decentralized, velocity-based algorithm for reactive collision avoidance among multiple agents. Each agent independently computes a set of permissible velocities that guarantee collision-free motion for a short time horizon, assuming other agents perform a similar reciprocal computation. It is often used as a safety layer alongside a higher-level planner like MPC.

• Decentralized: Each agent plans for itself, reducing central computation.
• Reciprocal Assumption: Agents share responsibility for avoiding collisions.
• Integration with MPC: ORCA can provide hard constraints to the MPC optimization, ensuring the resulting trajectory is collision-avoidant by design.

Kinodynamic planning is the problem of finding trajectories that satisfy both kinematic constraints (e.g., non-holonomic wheeled bases that cannot move sideways) and dynamic constraints (e.g., limits on acceleration, torque, or force). Unlike geometric planners, it plans directly in the state space of positions, velocities, and sometimes accelerations.

• MPC's Role: MPC is a premier method for solving kinodynamic planning problems in a receding horizon fashion.
• Constraints Handled: MPC's optimization framework naturally incorporates differential equations of motion (dynamics) and actuator limits.
• Example: Planning a smooth, executable turn for an autonomous forklift that accounts for its mass, wheel friction, and lift height.

The Timed Elastic Band is a local trajectory optimization method closely related to MPC. It starts with an initial, geometrically feasible path and optimizes the pose and time sequence of a robot along it. The "elastic band" is deformed using gradient-based optimization to minimize objectives like path length, proximity to obstacles, and acceleration, while adhering to kinematic constraints.

• Local vs. Global: TEB is typically used for local obstacle avoidance and path smoothing, whereas MPC can handle broader planning.
• Similarity to MPC: Both optimize over a future sequence of states. TEB can be viewed as a specific instance of a nonlinear MPC formulation.
• Common Use Case: Local navigation and obstacle avoidance for mobile robots in dense, dynamic environments.

A Control Barrier Function is a mathematical tool used to synthesize controllers that formally guarantee safety. A CBF defines a safe set of states (e.g., all positions outside obstacles). The controller is designed to ensure the CBF's value remains non-negative, which keeps the system within the safe set. CBFs are increasingly integrated with MPC to provide safety certifications.

• Safety Filter: A CBF can act as a runtime safety filter on top of an MPC-generated control input, minimally modifying it to ensure constraint satisfaction.
• Constraint Encoding: CBFs can encode complex safety constraints (e.g., maintaining safe distance) as simpler conditions in the MPC optimization.
• Formal Guarantees: Provides rigorous proofs of collision avoidance, critical for high-stakes deployments.

In optimization-based control like MPC, a warm start is a critical technique for achieving real-time performance. It involves initializing the numerical solver with the solution from the previous MPC iteration (slightly shifted in time). Since the robot's state and environment change incrementally, the previous solution is often very close to the new optimum.

• Performance Impact: A good warm start can reduce solver iterations by 50-90%, making high-frequency MPC (e.g., 100Hz) computationally feasible.
• Essential for Real-Time: Without warm starting, the solver starts "cold" from zero, drastically increasing solve time and jeopardizing control loop deadlines.
• Industry Standard: Universal best practice for deploying MPC in latency-sensitive robotics applications.