Infinite-Horizon Value Function Approximation for Model Predictive Control

Abstract: Model Predictive Control has emerged as a popular tool for robots to generate complex motions. However, the real-time requirement has limited the use of hard constraints and large preview horizons, which are necessary to ensure safety and stability. In practice, practitioners have to carefully design cost functions that can imitate an infinite horizon formulation, which is tedious and often results in local minima. In this work, we study how to approximate the infinite horizon value function of constrained optimal control problems with neural networks using value iteration and trajectory optimization. Furthermore, we demonstrate how using this value function approximation as a terminal cost provides global stability to the model predictive controller. The approach is validated on two toy problems and a real-world scenario with online obstacle avoidance on an industrial manipulator where the value function is conditioned to the goal and obstacle.

• The paper introduces a neural network-driven method to approximate the infinite-horizon value function within an MPC framework.
• It combines local gradient-based solvers with trajectory optimization to maintain global stability and avoid local minima.
• Experimental results in toy problems and industrial manipulators validate its effectiveness in achieving real-time, collision-free control.

Infinite-Horizon Value Function Approximation for Model Predictive Control

In the context of robotics and control systems, Infinite-Horizon Model Predictive Control (MPC) presents a theoretical framework offering enhanced stability over finite horizon approaches. The paper "Infinite-Horizon Value Function Approximation for Model Predictive Control" explores a strategy to approximate the infinite horizon value function using neural networks, addressing limitations associated with real-time implementation of MPC under constraints.

Overview of the Approach

The authors investigate a methodology that combines value iteration and trajectory optimization to approximate the infinite-horizon value function of constrained optimal control problems. By using neural networks to encapsulate these approximated value functions, the study underscores how embedding such approximations as terminal costs within an MPC framework can enhance the system's global stability.

Key Contributions

1. Local Gradient-Based Solver Usage: The paper details employing a local gradient-based solver in conjunction with value iteration to approximate the optimal value function pertinent to an infinite horizon constrained Optimal Control Problem (OCP).
2. Experimental Validation: Two primary experimental domains were explored: simple toy problems and a practical real-world example involving an industrial manipulator tasked with avoiding obstacles and reaching targets. These instances demonstrate the proposed method’s capability to circumvent local minima and uphold safety outside the training dataset distribution.
3. Collision-Free Real-Time Robotic Control: The study effectively demonstrates maintaining adherence to hard safety constraints during real-time execution, signifying the robustness of combining the use of neural networks to represent global value functions with local trajectory optimization.

Methodological Details

The authors develop a neural network architecture to approximate the infinite-horizon value function. This is achieved by iteratively refining neural network parameters through supervised learning, based on data generated by solving constrained OCPs, where the neural networks act as terminal cost functions.

Importantly, the value function is conditioned not only on the robot’s state but also on dynamic properties of the environment, such as positions of obstacles and goals. These considerations allow the trained model to exhibit adaptability across varying scenarios.

The technique of recurrently selecting optimal control inputs from a constructed horizon, while updating the expected value of the system’s future state, is central to achieving global stability in control outcomes. This setup effectively mitigates suboptimality arising from local minima in the optimization landscape.

Implications and Future Directions

The outcomes of this research highlight significant advancements in the field of MPC by bridging foundational theoretical concepts of infinite horizon control with practical computational techniques, offering a feasible alternative for on-the-fly control in complex environments.

Moving forward, this methodology's potential applications extend into areas involving complex robotic interactions such as multi-agent systems or contact-rich manipulation scenarios. Moreover, an important area for future exploration is enhancing neural network architectures' expressiveness and robustness, possibly investigating alternative formulations that can deal with partial observability and further reducing approximation errors.

The conducted research opens avenues for leveraging similar function approximation approaches in other control or planning frameworks where maintaining stability and handling constraints are critical for ensuring performance. As the landscape of AI and control systems continually evolves, methodologies that smartly balance computational intensity with theoretical rigor stand to deliver considerable utility in both classical and emerging domains.