⚡ electrical engineering

Real-Time Nonlinear MPC via Sequential Quadratic Programming with Structure-Exploiting ADMM and Interior-Point Methods for Underactuated Double-Pendulum Swing-Up

This paper presents a real-time nonlinear model predictive control strategy based on sequential quadratic programming that successfully achieves robust swing-up and stabilization of an underactuated double-pendulum system on unknown hardware for the "AI Olympics with RealAIGym" competition.

Original authors: Nick Karydakis, Konstantinos Chatzilygeroudis

Published 2026-08-11

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CC BY 4.0

Original authors: Nick Karydakis, Konstantinos Chatzilygeroudis

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to teach a robot to do a backflip. You can't just tell it "jump" and hope for the best; you have to figure out exactly how much force to apply, when to tuck its knees, and how to land without crashing. This is the world of robotics control, a field dedicated to giving machines the "muscle memory" to move through the real world. At the heart of this challenge is a concept called Model Predictive Control (MPC). Think of MPC as a super-advanced GPS for a robot. Instead of just looking at where the robot is right now, the GPS constantly looks ahead, simulating thousands of possible future paths in a split second to decide the very best next move. It's like a chess player who doesn't just think one move ahead, but calculates the entire game to ensure a win.

Now, imagine the robot you are controlling is a double-pendulum—a stick with another stick attached to its end, swinging freely. This system is notoriously tricky because it is underactuated, meaning it has more moving parts than it has motors. It's like trying to balance a broom on your hand while your hand is tied to a string; you have to swing it wildly to build up energy before you can finally balance it upright. This specific puzzle is the centerpiece of the "AI Olympics with RealAIGym," a competition where teams must program a robot to swing itself up from a hanging position and stay balanced, all while the robot's exact weight and friction are a mystery to the programmer.

The paper by Karydakis and Chatzilygeroudis presents a clever solution to this problem using a method called Sequential Quadratic Programming (SQP). Instead of guessing and checking, their controller acts like a master strategist who breaks the complex, wobbly problem of swinging a double-pendulum into a series of smaller, manageable puzzles. At every tiny fraction of a second, the robot asks itself: "If I move my motor this way, where will I be in the next few steps?" It solves a simplified math problem to find the best path, takes the first step, and then immediately recalculates for the next moment. This happens so fast—over 100 times a second—that the robot feels like it is thinking in real-time.

The researchers tested this "brain" on two different versions of the double-pendulum: the Pendubot (where the motor is at the shoulder) and the Acrobot (where the motor is at the elbow). They didn't just test it in a computer simulation; they uploaded their code to a physical robot in the cloud and let it run for real. The results were impressive. For the Pendubot, the robot successfully swung up and stayed balanced 100% of the time, even when someone gave it a random, unexpected push to knock it off balance. For the Acrobot, it succeeded 100% of the time without pushes, and 70% of the time when disturbed.

What makes this approach stand out is how it handles the "unknowns." The robot didn't know the exact weight of its own parts or how much friction was in its joints. Yet, by constantly re-evaluating its plan and using a special mathematical trick to solve the equations quickly, it managed to build up the necessary energy to swing up and then stabilize itself. The paper suggests that this method is robust enough to handle real-world chaos, proving that a model-based approach can compete with, and in some cases outperform, other methods that rely purely on trial-and-error learning. While the Acrobot still needed a little help with friction compensation to be perfect, the overall success shows that with the right mathematical tools, we can teach even the wobbliest robots to stand tall.

Technical Summary: Real-Time Nonlinear MPC via SQP for Underactuated Double-Pendulum Swing-Up

Problem Formulation
This work addresses the challenge of global swing-up and stabilization for underactuated two-link systems, specifically the Pendubot (actuated at the shoulder) and the Acrobot (actuated at the elbow). The task is framed within the context of the "AI Olympics with RealAIGym" competition, where controllers must be developed and evaluated directly on remotely accessible "CloudPendulum" hardware. Key constraints include limited interaction time, the absence of prior knowledge regarding system model parameters, and the requirement to stabilize an inherently unstable upright equilibrium using a single actuator. The problem is formulated as a discrete-time, finite-horizon nonlinear optimal control problem that must account for nonlinear dynamics, underactuation constraints, and actuator limits.

Methodology
The authors propose a real-time, nonlinear Model Predictive Control (NMPC) framework based on Sequential Quadratic Programming (SQP). The core methodology involves the following components:

Inverse-Dynamics Formulation: The system dynamics are described using the manipulator equation. The optimization variable is defined as the generalized acceleration ( $\dot{v}$ ), while joint torques are computed via inverse dynamics using the Recursive Newton–Euler Algorithm. This formulation explicitly handles underactuation by imposing equality constraints that force the torque of the passive joint to vanish, while inequality constraints enforce actuator torque limits.
SQP and QP Subproblems: At each control step, the nonlinear optimal control problem is approximated locally around a nominal trajectory to generate a structured Quadratic Program (QP). The cost function penalizes joint velocities, control effort, and task-space tracking errors (tip position for Pendubot; elbow and tip positions for Acrobot).
Structure-Exploiting Solvers: To achieve real-time performance, the QP subproblems are solved using solvers that exploit the temporal sparsity of the optimal control problem. The authors utilize two specific approaches:
1. A stagewise Alternating Direction Method of Multipliers (ADMM) solver, which uses LQR-based recursion to solve linear systems without factorizing a large sparse KKT system.
2. The HPIPM (High Performance Interior Point Method) solver, an alternative structure-exploiting interior-point method.
Real-Time Implementation: The controller operates in a receding-horizon fashion. The solution from the previous control instant is shifted to warm-start the state-control trajectory and solver variables for the next step. To meet computational budgets, the system performs a limited number of SQP iterations (4 for ADMM, 6 for HPIPM) per sampling instant, with QP subproblems limited to 100 iterations.

Key Contributions

Unified Control Policy: The paper presents a single optimization-based policy capable of handling both the nonlocal energy-building motions required for swing-up and the precise feedback control needed for stabilization.
Hardware-in-the-Loop Validation: Unlike many theoretical studies, this approach is validated on physical, remotely accessible hardware (CloudPendulum) without prior system identification, demonstrating robustness against model uncertainty and external disturbances.
Solver Comparison and Selection: The work evaluates the performance of structure-exploiting ADMM versus HPIPM solvers in a real-time embedded context, selecting the optimal solver for each system configuration (ADMM for Pendubot, HPIPM for Acrobot) based on tuning results.

Experimental Results
Experiments were conducted on 60-second episodes starting from the downward hanging configuration. Two scenarios were tested: nominal operation and operation with applied torque disturbances (random torques applied for 0.1s).

Pendubot Performance: The controller achieved a 100% success rate in both scenarios. The median uptime (time spent in the upright stabilization region) was 92.5% (55.5s) without disturbances and 87.3% (52.38s) with disturbances. The controller operated at an average frequency of approximately 400 Hz.
Acrobot Performance: The controller achieved a 100% success rate without disturbances and a 70% success rate with disturbances. Median uptime was 85.8% (51.48s) without disturbances and 69.7% (41.82s) with disturbances. The controller operated at an average frequency of approximately 240 Hz.
Robustness: In disturbance scenarios, the controller demonstrated the ability to recover from substantial deviations and restore the system to the upright equilibrium.

Significance and Claims
The paper claims that the proposed structure-exploiting SQP-based MPC approach is capable of generating the complex, nonlocal motions necessary for swing-up while simultaneously providing robust feedback stabilization and disturbance rejection. The authors emphasize that the method successfully meets the real-time computational requirements of the CloudPendulum platform despite the lack of prior model knowledge.

The work modestly acknowledges current limitations, specifically the need for friction compensation to achieve reliable performance on the Acrobot system. Future work is proposed to investigate online system identification to adapt dynamic parameters in real-time, thereby reducing model mismatch and potentially improving reliability and recovery performance without manual friction compensation tuning.

Technical Summary: Real-Time Nonlinear MPC via SQP for Underactuated Double-Pendulum Swing-Up

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