⚡ electrical engineering

Inertial-Based LQG Control: A New Look at Inverted Pendulum Stabilization

This paper proposes a novel inertial-based LQG control framework that leverages local differential flatness to incorporate accelerometer data and higher-order dynamics, thereby enabling robust attitude stabilization for mobile platforms in sensor-constrained outdoor environments where direct tilt measurement is unavailable.

Original authors: Daniel Engelsman, Itzik Klein

Published 2026-05-19

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CC BY 4.0

Original authors: Daniel Engelsman, Itzik Klein

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

The Big Picture: Balancing a Stick on Your Hand

Imagine you are trying to balance a long broomstick upright on the palm of your hand. This is the classic "inverted pendulum" problem. If the stick tilts even a little, gravity pulls it down, and you have to move your hand quickly to catch it before it falls.

For decades, engineers have used a sophisticated mathematical recipe called LQG Control to keep these systems (like Segways, hoverboards, or robot walkers) balanced. In a perfect lab setting, you have a camera and a ruler that tell you exactly where the stick is and how fast it's moving.

The Problem:
In the real world (outdoors), you often don't have a camera or a ruler. You only have inertial sensors (like the accelerometer and gyroscope in your smartphone). These sensors tell you how fast the system is accelerating and how fast it is tilting, but they don't tell you exactly where the system is on the map.

Because of this, the computer has to "guess" the position based on the acceleration. If the guess is wrong, the robot falls over. Previous methods tried to fix this, but they treated the acceleration data as a separate, messy piece of information that didn't quite fit into the main mathematical model.

The Solution: Adding "Jerk" to the Mix

The authors of this paper propose a new way to think about the problem. They realized that to make a better guess, the computer needs to understand not just where the robot is, but how the forces are changing moment-to-moment.

They used a mathematical concept called Differential Flatness. Think of this like a "master key" that unlocks the relationship between the robot's movement and the forces acting on it.

Instead of just tracking:

Position (Where is it?)
Velocity (How fast is it moving?)

They added higher-order steps to the model:
3. Acceleration (How fast is it speeding up?)
4. Jerk (How fast is the acceleration changing?)

The Analogy:
Imagine you are driving a car in the fog.

Old Method: You look at your speedometer (velocity). You guess where you are based on that. If you hit a bump, your guess gets wrong, and you drift off the road.
New Method (This Paper): You also feel the jerk—the sudden lurch when you hit a bump or the engine revs up. By feeling that "lurch" (jerk), your brain can predict exactly how your speed will change a split second later, even before you see the road. This keeps your mental map of where you are much more accurate.

What They Did

Built a Better Model: They created a new version of the "balancing stick" model (called A-IPoC) that includes these "jerk" states.
Integrated the Sensors: They figured out how to feed the raw acceleration data directly into this new model, rather than treating it as a separate, noisy signal.
Tested It: They ran thousands of computer simulations where they simulated "bad" conditions:
- Sensor Dropouts: Pretending the sensors stopped sending data for a few seconds (like losing GPS signal).
- Noise: Adding static and errors to the data.
- Tough Starts: Starting the robot in a very unstable, tilted position.

The Results: A Stronger Safety Net

The paper claims that their new method is significantly more robust than the old standard.

Wider Safety Zone: Imagine the "safety zone" is a circle on the floor where the robot can start and still stay upright. The new method made this circle 27% to 39% larger. This means the robot can start in a much more tilted or chaotic position and still recover.
Fewer Crashes: In their simulations, the new method crashed 10% to 15% less often than the old method when things got messy.
Better Recovery: When the sensors stopped sending data (the "fog" scenario), the new system could "guess" the position much more accurately for longer, preventing the robot from falling over.

Summary

The paper doesn't invent a new robot or a new type of battery. Instead, it invents a smarter brain for existing balancing robots. By teaching the robot's brain to pay attention to "jerk" (the rate of change of acceleration) and using a special mathematical trick called differential flatness, the robot can balance itself much better when it can't see where it is, making it safer and more reliable for real-world use.

Technical Summary: Inertial-Based LQG Control for Inverted Pendulum Stabilization

Problem Statement

The stabilization of an inverted pendulum on a cart (IPoC) is a fundamental benchmark in control theory, traditionally relying on direct measurements of configuration variables (cart position and pendulum angle) via displacement sensors and rotary encoders. However, in outdoor, dynamically stable mobile platforms (e.g., Segways, hoverboards, bipedal robots), such sensors are often unavailable or impractical. These systems rely heavily on inertial sensors (accelerometers and gyroscopes) for state estimation.

A critical gap exists in applying Linear Quadratic Gaussian (LQG) control to these sensor-limited scenarios. Standard LQG frameworks typically model system states up to velocity or acceleration but do not explicitly incorporate accelerometer data (jerk states) into the Newton-Euler formalism. Consequently, when direct configuration measurements are missing or intermittent, state estimation relies on sensor fusion that may lack the predictive power to handle high-order dynamics, leading to reduced robustness, increased drift during "dead reckoning" periods, and smaller stability regions.

Methodology

The authors propose an Inertial-aided LQG framework that integrates higher-order dynamics into the system model to improve state estimation and control robustness. The methodology proceeds as follows:

Differential Flatness Analysis: The authors demonstrate that the IPoC system is differentially flat. They define a flat output $\varepsilon = x + \ell \sin \theta$ (a combination of cart position and pendulum angle). Using this property, they derive explicit mappings where all system states and control inputs can be expressed as functions of the flat output and its derivatives up to the fifth order.
Augmented State-Space Model (A-IPoC): Leveraging the flatness property, the authors extend the standard state vector to include higher-order derivatives (accelerations and jerks). The new state vector is defined as $x = [x, \dot{x}, \ddot{x}, \theta, \dot{\theta}, \ddot{\theta}]^\top$ . This allows the system model to explicitly account for the dynamics of acceleration, which are directly measurable via inertial sensors.
Linearization and LQG Design: The nonlinear augmented model is linearized around the upright equilibrium point. The resulting Linear Time-Invariant (LTI) system is used to design a standard LQG controller, comprising a Linear Quadratic Regulator (LQR) for control and a Kalman Filter (KF) for state estimation. The KF utilizes the augmented model to predict states during periods of measurement dropout or delay.
Stability Analysis: Instead of relying solely on analytical stability guarantees (which assume continuous, noise-free feedback), the authors employ a Monte Carlo-based statistical sampling procedure (Algorithm 1). This method randomly samples initial conditions (linear and angular velocities) to empirically map the "basin of attraction" and determine the stability boundaries under various update ratios ( $\rho$ ) and noise conditions.

Key Contributions

The paper makes three primary contributions:

Inertial-aided LQG Framework: The authors extend the LQG framework with an acceleration-augmented process model. This allows for the direct incorporation of accelerometer data, improving disturbance rejection and state estimation accuracy in the absence of direct position sensors.
Higher-Order Dynamics Modeling: By exploiting differential flatness, the authors model linear and angular accelerations as explicit functions of control inputs. This refinement enhances system predictability, particularly during sensor-limited intervals.
Open-Source Implementation: To ensure reproducibility, the authors provide an interactive codebase implementation on GitHub.

Results and Analysis

The proposed A-IPoC model was compared against a conventional IPoC LQG implementation across various update ratios ( $\rho$ ), representing the frequency of measurement updates relative to prediction steps.

Transient Performance: Under conditions of measurement dropout ( $\rho < 1$ ), the A-IPoC model demonstrated significantly lower transient errors compared to the conventional model. Specifically, transient errors were reduced by 30%–80% in terms of Integral of Absolute Error (IAE) and Integral of Time-weighted Absolute Error (ITAE).
Stability Regions: Monte Carlo simulations revealed that the A-IPoC controller possesses a significantly larger basin of attraction. The stability region for the augmented model was found to be 27%–39% larger than that of the conventional IPoC model across different performance metrics (final position, final angle, control effort, and actuator saturation).
Robustness to Dropouts: At extreme update ratios (e.g., $\rho = 0.01$ ), the conventional IPoC controller exhibited divergence in both configuration variables. In contrast, the A-IPoC controller maintained marginal stability, successfully re-stabilizing the system from a wider range of initial perturbations.
Crash Rates: The improved stability translated to a 10%–15% reduction in crash rates (instances of instability) for the augmented model compared to the standard approach.

Significance and Claims

The paper claims to address a long-standing challenge in control theory: the integration of inertial sensor data directly into predictive motion models within the classical Newton-Euler framework. By leveraging differential flatness to incorporate higher-order dynamics, the authors establish a more robust control architecture for dynamically stable mobile platforms.

The significance of this work lies in its ability to enhance state extrapolation during sensor-limited periods, thereby strengthening the coupling between the observer and the controller. The authors posit that this approach bridges a crucial gap in existing research, offering a foundation for more reliable and efficient control of systems with non-minimum phase dynamics, which are often hindered by inherent delays and inverted control behaviors. The results suggest that for outdoor mobile platforms with limited sensor availability, augmenting the LQG model with inertial-derived higher-order dynamics is a viable strategy to expand stability margins and reduce failure rates.