⚡ electrical engineering

Dynamics and LQR Control of a Triple Inverted Pendulum on a Cart

This paper derives the nonlinear dynamics of a triple inverted pendulum on a cart using the Euler–Lagrange method, designs a Linear Quadratic Regulator (LQR) controller to stabilize the system around its upright equilibrium, and validates its robust performance through extensive numerical simulations and parametric sensitivity analyses.

Original authors: Ali Tebaan Hassan

Published 2026-06-30

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Ali Tebaan Hassan

Original paper licensed under CC BY 4.0 (https://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 a balancing act that sounds like a circus trick gone wrong: a small cart on a track, with not one, but three long sticks standing on top of each other, all trying to fall over at the same time. This is the "Triple Inverted Pendulum."

In the real world, this is incredibly hard to control. If you try to balance a single broomstick on your hand, it's tricky. If you stack three broomsticks on top of each other and try to balance them all with just your hand moving back and forth, it seems impossible. That's exactly what this paper does, but with math and computer simulations instead of a circus.

Here is the story of the paper, broken down into simple parts:

1. The Setup: A Tower of Wobbly Sticks

The researchers built a mathematical model of this system.

The Cart: A heavy block that can only move left and right on a straight track.
The Pendulums: Three rigid sticks (links) connected in a chain. The first stick is attached to the cart, the second to the first, and the third to the second.
The Goal: Keep all three sticks standing perfectly straight up (like a tower of pencils) while the cart moves to keep them from falling.

The paper starts by writing down the "rules of the game." They used a method called Euler–Lagrange (think of it as a super-precise recipe for physics) to calculate exactly how every part moves, how they push against each other, and how gravity tries to pull them down. They found that the math is very complex because the sticks are all connected; if the top stick wobbles, it pulls on the middle one, which pulls on the bottom one, which tugs the cart.

2. The Problem: It Wants to Fall

The system is naturally unstable. It's like trying to balance a pencil on its tip. Without any help, the sticks will fall over in less than a second. The paper shows that if you just let go, the whole tower collapses exponentially fast.

3. The Solution: The "Smart Brain" (LQR)

To stop the fall, the researchers designed a "smart brain" for the cart called an LQR Controller (Linear Quadratic Regulator).

How it works: Imagine a very fast-thinking coach standing next to the cart. This coach watches the angles of all three sticks. If the top stick leans even a tiny bit, the coach instantly tells the cart to move left or right to catch it.
The Math: The coach uses a special equation (the Riccati equation) to figure out the perfect amount of force to apply. It's not just about stopping the fall; it's about doing it with the least amount of wasted energy while keeping the sticks upright.
The Result: The coach is very aggressive. It calculates that to keep three sticks balanced, it needs to push the cart with a force about 12 times heavier than the weight of the heaviest stick.

4. The Simulation: The Test Run

Since building a physical triple-pendulum is dangerous and expensive, they ran a high-speed computer simulation (like a video game physics engine) to test their "coach."

The Test: They started the simulation with the sticks already leaning slightly (about 3 to 5 degrees off-center), as if someone had just nudged them.
The Outcome: The controller worked perfectly. Within about 3 seconds, the cart moved back and forth, correcting the wobbles, and all three sticks stood perfectly straight again.
The Cart's Journey: The cart didn't just sit still. It had to make a quick, sharp move (about 8 centimeters) to the left to counter the falling sticks, then smoothly return to the center.
The Force: The hardest push the cart had to make was about 18 Newtons (roughly the force of holding a 4-pound bag of apples), which is well within the limits of a standard motor.

5. What They Learned

The paper tested how "strict" the coach should be by changing the settings:

Strict Coach: If you tell the controller, "The sticks must be perfect right now," it moves the cart very fast and hard. The sticks stabilize in about 1.3 seconds, but the motor has to work very hard.
Laid-back Coach: If you say, "It's okay if they wobble a little longer," the cart moves more gently, but it takes about 4 seconds to stabilize.
The Sweet Spot: They found a middle ground where the sticks stabilize in about 3 seconds without overworking the motor.

6. Why This Matters (According to the Paper)

The authors explain that while this is a "toy" problem, the math behind it is the same as for much bigger, real-world engineering challenges. They mention that the same principles apply to:

Humanoid robots walking on two legs.
Tall buildings swaying in the wind or during earthquakes.
Rocket ships trying to land vertically (like SpaceX rockets).

Summary

This paper is a complete guide to solving one of the hardest balancing problems in physics. They wrote the math, built a "smart brain" to control it, and proved in a computer simulation that a single moving cart can successfully balance a tower of three falling sticks, keeping them upright in just a few seconds. They showed exactly how much force is needed and how to tune the system to be fast or gentle, depending on what you need.

Technical Summary: Dynamics and LQR Control of a Triple Inverted Pendulum on a Cart

Problem Statement
The paper addresses the control of a triple inverted pendulum (TIP) mounted on a cart constrained to linear motion. This system is identified as a highly challenging benchmark in nonlinear dynamics and classical control theory. The system is characterized by an 8-degree-of-freedom state space (cart position and three link angles), a single control input (horizontal force on the cart), and inherent instability. The upright equilibrium is unstable, with three pairs of unstable open-loop eigenvalues, and the dynamics involve significant nonlinear coupling terms in the mass matrix. The objective is to derive the complete nonlinear equations of motion, linearize the system around the unstable equilibrium, and design a Linear Quadratic Regulator (LQR) to stabilize the system from initial perturbations.

Methodology
The research follows a structured analytical and numerical approach:

Analytical Modeling: The authors derive the full nonlinear equations of motion using the Euler–Lagrange method. The system is modeled with a rigid cart of mass $M$ and three massless rigid links with point masses at their centers. The Lagrangian ( $L = T - V$ ) is formulated to account for the kinetic and potential energies of the cart and links, resulting in a configuration-dependent mass matrix $M(q)$ , a Coriolis–centripetal matrix $C(q, \dot{q})$ , and a gravitational vector $G(q)$ .
Linearization: The nonlinear equations are linearized around the unstable upright equilibrium point ( $q^* = 0, \dot{q}^* = 0$ ) using first-order Taylor expansions ( $\sin \theta \approx \theta, \cos \theta \approx 1$ ). This yields a constant-coefficient 8th-order linear state-space representation ( $\dot{z} = Az + Bu$ ).
Controllability Analysis: The Kalman controllability matrix is computed, confirming the system is completely state-controllable (rank 8) with the single cart force input.
Controller Design: An LQR controller is synthesized by solving the algebraic Riccati equation (ARE). The cost functional minimizes a weighted sum of state errors and control effort. Weighting matrices are selected based on Bryson's rule, prioritizing angular regulation over cart position.
Numerical Simulation: The closed-loop system is simulated using a fourth-order Runge–Kutta integration scheme (RK45) with adaptive step-size control. Simulations test the system's response to initial perturbations of up to 5° in the link angles with zero initial velocities.

Key Contributions
The paper claims the following specific contributions:

A self-contained derivation of the full nonlinear TIP equations of motion, retaining all coupling and geometric nonlinearities.
A rigorous linearization procedure resulting in an explicit 8×8 state-space model.
The synthesis of an LQR controller and verification of stability via eigenvalue analysis, demonstrating the relocation of all unstable poles to the left-half complex plane.
High-fidelity simulation results supported by ten diagnostic figures analyzing system states, energy, and phase-plane dynamics.
A parametric sensitivity study quantifying the trade-off between regulation performance (settling time) and control effort as the state-cost weighting matrices are varied.

Results
Numerical simulations demonstrate the following outcomes:

Stabilization: The LQR controller successfully stabilizes the system from initial angular perturbations of approximately ±3° to ±5°. The system converges to the upright equilibrium within approximately 3 seconds.
Control Effort: The peak control force required is approximately 18 N, occurring within the first 0.5 seconds. This remains well within the assumed saturation limit of ±50 N, validating the efficacy of the linear control strategy for these initial conditions.
System Dynamics: The cart undergoes a transient excursion of roughly 0.08 m before settling. The angular displacements of all three links converge to zero, with Link 3 (the top link) converging fastest due to high proportional gains.
Energy and Stability: Total mechanical energy decreases monotonically, indicating active energy extraction by the controller. Phase-plane analysis confirms global asymptotic stability with no limit cycles or chaotic attractors.
Sensitivity: A sensitivity study reveals that increasing the angular state weighting ( $Q_\theta$ ) from 10 to 1000 reduces the settling time from ~4.2 s to ~1.3 s. However, this improvement exhibits diminishing returns beyond $Q_\theta \approx 200$ , while requiring proportionally higher control forces.

Significance and Context
The paper positions the TIP as a critical testbed for advanced control techniques, noting its relevance to bipedal robotics, cable-driven manipulators, and the attitude control of launch vehicles. While acknowledging recent advances in data-driven methods (e.g., reinforcement learning, physics-informed neural networks) and their application to cart-pole systems, the authors assert that model-based LQR retains primacy for rapid, deterministic stabilization due to its global optimality guarantees within the linear regime and negligible online computational burden.

The work serves as a foundational reference for the theoretical underpinnings of TIP control. It contextualizes its results against recent literature (2021–2026) on learning-augmented control but focuses on the robustness and interpretability of classical optimal control. The authors conclude that while linear LQR is effective for small perturbations, future work will address nonlinear strategies (such as NMPC) for larger angles, experimental validation on physical testbeds, and comparisons with safe reinforcement learning and physics-informed neural ODE controllers.