⚡ electrical engineering

Error-State LQR Formulation for Quadrotor UAV Trajectory Tracking

This paper proposes a robust trajectory tracking control strategy for quadrotor UAVs that utilizes an error-state Linear Quadratic Regulator (LQR) with exponential coordinates for orientation errors, integrated with a cascaded bodyrate controller to ensure precise and stable performance in dynamic environments.

Original authors: Micah Reich

Published 2026-06-08

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CC0 1.0

Original authors: Micah Reich

Original paper dedicated to the public domain under CC0 1.0 (http://creativecommons.org/publicdomain/zero/1.0/). ✨ 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 tiny, four-rotor helicopter (a quadrotor) to fly perfectly along a specific path, like a tightrope walker following a rope in the sky. The problem is that the helicopter is naturally chaotic; it wobbles, spins, and reacts unpredictably to wind and its own movements. If you try to give it a single, giant set of instructions to follow the whole path at once, it gets confused and crashes.

This paper proposes a smarter way to control the drone, using a method called Error-State LQR. Here is how it works, broken down into simple concepts:

1. The "Ghost" vs. The "Real" Drone

Instead of trying to control the drone's actual, messy position directly, the controller creates a mental "Ghost Drone."

The Ghost Drone is flying perfectly along the ideal path you want. It knows exactly where it should be, how fast it should be going, and which way it should be facing.
The Real Drone is the one actually in the air, which might be slightly off-course.

The controller doesn't worry about the absolute position of the drone (like "I am at coordinates 10, 10, 10"). Instead, it only cares about the difference between the Real Drone and the Ghost Drone. This difference is called the Error State.

2. The "Tiny Angle" Trick (Exponential Coordinates)

One of the hardest parts of flying a drone is describing its orientation (which way it is tilted). Traditional methods use angles (like pitch, roll, and yaw), but these can get mathematically "stuck" or broken when the drone flips upside down (a problem called a singularity).

This paper uses a clever mathematical tool called Exponential Coordinates. Think of this like describing a tilt not by saying "it's 90 degrees," but by saying "it needs a tiny nudge in this specific direction to get back to straight."

As long as the drone is close to the path, these "tiny nudges" are very small and easy to calculate.
This allows the computer to treat the complex, twisting motion of the drone as a simple, straight-line problem, which is much faster to solve.

3. The "Perfect Balance" (LQR)

Once the problem is simplified to just "how far off are we?" and "how much do we need to nudge?", the controller uses a strategy called Linear Quadratic Regulator (LQR).

Imagine you are balancing a broomstick on your hand.

If the broomstick leans a little to the left, you move your hand a little to the left.
If it leans a lot, you move your hand a lot.
But you also don't want to move your hand too wildly, because that wastes energy and makes the broomstick shake.

The LQR is a mathematical formula that constantly calculates the perfect balance between:

Getting the drone back on the path quickly (minimizing the error).
Not using too much battery or motor power (minimizing the effort).

It solves this balancing act thousands of times per second. Every split second, it re-calculates the "Ghost Drone's" path and asks, "Okay, given where we are right now, what is the best tiny nudge to get back on track?"

4. The Two-Layer Team (Cascaded Control)

The paper describes a two-person team approach to actually move the drone:

The Brain (The LQR Controller): This high-level thinker decides, "We need to turn slightly to the right and speed up." It outputs a desired speed for the drone's rotation. It doesn't know how the motors work; it just knows the goal.
The Muscle (The Bodyrate Controller): This is the lower-level worker that actually controls the motors. It takes the Brain's goal ("Turn right at this speed") and figures out exactly how much power to send to each of the four motors to make that happen. It uses a simple feedback loop: "We are turning too slow? Give the motors more power. Too fast? Slow them down."

Summary

In short, this paper presents a system where a drone doesn't try to memorize a complex flight path. Instead, it constantly compares itself to a perfect "Ghost" version of the flight. It measures the tiny differences, uses a mathematical "balance scale" to figure out the most efficient way to correct those tiny differences, and then sends simple commands to its motors to execute the fix. This allows the drone to fly smoothly and accurately even in tricky, changing environments.

Technical Summary: Error-State LQR Formulation for Quadrotor UAV Trajectory Tracking

Problem Statement

The control of Quadrotor Unmanned Aerial Vehicles (UAVs) is complicated by their inherent nonlinear dynamics, underactuation, and the requirement for precise trajectory tracking within dynamic environments. Traditional control methods often face a trade-off between handling these nonlinearities efficiently and maintaining the computational tractability necessary for real-time applications. Specifically, representing orientation errors using standard parameterizations like Euler angles introduces singularities, while directly controlling full nonlinear dynamics can be computationally prohibitive for high-frequency feedback loops.

Methodology

The paper proposes an Error-State Linear Quadratic Regulator (LQR) formulation designed to address these challenges through a cascaded control architecture and a specific mathematical representation of errors.

1. Error-State Representation

The core of the methodology is the definition of an error-state vector ( $\delta x$ ) that captures deviations from a nominal trajectory.

State Definition: The full state includes position ( $p$ ), velocity ( $v$ ), and orientation (represented as a unit quaternion $q$ ).
Error Definition: The error state $\delta x = [\delta p, \delta \theta, \delta v]^T$ is constructed by subtracting the nominal state from the true state.
Orientation Handling: Crucially, the orientation error is not represented as a quaternion difference or Euler angles. Instead, it utilizes exponential coordinates ( $\delta \theta$ ), where the rotation error matrix is defined as $\delta R = \exp([\delta \theta]_\times)$ . This provides a compact, singularity-free representation of orientation errors.

2. System Dynamics and Linearization

The paper derives the kinematics and dynamics for the true state, the nominal state, and the error state based on a single rigid-body assumption with lumped mass.

Derivation: The time derivatives of the error states ( $\dot{\delta p}, \dot{\delta v}, \dot{\delta \theta}$ ) are derived. The derivation employs small-angle approximations and ignores second-order terms to simplify the relationship between the true and nominal dynamics.
Linearization: The nonlinear error-state dynamics are linearized about the current tracking error at a sufficiently high frequency. This iterative re-linearization allows the system to be represented in the form $\dot{\delta x} \approx A\delta x + B\delta u$ , where $A$ and $B$ are Jacobian matrices derived from the error dynamics. This approach enables the use of linear control theory on a nonlinear system by continuously updating the linearization point.

3. Control Architecture

The control strategy is implemented as a cascaded system:

Outer Loop (LQR): An LQR controller is designed to minimize a quadratic cost function $J = \int_0^\infty (\delta x^T Q \delta x + \delta u^T R \delta u) dt$ . The controller computes a feedback gain matrix $K$ (via the Algebraic Riccati Equation) to generate a corrective control input $\delta u = -K\delta x$ . The total control input is $u_t = u - K\delta x$ .
Inner Loop (Bodyrate Controller): The LQR treats the desired angular velocity as a virtual control input. A cascaded bodyrate controller is employed to track these desired velocities. This inner loop uses a Proportional (P) controller with feedback linearization, calculating the required torque $\tau$ based on the inertia tensor $J$ and the difference between desired and actual angular velocities.

Key Contributions

Singularity-Free Orientation Error: The paper demonstrates the use of exponential coordinates to represent orientation errors, avoiding the singularities associated with Euler angles.
Iterative Re-linearization: It provides a detailed derivation showing how error-state dynamics can be linearized about the current error, allowing an LQR controller to operate effectively on nonlinear quadrotor dynamics in real-time.
Cascaded Control Integration: The work integrates the LQR trajectory tracker with a bodyrate controller, explicitly handling the mapping from virtual angular velocity commands to physical actuator torques.
Comprehensive Derivation: The paper offers a complete mathematical derivation of the error-state kinematics, the linearization process, and the construction of the Jacobian matrices ( $A$ and $B$ ) required for the LQR formulation.

Results

The paper presents simulation results demonstrating the performance of the proposed controller.

Trajectory Tracking: The controller is shown to successfully track a lemniscate (figure-eight) trajectory.
Yaw Tracking: The system effectively tracks yaw angles along the trajectory.
Initial Conditions: The results note that tracking errors at the very beginning of the trajectory are attributed to the UAV starting with a flat initial orientation ( $q_0 = (1, 0, 0, 0)^T$ ), which the controller subsequently corrects.

Significance and Claims

The paper claims that this error-state LQR formulation offers a robust and computationally efficient solution for quadrotor trajectory tracking. By leveraging exponential coordinates and iterative linearization, the method achieves:

Precision and Stability: The ability to drive position, velocity, and orientation errors to zero in dynamic environments.
Real-Time Applicability: The linearized system representation allows for high-frequency control updates without the computational burden of solving full nonlinear optimal control problems at every step.
Robustness: The approach handles the nonlinearities of the quadrotor by focusing on minimizing deviations from a nominal trajectory rather than controlling the full nonlinear dynamics directly.

The work positions itself as a practical framework for precise quadrotor control, bridging the gap between classical linear control theory (LQR) and the complex nonlinear reality of UAV flight.