⚡ electrical engineering

Coordinated Trajectory Control Algorithm for Quadcopter Motion along a Smooth Spatial Trajectory

This paper presents a robust, geometrically synthesized control algorithm for quadcopters that ensures stable 3D trajectory tracking despite unmeasured disturbances and incomplete state information by utilizing an extended observer and a simplified tuning methodology.

Original authors: Stanislav Kim, Anton Pyrkin, Oleg Borisov

Published 2026-05-18

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Stanislav Kim, Anton Pyrkin, Oleg Borisov

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 a quadcopter (a drone with four propellers) not just as a flying toy, but as a very sensitive, high-speed train that needs to stay perfectly glued to an invisible, winding track floating in 3D space. This track isn't just a line on a map; it's a smooth, curving path that might twist up, down, and around obstacles.

The paper by Kim, Pyrkin, and Borisov is essentially a blueprint for a new "autopilot" system that keeps this drone on that invisible track, even when things get messy. Here is how they did it, broken down into simple concepts:

1. The Problem: "Shortest Path" vs. "Staying on the Track"

Usually, when we tell a robot to go from Point A to Point B, we tell it to take the shortest route, like a crow flying straight. But if there are walls or obstacles in the way, that straight line doesn't work.

The authors propose a different approach called Coordinated Control. Instead of just saying "Go to that specific spot," they say, "Follow this specific curvy line, at this specific speed, while facing this specific direction."

The Analogy: Imagine a race car driver. A "tracking" system tells the car, "Drive to the finish line as fast as possible." A "coordinated" system tells the car, "Stay exactly in the center of the lane, even if the lane curves sharply, and keep your speed steady." The goal isn't just to get there; it's to stay perfectly aligned with the path, even if that means making lots of turns to avoid hitting a wall.

2. The Challenge: The Drone is a "Messy" Machine

Controlling a drone is hard because it's a "nonlinear" system. This is a fancy way of saying the drone doesn't move in a straight, predictable line like a car on a road. If you tilt the drone forward, it doesn't just go forward; it also changes how it spins and how it reacts to gravity. All its movements (up/down, left/right, spinning) are tangled together like a ball of yarn.

Previous attempts to control drones often ignored the "yaw" (the spinning motion of the drone) or assumed the engineers could see every single internal part of the drone's brain (the full state vector). The authors wanted to solve this using only the information a standard drone can easily see: its position (x, y, z) and its spinning angle (yaw).

3. The Solution: A "Shadow" System and a "Smooth" Controller

To fix the tangled math, the authors created a new way of looking at the problem.

The Shadow System: Instead of trying to control the drone's raw position directly, they invented a "shadow" coordinate system. Imagine the drone is a dog on a leash, and the leash is the invisible track. They created a mathematical way to measure exactly how far the dog is from the leash (the error) and how fast it's drifting away. They turned this complex 3D problem into a simpler, straight-line problem that is much easier to solve.
The Smooth Controller: They designed a control algorithm that acts like a very skilled driver.
- The "Integrators": They added extra "memory" steps (integrators) to the controller. Think of this as the driver not just reacting to the road right now, but also remembering how the road felt a split second ago and a split second before that. This allows for smoother, more stable adjustments.
- The "Saturation" Safety: They included a "speed limiter" (saturation function) in the math. Just like a car has a maximum speed, this math ensures the drone's commands never get too extreme, preventing it from flipping out or breaking.

4. The "Black Box" Problem: Seeing Without Seeing

The most impressive part of their solution is how they handle missing information. In the real world, you can't always measure exactly how fast the drone is accelerating or spinning at every tiny fraction of a second.

The Observer: The authors built a mathematical "observer." This is like a detective who looks at the clues the drone does give (its position and spin) and guesses what the hidden clues are (acceleration, disturbances).
The Result: Even without seeing the "hidden" variables, the controller uses these guesses to keep the drone on the track. It's like driving a car with a foggy windshield; you can't see the road perfectly, but your brain (the observer) fills in the gaps so you don't crash.

5. The Goal

The paper claims that with this new algorithm:

The drone will stay very close to the invisible track (within a tiny margin of error).
It will match the desired speed along the track.
It will face the correct direction as it moves.
It can do all this even if there are small, unpredictable bumps or winds (robustness), using only basic position and angle sensors.

In a nutshell: The authors took a very complicated, tangled math problem of flying a drone in 3D space and untangled it. They created a "smart autopilot" that uses a clever mathematical trick to turn a 3D maze into a straight line, uses a "detective" to guess missing information, and keeps the drone glued to its path like a train on a track, all without needing expensive or complex sensors.

Technical Summary: Coordinated Trajectory Control Algorithm for Quadcopter Motion along a Smooth Spatial Trajectory

Problem Statement
The paper addresses the challenge of controlling a quadcopter to move along a smooth, three-dimensional spatial trajectory. The authors distinguish between two primary control paradigms: tracking control, where the robot attempts to reach a specific position at a specific time (often leading to shortest-path solutions that may fail in obstacle-rich environments), and coordinated control. In coordinated control, the trajectory and the speed profile along that curve are pre-specified. The control objective is to minimize the deviation from the path, rather than the distance to a specific point in time, allowing the vehicle to navigate around obstacles by adjusting its speed along the curve.

The specific problem tackled here involves a nonlinear dynamic model of a quadcopter where the relative degree of the control channels is non-identical, complicating model decomposition. Furthermore, the proposed solution must rely solely on output measurements (linear coordinates $x, y, z$ and yaw angle $\psi$ ) rather than full state vector measurements, which is a significant constraint compared to existing state-feedback approaches.

Methodology
The authors develop a control synthesis based on a geometric approach and robust output feedback:

Dynamic Modeling and Deviation Definition:
A complete nonlinear model of the quadcopter is established, accounting for linear coordinates and orientation (yaw, pitch, roll). The deviation from the desired trajectory $S$ is defined using a coordinate transformation. This involves:
- A path coordinate $s$ representing the closest point on the curve.
- Two linear error coordinates ( $e_1, e_2$ ) representing the distance to two planes intersecting at the desired velocity vector.
- Angular errors ( $\delta\phi, \delta\theta, \delta\psi$ ), where the desired pitch and roll are assumed zero, and the desired yaw is derived from the trajectory's projection.
Normal Form Transformation:
To facilitate control design, the system is transformed into a "normal form" using a sequence of differentiations and coordinate changes. The authors introduce auxiliary variables and integrators to handle the saturation of the thrust control input ( $u_1$ ). This transformation results in a system where the regulated variables ( $\xi_1$ ) and their derivatives ( $\xi_2, \xi_3, \xi_4$ ) form an extended state vector. The system is shown to be invertible, allowing the reconstruction of the original control inputs from the normal form variables.
Robust Output-Feedback Control Synthesis:
- Ideal State Feedback: Initially, a control law is synthesized assuming full state availability. This law uses a linear feedback structure with gains ( $\gamma_1$ to $\gamma_4$ ) chosen to form a Hurwitz polynomial, ensuring local asymptotic stability of the error dynamics.
- Output Feedback Implementation: Recognizing that full state measurement is not available, the authors employ an extended observer method. A high-gain observer is designed to estimate the unmeasured states ( $\hat{\xi}_2, \hat{\xi}_3, \hat{\xi}_4$ ) and the unknown disturbance term ( $\hat{\sigma}$ ).
- Control Law: The final control law utilizes these estimated states and includes a smooth saturation function to ensure the control inputs remain within physical limits. The thrust forces for the four propellers are calculated by inverting the transformation matrix derived from the normal form.

Key Contributions

Complete 3D Model: Unlike previous works that utilized incomplete models ignoring yaw dynamics or were restricted to planar motion, this paper presents a complete dynamic model for quadcopter motion in 3D space.
Output-Feedback Coordinated Control: The paper synthesizes a coordinated control law that relies only on linear coordinates and yaw angle measurements, addressing the difficulty of non-identical relative degrees in the control channels.
Simplified Tuning: By utilizing additional integrators and a dynamic control algorithm, the authors propose a methodology with simplified regulator tuning compared to previous tracking control algorithms that required complex parameter selection to satisfy proof conditions.
Robustness: The inclusion of an extended observer allows the system to handle unknown terms and disturbances, providing a robust solution for trajectory following.

Results and Claims
The paper provides a theoretical proof of local stability for the closed-loop system. The authors demonstrate that if the controller parameters are chosen as coefficients of a Hurwitz polynomial, the deviation variables ( $\tilde{\xi}$ ) asymptotically tend to zero. This guarantees that the quadcopter will:

Follow the prescribed spatial curve.
Maintain the prescribed velocity profile along the curve.
Keep orientation errors within bounded limits.

The mathematical derivation confirms that the control law achieves the objectives of minimizing deviation from the trajectory and maintaining the desired speed and orientation, provided the curvature of the trajectory is bounded and initial conditions are within the region of attraction.

Significance
The paper claims to bridge a gap in the literature by extending coordinated trajectory control from planar mobile robots to 3D quadcopter motion using only output feedback. It contrasts with prior works that either used incomplete models or required full state measurements. The significance lies in providing a theoretically grounded, robust algorithm for spatial path following that is practically implementable given standard sensor constraints (position and yaw), without the need for complex state estimation of angular rates or full orientation vectors. The authors note that while the theoretical framework is established, full-scale experimental studies are planned for future work to validate the synthesized control law.