⚡ electrical engineering

Practical Universal Tracking With Pivoted Unidirectional Actuation

This paper proposes a redesigned robust control law that enables practical universal tracking for robotic vehicles with pivoted unidirectional actuators by driving the actuator output toward a target ball, thereby recovering the performance guarantees of unconstrained input controllers.

Original authors: Ian J. Willebeek-LeMair, Craig A. Woolsey

Published 2026-04-08

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ian J. Willebeek-LeMair, Craig A. Woolsey

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: The "One-Way Thruster" Problem

Imagine you are driving a very special car. This car has a unique engine: it can only push you forward. It cannot push you backward. However, the engine is mounted on a swivel (like a turret on a tank). You can spin the engine around to point in any direction, but once it's pointing, it can only push in that direction.

This is exactly how many robots work:

Drones (Multirotors): Their propellers only push air down to go up. To go sideways, they have to tilt the whole drone.
Rockets: They have engines that only push forward. To turn, they gimbal (tilt) the engine.
Hovercraft: They push air backward to move forward. To turn, they change the angle of the push.

The Problem:
In the past, engineers tried to control these robots by pretending the engine could push in any direction (forward, backward, left, right) instantly. They treated the "swivel" as a magical input that could do anything.

But here's the catch: Physics doesn't work that way.
If a drone needs to stop falling, it must push up. If it needs to move down, it must push down. But if the drone is upside down, its engine is pointing at the ground. To push up (against gravity), the engine has to point down.

There is a specific moment in a complex maneuver (like a drone doing a flip or falling) where the math breaks. The computer tries to calculate a direction that requires the engine to push "backward" (which it can't do) or requires the engine to spin 180 degrees instantly (which takes time). This is called the "Free-Fall Singularity."

When this happens, standard controllers crash, the math explodes, and the robot might spin out of control or fail to follow the path.

The Paper's Solution: "Aiming for a Bullseye, Not a Point"

The authors, Ian and Craig, realized that trying to hit a single, perfect mathematical point is impossible with a one-way engine. Instead, they redesigned the controller to be practical.

Here is their clever strategy, broken down with analogies:

1. The "Target Ball" Analogy

Imagine you are playing darts.

Old Way: You try to hit the exact center of the bullseye (0,0). If your hand shakes or the wind blows, you miss, and the math says "Game Over."
New Way: The authors say, "Let's draw a small circle around the bullseye." As long as your dart lands anywhere inside that circle, you win.

In the robot's world, instead of forcing the engine to point at one exact angle, the controller says: "Just make sure the thrust vector lands inside this small, moving ball around where we want to be."

This tiny "ball" gives the robot a little wiggle room. It allows the robot to handle the moments where the math gets weird (like when the engine has to flip upside down) without crashing.

2. The "Double-Decker Bus" Analogy (Solving the Spin)

There is another tricky problem. Imagine you are standing in a room facing North. You want to face South.

You can turn 180 degrees clockwise.
Or, you can turn 180 degrees counter-clockwise.
Both get you to the same place, but they are different paths.

Standard controllers often get confused here. They might decide to spin the robot around the world three times (360 degrees) just to get to the same spot, because they got stuck in a "topological trap." It's like a GPS telling you to drive all the way around the globe to get to the next block.

The authors use a mathematical trick called "1D MRP coordinates" (Modified Rodriguez Parameters). Think of this as a double-decker bus map.

On the bottom deck, "South" is reached by turning right.
On the top deck, "South" is reached by turning left.
The controller is smart enough to switch decks if it realizes it's about to take a long, unnecessary spin. This ensures the robot always takes the shortest, smoothest turn.

3. The "Two-Layer" Control System

The paper proposes a system with two layers, like a manager and a worker:

The Outer Loop (The Manager): This part plans the route. It says, "We need to go from A to B." It calculates the ideal path, ignoring the fact that the engine is one-way. It gives a "wish list" of forces.
The Inner Loop (The Worker): This part actually controls the engine. It looks at the Manager's wish list and says, "Okay, I can't push exactly like that, but I can get close enough to land inside the target ball."

The Manager trusts the Worker to handle the messy details. The Worker ensures the robot stays within the "safe zone" (the ball), and the Manager ensures the robot stays on the general path.

Why This Matters

Before this paper, if a drone tried to do a very aggressive maneuver (like a fast flip or a sudden stop), the software would often panic because of the "Free-Fall Singularity." Engineers had to limit how fast the drone could move to avoid this, which made the robots sluggish and boring.

With this new method:

Robots can be more agile. They can flip, dive, and recover without the math breaking.
No more "Artificial Limits." We don't have to tell the robot, "You can't go faster than X." It can go as fast as the hardware allows.
It's "Practically Perfect." The robot won't follow the path with 100% mathematical perfection (because physics is messy), but it will follow it closely enough that a human observer would say, "Wow, that was perfect."

Summary

The authors took a problem where robots kept crashing because they tried to do the impossible (push backward with a one-way engine), and they solved it by saying, "Let's aim for a small target zone instead of a single point."

By accepting a tiny bit of imperfection, they unlocked the ability for robots to perform complex, high-speed maneuvers that were previously impossible, all while keeping the math stable and safe. It's the difference between trying to thread a needle while running a marathon (impossible) and threading a needle while walking briskly (doable and effective).

1. Problem Formulation

The paper addresses the control problem for robotic vehicles equipped with pivoted unidirectional actuators. These systems include multirotor UAVs, rockets, hovercraft, and ground vehicles constrained to forward motion.

System Dynamics: The system is modeled as an outer-loop vehicle subsystem (position/velocity) and an inner-loop pivot subsystem (attitude/angular velocity). The control input is a unidirectional force vector $\lambda(t)u_1(t)$ , where $\lambda(t)$ is a unit vector representing the pivot orientation and $u_1(t) \geq 0$ is the magnitude.
The Singularity Issue: Traditional control approaches often treat the actuator input as unconstrained (bidirectional). However, because $u_1(t)$ must be non-negative, the system cannot generate a force in the direction opposite to the pivot's orientation without rotating the pivot. This leads to a "free-fall singularity" (e.g., when a multirotor needs to accelerate downward faster than gravity).
Limitations of Existing Methods:
- Existing controllers often assume the singularity is never encountered or artificially limit acceleration to be less than gravity.
- These limitations compromise robustness (Input-to-State Stability, ISS) and performance.
- Theoretical Barrier: The authors prove that Exact Universal Tracking (EUT) is impossible for this class of systems. There exist smooth desired trajectories that the system physically cannot follow exactly due to the unidirectional constraint.

2. Methodology

The authors propose a Practical Universal Tracking (PUT) controller. Instead of demanding exact tracking ( $R=0$ ), the controller ensures the tracking error remains within a small, tunable bound ( $R > 0$ ).

A. Control Architecture

The design follows an inner-loop/outer-loop structure:

Outer-Loop: A baseline robust controller (designed for unconstrained inputs) generates a desired actuation command $a^*(t)$ .
Inner-Loop: A redesigned controller drives the actual actuator output $a(t) = \lambda(t)u_1(t)$ $a (t) = λ (t) u_{1} (t)$ to converge to a target ball $A(t)$ $A (t)$ centered at $a^*(t)$ $a^{*} (t)$ with a radius $r$ $r$ .
- The difference $a(t) - a^*(t)$ is treated as a bounded disturbance to the outer-loop.
- By ensuring the outer-loop is Input-to-State Stable (ISS) with respect to this disturbance, the system achieves practical tracking.

B. Key Technical Innovations

To handle the singularity at $a^*(t) = 0$ and the topological constraints of the pivot ( $S^1$ ), the authors introduce several novel mechanisms:

Set-Based Approach:
- Instead of forcing the pivot orientation $\lambda(t)$ to track a specific direction $\lambda^*(t) = a^*(t)/\|a^*(t)\|$ (which is undefined at zero), the controller drives the output into the ball $A(t) = a^*(t) + r\bar{B}_2$ .
- When $a^*(t)$ is near zero, the controller becomes independent of the specific orientation $\lambda^*(t)$ , avoiding the discontinuity that causes the singularity.
1D Modified Rodrigues Parameters (MRPs):
- To resolve the topological obstruction at $\lambda(t) = -\lambda^*(t)$ (where the shortest rotation path is ambiguous), the authors use a 1D analogue of MRPs, denoted as $\eta(t)$ .
- $\eta(t)$ provides a double covering of the circle $S^1$ , allowing for smooth control laws that avoid "unwinding" (performing unnecessary full rotations).
Smooth Switching and Initialization:
- The control law uses smooth step functions ( $\zeta$ ) to transition between control modes.
- When $\|a^*(t)\|$ is very small (near the singularity), the control terms driving the attitude are smoothly switched off, preventing discontinuities in the angular velocity command.
- An initialization rule prevents unwinding by selecting the MRP branch closest to the origin.
Lyapunov Analysis:
- A composite Lyapunov function is constructed to prove that the inner-loop state converges to the target set $A(t)$ .
- The analysis demonstrates that the outer-loop error is ultimately bounded by a function of the ball radius $r$ and the system's ISS gain.

3. Key Contributions

Proof of Non-Existence of EUT: The paper rigorously proves that no controller can achieve exact universal tracking for pivoted unidirectional systems, as there exist smooth trajectories that violate the unidirectional constraint.
Synthesis of a PUT Controller: The authors design a smooth, globally attractive controller that achieves Practical Universal Tracking. It guarantees that the tracking error converges to an arbitrarily small neighborhood of the desired trajectory.
Singularity Resolution: The proposed method overcomes the "free-fall singularity" without imposing artificial limits on acceleration or assuming the singularity is never encountered.
General Applicability: The framework applies to a wide class of systems, including multirotors, rockets, hovercraft, and ocean vehicles.

4. Simulation Results

The theory is validated using a planar multirotor simulation:

Scenario: The multirotor tracks an aggressive square trajectory requiring rapid changes in acceleration, including moments where the required vertical acceleration exceeds gravity (requiring the vehicle to flip upside down).
Outcome:
- The proposed controller successfully tracks the trajectory, flipping the vehicle when necessary to compensate for the unidirectional thrust constraint.
- Standard controllers (which assume unconstrained inputs) fail at the first vertical edge due to the singularity (division by zero).
- The tracking error remains bounded within the theoretical limits, demonstrating the effectiveness of the practical tracking approach.

5. Significance and Conclusion

This paper provides a fundamental shift in how unidirectional actuation is handled in control theory.

Theoretical Impact: It establishes that for unidirectional systems, the goal must shift from "Exact Tracking" to "Practical Tracking."
Practical Impact: The controller enables high-performance maneuvers (like aggressive multirotor flight) that were previously considered impossible or required conservative, performance-limiting assumptions.
Future Work: The authors suggest extending the results to 3D gimbaled systems, incorporating finite upper-bounds on actuation, and developing stronger integral ISS (SiISpS) guarantees for systems with finite control authority.

In summary, the paper offers a robust, mathematically rigorous solution to the long-standing problem of controlling vehicles with pivoted unidirectional actuators, enabling them to perform complex maneuvers without artificial constraints.