💻 computer science

Geometric Model Predictive Path Integral for Agile UAV Control with Online Collision Avoidance

This paper presents Geometric Model Predictive Path Integral (GMPPI), a novel sampling-based controller that integrates geometric SE(3) control and depth-image-based collision avoidance to enable agile UAVs to track precise trajectories while safely navigating obstacles at high speeds.

Original authors: Pavel Pochobradský, Ondřej Procházka, Robert Pěnička, Vojtěch Vonásek, Martin Saska

Published 2026-02-24

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CC BY 4.0

Original authors: Pavel Pochobradský, Ondřej Procházka, Robert Pěnička, Vojtěch Vonásek, Martin Saska

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 you are teaching a tiny, high-speed drone to fly through a dense, chaotic forest. The goal is to zip through the trees as fast as possible without crashing, while also following a specific, winding path perfectly.

This is a incredibly difficult balancing act. If the drone flies too carefully, it's slow. If it flies too fast and recklessly, it crashes. If it tries to follow the path too strictly, it might not see a tree coming and smash into it.

This paper introduces a new "brain" for the drone called GMPPI (Geometric Model Predictive Path Integral). Here is how it works, explained simply:

The Problem: The "Swing vs. Safety" Dilemma

Older drone controllers faced a tough choice:

The Precision Pilot: This drone is great at following a path perfectly but is "blind" to obstacles. If a tree appears, it might crash because it's too focused on the line.
The Safety Pilot: This drone is great at dodging trees, but it flies erratically. It takes wide, clumsy turns and can't keep up with fast, agile paths.

Most existing systems try to do both by randomly guessing thousands of paths, hoping one works. But this is like trying to find a needle in a haystack by throwing darts blindfolded; it's inefficient and often misses the mark.

The Solution: The "Hybrid Brain"

The authors created a controller that acts like a team of two pilots working together in real-time:

The "Math Genius" (Geometric SE(3)): A small group of the drone's "thoughts" are handled by a mathematically perfect pilot. This pilot knows exactly how to fly the drone smoothly and precisely along the desired path. It's like a professional race car driver who knows the perfect line through a corner.
The "Wild Explorer" (Random Sampling): The rest of the "thoughts" are handled by a chaotic, random explorer. This pilot throws the drone in random directions to see what happens. It's like a kid running around a playground, checking every nook and cranny to see if there are any trees or walls.

The Magic Trick:
Instead of choosing one pilot or the other, the GMPPI brain simulates hundreds of flight paths at once (using a powerful computer chip called a GPU).

It asks the "Math Genius" to fly the path perfectly.
It asks the "Wild Explorer" to fly wildly to check for obstacles.
Then, it averages all these results.

If the "Math Genius" says, "Go straight!" but the "Wild Explorer" says, "Wait, there's a tree there, go left!", the brain calculates a new, smooth path that goes left just enough to avoid the tree, but still stays close to the original perfect line.

The "X-Ray Vision" (Depth Cameras)

Most drones build a 3D map of the world before flying, which takes time and slows them down.

The Old Way: Like looking at a map of the forest, memorizing it, and then trying to fly.
The GMPPI Way: The drone looks at a live video feed from a depth camera (like a 3D eye) and projects its future flight paths directly onto that image. It's like holding a flashlight and seeing if the beam hits a tree before you actually walk there. This happens instantly, allowing for lightning-fast reactions.

The "Smart Timer"

The drone doesn't treat every second of its flight the same.

Near Future (Next 1 second): It looks at the path in tiny, detailed steps. This ensures it doesn't wobble or crash immediately.
Far Future (Next 5 seconds): It looks at the path in bigger, faster steps. This allows it to "see" further down the road without getting bogged down in too much detail.

The Results: Superhuman Flight

The team tested this new brain in two ways:

Simulations: They put the drone in a virtual forest with thousands of trees. The GMPPI drone flew faster and more accurately than any other method, including those trained by AI (which usually need massive amounts of data to learn).
Real World: They flew a real drone through a course of poles (simulating trees).
- It flew at speeds up to 17 meters per second (about 38 mph).
- It dodged obstacles at speeds up to 10 meters per second (about 22 mph).
- It never crashed.

The Bottom Line

Think of GMPPI as the ultimate co-pilot. It combines the precision of a surgeon with the adaptability of a jungle explorer. It allows drones to fly fast and agilely through dangerous, unknown environments without needing to be "taught" by years of trial and error. It's a step toward drones that can truly save lives in search-and-rescue missions or inspect power lines at high speeds, all while dodging obstacles on the fly.

1. Problem Statement

Autonomous Unmanned Aerial Vehicles (UAVs) operating in unknown, cluttered environments face a fundamental trade-off between agile trajectory tracking and obstacle avoidance.

Modular Approaches: Traditional methods separate mapping, planning, and control. This introduces latency, prevents the controller from accounting for full dynamics, and often leads to crashes if tracking errors occur on a collision-free path.
Learning-Based Approaches: End-to-end learning methods (RL/IL) unify planning and control but require massive training data, lack generalizability across different UAVs, and treat the system as a "black box."
Standard MPPI Limitations: Model Predictive Path Integral (MPPI) is a sampling-based controller that unifies planning and control. However, standard MPPI relies on adding random noise to a nominal trajectory. This creates a conflict: high-noise rollouts are necessary for exploring obstacle-free paths but are inefficient for precise trajectory tracking. Conversely, low-noise rollouts track well but fail to avoid unexpected obstacles.

2. Methodology: Geometric MPPI (GMPPI)

The authors propose GMPPI, a novel sampling-based controller that integrates geometric SE(3) control with MPPI to resolve the tracking vs. avoidance trade-off. The system runs on a GPU for parallel trajectory generation.

A. Hybrid Rollout Generation

Instead of relying solely on random noise, GMPPI generates $K$ candidate rollout trajectories per iteration using a hybrid approach:

Geometric Rollouts ( $K_{SE3}$ ): A subset of rollouts (e.g., 32 out of 768) are generated using a deterministic SE(3) geometric controller. These trajectories are optimized for precise reference tracking and smoothness.
Random Rollouts: The remaining rollouts are generated by adding noise to the nominal command sequence. These focus on exploration and finding collision-free paths in unknown environments.
Deterministic Yaw: To increase trajectory density without increasing sample count, the yaw axis is controlled deterministically using a proportional controller, removing one dimension from the stochastic search space.

B. Direct Depth Sensor Integration

Unlike previous MPPI implementations that rely on pre-built maps, GMPPI integrates depth camera data directly into the control loop:

Collision Checking: The UAV's bounding box corners and center are projected onto the latest depth image.
Optimistic Planning: The system assumes a safety depth ( $d_a$ ) behind detected obstacles, allowing the planner to "peek" around corners and plan return paths even if the full path isn't visible yet.
Latency Reduction: This approach bypasses heavy map construction, minimizing latency to the camera frame rate.

C. Dynamic Time-Step Sampling

To balance precision near the UAV with long-range obstacle avoidance:

Variable Time Steps: The rollout horizon uses a variable time-step vector. Early steps use small time steps ( $\Delta t_{near}$ ) for high precision and stability. Later steps use larger time steps ( $\Delta t_{far}$ ) scaled to the sensor range and UAV speed, ensuring the full sensor range is utilized without excessive computational cost.

D. Tailored Cost Function

The cost function is designed to prioritize obstacle avoidance while leveraging the strengths of both rollout types:

Tracking & Smoothness: Penalizes deviations in position, velocity, orientation, and angular velocity. It also penalizes excessive jerk (change in acceleration) to ensure smooth flight.
Time-Dependent Weights: Cost coefficients vary over the rollout horizon. Tracking precision is weighted heavily in the near term (to utilize SE(3) rollouts), while exploration is weighted more in the far term.
Obstacle Penalty: Collisions are penalized heavily. The penalty is scaled by $(N-j)$ , meaning collisions occurring sooner in the future are penalized more severely than distant ones.

3. Key Contributions

First Hybrid MPPI-SE(3) Controller: The first method to combine sampling-based MPPI with geometric SE(3) control, achieving high-precision tracking while maintaining obstacle avoidance capabilities.
Direct Depth Integration: The first UAV MPPI controller to utilize raw depth sensor data directly in the control loop without intermediate mapping, significantly reducing latency.
Dynamic Sampling Strategy: A variable time-step approach that optimizes the trade-off between near-term precision and far-range obstacle detection.
Ablation Study: Demonstrates that the combination of SE(3) rollouts, dynamic time steps, and time-varying cost/noise parameters is critical for performance.

4. Results

The authors validated GMPPI through simulations and real-world experiments.

Simulation Results

Tracking Performance: GMPPI achieved a 31% reduction in position RMSE compared to standard MPPI on agile trajectories. It matched the tracking error of a standalone SE(3) controller (within 20% RMSE) while adding obstacle avoidance.
Obstacle Avoidance: In simulated Poisson forests (tree density $\delta = 1/25$ ), GMPPI achieved a 100% success rate up to 11 m/s. This surpassed state-of-the-art learning-based controllers and the "Bubble" planner. At 13 m/s, it had 80% fewer failures than the Bubble planner.
Ablation: Removing SE(3) rollouts or using constant time steps/noise significantly degraded performance, confirming the necessity of the proposed architecture.

Real-World Experiments

Agile Flight: The UAV successfully tracked a figure-8 trajectory at 14 m/s with a position RMSE of 0.69 m, showing no performance loss compared to simulation.
Obstacle Avoidance: The controller successfully avoided obstacles in a real-world course (simulating trees with pylons) at speeds up to 10 m/s (42 flights, 0 collisions).
Maximum Speed: The system demonstrated flight capabilities up to 17 m/s and accelerations up to 35 m/s².

5. Significance

This work bridges the gap between geometric control (precision) and sampling-based control (robustness to uncertainty). By integrating SE(3) control into the MPPI framework, GMPPI allows UAVs to fly aggressively and precisely while reacting to dynamic, unknown obstacles in real-time. The direct use of depth sensors eliminates the latency associated with mapping, making it a viable solution for high-speed autonomous navigation in cluttered environments, outperforming both traditional modular planners and data-hungry learning-based approaches.