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Nonlinear MPC for Collision Avoidance and Controlof UAVs With Dynamic Obstacles

Björn Lindqvist, Sina Sharif Mansouri, Ali-akbar Agha-mohammadi, George Nikolakopoulos

arXiv:2008.00792v1cs.RO

TL;DR

The paper addresses UAV navigation among dynamic obstacles, where static-obstacle assumptions are insufficient for complex environments. It couples trajectory classification and prediction with NMPC, using PANOC and OpEn for constrained optimization. The scheme produced collision-free paths in the considered experiments while solving online within the 50 ms real-time restriction, although it depends on accurate trajectory classification and prediction.

  • Problem

    UAV obstacle-avoidance research often assumes static environments, leaving limited treatment of moving obstacles despite UAV interaction with vehicles, robotic agents, and inhabitants.

  • Method

    The method couples dynamic obstacle avoidance with NMPC, classifies obstacle trajectories, predicts future positions from discrete dynamics, and feeds those predictions to PANOC-based optimization through OpEn.

  • Results

    The proposed scheme successfully provided collision-free paths in all considered cases and solved the online optimization problem within the required 50 ms real-time restriction.

  • Takeaways & Limitations

    The control-layer NMPC framework supports online collision avoidance for the dynamic-obstacle scenarios evaluated in the laboratory experiments.

  • Takeaways & Limitations

    Performance relies on trajectory classification and prediction; prediction failure or large errors could cause the UAV to ignore an obstacle on a collision course.

Abstract

from arXiv · show

This article proposes a Novel Nonlinear Model Predictive Control (NMPC) for navigation and obstacle avoidance of an Unmanned Aerial Vehicle (UAV). The proposed NMPC formulation allows for a fully parametric obstacle trajectory, while in this article we apply a classification scheme to differentiate between different kinds of trajectories to predict future obstacle positions. The trajectory calculation is done from an initial condition, and fed to the NMPC as an additional input. The solver used is the nonlinear, non-convex solver Proximal Averaged Newton for Optimal Control (PANOC) and its associated software OpEn (Optimization Engine), in which we apply a penalty method to properly consider the obstacles and other constraints during navigation. The proposed NMPC scheme allows for real-time solutions using a sampling time of 50 ms and a two second prediction of both the obstacle trajectory and the NMPC problem, which implies that the scheme can be considered as a local path-planner. This paper will present the NMPC cost function and constraint formulation, as well as the methodology of dealing with the dynamic obstacles. We include multiple laboratory experiments to demonstrate the efficacy of the proposed control architecture, and to show that the proposed method delivers fast and computationally stable solutions to the dynamic obstacle avoidance scenarios.

I. INTRODUCTION

UAV obstacle avoidance must address moving obstacles in complex environments, motivating a control-layer NMPC approach that predicts obstacle trajectories and supports online collision avoidance.

  • Motivation: Dynamic obstacles matter because UAVs increasingly operate near moving vehicles, robotic agents, and inhabitants in complex environments.Earlier scenarios often assumed static surroundings, but urban and field operations require interaction with constantly moving obstacles.
  • Proposed approach: The proposed approach feeds predicted obstacle trajectories directly into NMPC to generate collision-free paths online.The method couples dynamic collision avoidance with the control layer and uses obstacle prediction as an optimizer parameter.
  • Optimization: PANOC and OpEn provide the nonlinear optimization framework, with low computational and memory requirements suited to real-time NMPC.OpEn uses a Newton-type method, simple algebraic operations, and a penalty method for equality constraints.
  • Trajectory prediction: The trajectory predictor classifies obstacles and models their future motion using discrete dynamics within the NMPC prediction horizon.The paper describes classification-based prediction as a way to represent changes in obstacle motion during optimization.
  • Evaluation: The paper evaluates the framework through multiple laboratory experiments covering different dynamic-obstacle avoidance scenarios.These experiments are presented as demonstrations of the method’s efficacy and applicability.

C. Outline

The paper introduces UAV coordinate frames and a yaw-compensated global representation, then presents the adopted kinematic model and its state variables before proceeding to the remaining control formulation.

  • C. Outline: The remaining outline covers the objective function, obstacle constraints and trajectories, optimization framework, trajectory classification, experiments, and results discussion.These topics are listed as the subsequent sections of the article.
  • A. UAV Kinematics: The UAV model uses global-frame position and linear velocity together with roll and pitch angles to describe its six degrees of freedom.The states are defined in a yaw-compensated global frame, with position p, velocity v, and attitude variables φ and θ.
  • A. UAV Kinematics: Figure 1 defines the world and body coordinate frames used to represent the UAV.W denotes the world frame and B denotes the body frame.

B. Cost Function

The NMPC combines set-point tracking and smooth control with fully parameterized dynamic-obstacle constraints. Obstacle positions, size, and safety margins enter the prediction across future time steps.

  • Cost function: The NMPC cost penalizes state-reference deviation, input deviation from hovering, and successive input changes.The three terms use positive-definite weight matrices for states, inputs, and input rates.
  • Obstacle constraints: The safety radius increases linearly along the prediction horizon to account for growing trajectory uncertainty.The paper attributes this need to imperfect measurements and prediction models at longer prediction distances.
  • Obstacle constraints: Obstacle constraints use spherical geometry with obstacle position, obstacle radius, and an additional safety radius.The constraint is zero when the UAV remains outside the sphere.
  • Obstacle constraints: The obstacle trajectory is supplied as a parameter and checked at every predicted UAV position over the horizon.The formulation remains agnostic to the method used to predict the obstacle trajectory.
  • Control-rate constraints: Control-rate constraints bound successive changes in the reference roll and pitch inputs.Upper and lower bounds limit aggressive or oscillatory control behavior when avoiding an incoming obstacle.

2) Control Input Rate:

The controller limits how quickly the reference roll and pitch inputs may change between successive prediction steps.

  • Control Input Rate: Reference-angle rate constraints impose maximum per-step changes on φref and θref.The same constraint structure is applied to θ, using separate maximum changes for roll and pitch.

3) Input constraints:

The input formulation combines hard actuator and attitude bounds with PANOC-based constrained optimization using OpEn and quadratic penalties for equality constraints.

  • Input constraints: Hard bounds constrain the reference roll, reference pitch, and thrust inputs to reflect attitude-stabilization and thrust limits.The control vector is restricted to feasible values for a real UAV.
  • Optimization: The NMPC is solved with PANOC, while OpEn provides the associated framework for general parametric optimization.Equality constraints are handled using a penalty method.
  • Optimization: The NMPC is formulated for multiple spherical obstacles with parameterized trajectories.The formulation combines the cost and constraints into an optimization problem for Ns obstacles.
  • Optimization: The optimization parameter includes initial conditions, references, and the obstacle trajectory.Single shooting uses the control sequence as the decision variable and input constraints define its feasible set.

E. Trajectory Classification

Obstacle trajectories are classified from recent position and velocity measurements, then propagated to generate the future trajectory supplied to the NMPC. The paper considers linear, projectile, and static motion classes.

  • Trajectory classes: The trajectory classifier considers linear motion, projectile motion, and static obstacles, although the NMPC formulation permits arbitrary trajectory shapes.The selected classes support direct prediction of future obstacle positions when the trajectory class can be identified.
  • Prediction models: Projectile prediction includes linear aerodynamic damping, ignores buoyancy, and is discretized with forward Euler at the controller sampling time.The projectile model also includes simplified ground bouncing with a coefficient of restitution.
  • Classification: Classification compares recent measured position and velocity terms with backwards predictions generated from the candidate trajectory models.The comparison starts from the current measured obstacle state.
  • Classification: The class with the lowest trajectory error is selected for future prediction.The error is evaluated for all three trajectory classes using the measured history.
  • Online prediction: The classification runs after each new obstacle measurement, allowing a single obstacle’s predicted trajectory class to change during motion.Future positions are generated by iterating the selected discrete prediction model.
  • Obstacle parameterization: Obstacle radii are provided by the operator, while predicted positions are computed over the NMPC horizon from the current obstacle state.The full obstacle parameter includes obstacle radius, safety radius, and predicted positions.

A. Experimental Set-up

Experiments use a motion-captured Crazyflie platform, with obstacle and UAV states supplied to an NMPC controller configured for short-horizon prediction. The evaluation includes repeated obstacle-avoidance trials, including incoming projectiles.

  • The Crazyflie 2.0 Nano Quadcopter was selected because its small, lightweight platform improves safety when obstacles are thrown at the UAV.
  • Vicon tracks UAV and obstacle states, while a complementary filter estimates velocities for the NMPC.The NMPC uses UAV state (p, v, θ, φ) and obstacle state (pobs, vobs).
  • The controller uses a 40-step horizon with Ts = 50 ms, corresponding to a two-second prediction.The horizon balances computational complexity against allowing time to avoid collisions.
  • Obstacle safety radius increases linearly from 0.0 m initially to 0.2 m at the end of the horizon, with obstacle radius specified per experiment.
  • Each proposed-method experiment was repeated 5-10 times without collisions, but reported data used only the best-looking trajectories.
  • The projectile experiment asks the UAV to hold position while avoiding an incoming thrown obstacle.It evaluates methods that do not predict the obstacle trajectory.

1) Position hold while avoiding projectile:

The predictive NMPC identifies obstacle motion and initiates avoidance using predicted future positions, succeeding in projectile and pedestrian scenarios while maintaining real-time solver behavior. Baseline methods without trajectory prediction fail against the projectile, and the experiment exposes a brief classification failure and input delay.

  • Projectile avoidance: Methods without trajectory prediction fail to avoid the projectile because they react too late and may maneuver along its future path.Both the potential-field method and static-obstacle NMPC avoid slowly moving obstacles but collide with the projectile.
  • Projectile avoidance: 0.51 m was the proposed controller’s minimum distance from the projectile, compared with an approximately 0.17 m initial distance that would result in collision.The obstacle radius was set to 0.4 m.
  • Projectile avoidance: The classifier correctly identifies projectile motion and triggers avoidance in time, with solver time peaking at 40 ms.The controller computes avoidance inputs as soon as the trajectory is classified as projectile motion.
  • Observed limitations: The projectile classifier briefly labels the trajectory as linear at exactly one second, while remote control introduces an approximately 0.1-0.15 s delay between input and state change.
  • Control architecture: The predicted trajectory is generated from measured obstacle motion and supplied to the NMPC together with solver parameters.The NMPC outputs φref, θref, and T for the low-level attitude controller.
  • Pedestrian avoidance: During the pedestrian scenario, the controller begins avoidance around 0.4 s, maintains a 0.59 m minimum distance, and reaches a maximum solver time of 29 ms.The pedestrian begins moving at 0.3 s toward the UAV on a direct collision course.

3) Bounce condition:

The bouncing-ball experiment evaluates dynamic obstacle avoidance when the obstacle trajectory includes a bounce. The proposed method maintains a close safety distance while solving within real-time limits.

  • Bounce condition:: 0.38 m was the minimum distance during the bouncing-ball experiment, indicating a small constraint violation relative to the configured safety radius.The authors attribute the small violation to solver tolerances and imperfect measurements.
  • Bounce condition:: 33 ms was the peak solver time in the bouncing-ball experiment.
  • Bounce condition:: The multi-obstacle experiment maintained minimum distances of 0.45 m and 0.42 m while avoiding an approaching UAV and projectile simultaneously.Its solver time peaked at 35 ms, similar to the single-obstacle case.

4) Multiple Obstacles:

The proposed controller handles multiple dynamic obstacles by combining trajectory classification with NMPC constraints. Experiments report collision-free paths, maintained safety distances, and solution times within the 50 ms real-time requirement, while highlighting reliance on prediction accuracy and scalability as open boundaries.

  • 4) Multiple Obstacles:: The proposed NMPC and trajectory-classification scheme produced collision-free paths in all considered cases and solved within the required 50 ms without violating obstacle or input constraints.
  • 4) Multiple Obstacles:: The framework predicts dynamic obstacles through a classification scheme, while its NMPC formulation permits parameterized obstacle trajectories.
  • 4) Multiple Obstacles:: The method’s performance relies on trajectory classification and explicit prediction of future obstacle positions, which may fail when prediction errors are too large.The paper reports that the trajectory was momentarily classified incorrectly in the limited study.
  • 4) Multiple Obstacles:: Further analysis is needed on classification performance in more difficult scenarios and on how NMPC complexity scales with additional obstacles.
  • 4) Multiple Obstacles:: The solver-time figure concerns the four obstacle-avoidance scenarios, while the prediction-error figure identifies the lowest-error class used for obstacle-trajectory prediction.
  • 4) Multiple Obstacles:: The authors frame control-layer collision avoidance as a potential path toward UAV operation in urban dynamic environments where safety is important.
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