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Linear vs Nonlinear MPC for Trajectory Tracking Applied to Rotary Wing Micro Aerial Vehicles
Mina Kamel, Michael Burri, Roland Siegwart
TL;DR
Precise trajectory tracking is important for MAV operation under disturbances and in cluttered environments. The paper compares cascaded LMPC and full-dynamics NMPC controllers across hovering, step response, and aggressive tracking experiments. NMPC shows slightly better disturbance rejection, step response, tracking performance, and computational effort, while both controllers behave comparably in hovering.
Problem
MAVs need precise trajectory tracking for operation under external disturbances and for agile tasks such as following moving objects.
Method
The paper compares a linearized-model LMPC with a full-system-dynamics NMPC in a cascaded control architecture.
Results
NMPC shows slightly better disturbance rejection, step response, tracking performance, and computational effort, while hovering performance is comparable between controllers.
Takeaways & Limitations
Considering full system dynamics provides a modest overall advantage for MAV trajectory tracking across the evaluated experiments.
Takeaways & Limitations
The linearized vehicle model assumes small attitude angles and a heading aligned with the first inertial axis.
Abstract
from arXiv · showhide
Precise trajectory tracking is a crucial property for \acp{MAV} to operate in cluttered environment or under disturbances. In this paper we present a detailed comparison between two state-of-the-art model-based control techniques for \ac{MAV} trajectory tracking. A classical \ac{LMPC} is presented and compared against a more advanced \ac{NMPC} that considers the full system model. In a careful analysis we show the advantages and disadvantages of the two implementations in terms of speed and tracking performance. This is achieved by evaluating hovering performance, step response, and aggressive trajectory tracking under nominal conditions and under external wind disturbances.
1. INTRODUCTION
MAVs require precise, agile trajectory tracking for real-world tasks under disturbances. The paper compares linearized and full-dynamics predictive controllers within a cascaded architecture, including computational and inner-loop considerations.
- Motivation: MAV trajectory tracking matters for operations near structures, under disturbances, and for following moving objects while filming aerial footage.The motivation spans infrastructure inspection, exploration, search and rescue, forest monitoring, and personal drones.
- Motivation: Optimization-based control techniques are becoming suitable for real-time MAV control as onboard computation advances.
- Controller comparison: The study compares LMPC using a linearized MAV model with NMPC using the full system dynamics.The comparison targets performance improvement, disturbance rejection, and computation effort.
- Control architecture: Both controllers operate in a cascade with a low-level attitude controller as the inner loop and a model-based trajectory tracker as the outer loop.The architecture separates critical flight control from higher-level computation.
- Control architecture: System identification is used to account for closed-loop attitude dynamics in the trajectory-tracking controller.
Paper Contributions
The paper contributes a thorough LMPC–NMPC comparison for dynamic MAV trajectory tracking and provides open-source implementations of both controllers. The paper is organized around modeling, controller design, disturbance observation, and experiments.
- Contribution: The main contribution is a thorough comparison of LMPC and NMPC for dynamic MAV trajectory tracking.The comparison examines the benefits of including full system dynamics and its effect on the flight envelope.
- Contribution: The work includes an open-source C++ implementation of both controllers.
- Paper structure: The paper presents the MAV model, both predictive controllers, an external disturbance observer, and experimental results.
2. RELATED WORK
Related work covers attitude, feedback, hierarchical, robust, and learning-based control for MAVs. It motivates MPC through constraint handling while noting experimental, disturbance-evaluation, and computational limitations in prior approaches.
- Prior MAV controllers: A prior PID–LQ comparison found PID outperforming LQ for attitude stability, attributed to an imperfect model.
- Prior MAV controllers: Nonlinear and high-gain feedback controllers achieved strong tracking but could not guarantee state or input constraints and required careful trajectory selection.
- MPC motivation: MPC can directly include constraints in its optimization, making constraint handling important for real systems.
- NMPC limitations: A previous geometric NMPC attitude controller handled configuration recovery and propeller failure experimentally, but computation cost limited practical use.
- MPC studies: Earlier hierarchical and robust MPC studies addressed autonomous navigation or disturbances, but reported missing experimental validation or unevaluated disturbance performance.
- Learning-based robustness: A Gaussian-Process-based approach updated a linear robust controller online to reduce conservativeness under learned uncertainty.
3. MAV MODEL
The MAV model describes vehicle motion using inertial and body-fixed frames, position, velocity, orientation, and angular rate. It provides the foundation for the controllers’ force, moment, and attitude modeling.
- Model overview: The model introduces the full vehicle dynamics and the closed-loop attitude model used by the trajectory-tracking controller.
- State representation: The vehicle configuration comprises CoG position p in the inertial frame, inertial velocity v, orientation RIB ∈SO(3), and body angular rate ω.
System model
The MAV model accounts for propeller-generated thrust and drag moments, aerodynamic damping effects, and vehicle motion under external forces such as wind.
- Propeller forces: Each propeller generates thrust proportional to the square of its rotation speed and a drag-induced angular moment.The thrust and moment are parameterized by rotor speed and positive thrust and moment constants.
- Aerodynamic effects: Blade flapping and induced drag are combined into a lumped drag coefficient that adds damping in the rotor plane.These effects introduce additional force in the x-y rotor plane during dynamic maneuvers.
- Aerodynamic effects: The aerodynamic force is modeled using a drag matrix with equal x- and y-axis coefficients and zero z-axis coefficient.The model uses Kdrag = diag(kD, kD, 0) and the thrust-force z component.
- Vehicle dynamics: Vehicle motion is described using equations involving mass, inertia, propeller allocation, and external forces such as wind.The external-force term represents disturbances acting on the vehicle.
Attitude model
The trajectory controller uses a cascaded architecture with a first-order closed-loop attitude model identified from the vehicle’s inner-loop response. Heading-rate tracking is assumed instantaneous because heading does not affect position.
- Attitude model: The controller follows a cascaded approach in which an attitude controller regulates the vehicle’s inner loop.The trajectory-tracking controller models the attitude response rather than treating the vehicle as a purely kinematic system.
- Attitude model: A first-order closed-loop attitude response is identified using system identification.This identified response is used to represent the inner-loop attitude dynamics.
- Attitude dynamics: The roll and pitch dynamics use gains and time constants, while commanded heading angular velocity is tracked directly.The model explicitly defines separate roll and pitch gains and time constants.
- Model assumption: Vehicle heading angular rate is assumed to track its command instantaneously because heading angle has no effect on MAV position.This is an explicit modeling assumption in the trajectory controller.
4. LINEAR MPC
The LMPC linearizes the MAV around hover, uses a discrete linear state-space model, and repeatedly solves a constrained finite-horizon OCP with disturbance estimation and optional acceleration feed-forward.
- 4. LINEAR MPC: The LMPC uses a simplified MAV model linearized around hovering conditions, with nonlinearity compensation discussed in its formulation.The linearization assumes small attitude angles and heading aligned with the first inertial-frame axis.
- 4.1 Model Linearization: The linearization assumes small attitude angles and vehicle heading ψ = 0.This fixes the heading alignment used to obtain the simplified model.
- 4.1 Model Linearization: The control input contains heading-free roll and pitch commands and commanded thrust, with thrust assumed instantaneous because motor dynamics are fast.Roll and pitch are represented in the inertial frame to remove vehicle heading from the model.
- 4.2 Controller Formulation: After linearization and discretization, the controller uses a linear state-space model and repeatedly solves a constrained OCP.The OCP includes linear dynamics, input constraints, and initial-state conditions.
- 4.2 Controller Formulation: The OCP penalizes state and control-input errors relative to target trajectories and includes a terminal state-error penalty.The weighting matrices specify penalties for state error, input error, and terminal state error.
- 4.2 Controller Formulation: The LMPC assumes constant disturbances across the prediction horizon and applies only the first optimized control input before repeating the process.This is the stated receding-horizon implementation.
- 4.2 Controller Formulation: Dynamic-trajectory tracking can be improved by including a feed-forward reference control based on desired trajectory accelerations expressed in the MAV body frame.The feed-forward term sets the reference control input using body-frame accelerations.
5. NONLINEAR MPC
The NMPC formulates a continuous-time optimal control problem using the full MAV dynamics and solves it in receding-horizon operation for real-time control.
- 5. NONLINEAR MPC: The NMPC controller considers the full system dynamics in a continuous-time formulation.The formulation is organized around an optimal control problem and a real-time solution method.
- 5.1 Optimal Control Problem: The NMPC first defines a state vector and control input vector for the full vehicle model.These vectors provide the variables used in the subsequent optimal control problem.
- 5.1 Optimal Control Problem: The optimal control problem imposes nonlinear dynamics, admissible inputs, and an initial-state condition.The dynamics are represented by ẋ = f(x, u), with f composed of the vehicle-model equations.
- 5.1 Optimal Control Problem: The NMPC repeatedly solves the optimization problem at each time step but applies only the first control input.This receding-horizon procedure is required for real-time implementation.
- 5.2 OCP Solution: Multiple shooting discretizes dynamics and constraints over a coarse time grid and solves interval boundary-value problems with continuity constraints.An implicit fourth-order Runge–Kutta integrator forward-simulates the dynamics, producing a nonlinear program.
6. EXTERNAL DISTURBANCES ESTIMATION
The paper estimates external disturbances with an augmented-state EKF using the controller model, vehicle heading, and inner-loop attitude dynamics to support offset-free tracking.
- External disturbances estimator: The augmented-state EKF estimates external forces for offset-free trajectory tracking.The estimator includes external forces in its state.
- Hovering evaluation: Table 1 reports RMSE for individual LMPC and NMPC axes during hovering with and without external disturbances, in centimeters.The table provides the axis-wise error measure used for the hovering comparison.
- External disturbances estimator: The estimator uses the same model as the controller and includes vehicle heading angle ψ.Including heading allows estimated external forces to capture modeling error and achieve zero steady-state tracking error.
7. IMPLEMENTATION AND RESULTS
Experiments compare LMPC and NMPC on a hexacopter across hovering, step response, and aggressive tracking, including nominal and windy conditions. NMPC generally improves response, disturbance rejection, tracking, and computation effort, while nominal hovering performance remains comparable.
- Implementation: The controllers were evaluated on an Asctec NEO hexacopter using onboard optimization and state estimation infrastructure.LMPC used CVXGEN, while NMPC used ACADO with a Gauss–Newton real-time iteration scheme.
- Hovering Performance: 1.84 cm and 2.05 cm RMSE were measured for LMPC and NMPC, respectively, during nominal hovering.Both controllers showed very comparable performance because LMPC was linearized around hovering.
- Hovering Performance: 2.7 cm and 2.5 cm RMSE were measured for LMPC and NMPC, respectively, during hovering under approximately 11 m/s wind.The wind direction is indicated by yellow arrows in Figure 4.
- Step Response: 1.6 s and 1.0 s rise times were measured for LMPC and NMPC, respectively, while both had 1.98% overshoot in the x-direction step response.NMPC’s faster response is attributed to exploiting full system dynamics, especially thrust-command coupling with lateral motion.
- Aggressive Trajectory Tracking: 10.8 cm and 7.1 cm RMSE were measured for LMPC and NMPC, respectively, during aggressive polynomial trajectory tracking under external disturbances.The corresponding trajectory error and tracking plots are shown in Figures 7 and 8.
- Computation Effort: 0.45 ms versus 2.35 ms average solver time was achieved by NMPC and LMPC, respectively, during aggressive trajectory tracking.The NMPC real-time iteration scheme improved computation effort by a factor of 5.
8. CONCLUSION
The LMPC and NMPC showed comparable behavior across multiple trajectory-tracking tests, while the NMPC performed slightly better overall, including under external disturbances.
- The controllers were compared during hovering, step response, and aggressive trajectory tracking under external disturbances.
- The NMPC showed slightly better disturbance rejection capability, step response, tracking performance, and computational effort.
- An open-source implementation of both controllers is available.