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Robust Trajectory Tracking Error Model-Based Predictive Control for Unmanned Ground Vehicles
Erkan Kayacan, Herman Ramon, Wouter Saeys
TL;DR
Autonomous tractor-trailer guidance requires control of coupled vehicle dynamics across straight and curvilinear trajectories, where conventional PID and linear MPC have limitations. This paper develops a trajectory tracking error-based model for linear MPC and combines its feedback action with feedforward and robust actions. Field experiments show tracking of both straight and curved trajectories with reported mean errors and real-time computation.
Problem
Conventional PID guidance is limited by complex vehicle dynamics, while linear MPC has difficulty tracking curvilinear trajectories and operating when the system is off-track.
Method
A trajectory tracking error-based model incorporating speed and yaw models is used to design linear MPC combined with feedforward and robust control actions.
Results
23.49 cm and 21.21 cm mean Euclidean errors were obtained for tractor and trailer on straight lines, versus 39.82 cm and 36.21 cm on curved lines; LMPC computation averaged 1.1 ms.
Takeaways & Limitations
The proposed controller tracked straight and curvilinear target trajectories with reasonable accuracy despite modeling errors and disturbances, while remaining feasible in real time.
Abstract
from arXiv · showhide
This paper proposes a new robust trajectory tracking error-based control approach for unmanned ground vehicles. A trajectory tracking error-based model is used to design a linear model predictive controller and its control action is combined with feedforward and robust control actions. The experimental results show that the proposed control structure is capable to let a tractor-trailer system track both linear and curvilinear target trajectories with low tracking error.
I. INTRODUCTION
Autonomous agricultural guidance has expanded with GPS, but conventional PID control is limited for the coupled, multi-input multi-output vehicle dynamics of tractor-trailer systems. The study therefore targets a robust linear MPC approach that retains low computation while addressing straight and curved trajectory tracking.
- GPS guidance improves agricultural navigation accuracy and supports operation without precise manual steering, including at night or in fog.
- PID-based machine guidance is limited by complex vehicle dynamics and requires guarded or adaptive tuning.
- Tractor-trailer guidance is a multi-input multi-output problem involving tractor and trailer steering, speed, positions, velocities, yaw rates, and yaw angles.
- Linear MPC can work near references and on straight lines, but reported linear methods struggle with off-track operation and curvilinear trajectories.
- NMPC can track straight and curvilinear lines but has higher computational burden, whereas LMPC is computationally efficient but less accurate on curved trajectories.
II. AUTONOMOUS TRACTOR-TRAILER SYSTEM AND KINEMATIC TRICYCLE MODEL
The experimental platform is a small autonomous tractor-trailer whose time-based trajectory is tracked using a kinematic tricycle and speed model. RTK GPS, steering sensors, actuators, and a real-time PXI controller support field operation.
- The study tracks a time-based trajectory with a small agricultural tractor-trailer operating in bumpy and wet grass-field conditions.
- The system uses RTK GPS with two antennas positioned over the tractor and trailer centers to obtain positional information.
- Electro-hydraulic steering actuators control the tractor and trailer, while an electromechanical actuator controls system speed through the hydrostat pedal.
- The tractor-trailer model is kinematic and neglects dynamic force balances, with tractor and trailer linked through a drawbar and two revolute joints.
- The model includes tractor and trailer positions, yaw angles, longitudinal speed, steering angles, hitch geometry, and hydrostat position.
III. TRAJECTORY TRACKING ERROR-BASED SYSTEM MODEL
The paper derives a new trajectory tracking error model by incorporating the vehicle’s speed and yaw models, rather than treating speed and yaw rates as direct inputs. Linearization around the reference trajectory produces a linear time-varying model with actuator-level inputs.
- Trajectory tracking is nonlinear, but linearization around the reference trajectory yields a linear time-varying system suitable for linear control.
- The new error model incorporates the speed and yaw models, making gas pedal position and steering angles its control inputs.
- The reference trajectory is represented by a reference state vector, and tracking errors are expressed in tractor- and trailer-attached frames.
- The error dynamics are derived by differentiating the error state and incorporating the tractor-trailer system model.
- The error model is linearized around zero position, yaw-difference, and hydrostat-position errors to obtain a state-space form.
- The model is controllable when either the reference longitudinal velocity or the reference yaw rates is nonzero, which is sufficient.
IV. DESIGN OF THE ROBUST TRAJECTORY TRACKING ERROR-BASED CONTROLLER
The controller combines feedback MPC with feedforward and robust control actions. Its linear constrained formulation is convex, enabling quadratic programming with substantially lower computational burden than nonlinear MPC.
- The applied control input is calculated from feedforward and robust actions combined with the feedback control action.
- The controller separately formulates feedback, feedforward, and robust control actions.
A. Feedback Control Action: Model Predictive Control
The feedback controller uses LMPC to optimize trajectory-tracking errors and control-input changes online under linear constraints, with receding-horizon implementation and lower computational burden than NMPC.
- Model formulation: The system is represented by a linear discrete-time model whose state and input matrices are obtained from the continuous-time trajectory tracking error model.The matrices are discretized using the real-time system sampling time.
- Optimization objective: The cost function uses prediction horizon Np = 8 and control horizon Nc = 3 with a 200 ms sampling time, corresponding to 1.6 s and 0.6 s horizons.The optimization penalizes input changes through weighting matrices Q and R.
- Online implementation: At each sampling time, the online controller measures or estimates the current state, solves the optimization, applies the first input, and repeats over a shifted horizon.This is the standard receding-horizon implementation described for the LMPC.
- Optimization objective: The LMPC minimizes tractor and trailer position errors on the x- and y-axes together with differences between reference and actual control inputs.Its outputs are feedback corrections rather than the actual system inputs.
- Computational properties: The LMPC optimization has linear equality and inequality constraints, making it convex and solvable by quadratic programming with substantially lower computational burden than NMPC.The comparison is attributed to LMPC’s convex formulation versus NMPC’s constrained nonlinear, non-convex formulation.
B. Feedforward Control Action
The feedforward controller derives reference control actions from the target trajectory and system model, complementing LMPC feedback corrections under stated trajectory and modeling assumptions.
- Feedforward design: Feedforward inputs are combined with LMPC outputs because the LMPC generates differences between reference and actual control variables rather than actual system inputs.The resulting applied inputs include feedforward references and feedback corrections.
- Feedforward design: The feedforward actions comprise reference steering angles and reference hydrostat position derived from the reference trajectory using the system model.Reference longitudinal velocity and yaw rates are also derived for the tractor-trailer system.
- Feedforward assumptions: The derivation assumes small steering angles and incorporates the steady-state relation between longitudinal velocity and hydrostat position.These assumptions are used to derive the feedforward actions from the system model.
- Scope: Feedforward alone can drive the tractor-trailer on the reference trajectory only without disturbances, uncertainties, or initial-state errors.Feedback and robust actions therefore supplement the calculated feedforward references in the control structure.
- Feedforward assumptions: The trajectory must be twice-differentiable, with nonzero reference velocity vr and nonzero speed-model gain K.The sign convention distinguishes forward and reverse driving directions.
C. Robust Control Action
The robust control action addresses mismatch between the linearized error model and the real system, using uncertainty feedback, invariant-set constraints, saturation, and derivative action.
- Motivation: Linearization can produce poor control performance when the system is not close to the reference, motivating a robust action that brings and keeps it near the reference.The robust term addresses mismatch between the trajectory tracking error model and the real system.
- Robust law: The robust control law uses the modeling error between the nominal linearized model and the real system together with a feedback gain K.The modeling error is explicitly calculated as the difference between the linearized model and the real system.
- Constraints: The robust design assumes the invariant set Zm is contained in the state constraints and that KZm is contained in the input constraints.Nominal state and input constraints are tightened using set differences.
- Implementation: A tanh function saturates the robust action to respect actuator limits, while the derivative of the uncertainty vector is included to reduce overshoots and oscillations.The proportional, derivative, and saturation gains define the robust control term.
- Invariant-set design: The nominal controller is computed online, while offline ancillary gains maintain system-error trajectories inside a robust control invariant set centered on the nominal trajectory.The invariant-set formulation supports robust tube-based MPC for constrained linear systems with disturbances.
V. EXPERIMENTAL RESULTS
The robust trajectory tracking controller was evaluated on a time-based 8-shaped path containing straight and curved segments. It tracked the tractor-trailer near the target without oscillatory system behavior, while curved segments produced larger errors and greater robust-control contribution.
- Reference trajectory: The time-based reference was an 8-shaped trajectory comprising two straight lines and two smooth curves with 10 m radius and curvature 0.1.The curvature is the inverse of the curve radius.
- Tracking performance: The robust controller kept the autonomous tractor-trailer system close to the target trajectory without oscillatory behaviour.Actual trajectories and zoomed views were provided for the tractor and trailer.
- Tracking error: 23.49 cm and 21.21 cm were the mean Euclidian distance errors for the tractor and trailer on straight lines.These errors were lower than the corresponding curved-line means.
- Tracking error: 39.82 cm and 36.21 cm were the mean Euclidian distance errors for the tractor and trailer on curved lines.The reported tracking error was higher on curved lines than on straight lines.
- Control signals: The total control inputs stayed within bounds, with feedback, feedforward, and robust control actions contributing to the applied control.Robust control contributed more on curved lines, whereas feedback control dominated on straight lines.
- Real-time computation: 1.1 ms was the average LMPC computation time, which was feasible for real-time operation.The reported NMPC comparison was 6.8 ms, six times larger.
VI. CONCLUSIONS
The study developed and field-evaluated a robust trajectory tracking error-based model predictive controller for an autonomous tractor-trailer system. The controller achieved reasonable tracking accuracy under modeling errors and disturbances, with 1.1 ms LMPC computation time.
- Controller and evaluation: The controller used a tube-based approach to increase robustness and was evaluated in real time for computation time and tracking accuracy.The field experiments addressed modeling errors and disturbances.
- Tracking accuracy: 23.49 cm and 21.21 cm were the mean Euclidian distance errors for the tractor and trailer on straight lines.The corresponding curved-line means were 39.82 cm and 36.21 cm.
- Tracking accuracy: 39.82 cm and 36.21 cm were the mean Euclidian distance errors for the tractor and trailer on curved lines.These values were higher than the respective straight-line errors.
- Computation time: 1.1 ms was the LMPC computation time, significantly smaller than the reported NMPC computation time.The paper reports approximately 6.8 ms for NMPC.