My research mainly focuses on Model Predictive Control (MPC). In MPC a sequence of inputs is optimized to minimize a given cost function, while satisfying constraints. Then, only the first input of the optimized input sequence is applied to the system. Subsequently, the MPC optimal control optimal control problem is solved again with updated system and environment data.

For systems and environments subject to uncertainty, it is required to account for this uncertainty within MPC. Let us consider the example of automated driving, where it is impossible to perfectly predict the future behavior of other vehicles. Collisions with other vehicles may be avoided robustly by accounting for worst-case behavior, but this limits performance. By using probabilistic constraints, conservatism may be reduced by neglecting highly unlikely events. However, this comes with a non-zero probability of collision.

My work aims at developing new MPC methods, which enable efficient and safe planning in uncertain environments. Many of the proposed MPC methods are suitable for trajectory planning in automated driving.

Google Scholar profile

Stochastic Model Predictive Control (for Automated Driving)

Model Predictive Control (MPC) has proved to be effective for trajectory planning. Regarding the example of autonomous driving, an optimization problem is solved to generate a vehicle trajectory for the near future. After the optimized input is applied to only the next time step, the optimization problem is solved again – but now the horizon is shifted by one step. In Stochastic MPC, probabilistic constraints are formulated to efficiently account for system uncertainty.

As driving takes place in environments that are not deterministic, it is necessary to account for these uncertainties, e.g., due to multiple possible maneuvers by other vehicles. For this purpose, we are working on Stochastic MPC for autonomous driving, which included a cooperation with BMW.

One of the major outcomes of our research is a safe Stochastic MPC framework. By combining Stochastic MPC with failsafe trajectory planning (FT), the advantage of optimistic vehicle behavior with Stochastic MPC is combined with the safety guarantees of failsafe trajectory planning. This approach (SMPC+FT) yields efficient and safe vehicle behavior. (see early version video here)

Further contributions include interactive autonomous racing, grid-based Stochastic MPC, and Stochastic MPC specifically accounting for maneuver uncertainty and maneuver execution uncertainty of other vehicles.

Compared to SMPC, failsafe SMPC (SMPC+FT) guarantees safety even in challenging situations.

Brüdigam T, Zhan, J, Wollherr D, Leibold M. Collision Avoidance with Stochastic Model Predictive Control for Systems with a Twofold Uncertainty Structure. 24th IEEE International Conference on Intelligent Transportation Systems. 2021. [PDF]

Brüdigam T, Capone A, Hirche S, Wollherr D, Leibold M. Gaussian Process-based Stochastic Model Predictive Control for Overtaking in Autonomous Racing. 2021. arXiv: 2105.12236. [PDF]

Brüdigam T, Olbrich M, Wollherr D, Leibold M. Stochastic Model Predictive Control with a Safety Guarantee for Automated Driving. IEEE Transactions on Intelligent Vehicles, pages 1–1, 2021. doi: 10.1109/TIV.2021.3074645 [PDF]

Brüdigam T, Olbrich M, Wollherr D, Leibold M. Stochastic Model Predictive Control with a Safety Guarantee for Automated Driving: Extended Version. 2020. arXiv: 2009.09381. [PDF]

Brüdigam T, di Luzio F, Pallottino L, Wollherr D, Leibold M. Grid-Based Stochastic Model Predictive Control for Trajectory Planning in Uncertain Environments. 23rd IEEE International Conference on Intelligent Transportation Systems. 2020. doi: 10.1109/ITSC45102.2020.9294388 [PDF]

Brüdigam T, Olbrich M, Leibold M., Wollherr D. Combining Stochastic and Scenario Model Predictive Control to Handle Target Vehicle Uncertainty in an Autonomous Driving Highway Scenario. 21st IEEE International Conference on Intelligent Transportation Systems. 2018. doi: [PDF]

Minimizing Constraint Violation Probability in MPC

System uncertainty can be handled in different ways within MPC. Robust MPC, as the name indicates, robustly accounts for the uncertainty, often resulting in conservative solutions. While Stochastic MPC yields efficient solutions, a small probability of constraint violation is permitted, based on a predefined risk parameter.

In contrast to Robust MPC and Stochastic MPC, we propose an MPC method (CVPM-MPC), which minimizes the probability that a constraint is violated while also optimizing other control objectives. The proposed method is capable of dealing with changing uncertainty and does not require choosing a risk parameter. CVPM-MPC can be regarded as a link between Robust and Stochastic MPC.

When the bound of possible next-step locations for the obstacle (orange) increases, the controlled agent (blue) performs an evasion maneuver with minimal risk.

Brüdigam T, Gaßmann V, Wollherr D, Leibold M. Constraint Violation Probability Minimization in Model Predictive Control. Int J Robust Nonlinear Control. 2021;1–33. doi: 10.1002/rnc.5636 [PDF]

Extending the MPC Prediction Horizon

A long prediction horizon in MPC is often beneficial. However, a long prediction horizon with a detailed prediction model quickly becomes computationally challenging. We provide different adaptations to MPC in order to take advantage of long prediction horizons while keeping the computational effort manageable.

These adaptations are based on two ideas:

  1. A simple system model is used for long-term predictions (with a detailed short-term prediction model).

  2. The sampling time is increased along the horizon, resulting in a non-uniformly spaced MPC prediction horizon.

In addition, these adaptations are combined with methods from Robust MPC and Stochastic MPC to account for potential model uncertainty and disturbances.

Brüdigam T, Prader D, Wollherr D, Leibold M. Model Predictive Control with Models of Different Granularity and a Non-uniformly Spaced Prediction Horizon. American Control Conference (ACC). 2021.

Brüdigam T, Teutsch J, Wollherr D, Leibold M. Combined Robust and Stochastic Model Predictive Control for Models of Different Granularity. 21st IFAC World Congress. 2020. doi: 10.1016/j.ifacol.2020.12.515 [PDF]

RMPC on a short horizon (red prediction) is too conservative to pass a moving obstacle.

(Red obstacle boundary for nominal states only)

Extending the horizon with a simple model and chance constraints (pink) allows passing.

(Red obstacle boundary for nominal states only)

Legible Model Predictive Control for Automated Driving

Autonomous vehicles can assist other traffic participants in correctly predicting the autonomous vehicle’s future maneuvers. This cooperative behavior has a positive effect on traffic flow, while not increasing risk. For this reason, we developed and investigate Legible MPC to generate readable vehicle trajectories.

Brüdigam T, Ahmic K, Leibold M, Wollherr D. Legible Model Predictive Control for Autonomous Driving on Highways. 6th IFAC Conference on Nonlinear Model Predictive Control. 2018. doi: 10.1016/j.ifacol.2018.11.016 [PDF]

Interpretation of the maneuver of the automated red vehicle.