Automated vehicles require efficient and safe planning to maneuver in uncertain environments. Largely this uncertainty is caused by other traffic participants, e.g., surrounding vehicles. Future motion of surrounding vehicles is often difficult to predict. Whereas robust control approaches achieve safe, yet conservative motion planning for automated vehicles, Stochastic Model Predictive Control (SMPC) provides efficient planning in the presence of uncertainty. Probabilistic constraints are applied to ensure that the maximal risk remains below a predefined level. However, safety cannot be ensured as probabilistic constraints may be violated, which is not acceptable for automated vehicles. Here, we propose an efficient trajectory planning framework with safety guarantees for automated vehicles. SMPC is applied to obtain efficient vehicle trajectories for a finite horizon. Based on the first optimized SMPC input, a guaranteed safe backup trajectory is planned using reachable sets. This backup is used to overwrite the SMPC input if necessary for safety. Recursive feasibility of the safe SMPC algorithm is proved. Highway simulations show the effectiveness of the proposed method regarding performance and safety.

In order to satisfy the demand for the high functionality of future microdevices, research on new concepts for multistable microactuators with enlarged working ranges becomes increasingly important. A challenge for the design of such actuators lies in overcoming the mechanical connections of the moved object, which limit its deflection angle or traveling distance. Although numerous approaches have already been proposed to solve this issue, only a few have considered multiple asymptotically stable resting positions. In order to fill this gap, we present a microactuator that allows large vertical displacements of a freely moving permanent magnet on a millimeter-scale. Multiple stable equilibria are generated at predefined positions by superimposing permanent magnetic fields, thus removing the need for constant energy input. In order to achieve fast object movements with low solenoid currents, we apply a combination of piezoelectric and electromagnetic actuation, which work as cooperative manipulators. Optimal trajectory planning and flatness-based control ensure time- and energy-efficient motion while being able to compensate for disturbances. We demonstrate the advantage of the proposed actuator in terms of its expandability and show the effectiveness of the controller with regard to the initial state uncertainty.

A current goal for microactuators is to extend their usually small working ranges, which typically result from mechanical connections and restoring forces imposed by cantilevers. In order to overcome this, we present a bistable levitation setup to realise free vertical motion of a magnetic proof mass. By superimposing permanent magnetic fields, we imprint two equilibrium positions, namely on the ground plate and levitating at a predefined height. Energy-efficient switching between both resting positions is achieved by the cooperation of a piezoelectric stack actuator, initially accelerating the proof mass, and subsequent electromagnetic control. A trade-off between robust equilibrium positions and energy-efficient transitions is found by simultaneously optimising the controller and design parameters in a co-design. A flatness-based controller is then proposed for tracking the obtained trajectories. Simulation results demonstrate the effectiveness of the combined optimisation.

This work presents an analytical modeling approach for the semi-automated design of magnetic actuators with multiple pre-defined stable positions. The analyzed system allows for large out-of-plane displacements of a freely movable permanent magnetic plunger that is actuated by electromagnetic coils for switching between multiple stable resting positions, in which the plunger can be held without the need for further continuous energy consumption. These equilibrium positions are generated by the reluctance force between several mu-metal rings and the permanent magnet. Instead of computationally expensive finite element modeling, a phenomenological model is derived that describes the force contribution of single mu-metal rings depending on their dimensions. We exploit this analytical computation to accurately describe arbitrary multi-ring configurations, enabling an efficient analysis and design procedure. This model then provides the foundation for a requirements-driven design approach, in which pre-defined stable positions are used to derive the corresponding mu-metal ring configuration. In this context, the robustness of the system in terms of force barriers between the stable states is also considered. This eventually enables an automated design process that significantly reduces the complexity compared to manually designing application-specific setups. The modeling and design approaches are validated experimentally, showing good agreement between the predicted and measured force-displacement curves.

The performance of micromirrors in terms of their maximum deflection is often limited due to mechanical constraints in the design. To increase the range of achievable deflection angles, we present a novel concept in which a free-lying sphere with a flat side as reflector can be rotated. Due to the large forces needed to move the sphere, multiple electrostatic actuators are used to cooperatively rotate the sphere in iterative steps by impacts and friction. A parameterized system-level model of the configuration is derived, which considers arbitrary multi-contact scenarios and can be used for simulation, analysis, and control design purposes. Due to the complex, indirect relation between the actuator voltages and the sphere motion, model-based numerical optimization is applied to obtain suitable system inputs. This results in rotation sequences, which can be understood as a sequence of motion primitives, thus transforming the continuous time model into an abstract discrete time model. Based on this, we propose a feedback control strategy for trajectory following, considering model uncertainties by a learning scheme. High precision is achieved by an extension controlling the angular change of each rotation step. The suitability of the overall approach is demonstrated in simulation for maximum angles of 40°, achieving angular velocities of approximately 10°/s.