Verification of control systems often involves the computation of reachable sets. In this context, reachable sets are used to verify that a system is capable of reaching a desired set of states or that it will not reach a set of undesired states. This contribution presents a novel method for computing convex inner approximations of reachable sets for non-linear input-affine systems with constrained inputs. The inner approximation is obtained using the reachable set of the linearized system that is shrunk by a factor depending on the linearization errors. The proposed method is integrated into a motion planning concept for mobile robots. In this way, the motion planning concept guarantees, that motions that are planned based on a simplified model of a robot can be reached by an accurate model of the robot. The application example demonstrates the computational efficiency of the proposed method as it is utilized to compute inner approximated reachable sets of a system that consists of 18 states and features eight inputs. 

Efficient Computation of Inner Approximations of Reachable Sets for a Verified Motion Planning Concept

Over-actuated vehicles featuring all-wheel inde-pendent drive and all-wheel independent steering with extended steering angles are able to achieve benefits for logistic and transportation systems due to the improved maneuverability. For such multi-wheeled systems, the paper at hand proposes a model predictive reference generation (mprg) of the steering angles and the rotational speeds of the vehicle's wheels. As an example system, a four-wheel drive and four-wheel steered electric vehicle is discussed. Based on the state of the art as well as previously published results [1], requirements of the algorithm are specified comprehensively. Hence, the proposed algorithm satisfies hard constraints due to limited steering angles and considers the actuator dynamics leading to smooth references for the underlying low-level controllers. Further-more, wear and tear of the tires are reduced and desired final states can be specified, such as turning the wheels back to their initial directions during parking procedures. From a control perspective, control signals given by motion control as well as control allocation are tracked without restrictions on the wheel's translational velocities. Therefore, the weightings of the algorithm's cost function are time-varying depending on the vehicle's velocity and an estimation of the wheel's current slip. Results are discussed with a slalom and a parking maneuver requiring steering angles that are larger as achieved by a conventional vehicle as well as the ISO double lane change challenging the velocity independence of the approach.

Model Predictive Reference Generation of Wheel-Individually Controlled Vehicles

Formal verification of dynamic control systems often involves reachability analysis to ensure safety and performance characteristics. Hereby, Hamilton-Jacobi-based methods are beneficial as they can be applied to non-linear, continuous systems under the influence of bounded disturbances. Furthermore, they can consider input- and state constraints. These benefits come with the computational effort of solving a Hamilton-Jacobi partial differential equation. State-of-the-art methods numerically determine a viscosity solution at the vertices of a static grid that is used to discretize the state space. This becomes particularly costly if the reachable set propagates fast and needs to be determined precisely, as this requires a grid of many vertices. This contribution proposes a method that computes the solution successively on small adaptive grids instead of on one static grid to reduce the computational effort of Hamilton-Jacobi reachability analysis. Using the proposed method, changes between grids can be performed in an outer or inner approximative manner. The performance of the proposed method is demonstrated in a numerical example computing a forward reachable set of a Dubins car model. While increasing the accuracy of the resulting set, the method proposed saves 73 % of computation time, 76 % of average memory usage, and 43 % of maximum memory usage in the presented scenario.

Time and Memory-Efficient Computation of Hamilton-Jacobi Reachable Sets Based on a Level Set Method Employing Adaptive Grids

Captivity-Escape Games as a Means for Safety in Online Motion Generation

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Model Predictive Reference Generation of Wheel-Individually Controlled Vehicles

Time and Memory-Efficient Computation of Hamilton-Jacobi Reachable Sets Based on a Level Set Method Employing Adaptive Grids

Efficient Computation of Inner Approximations of Reachable Sets for a Verified Motion Planning Concept

Captivity-Escape Games as a Means for Safety in Online Motion Generation