# Applications

## Selected Examples

The following examples have been investigated by the Engineering Mathematics group of the Department of Aerospace Engineering at the Bundeswehr University Munich.

## Optimal Obstacle Avoidance Maneuver

 For a predefined setting, an optimal obstacle avoidance maneuver (double lane change) was computed using OCPID-DAE1. The derived control is then implemented on the model car.

## Online NMPC with obstacle avoidance

 The video shows the online control of a scale car with nonlinear model-predictive control (NMPC) and obstacle avoidance. The position is obtained by an indoor GPS system. The online optimization within the NMPC is done by the software OCPID-DAE1.

## Distributed Model-Predictive Control with Car-to-Car Communication

 Modern vehicles offer a number of passive and active driver assistance systems, which shall support the driver in crucial situations. The degree of automation of such systems increases continuously and eventually ends in systems that act fully autonomously within given bounds. Such assistance systems become more and more important in controlling cars in all-day road traffic. In future cars will communicate and negotiate driving strategies. The following video illustrates a distributed model-predictive control algorithm for autonomous cars in typical traffic scenarios using car-to-car communication. The control algorithm uses a dynamic hierarchy to coordinate the cars.

## Ā Docking Maneuver to a Tumbling Target

 As the number of uncontrollable objects in low earth orbit is rising, the thread of collisions and thus the breakdown of working satellites becomes worth analyzing. Consequently, space debris removal becomes an issue. In this study an optimal docking maneuver of a service satellite to an uncontrolled tumbling target is modeled and solved numerically. After deriving the system dynamics, we introduce boundary conditions to ensure a safe and realizable maneuver and a general Bolza type cost functional to incorporate different optimization goals. In order to solve the resulting problem, we transform the dynamics to a set of diļ¬erential algebraic equations involving quaternions.

## Automatic testdrive along a sloped road

 The animation shows the automatic testdrive along a sloped road. The mathematical model is based on the single track model, which was augmented by dynamic tyre loads and force terms taking into account the longitudinal and lateral slope of the road. The trajectory was computed using OCPID-DAE1 using a model-predictive control scheme.

## Dynamic Robot Interaction

 Automatic path planning tasks in robotics can be modeled by suitable optimal control problems. A particular challenge is the computation of optimal paths of robots that interact dynamically. We use multiple phases to model the maneuver, i.e. an approach phase, an interaction phase, and a return phase. This multiphase optimal control problem is then transformed by standard techniques to a single stage optimal control problem, which can be solved by a direct shooting method.

## Collision Avoidance with KUKA youBot

 Automatic path planning tasks in robotics can be modeled by suitable shortest path problems. A particular challenge is the computation of optimal paths in the presence of obstacles. For the detection of collisions of geometric bodies we use a linear programming approach, which exploits the Lemma of Gale.

## Object Detection with XTION PRO LIVE and KUKA youBot

 This video shows an KUKA youBot detecting a red ball with a stereo camera module XTION PRO LIVE.The inverse positioning is realized using the sqpfiltertoolbox.

## youBot High Precision Marking

 The video shows an application with the KUKA youBot robot. The task is to mark predefined positions automatically with a high precision. The high accuracy is achieved with an external laser based positioning system.