Trajectory Planning for Concrete Element Fabrication with Optimal Control

Functionally graded concrete (FGC) is a novel technology in the building industry, allowing for savings of up to 50% mass in standard concrete elements like slabs and beams. This is achieved by cavities in the elements interior, which are generated by placing mineral hollow spheres inside the formwork before casting. The concrete is then casted with a conveying system (CS), consisting of a pump and an extrusion unit, mounted on a manipulator. An important challenge for automation is trajectory generation for the CS and the manipulator, which accounts for physical system restrictions and the fabrication process. In this paper, the problem is described for the case of rectangular elements with hollow spheres in a simple cubic pattern. As buoyancy forces would lead to floating hollow spheres in a one-shot production, suitable layer heights are calculated by an optimization problem in a first step. Then, a boustrophedon path is planned, which neatly covers the element while avoiding the spheres. Using the element configuration, the volume demand along the path can be calculated, which serves as a cost function. This allows the formulation of an optimal control problem, which is solved with multiple shooting, in order to obtain the input trajectorys.