The convex hull of a set of points is the smallest convex set that contains the points. This article presents a practical convex hull algorithm that combines the two-dimensional Quickhull Algorithm with the general-dimension Beneath-Beyond Algorithm. It is similar to the randomized, incremental algorithms for convex hull and Delaunay triangulation. We provide empirical evidence that the algorithm runs faster when the input contains nonextreme points and that it uses less memory. Computational geometry algorithms have traditionally assumed that input sets are well behaved. When an algorithm is implemented with floating-point arithmetic, this assumption can lead to serious errors. We briefly describe a solution to this problem when computing the convex hull in two, three, or four dimensions. The output is a set of “thick ” facets that contain all possible exact convex hulls of the input. A variation is effective in five or more dimensions.

Abstract — Control allocation problems for marine vessels can be formulated as optimization problems, where the objective typically is to minimize the use of control effort (or power) subject to actuator rate and position constraints, power constraints as well as other operational constraints. In addition, singularity avoidance for vessels with azimuthing thrusters represent a challenging problem since a non-convex nonlinear program must be solved. This is useful to avoid temporarily loss of controllability in some cases. In this paper, a survey of control allocation methods for overactuated vessels are presented. I.

Abstract — The task in control allocation is to determine how to generate a specified generalized force from a redundant set of control effectors where the associated actuator control inputs are constrained, and other physical and operational constraints and objective should be met. In this paper we consider a convex approximation to a control allocation problem for a thrustercontrolled floating platform. The platform has eight rotatable azimuth thrusters and the high level controller is assumed to specify three generalized forces; surge, sway and yaw. The control allocation problem is formulated as a convex quadraticor linear program, where the constraints are dependent on the specified generalized force. The problem is solved explicitly by viewing the generalized forces as a vector of parameters and utilizing parametric programming techniques. For convex parametric quadratic programs (pQP) or parametric linear programs (pLP) with a linear parametrization of the constraints, there always exists a continuous piecewise affine (PWA) minimizer function. Consequently, the conventional on-line optimization can be replaced by a simple evaluation of a PWA function. Experimental results for a scale model of a platform are presented. It is shown how thruster failure scenarios can be handled by automatic reconfiguration of the control allocation, exploiting symmetry of the thruster configuration.