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Abstract-In this paper we study the problem of computing smooth planar paths in the presence of obstacles where we have an a priori knowledge about the environment. We investigate how the smoothness of a path and the total travel time required by the path are related for paths used by a four-wheel fourgear articulated vehicle. A path is considered smooth if the variation of its curvature, i.e., the integral of the square of the derivative of curvature along the path, is minimal. Paths are defined by quartic B-splines and obstacles are represented by polygonal chains. Quartic B-splines have a continuous derivative of curvature. Obstacle-avoidance is achieved by means of the envelope of the B-splines.We present a study of eight cases based on real-world application data from the Swedish mining company LuossavaaraKiirunavaara AB (LKAB). The results indicate that a minimum curvature variation B-spline path-planning algorithm we have developed yields paths that are substantially better than the ones used by LKAB today. Our simulations shows that the new paths are up to 39% faster to travel along than the paths currently in use. They even decrease the wear on the vehicle. Preliminary results from the production at LKAB show an overall 5-10% decrease in the total time. The total time includes both travel on the path and ore loading and unloading. Note to Practitioners -This article was motivated by the problem of how to automatically produce drive-paths of high quality for autonomous transportation vehicles in mines.The vehicles are heavy (> 100 tonnes) but are still expected to run at speeds up to 20 km/h to be productive. To reach these speeds without destroying the steering gear and the mechanics of the vehicles, their paths must be smooth. It turns out that visual inspection is often not sufficient to distinguish a smooth path from a harmful one. We suggest a method for computing paths, requiring an a priori knowledge about the environment, that minimizes the amount of steering needed during a transport. The computed paths are also safe in that they guarantee that the vehicle will not collide with a tunnel wall. We present a study of eight cases based on real-world application data from the Swedish mining company Luossavaara-Kiirunavaara AB (LKAB). Our simulations show that our paths are up to 39% faster to travel along than the paths currently in use while they cause no more wear on the vehicles. Tests performed in the production at LKAB show an overall 5-10% decrease in the total time if not only travel but also loading and unloading is included.

We study the problem of computing a planar curve, restricted to lie between two given polygonal chains, such that the integral of the square of arc-length derivative of curvature along the curve is minimized. We introduce the Minimum Variation B-spline problem which is a linearly constrained optimization problem over curves defined by Bspline functions only.An empirical investigation indicates that this problem has one unique solution among all uniform quartic B-spline functions. Furthermore, we prove that, for any B-spline function, the convexity properties of the problem are preserved subject to a scaling and translation of the knot sequence defining the B-spline.

Using a set of landmarks to represent a rigid body, a rotation of the body can be determined in least-squares sense as the solution of an orthogonal Procrustes problem. We discuss some geometrical properties of the condition number for the problem of determining the orthogonal matrix representing the rotation. It is shown that the condition number critically depends on the configuration of the landmarks. The problem is also reformulated as an unconstrained nonlinear leastsquares problem and the condition number is related to the geometry of such problems. In the common 3-D case, the movement can be represented by using a screw axis. Also the condition numbers for the problem of determining the screw axis representation are shown to closely depend on the configuration of the landmarks. The condition numbers are finally used to show that the used algorithms are stable.

A hybrid algorithm consisting of a Gauss-Newton method and a second order method for solving constrained and weighted nonlinear least squares problems is developed, analyzed and tested. One of the advantages of the algorithm is that arbitrary large weights can be handled and that the weights in the merit function do not get unnecessary large when the iterates diverge from a saddle point. The local convergence properties for the Gauss-Newton method is thoroughly analyzed and simple ways of estimating and calculating the local convergence rate for the Gauss-Newton method are given. Under the assumption that the constrained and weighted linear least squares subproblems attained in the Gauss-Newton method are not too ill-conditioned, global convergence towards a rst order KKT point is proved.

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