Hazardous terrains pose a crucial challenge to mobile robots. To operate safely and efficiently, it is necessary to detect the terrain type and modify operation strategies in real-time. Fast analytical models of wheeled locomotion on deformable terrains are thus important. Based on classic terramechanics, a closed-form wheel–soil interaction model was derived by quadratic approximation of stresses along the wheel–soil interface. The bulldozing resistance and the effects of grousers were also added for more accurate prediction of wheel contact forces. A non-iterative method was proposed to estimate the entry angle, by using approximated vertical pressure acting on the wheels. The computational efficiency was improved by avoiding traditional recursive search. Real-time computation of the wheel contact forces is achieved by the terramechanics-based formula (TBF), which was developed by integrating the wheel–soil interaction model and the entry angle estimator. In addition, an automotive-inspired approach was used to integrate the TBF and the simplified vehicle dynamics model for fast simulation of mobile robots. Stability problems in numerical simulation could be avoided by this method. The above models were verified by comparing simulation results and experiment data, including single-wheel experiments and full-vehicle experiments.

We define a variant of the crossing number for an embedding of a graph G into ℝ3, and prove a lower bound on it which almost implies the classical crossing lemma. We also give sharp bounds on the rectilinear space crossing numbers of pseudo-random graphs.

A graph H is called common if the sum of the number of copies of H in a graph G and the number in the complement of G is asymptotically minimized by taking G to be a random graph. Extending a conjecture of Erdős, Burr and Rosta conjectured that every graph is common. Thomason disproved both conjectures by showing that K4 is not common. It is now known that in fact the common graphs are very rare. Answering a question of Sidorenko and of Jagger, Št'ovíček and Thomason from 1996 we show that the 5-wheel is common. This provides the first example of a common graph that is not three-colourable.

Pure Type Systems are usually described in two different ways, one that uses an external notion of computation like beta-reduction, and one that relies on a typed judgment of equality, directly in the typing system. For a long time, the question was open to know whether both presentations described the same theory. A first step towards this equivalence has been made by Adams for a particular class of Pure Type Systems (PTS) called functional. Then, his result has been relaxed to all semi-full PTSs in previous work. In this paper, we finally give a positive answer to the general question, and prove that equivalence holds for any Pure Type System.