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A transition state for a Hamiltonian system is a closed, invariant, oriented, codimension-2 submanifold of an energy-level that can be spanned by two compact codimension-1 surfaces of unidirectional flux whose union, called a dividing surface, locally separates the energy-level into two components and has no local recrossings. For this to happen robustly to all smooth perturbations, the transition state must be normally hyperbolic. The dividing surface then has locally minimal geometric flux through it, giving an upper bound on the rate of transport in either direction. Transition states diffeomorphic to $\mathbb S^{2m-3}$ are known to exist for energies just above any index-1 critical point of a Hamiltonian of $m$ degrees of freedom, with dividing surfaces $\mathbb S^{2m-2}$. The question addressed here is what qualitative changes in the transition state, and consequently the dividing surface, may occur as the energy or other parameters are varied? We find that there is a class of systems for which the transition state becomes singular and then regains normal hyperbolicity with a change in diffeomorphism class. These are Morse bifurcations. Various examples are considered. Firstly, some simple examples in which transition states connect or disconnect, and the dividing surface may become a torus or other. Then, we show how sequences of Morse bifurcations producing various interesting forms of transition state and dividing surface are present in reacting systems, by considering a hypothetical class of bimolecular reactions in gas phase., 36 pages, final version - some improvements to clarity and notation, fixed typos, added some references

VDI article 2617 specifies characteristics to describe the accuracy of articulated arm coordinate measuring machines (AACMMs) and outlines procedures for checking them. However the VDI prescription was written with a former generation of machines in mind: manual arms exploiting traditional touch probe technologies. Recent advances in metrology have given rise to noncontact laser scanning tools and robotic automation of articulated arms – technologies which are not adequately characterised using the VDI specification. In this paper we examine the “guidelines” presented in VDI 2617, finding many of them to be ambiguous and open to interpretation, with some tests appearing even to be optional. The engineer is left significant flexibility in the execution of the test procedures and the manufacturer is free to specify many of the test parameters. Such flexibility renders the VDI tests of limited value and the results can be misleading. We illustrate, with examples using the Nikon RCA, how a liberal interpretation of the VDI guidelines can significantly improve accuracy characterisation and suggest ways in which to mitigate this problem.\ud We propose a series of stringent tests and revised definitions, in the same vein as VDI 2617 and similar US standards, to clarify the accuracy characterisation process. The revised methodology includes modified acceptance and reverification tests which aim to accommodate emerging technologies, laser scanning devices in particular, while maintaining the spirit of the existing and established standards. We seek to supply robust re-definitions for the accepted terms “zero point” and “useful arm length”, pre-supposing nothing about the geometry of the measuring device.\ud We also identify a source of error unique to robotised AACMMs employing laser scanners – the forward-reverse pass error. We show how eliminating this error significantly improves the repeatability of a device and propose a novel approach to the testing of probing error based on statistical uncertainty.

In partnership with Metris UK we discuss the utilisation of modelling and simulation methods in the development of a revolutionary 7-axis Robot CMM Arm (RCA). An offline virtual model is described, facilitating pre-emptive collision avoidance and assessment of optimal placement of the RCA relative to scan specimens. Workspace accessibility of the RCA is examined under a range of geometrical assumptions and we discuss the effects of arbitrary offsets resulting from manufacturing tolerances. Degeneracy is identified in the number of ways a given pose may be attained and it is demonstrated how a simplified model may be exploited to solve the inverse kinematics problem of finding the “correct” set of joint angles. We demonstrate how the seventh axis may be utilised to avoid obstacles or otherwise awkward poses, giving the unit greater dexterity than traditional CMMs. The results of finite element analysis and static force modelling on the RCA are presented which provide an estimate of the forces exerted on the internal measurement arm in a range of poses.

We present a new stochastic framework for studying ship capsize. It is a synthesis of two strands of transition state theory. The first is an extension of deterministic transition state theory to dissipative non-autonomous systems, together with a probability distribution over the forcing functions. The second is stochastic reachability and large deviation theory for transition paths in Markovian systems. In future work we aim to bring these together to make a tool for predicting capsize rate in different stochastic sea states, suggesting control strategies and improving designs., 8 pages, 3 figures, paper for STAB&S 2021 proceedings