A bounded curvature path corresponds to a $C^1$ and piecewise $C^2$ path lying in a manifold $M$. A bounded curvature path connects two elements of the tangent bundle ${TM}$ and has its curvature bounded by a positive constant. Throughout the papers: A geometric approach to shortest bounded curvature paths, Length minimising bounded curvature paths in homotopy classes (watch video), Non-uniqueness of the homotopy class of bounded curvature paths and The classification of homotopy classes of bounded curvature paths I developed techniques to answer questions about the connectivity of the spaces of planar bounded curvature paths and to establish the minimal length elements in these spaces. In 1961 in the paper On plane curves with curvature, Lester Dubins raised important questions initiating the study of the homotopy classes of bounded curvature paths. "Here we only begin the exploration, raise some questions that we hope will prove stimulating, and invite others to discover the proofs of the definite theorems, proofs that have eluded us" (compare On plane curves with curvature page 471). In my Ph.D. dissertation, I managed to close the program proposed by Lester Dubins. |

Let $\Gamma(n)$ be the space of bounded curvature paths having fixed initial and final points in $T{\mathbb R}^2$ and having winding number $n$. We define the winding number by closing up the paths using a fixed path from the final to the initial point in $T{\mathbb R}^2$. By analysing the distances between the points we obtain four conditions of distances called proximity conditions. The proximity conditions give qualitative insight to understand some of the topological features of $\Gamma(n)$, as for example, the number of homotopy classes such space contains. In the papers A geometric approach to shortest bounded curvature paths and Length minimising bounded curvature paths in homotopy classes we classified the minimal length elements in $\Gamma(n)$ for all $n$ and all initial and final elements in $T{\mathbb R}^2$. In particular, in A geometric approach to shortest bounded curvature paths we characterised the bounded curvature paths that are candidates for being of minimum length via a normalisation process. This is then followed by a reduction process performed on bounded curvature paths to obtain the well known CSC, CCC characterisation as first obtained in On curves of minimal length with a constraint on average curvature and with prescribed initial and terminal positions and tangents by Lester Dubins in 1957. In Length minimising bounded curvature paths in homotopy classes we characterised the bounded curvature paths of minimal length in $\Gamma(n)$ for all $n$. This is achieved by using some of the ideas developed in A geometric approach to shortest bounded curvature paths. The normalisation and reduction processes are intimately related to the concept of fragmentation of a path and this concept is developed in a way that leaves the winding number of the paths invariant. |

In Non-uniqueness of the homotopy class of bounded curvature paths we analysed the extent to which paths in $\Gamma(n)$ can be made homotopic via a one-parameter family of bounded curvature paths, paying special attention on the proximity condition the initial and terminal points satisfy. We concluded that under certain conditions it is not possible to continuously deform embedded bounded curvature paths lying in a planar region $\Omega$ to paths having at least one point in the complement of $\Omega$ while keeping the curvature bounded at each state of the deformation (see the grey regions in the figures above). In particular, the regions $\Omega$ are compact and paths in the homotopy class of the length minimiser cannot be homotopic via a one-parameter family of bounded curvature paths to a path outside of $\Omega$. In addition, no path in $\Omega$ can be made bounded homotopic to a path in $\Omega$ having a self-intersection. We conclude the existence of homotopy classes exclusively conformed by embedded paths. This is achieved by an applying an important principle bounded curvature paths satisfy, the S-lemma (see the S-shaped paths in the figures above). In The classification of homotopy classes of bounded curvature paths, we developed a continuity argument in order to make sure that when deforming bounded curvature paths the curvature bound is never violated. Then we put together the ideas above to obtain the classification of homotopy classes of bounded curvature paths. These ideas can be found in detail in the articles cited above. |