Mathematicians crack the geometry of smooth curves, opening doors for robotics
Researchers have solved a longstanding mathematical problem about creating smooth, efficient curved paths without sharp reversals—the kind needed for robotic arms, autonomous vehicles, and surgical instruments. The explicit formulas they derived could let engineers design safer, more precise movement systems that avoid sudden direction changes that damage equipment or harm delicate objects.
Originaltitel: On Sub-Riemannian Geodesics in SE(3) Whose Spatial Projections do not Have Cusps
<p>We consider the problem P (c u r v e) of minimizing for a curve x in with fixed boundary points and directions. Here, the total length Laeyen0 is free, s denotes the arclength parameter, kappa denotes the absolute curvature of x, and xi amp;gt; 0 is constant. We lift problem P (c u r v e) on to a sub-Riemannian problem P (m e c) on SE(3)/({0}xSO(2)). Here, for admissible boundary conditions, the spatial projections of sub-Riemannian geodesics do not exhibit cusps and they solve problem P (c u r v e) . We apply the Pontryagin Maximum Principle (PMP) and prove Liouville integrability of the Hamiltonian system. We derive explicit analytic formulas for such sub-Riemannian geodesics, relying on the co-adjoint orbit structure, an underlying Cartan connection, and the matrix representation of SE(3) arising in the Cartan-matrix. These formulas allow us to extract geometrical properties of the sub-Riemannian geodesics with cuspless projection, such as planarity conditions, explicit bounds on their torsion, and their symmetries. Furthermore, they allow us to parameterize all admissible boundary conditions reachable by geodesics with cuspless spatial projection. Such projections lay in the upper half space. We prove this for most cases, and the rest is checked numerically. Finally, we employ the formulas to numerically solve the boundary value problem, and visualize the set of admissible boundary conditions.</p>