This paper leverages the higher-order kinematic derivatives of serial robots to achieve efficient, aspect-preserving real-time inverse kinematics. First, a family of higher-order solvers is derived, with a necessary orientation-error correction map, yielding the QuIK4 and QuIK5 methods with fourth- and fifth-order convergence. Second, an aspect-preserving step limiter extrapolates the Jacobian determinant along the step to predict and avoid singularities, shortening steps only when they head toward a singularity rather than when the current iterate is ill-conditioned. Third, a Hermite-interpolation certifier verifies the aspect of each answer, and a certified restart algorithm recovers failures from proven same-aspect seeds. The proposed methods are shown to outperform prior methods, reducing failures by up to four orders of magnitude and nearly eliminating aspect switching at tracking-level errors. A C++ library is released with ROS 2 integration.

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