When exploring glacier ice it is often necessary to take samples or implement sensors at a certain depth underneath the glacier surface. One way of doing this is by using heated melting probes. In their common form these devices experience a straight one-dimensional downwards motion and can be modeled by standard close-contact melting theory. A recently developed melting probe however, the IceMole, achieves maneuverability by simultaneously applying a surface temperature gradient to induce a change in melting direction and controlling the effective contact-force by means of an ice screw to stabilize its change in attitude. A modeling framework for forced curvilinear melting does not exist so far and will be the content of this paper. At first, we will extend the existing theory for quasi-stationary close-contact melting to curved trajectories. We do this by introducing a rotational mode. This additional unknown in the system implies yet the need for another model closure. Within this new framework we will focus on the effect of a variable contact-force as well as different surface temperature profiles. In order to solve for melting velocity and curvature of the melting path we present both an inverse solution strategy for the analytical model, and a more general finite element framework implemented into the open source software package ELMER. Model results are discussed and compared to experimental data conducted in laboratory tests. (C) 2015 Elsevier Ltd. All rights reserved.