The field of robotic surgery increasingly advances towards highly articulated and continuum robots, requiring new kinematic strategies to enable users to perform dexterous manipulation in confined workspaces. This development is driven by surgical interventions accessing the surgical workspace through natural orifices such as the mouth or the anus. Due to the long and narrow nature of these access pathways, external triangulation at the fulcrum point is very limited or absent, which makes introducing multiple degrees of freedom at the distal end of the instrument necessary. Additionally, high force and miniaturization requirements make the control of such instruments particularly challenging. This letter presents the kinematic considerations needed to effectively manipulate these novel instruments and allow us their dexterous control in confined spaces. A nonlinear calibration model is further used to map joint to actuator space and improve significantly the precision of the instrument's motion. The effectiveness of the presented approach is quantified with bench tests, and the usability of the system is assessed by three user studies simulating the requirements of a realistic surgical task.