Finite-Time Steering for a Robotic Surgical Needle in Prostate Tissue Based on the Adaptive-Robust Nonlinear Control Approach

In this paper, three different finite-time nonlinear controllers are proposed to steer a robotic surgical needle in prostate tissue subject to parametric and modeling uncertainties. The torque generated by each type of these controllers is injected to the surgical needle’s closed-loop structure and, in consequence, the system’s state variable precisely converges to the desired path in prostate tissue within an adjustable finite time. The mentioned controllers are constructed based on the developed terminal sliding mode control method (as the main approach of robust-nonlinear control) incorporated with the adaptive control technique (for designing adaptation laws and estimation of unknown physical constants). It is worth noting that the basic difference between these controllers is in the definition of their nonlinear sliding manifolds. By utilizing the Lyapunov stability theory and several applicable lemmas, it is mathematically proven that all types of the introduced control approaches are able to accomplish the finite-time steering objective and guarantee the global finite-time stability for the needle-tissue dynamical system. Adaptation laws (existing in the proposed nonlinear controllers) continuously estimate the unknown physical constants and it is demonstrated that time responses of these estimations exactly reach the constants values over the finite time. Finally, by using MATLAB software, three types of the proposed controllers are separately simulated onto a second-order needle-tissue system to illustrate their proper performance.