Modeling of Looping Behavior in a Flexible Instrument Using a Bézier Curve

Flexible medical instruments undergo looping during insertion and navigation inside the human body. It makes the control of the distal end difficult and raises safety concerns. This paper proposes the minimum strain energy concept to get the deformed shape of a flexible instrument in three-dimensional space. A Bézier curve is used to define the trajectory of the deformed shape under different loading conditions and constraints. Looping behavior is studied for different end shortening conditions. The effect of end twist on looping behavior is studied. It is observed that end twist leads to early onset of out-of-plane deformation leading to looping. The strain energy plot gives an insight into the behavior of these instruments with respect to end shortening and twist. The strain energy plot shows the minimum value for $2π$ end twist. Therefore, the instrument tends to go for looping if the end twist is present. Force and torque characteristics are obtained which will lead to the design and control of these instruments. Force and torque plots show negative stiffness when the instrument is going for looping. The unlooping phenomenon is also discussed and a strategy is proposed to mitigate looping. The proposed modeling approach can be utilized to address the complex behavior of a flexible instrument in medical as well as in other industrial applications. The insight developed will help in designing and developing control for safe and reliable usage of flexible instruments in various domains.