Learning Inverse Kinematics Multiplicity of Concentric Tube Robots Using Invertible Neural Networks

Concentric tube robots (CTR) are a promising technology for medical applications due to their small size, flexibility, and ability to make complex shapes. These robots are built from a series of pre-curved, super-elastic tubes that are arranged concentrically and manipulated through rotational and translational movements at their proximal ends. Achieving accurate kinematics is essential in making CTRs useful in minimally invasive surgical procedures where precision and safety is paramount. Due to the difficulty of incorporating nonlinear effects like friction and tube clearances into analytical models, previous works have investigated machine learningbased models for CTR kinematics, leading to higher kinematic accuracies. We present a kinematic model for CTRs using an invertible neural network architecture that, unlike other learningbased models, can generate multiple inverse kinematic solutions. Our model achieved a mean forward kinematic tip error of 2.86 mm (<tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathbf{3. 4 3 \%}$</tex> normalized to arclength), outperforming a Cosserat rod-based analytical CTR model. In three test cases (two static points and a circle trajectory), our model achieved mean inverse kinematic errors of <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$6.19,3.33$</tex>, and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$3.86 ~\text{mm}(5.66 \%, 4.12 \%$</tex>, and 4.16 % normalized to arclength). We additionally present a robust data capture pipeline that is able to reconstruct the CTR's shape that uses state-of-the-art segmentation models.