A projection-based inverse kinematic model for extensible continuum robots and hyper-redundant robots with an elbow joint

Inverse kinematics is a core problem in robotics, involving the use of kinematic equations to calculate the joint configurations required to achieve a target pose. This study introduces a novel inverse kinematic model (IKM) for extensible (i.e., length-adjustable) continuum robots (CRs) and hyper-redundant robots (HRRs) featuring an elbow joint. This IKM numerically solves a set of equations representing geometric constraints (abbreviated as NSGC). NSGC can handle target poses $X_t=\left[x_t,y_t,z_t,{\psi }_t\right]$ in 3D space, which are projected onto a 2D plane and solved numerically. NSGC is capable of real-time operation and accounts for elbow joint limits. Extensive simulations and empirical tests confirm the reliability, performance, and practical applicability of NSGC.