Algorithm 1: Kinematic Parameter Self-Calibration Algorithm for Cable-Driven Hybrid Robots

Input: Structural height $h_m$, measured cable length set $\left\{{\widehat{l}}_i^{(k)}\right\}$, convergence threshold $ϵ$, $N_c$     Output: Calibrated parameters ${}^{b}X_m,{}^{v}X_e,{}^{g}p_{bi}$.     Initialization: Initialize parameter vector ${\Theta }^{(0)}=[{}^b\tilde{X}_m^{(0)},{}^v\tilde{X}_e^{(0)},{}^g\tilde{p}_{bi}^{(0)}]$$, \mathrm{with} {}^b\tilde{X}_m^{(0)}$$\mathrm{set} \mathrm{to}{[0,0,\pm h_m/2,0,0,0]}^{\mathrm{T}}$.     Data Collection: Collect cable length data for $N_{calib}$ poses, satisfying $N_{calib}>\mathrm{floor}((12+3D_c)/D_c)$.     $\mathrm{While} t<N_{max}$$\mathrm{and} \mathrm{error} >ϵ$ do:         $\mathrm{Calculate} \mathrm{theoretical} \mathrm{cable} \mathrm{lengths} {\tilde{l}}_i^{(k)}$ using Equation (43) with current parameters.         Construct the least-squares objective function: $\min {\displaystyle \sum _k{\displaystyle \sum _i\Vert {\widehat{l}}_i^{(k)}-{\tilde{l}}_i^{(k)}\Vert }}$.         $\mathrm{Update} \mathrm{parameter} \mathrm{vector} {\Theta }^{(t+1)}$ using the Levenberg–Marquardt (LM) algorithm.         Calculate residuals and check for convergence.     End   Return Optimized parameter set ${\Theta }^{*}$.