Forward kinetostatics of spatial tendon-driven continuum robots typically requires a nonlinear equilibrium solve for each actuation input. This paper develops a force-to-Cartesian-configuration model with a closed-form solution in quadratures for spatial multi-segment robots under tendon actuation. The Cartesian backbone centerline and accumulated material twist serve as generalized coordinates, from which the strain measures and tendon geometry are derived. Variational equilibrium yields explicit axial and bending relations and establishes zero equilibrium material twist within the proposed model for admissible longitudinal non-helical routing. The solution is propagated segment by segment without an iterative equilibrium solve, while retaining axial deformation, spatially varying axial and bending stiffnesses and tendon-routing diameter, and segment-dependent tendon participation. Numerical comparisons with a full-strain geometric variable-strain model (GVS) yield maximum length-normalized tip-position discrepancies of 8.91 x 10^-6 and 1.01 x 10^-5 for the single- and three-segment robots, respectively. Mean evaluation times of 1.52 μs and 2.94 μs, with corresponding speedups of approximately 1864x and 3348x over the baseline, demonstrate the computational advantage of the explicit force-to-configuration mapping in the reported benchmark.

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