Visual self-modelling lets robots learn body models from observations, but such models usually encode kinematics implicitly rather than exposing metric joint parameters for computation and control. Moreover, a good fit to images does not by itself establish accurate kinematics. Here we present an explicit analytic robot self-model that requires neither deformation networks nor URDF/CAD templates. Multi-view images and joint readings establish Gaussian appearance and provide motion evidence for rigid link assignments, whereas sparse end-effector poses independently identify metric screw axes. The resulting closed-form forward kinematics and analytic Jacobian can be evaluated beyond the fitting ranges. We validate the model on a simulated seven-axis arm, a simulated twelve-joint quadruped and a real six-axis arm. In simulation, it achieves millimetre-level reaching and pick-and-place, as well as closed-loop quadruped walking. On the real arm, it renders held-out configurations, predicts end-effector positions with a median error of 6.4 mm, compared with 11.7 mm for a calibrated nominal model, and supports reaching with external marker feedback. A controlled comparison shows that image fit alone is not enough: two Panda models differing only in their joint axes are 0.54 dB apart in PSNR, yet their end-effector prediction errors differ 13.7-fold.

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