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Traditional approaches to quadruped control frequently employ simplified, hand-derived models. This significantly reduces the capability of the robot since its effective kinematic range is curtailed. In addition, kinodynamic constraints are often non-differentiable and difficult to implement in an optimisation approach. In this work, these challenges are addressed by framing quadruped control as optimisation in a structured latent space. A deep generative model captures a statistical representation of feasible joint configurations, whilst complex dynamic and terminal constraints are expressed via high-level, semantic indicators and represented by learned classifiers operating upon the latent space. As a consequence, complex constraints are rendered differentiable and evaluated an order of magnitude faster than analytical approaches. We validate the feasibility of locomotion trajectories optimised using our approach both in simulation and on a real-world ANYmal quadruped. Our results demonstrate that this approach is capable of generating smooth and realisable trajectories. To the best of our knowledge, this is the first time latent space control has been successfully applied to a complex, real robot platform.