Legged robots come with unprecedented mobility, potentially unleashing many applica-
tions such as autonomous rescue in disaster zones. Such applications require robots to be capable of high mobility and adaptability. To meet these demands, modern robot designs
have shifted toward parallel and hybrid architectures, which drastically reduce limb inertia
and improve impact absorption compared to classical serial architectures. However, these advanced platforms introduce significant mathematical bottlenecks: their closed-loop kine-
matics yield complex algebraic constraints, while their environmental interactions involve frictional contact forces, challenging the limits of current methods. Traditional control ap-
proaches struggle to integrate these dynamics, relying instead on simplified physical models and rigid, pre-defined contact sequences that lead to brittleness when facing real-world uncertainties. This thesis addresses these two issues through the scope of optimization-
based control.

First, we present a modeling framework for closed-loop systems along with its analytical
derivatives, which we apply to standard gradient-based algorithms to generate various
locomotion behaviors. Second, we demonstrate the limitations of purely gradient-based optimization when handling frictional contact models, an inherently non-smooth phenom-
enon. To address the challenge of optimizing through the contact sequence, we propose leveraging zero-th order optimization algorithms. This provides an efficient and versatile
framework for motion retargeting and imitation learning on humanoid robots. Finally, we
introduce a multiple-shooting formulation for the Model Predictive Path Integral (MPPI)
algorithm, mitigating the drawbacks of the sampling approach and improving performance
in terms of stability and prediction horizon.

We expect that this work will guide research toward contact based trajectory optimiza-
tion enabling robust loco-manipulation for legged robots in unpredictable environments.