Book excerpt: The interest in using legged robots for a variety of terrestrial and space applications has grown steadily since the 1960s. At the present time, a large fraction of these robots relies on electric motors at the joints to achieve mobility. The load distributions inherent to walking, coupled with design constraints, can cause the motors to operate near their maximum torque capabilities or even reach saturation. This is especially true in applications like space exploration, where critical mass and power constraints limit the size of the actuators. Consequently, these robots can benefit greatly from motion optimization algorithms that guarantee successful walking with maximum margin to saturation. Previous gait optimization techniques have emphasized minimization of power requirements, but have not addressed the problem of saturation directly. This dissertation describes gait optimization techniques specifically designed to enable operation as far as possible from saturation during walking. The benefits include increasing the payload mass, preserving actuation capabilities to react to unforeseen events, preventing damage to hardware due to excessive loading, and reducing the size of the motors. The techniques developed in this work follow the approach of optimizing a reference gait one move at a time. As a result, they are applicable to a large variety of purpose-specific gaits, as well as to the more general problem of single pose optimization for multi-limbed walking and climbing robots. The first part of this work explores a zero-interaction technique that was formulated to increase the margin to saturation through optimal displacements of the robot's body in 3D space. Zero-interaction occurs when the robot applies forces only to sustain its weight, without squeezing the ground. The optimization presented here produces a swaying motion of the body while preserving the original footfall locations. Optimal displacements are found by solving a nonlinear optimization problem using sequential quadratic programming (SQP). Improvements of over 20% in the margin to saturation throughout the gait were achieved with this approach in simulation and experiments. The zero-interaction technique is the safest in the absence of precise knowledge of the contact mechanical properties and friction coefficients. The second part of the dissertation presents a technique that uses the null space of contact forces to achieve greater saturation margins. Interaction forces can significantly contribute to saturation prevention by redirecting the net contact force relative to critical joints. A method to obtain the optimal distribution of forces for a given pose via linear programming (LP) is presented. This can be applied directly to the reference gait, or combined with swaying motion. Improvements of up to 60% were observed in simulation by combining the null space with sway. The zero-interaction technique was implemented and validated on the All Terrain Hex-Limbed Extra-Terrestrial Explorer (ATHLETE), a hexapod robot developed by NASA for the transport of heavy cargo on the surface of the moon. Experiments with ATHLETE were conducted at the Jet Propulsion Laboratory in Pasadena, California, confirming the benefits predicted in simulation. The results of these experiments are also presented and discussed in this dissertation.