Book excerpt: As legged robots have demonstrated versatility, they are more and more favorable for many applications, such as logistics, surveillance, disaster relief, and even home service. Legged robots have the potential to explore and interact with the environment around humans but cannot be handled by robots of other types. A key difficulty in legged locomotion control is that the movement of the floating base cannot be commanded directly, but instead results from the contact forces between the robot and the environment. The contact forces introduce some physical constraints, such as friction cones and unilateral features. Additionally, the hybrid and highly nonlinear dynamics further complex the motion generation and also the motion execution. For tackling legged locomotion, the control framework is often designed hierarchically, in which the high level is in charge of planning reference motion trajectories, and the low level is responsible for tracking this reference trajectory under disturbances. The ideal case is that the reference motion from the high-level planner can be executed by the low-level controller perfectly. However, the discrepancy is always presented given model simplifications and task assumptions. The main objective of this dissertation is to make contributions to mitigate this discrepancy by focusing on high-level motion planning. In motion planning for legged robots, the motion can be categorized into two main types, quasi-static and dynamic motions. Quasi-static motions are defined with a series of discrete contact sequences while the acceleration is kept zero in every time instance. Although energy inefficient, it is often considered a high-risk task. In this dissertation, two motion planners are presented for a six-legged wall-climbing robot given a unique combination of constraints on contact points, contact forces, and body posture. For the first on-wall planner that decouples contact and force planning, on-wall contact points are generated using a mixed-integer convex programming (MICP) with a pre-specified contact sequence while contact forces are optimized subsequently with convex programming. For the second planner, the unscheduled contact sequence is optimized by solving nonlinear programming (NLP). We consider various motions on different environment setups via modeling contact constraints and limb switchability as complementarity conditions. With presented planners, the robot is able to overcome the transition phase between the ground and walls, and also climb vertically between two walls with irregular profiles using pure friction. As for dynamic motions which are seen more commonly in legged animals, trajectory optimization can be utilized to generate a more continuous motion while acceleration resulting from the model dynamics plays a key role. In this dissertation, a jumping planner is presented for a miniature bipedal robot with proprioceptive actuation. The algorithm adopts centroidal dynamics to consider whole-body mass and inertia distribution and generates various motions, directional jumps, twisting jumps, step jumps, and somersaults. The optimized motion can not only mimic human jumping behaviors but also compensate for undesired angular momentum. To prepare a more accurate model for the planner, optimization-based system identification is applied here. Additionally, a heuristic landing location planner based on real-time momentum feedback in the air phase is presented to improve landing stability when executing the jumping reference trajectory.