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Template models, such as the Bipedal Spring-Loaded Inverted Pendulum and the Virtual Pivot Point, have been widely used as low-dimensional representations of the complex dynamics in legged locomotion. Despite their ability to qualitatively match human walking characteristics like M-shaped ground reaction force (GRF) profiles, they often exhibit discrepancies when compared to experimental data, notably in overestimating vertical center of mass (CoM) displacement and underestimating gait event timings (touchdown/ liftoff). This paper hypothesizes that the constant leg stiffness of these models explains the majority of these discrepancies. The study systematically investigates the impact of stiffness variations on the fidelity of model fittings to human data, where an optimization framework is employed to identify optimal leg stiffness trajectories. The study also quantifies the effects of stiffness variations on salient characteristics of human walking (GRF profiles and gait event timing). The optimization framework was applied to 24 subjects walking at 40% to 145% preferred walking speed (PWS). The findings reveal that despite only modifying ground forces in one direction, variable leg stiffness models exhibited a >80% reduction in CoM error across both the B-SLIP and VPP models, while also improving prediction of human GRF profiles. However, the accuracy of gait event timing did not consistently show improvement across all conditions. The resulting stiffness profiles mimic walking characteristics of ankle push-off during double support and reduced CoM vaulting during single support. 

For a given endpoint position, a five-bar manipu-lator may assume several separate configurations, with each offering distinct differential kinematics. The corresponding configurations are separated by output singularities and are said to belong to different output modes. In this work, a procedure for dynamically switching between output modes is proposed, with each mode offering different directional force/velocity transmission ratios. The procedure involves solving an optimal control problem using a projection-based direct collocation method for constrained mechanisms to find an optimal trajectory along which the mechanism changes output modes. Using this procedure, a five-bar mechanism configured at a given end-effector position is shown to switch to another output mode where the electrical energy consumed by the actuators to statically hold the mechanism reduces by 80%. Furthermore, the computed trajectories are seen to cross input singularities, a maneuver made possible by momentum planning since actuator authority is impaired at these configurations. 

In recent years, the increasing complexity and safety-critical nature of robotic tasks have highlighted the importance of accurate and reliable robot models. This trend has led to a growing belief that, given enough data, traditional physics-based robot models can be replaced by appropriately trained deep networks or their variants. Simultaneously, there has been a renewed interest in physics-based simulation, fueled by the widespread use of simulators to train reinforcement learning algorithms in the sim-to-real paradigm. The primary objective of this review is to present a unified perspective on the process of determining robot models from data, commonly known as system identification or model learning in different subfields. The review aims to illuminate the key challenges encountered and highlight recent advancements in system identification for robotics. Specifically, we focus on recent breakthroughs that leverage the geometry of the identification problem and incorporate physics-based knowledge beyond mere first-principles model parameterizations. Through these efforts, we strive to provide a contemporary outlook on this problem, bridging classical findings with the latest progress in the field.