Trajectory optimization with direct collocation are widely used in various bipedal walking studies, from dynamic simulation in biomechanics to efficient bipedal walking motion generation with multiple contact domains. Although the latter has gained popularity, most of the approaches in this field in general rely on pre-determined contact sequence (domains). This motivates us to use trajectory optimization through contact for generating an efficient and human-like walking gait, because this approach can automatically generate the contact sequence by solving a nonlinear program (NLP) with complementary constraints. However, in this approach the initial guess affects the result significantly, and the direct collocation with Euler method may not be accurate enough for the system dynamics. Therefore, we propose a modified framework and constraints to improve the generated results. We used a zero moment point (ZMP)-based flat-feet walking gait as an initial guess. We also show how to add virtual components like springs at ankle joints to alter the behavior of the resultant walking gait. In addition, considering the one-sided springs at the passive toe joints of the bipedal robot AMBER 3, additional complementary constraints are introduced for a better match of the full dynamics. The results of our modified approach with different constraint conditions are presented and discussed.
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