Dynamic modeling and closed-loop control design for humanoid robotic systems: Gibbs–Appell formulation and SDRE approach

Analyzing the walking motion of the bipedal robots that have upper-body parts as well as lower-body legs and exhibit a human-like gait is a challenging task. One of the main objectives of this paper is to present a new and systematic method for designing a desired movement trajectory for a bipedal robot such that it has the greatest conformity with the system dynamics and makes the gait of a bipedal robot similar to the configuration of a human being walking on a sloping surface. To this end, first, the kinematics and the dynamics of a bipedal robot walking down a ramp of shallow slope are investigated. Using the recursive Gibbs–Appell (G-A) methodology and Newton’s kinematic impact law, the governing dynamic equations of this bipedal robot in the two single-support and double-support phases are derived so that we can alter the system’s degrees of freedom without having to perform manual computations. Based on the dynamic equations obtained in the process, an eigenvalue problem is achieved, which can be solved to determine the suitable initial conditions needed for the passive gait of the bipedal robot. Then, having the initial and final conditions (before an impact with the inclined surface), a new method called “passive gait-based trajectory design (PGBTD)” is employed to determine the desired walking trajectory of the robot for one step. Considering the nonlinearity of the examined system, an optimal control method based on the state-dependent Riccati equation (SDRE) is employed to track the desired trajectory obtained. The performed simulations show that by just using a small amount of control energy at the beginning of each step, the steady and continuous gait of the bipedal robot on sloping surfaces can be controlled.