Finite Difference-Based Suboptimal Trajectory Planning of Biped Robot with Continuous Dynamic Response

Abstract

Abstract-One of the clear problems encountered in the dynamic response of the biped robot is the discontinuity of the actuating torques/ground reaction forces at the transition instances during transferring form single support phase to double support phase and vice versa. Therefore, this paper suggests the linear transition function used in the biomechanics field for estimating the ground reaction forces during the double support phase such that gradual increase/decrease of the ground reaction forces can occur. The closure loop of the biped robot at the transition instances during DSP can be broken using the mentioned strategy. Consequently, the continuous dynamic response can be achieved. Two cases are simulated using the optimal control theory. The inverse dynamics-based optimization is preferred as a direct suboptimal tool because it can show less computation and easinessthan other optimal control approaches. Due to easiness of the finite difference approach, it is used for discretization of the dynamic equations and the imposed constraints to convert the dynamic optimal control problem into parameter optimization. The simulated case 1 have been used repeatedly in the literature, whereas the case 2 adopts the linear transition function of the ground reaction forces keeping the same generalized coordinates of the biped configuration at the transition instances. The results show the superiority of the suggested method to guarantee continuous actuating torques/ground reaction forces at the transition instances.

Decentralized adaptive partitioned approximation control of high degrees-of-freedom robotic manipulators considering three actuator control modes

“…See [48,49] for more details. Figure 9 shows the configuration of the biped robot during the SSP; for more details on modeling and control of biped robots see [47][48][49][50][51][52][53][54][55][56][57]. The physical parameters are borrowed from [58].…”

“…The ZMP location coincides with the location of the center of pressure for the balance walking; however, this is not the case for unbalanced walking. See[47][48][49][50][51][52][53][54][55][56][57] and the references therein for more details.…”

Dynamic Modeling of Biped Robot using Lagrangian and Recursive Newton-Euler Formulations

“…Proof. Let (22) By substituting (19) into (22) we get (23) Since is skew-matrix according to Property 1, then is also skew-matrix. Property 5.…”

A Simple Algorithm for Generating Stable Biped Walking Patterns

“…Remark 7. Since the biped robot does not have a unique solution during the DSP, we assume a linear transition function for the ground reaction forces of the front foot (right foot) as follows [7] (18) where , denote the absolute time of the SSP and DSP respectively, is the mass of the center of gravity and represents the acceleration of the biped COG. Whereas the rear foot (left foot) has the following ground reaction forces where is walking step length, is the mass of link (i), ( is the position of COG of link (i), is the number of links, is the moment of inertia about COG of link (i), and denotes the angular acceleration of link (i).…”

“…Numerous approaches have been used to generate the motion of biped robot. However, there are three efficient methods have been used successfully for this purpose: Optimization-based gait, COG-based gait and the interpolation-based gait [5][6][7][8]; see the references therein. Despite the miscellaneous walking pattern generation and stabilization approaches, it is difficult to find a thorough method that can tune the walking parameters to satisfy the kinematic and dynamic constraints: singularity condition at the knee joint, zero-moment point (ZMP) constraint, and unilateral contact constraints.…”