A novel projected Fletcher‐Reeves conjugate gradient approach for finite‐time optimal robust controller of linear constraints optimization problem: Application to bipedal walking robots

Abstract

SummaryFor finite-time optimal robust control problem of bipedal walking robot, a class of global and feasible projected Fletcher-Reeves conjugate gradient approach is proposed based on an online convex optimization algorithm. The optimal robust controllers are solved by projected Fletcher-Reeves conjugate gradient approach.The approach can rapidly converge to a stable gait cycle by selecting an initial gait. Under some suitable conditions, we provide a rigorous proof of global convergence and well-defined properties for projected Fletcher-Reeves conjugate gradient approach. To demonstrate the effectiveness of the bipedal walking robot, we will conduct numerical simulations on the model of 3-link robot with nonlinear, impulsive, and underactuated dynamics. Furthermore, to indicate the availability of high-dimensional robotic system, the main result is illustrated on a nonlinear impulsive model of a bipedal walking robot through simulations via finite-time optimal robust controller. Numerical results show that the projected Fletcher-Reeves conjugate gradient approach is feasible and effective for bipedal walking robots. Therefore, it is reasonable to infer that the optimal robust control approach can be used in practical systems.

A Dai–Yuan conjugate gradient algorithm of linear equation constrained optimization approach for optimal robust controller of bipedal robots

“…Second, the computational effect is more superior than the Hestenes-Stiefel method and Fletcher-Reeves method. 20,23 Last, the proposed approach of this article is more robust than the classical methods for all numerical problems. 20,23 It can be seen from numerical results that the proposed Dai-Yuan conjugate gradient method is feasible, effective, and superior than the classical approaches which can be represented as the Hestenes-Stiefel method and Fletcher-Reeves method.…”

“…20,23 Last, the proposed approach of this article is more robust than the classical methods for all numerical problems. 20,23 It can be seen from numerical results that the proposed Dai-Yuan conjugate gradient method is feasible, effective, and superior than the classical approaches which can be represented as the Hestenes-Stiefel method and Fletcher-Reeves method. 20,23 In addition, on the one hand, the bipedal robots are high dimensional nonlinear dynamic multibody system, and furthermore, the walking legs are continued to accept to irregular shock.…”

“…20,23 It can be seen from numerical results that the proposed Dai-Yuan conjugate gradient method is feasible, effective, and superior than the classical approaches which can be represented as the Hestenes-Stiefel method and Fletcher-Reeves method. 20,23 In addition, on the one hand, the bipedal robots are high dimensional nonlinear dynamic multibody system, and furthermore, the walking legs are continued to accept to irregular shock. On the other hand, the motion control of bipedal robots involves the internal dynamics, the high order, nonlinear and strong coupling characteristic.…”

“…Taking the energy consumption as the performance functional, the optimal robust controller is solved via a nonlinear optimization method, and then the dynamic walking of bipedal robot is realized and analyzed, the details are similar to the literature. 20 However, in this article, the algorithm is designed by fully considering the saturation constraints, unilateral constraint condition and maximum ratio and minimum normal ground reaction force of the control torques in the solving solution process. Therefore, the nonlinear optimization model is considered as follows…”

“…Three-link bipedal robot. [20][21][22] for control input versus time. Let u 1 ¼ t 1 ; u 2 ¼ t 2 ; as shown in Figure 3, and using the Dai-Yuan projection conjugate gradient optimization algorithm to solve the optimal robust controller makes the control input increase with the number of walking steps of the robot, and the robot can achieve dynamic walking with minimum energy.…”