Ditch crossing is one of the essential capabilities required for a biped robot in disaster management and search and rescue operations. Present work focuses on crossing a wide ditch with landing uncertainties by an under-actuated planar biped robot with five degrees of freedom. We consider a ditch as wide for a robot when the ankle to ankle stretch required to cross it is at least equal to the leg length of the robot. Since locomotion in uncertain environments requires real-time planning, in this paper, we present a new approach for generating real-time joint trajectories using control constraints not explicitly dependent on time, considering impact, dynamic balance, and friction. As part of the approach, we introduce a novel concept called the point of feasibility for bringing the biped robot to complete rest at the end of ditch crossing. We present a study on the influence of initial posture on landing impact and net energy consumption. Through simulations, we found the best initial postures to efficiently cross a wide ditch of width 1.05 m, with less impact and without singularities. Finally, we demonstrate the advantage of the proposed approach to cross a wide ditch when the surface friction is not same on both sides of the ditch.

In the article an optimal control problem subject to a stationary variational inequalityis investigated. The optimal control problem is complemented with pointwise controlconstraints. The convergence of a smoothing scheme is analyzed. There, the variationalinequality is replaced by a semilinear elliptic equation. It is shown that solutions ofthe regularized optimal control problem converge to solutions of the original one. Passingto the limit in the optimality system of the regularized problem allows to proveC-stationarity of local solutions of the original problem. Moreover, convergence rateswith respect to the regularization parameter for the error in the control are obtained,which turn out to be sharp. These rates coincide with rates obtained by numericalexperiments, which are included in the paper.

We present an a posteriori error analysis of adaptive finiteelement approximations of distributed control problems for secondorder elliptic boundary value problems under bound constraints onthe control. The error analysis is based on a residual-type a posteriori error estimator that consists of edge and elementresiduals. Since we do not assume any regularity of the data ofthe problem, the error analysis further invokes data oscillations.We prove reliability and efficiency of the error estimator andprovide a bulk criterion for mesh refinement that also takes intoaccount data oscillations and is realized by a greedy algorithm. Adetailed documentation of numerical results for selected testproblems illustrates the convergence of the adaptive finiteelement method.

In this paper sufficient optimality conditions are established for optimal control ofboth steady-state and instationary Navier-Stokes equations. The second-order condition requirescoercivity of the Lagrange function on a suitable subspace together with first-order necessaryconditions. It ensures local optimality of a reference function in a Ls -neighborhood, whereby the underlying analysis allows to use weaker norms than L∞ .