The paper presents application of direct pseudospectral Chebyshev method for solving a commercial airplane trajectory optimization problem. This method employs Nth-degree Lagrange polynomial approximations for the state and control variables with the values of these variables at the Chebyshev-Gauss-Lobatto (CGL) points as the expansion coefficients. This process is converted to a nonlinear programming problem (NLP) with the state and control values at the CGL points as unknown NLP parameters. The kinetic model of flight is formulated, where it is assumed that an airplane is a particle and the motion takes place in the vertical plane. The method is implemented in Matlab using sequential quadratic programming algorithm (SQP) as an efficient solver. Sensitivity analyses are performed concerning the influence of the degree of discretization and the initial approximation on the solution. Three examples of optimized trajectories in presence of wind are shown.

Adaptive locomotion over difficult or irregular terrain is considered as a superiority feature of walking robots over wheeled or tracked machines. However, safe foot positioning, body posture and stability, correct leg trajectory, and efficient path planning are a necessity for legged robots to overcome a variety of possible terrains and obstacles.Without these properties, anywalking machine becomes useless. Energy consumption is one of the major problems for robots with a large number of Degrees of Freedom (DoF). When considering a path plan ormovement parameters such as speed, step length or step height, it is important to choose the most suitable variables to sustain long battery life and to reach the objective or complete the task successfully.We change the settings of a hexapod robot leg trajectory for overcoming small terrain irregularities by optimizing consumed energy and leg trajectory during each leg transfer. The trajectory settings are implemented as a part of hexapod robot simulation model and tested through series of experiments with various terrains of differing complexity and obstacles of various sizes. Our results show that the proposed energy-efficient trajectory transformation is an effective method for minimizing energy consumption and improving overall performance of a walking robot.