force was considered in Refs. [24, 25] in second order perturbation theory. In the case of two ideal metal parallel plates covered with periodic uniaxial corrugations of rectangular shape, some results for the normal Casimir force were found in Refs. [26, 27]. It is pertinent to note that the use of normal and lateral Casimir forces gives the possibility to actuate both normal and lateral translations of corrugated surfaces in micromachines by means of the electromagnetic zero-point oscillations. The lateral Casimir force might be used to solve the tribological problems plaguing the microdevice industry. Specifically, it was suggested [28– 30] to use this effect for frictionless transmission of lateral motion. There are also proposals [31, 32] to measure the Casimir torque arising between the anisotropic bodies due to the lateral Casimir force. Both the normal and lateral Casimir forces are in general nonadditive and possess a complicated dependence on the geometrical shape of the boundary surfaces connected with diffraction effects. In the configuration of a smooth Al coated sphere above a sinusoidally corrugated Al coated plate the nontrivial behavior of the normal Casimir force was exper- imentally demonstrated [33]. This was done in an additive regime when the sphere-plate separation is much smaller than the period of corrugations [23]. Recently [34] the nontrivial behavior of the normal Casimir force caused by the complicated geometry of a boundary surface was observed in the configuration of a smooth Au coated sphere above a Si plate covered with rectangular corrugations (trenches). In this case the nonadditive regime was used where separation distances are of the order of corrugation period. The deviations of the measurement data from the additive theory were reported [34]. These deviations are, however, about 50% less than those expected for ideal metal boundaries in the theory taking exact account of the geometrical shape [27]. The remaining discrepancy might be explained by the role of a nonzero skin depth which was not accounted for in Ref. [27]. Thus, the scattering approach for grating geometries taking the exact account of surface geometry at zero temperature and describing metallic properties by means of the simple plasma model [35, 36] was found to be in better agreement with the data of Ref. [34]. The first measurement of the lateral Casimir force was performed between the surfaces of an Au coated sphere and an Au coated plate both covered with aligned uniaxial corrugations of sinusoidal shape [37, 38]. The corrugation amplitudes were A$_{1}$ = 59 nm on the plate and A$_{2}$ = 8 nm on the sphere. The period of corrugations was equal to 1200 nm, i.e., much larger than separation distances between the sphere and plate. This means that the experiments of 4