We report an experimental and numerical study of the vortex matter moving in a very thin type II superconducting strip with asymmetrically nanostructured profile. The asymmetric thickness profile generates a geometrical force landscape that sets a preferential direction for vortex motion, resulting in an uncommon voltage-current curve of the strip, with vanishingly small asymmetry in the positive and negative critical currents but appreciably large asymmetry in the voltages. Experimental results as well as the geometrical force affecting the moving vortex matter are discussed in the framework of time dependent Ginzburg-Landau model for superconductors with variable thickness.