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The one-dimensional crawling movement of a cell is considered in this theoretical study. Our active gel model shows that a moving cell with weakly mechanosensitive adhesion complexes tends to move at constant velocity. As the mechanosensitivity of the adhesion complexes increases, a cell with sufficiently strong myosin contractile or high actin polymerization rate can exhibit stick-slip motion. Finally, a cell with highly mechanosensitive adhesion complexes exhibits periodic back-and-forth migration. A simplified model that assumes that the cell crawling dynamics are controlled by the evolution of the myosin density dipole and the asymmetry of adhesion complex distribution captures the motility behaviors of crawling cells qualitatively. It suggests that complex cell crawling behaviors could result from the interplay between the distribution of contractile force and mechanosensitive bonds.