Abstract: GOCE has released gravity gradients,but there aren't data in polar regions because the inclination of the orbit is about 96.7°.We need to simulate data with known gravity field model in this area,whereas many formulas to compute the gravity gradients are singular at poles,which can cause a lot of difficulties in gravity field recovery using GOCE data.In order to overcome the so-called singularities,we firstly analyze the singular term,then derives a new nonsingular expression for gravitational gradients calculation based on properties of Legendre function.At last we compare several different formulas through calculation.The proposed formulas is more accurate than conventional methods.