This paper describes a new technique for the calculation of gradients of constraints associated with the moderate amplitude criteria of the ADS-33 helicopter handling qualities speci cations. The gradients are calculated using low-order linear approximations to the full nonlinear model of the helicopter. The low-order models approximate the gradients well and reduce the additional cost of calculating the gradient by a factor of about 50. Most of the reduction in the objective function obtainable using the exact gradients are retained, with no additional infeasible intermediate designs. The accuracy of linear Taylorseries expansions of the constraint in terms of the design variables is found to depend on the size of the changes of each design variable. The accuracy is not improved by using intermediate design variables, such as reciprocals and cubes of the design variables, but it improves if the bandwidth or derivative ratios such as are used as intermediate variables in the expansions. L /L u p 1c Nomenclature F(X ) = objective function value G (s), G (s) Lbase Lpert= baseline and perturbed low-order transfer functions G(s), G pert (s) = baseline and perturbed transfer functions g j (X ) = jth constraint k b, k z = ap and lag spring stiffnesses for main rotor blades k f , k p = roll attitude and rate feedback gains L p, L u 1c = roll stability and control derivatives p pk = peak roll rate p, q, r = aircraft angular velocities in roll, pitch, and yaw R = ap-lag blade elastic coupling parameter u = input displacement vector u, v, w = aircraft velocity components along the body axes W 1 , W 2 = frequency response weighting functions X = vector of design variables (X i ) U , (X i ) L = upper and lower bound on ith component of design vector x = vector of states ( j ) x = ight control system state b 0 , b 1c , b 1s , b 2 = rotor ap degrees of freedom in nonrotating coordinate system Df min , f pk = minimum and peak roll attitudes z0, z1c, z1s, z2 = rotor lag degrees of freedom in nonrotating coordinate system lj(?) = jth eigenvalue l 0 , l 1c , l 1s = in ow degrees of freedom f, u, c = aircraft attitudes in roll, pitch, and yaw