A quasi-sliding mode model-free adaptive controller(SM-MFAC) is designed for trajectory tracking control of a class of second-order nonlinear systems. First, the model is decomposed into two discrete subsystems in series, and the expression of the whole system pseudo partial derivative(PPD) is given, then we use the output data of the two subsystems to design the expected state of the output of subsystem 2. By using the MFAC to continuously track the expected status of online updates, the tracking control of the overall goal is realized. Then the stability of the SM-MFAC control system is analyzed, it has been proven that the output error asymptotically approaches some neighborhood of zero and the equivalent control input is bounded. Finally, the SM-MAFC control theory is verified by taking the joint trajectory tracking control of a free-floating space manipulator as an example. A two-linked free-floating space manipulator is built in multi-body kinematics and dynamics simulation software MBdyn. There are dead zones, input saturation and friction characteristics in the joint system. The joint simulation in Matlab-simulink shows that the MFAC control scheme can not accurately estimate the overall value of PPD of this class of second-order nonlinear systems, resulting in reduced control performance, the designed SM-MFAC controller can track the target curve faster and more accurately than the traditional PID(proportional-integral-differential), PD(proportional-differential)-MFAC.