The kernel of nonlinear system analysis is the solving of system state equation.Therefore

for a general nonlinear control system

the concept of general time-state space comprising of state variables

control variable

and time t is introduced.In order to solve the state equation of nonlinear control systems

at the operation point of general time-state space

the right side of the state equation can be expanded as Taylor series about time.Then the series solution of the nonlinear control state equation

for which the solution is expressed in time series

can be obtained by using direct-integrating approach.Sliding mode controller is established to control the typical coupling nonlinear system model of the spherical robot.Then we obtain the analytical solution of control and controlled variable by the direct-integrating method.The validity of this method is verified by experiment.