The initial value problem for the mathematical model of a multiple-input multiple-output flexible parallel mechanism

• The mathematical model of a multiple-input multiple-output flexible parallel mechanism is established using the catenary equations.
• The layer-wise substitution method has been proposed to deal with the initial value of the mathematical model.
• The symmetry of cable-net has been used to simplify the mathematical model of the multiple-input multiple-output flexible parallel mechanism.
• The number of unknown parameters in the equations is reduced to about one-tenth of the whole cable-net.

The cable-net support structure of the Five-hundred-meter Aperture Spherical Telescope (FAST) is a multiple-input multiple-output (MIMO) flexible parallel mechanism. The mathematical model of the cable-net structure is set up with the catenary equation, which produces a large number of non-linear equations. Finding the initial value plays an important role in the working state of the FAST. In this paper, a novel method is proposed to solve the equations of the model. This method uses the solution of the (n − 1)-th layer cable-net as the initial value of the n-th layer cable-net. The n-layer cable-net equations are then solved simultaneously and the tension of each cable and the coordinate of each node in the cable-net can be obtained. With the introduction of the symmetry of cable-net, the number of unknown parameters in the equations is reduced to about one-tenth of the whole cable-net. The initial value of the cable-net support structure of the FAST is provided to show the validity of the method. The results of this study are of practical value for the FAST's new type of flexible cable driving mechanism design and control method.