Dynamic modelling of a rigid-flexible manipulator for constrained motion task control

A mathematical model governing the dynamics of a constrained rigid-flexible manipulator moving in a horizontal plane is derived using Hamilton's principle. A new variable is introduced before the procedure of modal expansion in order to convert the non-homogeneous boundary condition into a homogeneous one. The static tip deflection of the flexible link is allowed in order to maintain the contact force between the end effector and the constrained path and this tip deflection is considered in both the inverse kinematics and the order reduction procedures. The state vector of the proposed controller consists of joint angle of the rigid link, its derivative and integral, the first deflection mode and its derivative, and the integral of contact force. A multivariable controller is proposed for the simultaneous motion and force control of the manipulator. The controller consists of a feedforward term which contributes the torque for the expected joint angles and the contact force, and a feedback term with the time varying optimal gains obtained from the Matrix Riccati equation. Computer simulation results show that this proposed controller is capable of performing the straight line tracking task satisfactorily under four different conditions.