ロボット機構制御特論（田中担当分）レポート課題HomeworkConsider the mechanical system, where all the frictions in the mechanical system are negligible. The spring is unstretched when x=0. I denotes the moment of...

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ロボット機構制御特論（田中担当分）レポート課題 Homework Consider the mechanical system, where all the frictions in the mechanical system are negligible. The spring is unstretched when x=0. I denotes the moment of inertia around the center of gravity G of the rigid bar. The point G is located at the center of the rigid bar. Hence, note that ?? = 1 3 ????2. The system is driven by the force F(t) that acts on the bottom end of the rigid bar in the horizontal direction. (1) Show the total kinetic energy of the mechanical system. (2) Show the total potential energy of the mechanical system. (3) Find the equations of motion (apply the Lagrange method). (4) Rewrite the equations of motion as the following matrix equality form that is, specify the matrices ?? ?? ?? ,??(??(??), �̇�?(??)) and the vector ??(??(??)). ?? ?? denotes the external force (the generalized force) vector defined with the force F(t) and (5) In the matrix equality form, show that ??(??) is a positive definite matrix in ??(??) and �̇�?(??) − 2??(??, �̇�?) is a skew symmetric matrix. (6) Assume that F(t)=0 for all t and ?? = ?? = ?? = ?? = 1. Also assume that ?? ?? is small enough for all t. The latter assumption may mean that sin ?? ≃ ??, cos?? ≃ 1 ?????? �̇�?2 ≃ 0. Then, linearize the equations of motion around the equilibrium point and solve the linearized system, i.e., find the general solution for the linearized system. ??(??(??))�̈�?(??) + ??(??(??), �̇�?(??))�̇�?(??) + ??(??(??)) = ??(??), ?? ?? = ?? ???? ?? . ロボット機構制御特論（田中担当分）レポート課題 Homework Homewrok スライド番号 4