Consider the twolink planar arm in the figure below for whic

Consider the two-link planar arm in the figure below, for which the vector of generalized coordinates is q = [theta_1, theta_2]^T. Let l_1, l_2 be the distances of the centers of mass of the two links from the respective joint axes and a_1, a_2 be the length of the two links. Also, let m_1, m_2 be the masses of the two links, and finally, let I_1, I_2 be the moments of inertia relative to the centers of mass of the two links, respectively. Derive dynamical model of the system, tau = M(q) doubledot q + C(q, dot q) dot q + g(q), using Lagrange\'s equation. Explore the use of symbolic software, such as Maple or Mathematica, for this problem. Write out the equations of motion i.e. doubledot q. Derive a state-space model, dot x = f(x, u), for the system by introducing appropriate state variables. - Remember from the lecture, when you have n many 2^nd-order differential equations. you will have 2 n many state variables hence 2n many first order differential state equations, where n, in our cases of study i.e. arm robots, usually refers to the number of actuators/joints which is the same as the number of equations of motion too i.e. the size of vector doubledot q or tau. Is this system linear? When the input u is zero, find all the equilibrium states of the system. Remember from the lecture, when the torque input is zero, equilibrium states can be found from dot x = f(x, u) = 0.

Solution

To motivate the subsequent derivation,we show first how the eulers lagrange equations canbe derived from Newtons second law for a single degree of freedom system consisting of a particle of constant mass m , constrained to move in the y- direction , and subject to a force f and the graitatinal force mg . by newtons second law.

F= ma, The equatin of motionof particle

My = f -mg