Problem-1: Assume a single-DOF motion control system whose mathematical model is given as: a(q)q+b(q,q)+9(q) + text...

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Problem-1: Assume a single-DOF motion control system whose mathematical model is given as: a(q)q+b(q,q)+9(q) + text = Kiref For this system, the torque constant and the inertia can respectively be given as a(q) = a + Aa and K = K+ AK. a) Express this system using the nominal parameters and lumping all of the remaining terms as disturbance. Clearly show the content of the disturbance in the parametric form. b) Find out the necessary current to compensate the disturbance term formulated in part a). c) The system is supposed to trace a twice differentiable continuous reference qef (t). Formulate a generalized error for this system. (Hint: The generalized error is a linear combination of the position and the velocity errors). d) Write down a differential equation for the error whose solution results in exponential convergence to zero from any initial value. e) Write down the expression for the desired current which enforces the tracking of the given reference qef (t). Problem-2: Assume that the system given in Problem-1 will be used in force control application. Further, assume also that the disturbances in the system are compensated and the system is initially positioned at the contact point of the environment. a) If the environment is a lossless environment for which the contact force can be given as Fe(t) Keq(t) and the system is supposed to trace a twice differentiable continuous force reference Fref (t); 1) Write down the tracking error for this system. 2) Propose a differential equation for the error given in part 1) whose solution results in exponentially decaying error from any initial value to zero. 3) Write down the condition for the differential equation given in part 2) for convergence of error without oscillations. 4) Using the error dynamics in 2), write down the expression of the desired acceleration for the tracking of Fref (t). b) If the environment has losses (i.e. friction) for which the contact force can be given as Fe(t) = Keq(t) + Deq(t) and the system is supposed to trace continuous force reference Fref (t), whose first derivative exists; 1) Write down the tracking error for this system. 2) Propose a differential equation for the error given in part 1) whose solution results in exponentially decaying error from any initial value to zero. Problem-1: Assume a single-DOF motion control system whose mathematical model is given as: a(q)q+b(q,q)+9(q) + text = Kiref For this system, the torque constant and the inertia can respectively be given as a(q) = a + Aa and K = K+ AK. a) Express this system using the nominal parameters and lumping all of the remaining terms as disturbance. Clearly show the content of the disturbance in the parametric form. b) Find out the necessary current to compensate the disturbance term formulated in part a). c) The system is supposed to trace a twice differentiable continuous reference qef (t). Formulate a generalized error for this system. (Hint: The generalized error is a linear combination of the position and the velocity errors). d) Write down a differential equation for the error whose solution results in exponential convergence to zero from any initial value. e) Write down the expression for the desired current which enforces the tracking of the given reference qef (t). Problem-2: Assume that the system given in Problem-1 will be used in force control application. Further, assume also that the disturbances in the system are compensated and the system is initially positioned at the contact point of the environment. a) If the environment is a lossless environment for which the contact force can be given as Fe(t) Keq(t) and the system is supposed to trace a twice differentiable continuous force reference Fref (t); 1) Write down the tracking error for this system. 2) Propose a differential equation for the error given in part 1) whose solution results in exponentially decaying error from any initial value to zero. 3) Write down the condition for the differential equation given in part 2) for convergence of error without oscillations. 4) Using the error dynamics in 2), write down the expression of the desired acceleration for the tracking of Fref (t). b) If the environment has losses (i.e. friction) for which the contact force can be given as Fe(t) = Keq(t) + Deq(t) and the system is supposed to trace continuous force reference Fref (t), whose first derivative exists; 1) Write down the tracking error for this system. 2) Propose a differential equation for the error given in part 1) whose solution results in exponentially decaying error from any initial value to zero.