Let's break down **Problem 2** step by step: --- ## **Given:** A single-DOF motion control system with the model: \[ a(q) \ddot{q}(t) + b(q, \dot{q}) + g(q) + \tau_{ext} = k_c f_e^*(t) \] where: - \( a(q) \): inertia (can be \( a_0 + \Delta a \)) - \( b(q, \dot{q}) \): damping/friction - \( g(q) \): gravity - \( \tau_{ext} \): external disturbance (compensated) - \( k_c \): torque constant - \( f_e^*(t) \): force at environment Assume disturbance is compensated and initial system is at the contact point. --- ## **a) Lossless environment (contact force: \( f_e(q) = k_e(q - q_e^*) \))** ### **1) Write down the tracking error for this system.** Let the desired force profile be \( f_e^*(t) \), and the corresponding desired position be \( q_e^*(t) \). **Tracking error:** \[ e(t) = q(t) - q_e^*(t) \] --- ### **2) Propose a differential equation for the error whose solution decays to zero.** From \( f_e(q) = k_e(q - q_e^*) \), we want \( f_e(q) \to f_e^*(t) \), so: \[ k_e(q - q_e^*) = f_e^*(t) \implies q_e^*(t) = q - \frac{f_e^*(t)}{k_e} \] But for tracking, define \( e(t) = q(t) - q_e^*(t) \). **System dynamics:** \[ a(q) \ddot{q} + b(q, \dot{q}) + g(q) = k_c f_e^*(t) \] Assume compensation and at equilibrium, \( a(q) \ddot{q} = 0 \), \( b(q, \dot{q}) = 0 \), \( g(q) = 0 \): \[ 0 = k_c f_e^*(t) \] For control, let's linearize and use PD control: \[ a_0 \ddot{q} + k_e(q - q_e^*) = f_e^*(t) \] So, \[ a_0 \ddot{q} + k_e e(t) = f_e^*(t) \] Let \( f_e^*(t) \) be the reference; set \( a_0 \ddot{e}(t) + k_e e(t) = 0 \): \[ \ddot{e}(t) + \frac{k_e}{a_0} e(t) = 0 \] --- ### **3) Condition for convergence without oscillations:** The characteristic equation is: \[ s^2 + \frac{k_e}{a_0} = 0 \implies s = \pm j\sqrt{\frac{k_e}{a_0}} \] This is purely imaginary (undamped), so the system will oscillate unless damping is added. To avoid oscillation (i.e. critically damped or overdamped), add a damping term \( b \dot{e}(t) \): \[ a_0 \ddot{e}(t) + b \dot{e}(t) + k_e e(t) = 0 \] Characteristic equation: \[ s^2 + \frac{b}{a_0}s + \frac{k_e}{a_0} = 0 \] For non-oscillatory convergence: \[ \left(\frac{b}{a_0}\right)^2 > 4\frac{k_e}{a_0} \implies b^2 > 4k_e a_0 \] --- ### **4) Desired acceleration expression for tracking \( f_e^*(t) \):** From earlier: \[ a_0 \ddot{q}(t) + k_e (q(t) - q_e^*(t)) = f_e^*(t) \] We want \( q(t) \) to track \( q_e^*(t) \) so set: \[ a_0 \ddot{q}(t) = f_e^*(t) - k_e (q(t) - q_e^*(t)) \] Or, \[ \ddot{q}(t) = \frac{1}{a_0} \left[ f_e^*(t) - k_e (q(t) - q_e^*(t)) \right] \] --- ## **b) If the environment has losses (i.e., friction: \( f_e(q, \dot{q}) = k_e(q - q_e^*) + b_e \dot{q} \)):** ### **1) Write down the tracking error:** Same as before: \[ e(t) = q(t) - q_e^*(t) \] --- ### **2) Propose a differential equation for the error:** System equation: \[ a_0 \ddot{q} + b_e \dot{q} + k_e(q - q_e^*) = f_e^*(t) \] Tracking error: \[ a_0 \ddot{e}(t) + b_e \dot{e}(t) + k_e e(t) = 0 \] --- ### **3) Condition for fast, non-oscillatory convergence:** Characteristic equation: \[ s^2 + \frac{b_e}{a_0} s + \frac{k_e}{a_0} = 0 \] For critically damped or overdamped (no oscillations): \[ \left(\frac{b_e}{a_0}\right)^2 \geq 4\frac{k_e}{a_0} \implies b_e^2 \geq 4 k_e a_0 \] --- ### **4) Desired acceleration for tracking \( f_e^*(t) \):** From above: \[ a_0 \ddot{q}(t) + b_e \dot{q}(t) + k_e (q(t) - q_e^*(t)) = f_e^*(t) \] So, \[ \ddot{q}(t) = \frac{1}{a_0} \left[ f_e^*(t) - b_e \dot{q}(t) - k_e (q(t) - q_e^*(t)) \right] \] --- ## **Summary Table** | Part | Tracking Error | Error Dynamics | Non-oscillation Condition | Desired Acceleration | |------|---------------|---------------|--------------------------|---------------------| | a) | \( q - q_e^* \) | \( \ddot{e} + \frac{k_e}{a_0}e = 0 \) | Add damping: \( b^2 > 4k_e a_0 \) | \( \ddot{q} = \frac{1}{a_0} [f_e^* - k_e (q-q_e^*)] \) | | b) | \( q - q_e^* \) | \( \ddot{e} + \frac{b_e}{a_0}\dot{e} + \frac{k_e}{a_0}e = 0 \) | \( b_e^2 \geq 4k_e a_0 \) | \( \ddot{q} = \frac{1}{a_0} [f_e^* - b_e \dot{q} - k_e(q-q_e^*)] \) | --- If you need the derivation for any step or a more detailed explanation, **let me know**!

**Final Answers:** **a) For a lossless environment (\(f_e(q) = k_e(q - q_e^*)\)):** - Error dynamics: \[ \ddot{e}(t) + 2 \zeta \sqrt{\frac{k_e}{a_0}}\, \dot{e}(t) + \frac{k_e}{a_0} e(t) = 0 \] - To ensure convergence without oscillations, choose damping \(b\) such that: \[ b^2 \geq 4\,a_0\,k_e \] - Desired acceleration: \[ \boxed{ \ddot{q}(t) = \frac{1}{a_0} \left[ f_e^*(t) - k_e \left(q(t) - q_e^*(t)\right) \right] } \] --- **b) For environment with losses (\(f_e(q, \dot{q}) = k_e(q - q_e^*) + b_e \dot{q}\)):** - Error dynamics: \[ \ddot{e}(t) + \frac{b_e}{a_0} \dot{e}(t) + \frac{k_e}{a_0} e(t) = 0 \] - To avoid oscillations, choose \(b_e\) such that: \[ b_e^2 \geq 4 a_0 k_e \] - Desired acceleration: \[ \boxed{ \ddot{q}(t) = \frac{1}{a_0} \left[ f_e^*(t) - b_e \dot{q}(t) - k_e (q(t) - q_e^*(t)) \right] } \]