5.3.2 Model in Compact Form For control purposes, it is more practical to rewrite the Lagrangian...

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5.3.2 Model in Compact Form For control purposes, it is more practical to rewrite the Lagrangian dynamic model of the robot, that is, Equations (5.3) and (5.4), in the compact form (3.18), i.e. q+ [Mu(q) Mz(9) C11(9,9) C12(9,9) M21 (9) M22(q). C21 (9,9) C22(9,9) + [91 (9)] =T 92(9) M(g) C(q,) g(9) where M11(9) ml+m2 [12+12+211l2 cos(92)] + 11+ 12 M12(9) = = m2 [2+12 cos(92)] + I2 M21 (9) = m2 [2+1112 cos(92)] + 12 M22(9) m2l2+12 == = C11(9,9) mal1lc2 sin(92)92 C12(9,9) m2l1le2 sin(92) [1 +42] C21(9, 9)=malle2 sin(92)91 == C22(9,9) = 0 = 91(9) [mile1+m2l1]g sin(91) + m2lc29 sin(91 +92) 92(9) m2lc29 sin(91 + 92). == We emphasize that the appropriate state variables to describe the dynamic model of the robot are the positions q and q2 and the velocities 1 and 42. In terms of these state variables, the dynamic model of the robot may be written as 91 d 92 dt i -92 92 M(a)(t)-C(q, q)9-9(a)]. PD Control: Consider the three-DOF arm in the figure below. In this problem, you will design a PD- control law for this arm, with and without gravity compensation. at) 9,0 a. Write the dynamic model of the robot arm in the general matrix form of 7 = M(q) + C(q, q)q+g(q). Write out M, C, g matrices similar to what you see in page 123 of the Kelly's book. (step 1) b. Consider two PD Set-Point Control laws; u = ke + ke and u = ke + k + g(q) where e = qd-q and = qd -q (step 2). Note that u is PD control (Ref: Chapter 6, Kelly) and u2 is PD control with gravity compensation (Ref: Chapter 7, Kelly). Do the following parts considering u (PD control with gravity compensation) as the controller. c. Close the loop! Replace 7 (in the dynamic model derived from part a) with the control law u shown in part b (step 3). Note that when you use uz, g(q) will cancel out from both sides of the closed- loop system as described in Chapter 7 of Kelly. This is actually the same as if you use u and remove the gravity term, g(q), from the dynamic model! d. Represent the closed-loop control system derived from part c in the state-space format. (step 4) e. Is the closed-loop system linear? Why or why not? f. Find the equilibrium/equilibria of the system. Show your work! (step 5) g. Show that the origin is an equilibrium. (step 5) h. Using the Lyapunov's Direct Method, show that the origin is stable. (step 6) 5.3.2 Model in Compact Form For control purposes, it is more practical to rewrite the Lagrangian dynamic model of the robot, that is, Equations (5.3) and (5.4), in the compact form (3.18), i.e. q+ [Mu(q) Mz(9) C11(9,9) C12(9,9) M21 (9) M22(q). C21 (9,9) C22(9,9) + [91 (9)] =T 92(9) M(g) C(q,) g(9) where M11(9) ml+m2 [12+12+211l2 cos(92)] + 11+ 12 M12(9) = = m2 [2+12 cos(92)] + I2 M21 (9) = m2 [2+1112 cos(92)] + 12 M22(9) m2l2+12 == = C11(9,9) mal1lc2 sin(92)92 C12(9,9) m2l1le2 sin(92) [1 +42] C21(9, 9)=malle2 sin(92)91 == C22(9,9) = 0 = 91(9) [mile1+m2l1]g sin(91) + m2lc29 sin(91 +92) 92(9) m2lc29 sin(91 + 92). == We emphasize that the appropriate state variables to describe the dynamic model of the robot are the positions q and q2 and the velocities 1 and 42. In terms of these state variables, the dynamic model of the robot may be written as 91 d 92 dt i -92 92 M(a)(t)-C(q, q)9-9(a)]. PD Control: Consider the three-DOF arm in the figure below. In this problem, you will design a PD- control law for this arm, with and without gravity compensation. at) 9,0 a. Write the dynamic model of the robot arm in the general matrix form of 7 = M(q) + C(q, q)q+g(q). Write out M, C, g matrices similar to what you see in page 123 of the Kelly's book. (step 1) b. Consider two PD Set-Point Control laws; u = ke + ke and u = ke + k + g(q) where e = qd-q and = qd -q (step 2). Note that u is PD control (Ref: Chapter 6, Kelly) and u2 is PD control with gravity compensation (Ref: Chapter 7, Kelly). Do the following parts considering u (PD control with gravity compensation) as the controller. c. Close the loop! Replace 7 (in the dynamic model derived from part a) with the control law u shown in part b (step 3). Note that when you use uz, g(q) will cancel out from both sides of the closed- loop system as described in Chapter 7 of Kelly. This is actually the same as if you use u and remove the gravity term, g(q), from the dynamic model! d. Represent the closed-loop control system derived from part c in the state-space format. (step 4) e. Is the closed-loop system linear? Why or why not? f. Find the equilibrium/equilibria of the system. Show your work! (step 5) g. Show that the origin is an equilibrium. (step 5) h. Using the Lyapunov's Direct Method, show that the origin is stable. (step 6)