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Homework 10

Due date: 4/22/2023

1.   The Lagrange equation of motion for the double pendulum discussed in Lecture 12B in the generalized coordinates 91  and  92   is

「91 ]                               「  922 ]   「(m1 + m2 )L1 sin(91 )]      「FL1 cos(91 −0) ]

M 92  + m2L1L2 sin(91 −92 ) −912  + |L    m2L2 sin(92 )     」| g = |LFL2 cos(92  −0)」|

with

「(m1 + m2 )L1(2)                                  m2L1L2 cos(91 − 92 )]

M = |                                                                           | .

|Lm2L1L2 cos(91 − 92 )                    m2L2(2)                    」|

A mechanism described by this equation is used in a robotic arm in which it is required that large angle motions of the robotic arm be accurately tracked. Thus, linearizing this equation and finding a controller for the linearized system would not work. The full nonlinear description of the system given by the above equation must be used.

It is required that starting from  91 (t = 0) = 92 (t = 0) = 1,91 (t = 0) = 1,92 (t = 0) = 1 this robotic arm needs to exactly track the trajectory

92 (t) = 91 (t) + sin(t) .

What control force would be needed to be applied to generate this motion of this nonlinear system that exactly tracks this trajectory?

Hint: Find Fc . Use Matlab (symbolic).                                                                     (10)

2.   Consider the Lagrangian, L, for a Lagrangian System discussed in Lecture 13 B with one degree of freedom described by the generalized coordinate q. Let

L(q, q) = q2  + q2

Show that adding the term (f (q, t)) to L, where f (q, t) is any smooth function (i.e.,

has the requisite derivatives) would leave the Largrange equation for q unchanged. This

is then a particular case of the general result we gave in Lecture 13B.                    (10)