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Controls Final

# Controls Final - MEM 255 Introduction to Controls Final...

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MEM 255: Introduction to Controls Final Examination September 4, 2007 Summer 2006 - 2007 Team J Jephte Augustin Basil Milton Peter Fink Jason Hollenstein John Gunn

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ABSTRACT We are required to study the means of transportation of astronauts during their EVAs. We intend to evaluate the feasibility of using a thruster for linear translation and a torquer to maintain an erect posture, both mounted on a backpack. Below we have provided with a mathematical model of the astronaut-backpack system, schematics, and relevant numerical values. The equations of motion for an astronaut in deep space are reduced to the following differential equation for ease of calculation: = Γ + = + + ψ θ q f BT Kq q D q M c ; The following assumptions have been made: 1. The body/backpack system is idealized as two rigid bodies connected by a line hinge at the hip joint. The hinge has spring restraint k and viscous damping c . 2. The control torque applied from a reaction wheel is T c and the force f is obtained from the thruster. 3. An angular position sensor measures the pitch θ of the torso. The following constants are used: J 1 = 33 kg.m 2 , J 2 = 10 kg.m 2 m 1 = 6 m 2 (=120 kg) m 2 = 20 kg. l 1 = 0.5 m l 2 = 0.5 m ε = 0.2 k = 1000 N.m.rad -1 c = 2N.m.rad -1 .sec MODEL PARAMETERS 2 2 1 2 1 2 1 2 11 1 2 1 2 12 2 1 2 2 22 2 2 1 2 1 2 1 2 ( ) ; ( ) ; m m m m m m M l l J J M l l l J M l J m m m m m m
11 12 12 22 2 2 2 1 1 2 1 2 2 2 1 2 0 0 0 0 1 ; ; ; ; 0 0 0 M M M D K B M M c k m m l l m m m m m l m m   Below is a schematic of proposed design l 2 l 1  l 1 f T c schematic : Center of masses reference

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1. Obtain a state space representation of the system, and identify A, B (treat both T c and f as inputs), and M matrices. Use = q q x as the states.
MatLAB Code %Part1 %Calculation of new damping, stiffness and input matrices clc M=[60.1429 18.5714;18.5714 14.2857] D=[0 0;0 2 ] K=[0 0;0 1000] I=[1 -0.0429;0 -0.0714] D1=inv(M)*D K1=inv(M)*K I1=inv(M)*I Output M = 60.1429 18.5714 18.5714 14.2857 D = 0 0 0 2 K = 0 0 0 1000 I = 1.0000 -0.0429 0 -0.0714 D1 = 0 -0.0722 0 0.2339 K1 = 0 -36.1110 0 116.9443 I1 = 0.0278 0.0014 -0.0361 -0.0068 2a. Compute the eigenvalues and eigenvectors of A . Is this system asymptotically stable? Why, or why not ? Explain.

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MatLAB Code A= [0 0 1 0 ; 0 0 0 1 ; 0 36.111 0 0.0722; 0 -116.9443 0 -0.2339]; B=[0 0 ;0 0; 0.0278 0.0014; -0.0361 -0.0068]; C=[1 0 0 0]; D=[0 0]; [Vect, val]=eig(A) Output val = 0 0 0 0 0 0 0 0 0 0 -0.1170 +10.8134i 0 0 0 0 -0.1170 -10.8134i Explanation: This system is not asymptotically stable because two of the roots are zero.
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