HW3_sol_2022S_v1 (3)

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Robot Devices, Kinematics, Dynamics, and Control (EN.530.646) Homework 3 Solution Jin Seob Kim, Ph.D. This Robot DKDC Homework/Exam Solution is exclusively use of Spring 2022 EN.530.646 RDKDC students, and is not to be posted, shared, or otherwise distributed. This document is copyrighted by the Johns Hopkins University and may not be distributed or transmitted without written permission from the University. 1
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(* HW3 Problem 2 Solution - Jin Seob Kim *) (* Part ( a ) *) (* Basic rotations *) Rotz [ a _] : = {{ Cos [ a ] , - Sin [ a ] , 0 } , { Sin [ a ] , Cos [ a ] , 0 } , { 0, 0, 1 }} Rotx [ a _] : = {{ 1, 0, 0 } , { 0, Cos [ a ] , - Sin [ a ]} , { 0, Sin [ a ] , Cos [ a ]}} Roty [ a _] : = {{ Cos [ a ] , 0, Sin [ a ]} , { 0, 1, 0 } , {- Sin [ a ] , 0, Cos [ a ]}} (* R _ xyz *) Rxyz = Rotx [ ϕ ] .Roty [ θ ] .Rotz [ ψ ] // Simplify; Rxyz // MatrixForm Cos [ θ ] Cos [ ψ ] - Cos [ θ ] Sin [ ψ ] Sin [ θ ] Cos [ ψ ] Sin [ θ ] Sin [ ϕ ] + Cos [ ϕ ] Sin [ ψ ] Cos [ ϕ ] Cos [ ψ ] - Sin [ θ ] Sin [ ϕ ] Sin [ ψ ] - Cos [ θ ] Sin [ ϕ ] - Cos [ ϕ ] Cos [ ψ ] Sin [ θ ] + Sin [ ϕ ] Sin [ ψ ] Cos [ ψ ] Sin [ ϕ ] + Cos [ ϕ ] Sin [ θ ] Sin [ ψ ] Cos [ θ ] Cos [ ϕ ] (* Then R _ yxz *) Ryxz = Roty [ θ ] .Rotx [ ϕ ] .Rotz [ ψ ] // Simplify; Ryxz // MatrixForm Cos [ θ ] Cos [ ψ ] + Sin [ θ ] Sin [ ϕ ] Sin [ ψ ] Cos [ ψ ] Sin [ θ ] Sin [ ϕ ] - Cos [ θ ] Sin [ ψ ] Cos [ ϕ ] Sin [ θ ] Cos [ ϕ ] Sin [ ψ ] Cos [ ϕ ] Cos [ ψ ] - Sin [ ϕ ] - Cos [ ψ ] Sin [ θ ] + Cos [ θ ] Sin [ ϕ ] Sin [ ψ ] Cos [ θ ] Cos [ ψ ] Sin [ ϕ ] + Sin [ θ ] Sin [ ψ ] Cos [ θ ] Cos [ ϕ ] (* As you can see Rxyz and Ryxz are different. The reason is that the rotations do not commute. *) (* Part ± b ² and ( c ) : refer to the next page *)
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Problem 3
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Problem 5
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Problem 6
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