tailieunhanh - Ebook Essentials of mechatronics: Part 2
(BQ) Part 2 book "Essentials of mechatronics" has contents: Vectors, matrices, and tensors; mathematics for control; robotics, dynamics, and kinematics; further control theory; computer implementation, machine vision, case studies, the human element. | 7 Vectors, Matrices, and Tensors For both state space control theory and kinematics, we can take advantage of matrix methods. There is a tendency among mathematicians to regard matrices as arcane and mystic entities, with cryptic properties that reward a lifetime of study. Engineers can be duped into this point of view if they are not careful. MEET THE MATRIX Matrices are, in fact, just a form of shorthand that can come in very useful when a lot of calculating operations are involved. There are strict rules to observe, but when used properly matrices, vectors, and tensors are mere tools that are the servant of the engineer. You will probably have fi rst encountered matrices in the solution of simultaneous equations. To take a simple example, the equations 5x + 7 y = 2 2 x + 3y = 1 can be “tidied up” by separating the coefficients from the variables in the form Essentials of Mechatronics, by John Billingsley Copyright © 2006 John Wiley & Sons, Inc. 131 132 VECTORS, MATRICES, AND TENSORS 5 7 x 2 2 3 y = 1 where the variables x and y are now conveniently grouped as a vector. Now the multiplication rule has defined itself. We move across the top row of the matrix, multiplying each element by the corresponding component as we move down the vector to its right, adding up these products as we go. We put the resulting total in the top element, here 5x + 7y. Then we do the same for the next row, and so on. MORE ON VECTORS What does a vector actually “mean”? The answer has to be “anything you like.” Anything, that is, that cannot be represented by a single number but requires a string of numbers to defi ne it. It could even be a shopping list: 5 oranges + 3 lemons + 2 grapefruit can be written in matrix format as 5 [ orange lemon grapefruit ] 3 2 which we might write in a line of text as (orange, lemon, grapefruit) (5,3,2)′ or else place the dot between them that we use for scalar product. The numbers on the right
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