Matrices

Question 1

What is a vector?

Answer 1

A Parameter with a magnitude and a directioin

Question 2

What is a row vector?

Answer 2

[1,2]

Question 3

What is a column vector?

Answer 3

1

2

Question 4

How are two vectors summed?

Answer 4

By adding the number together

Question 5

What is a Matrix?

Answer 5

This is a set of vectors arranged in a square

Question 6

What are the parameters of the size of a matrix and what do they mean?

Answer 6

M and N
> M is the height
> N is the width

Question 7

How is matrix addition done?

Answer 7

Add the number in the same position on each vector together.
1 2 + 3 2 = 4 4
2 3 4 1 6 4

Question 8

How does matrix multiplication work?

Answer 8

By multiplying and summing along rows and collumns
a b + v x = av+by ax+bz
c d c d cv+dy cx+dz

Question 9

Do matrices have to be the same size to multiply?

Answer 9

No

Question 10

How is the determinant calculated?

Answer 10

ad-bc
a b ==> ad-bc
c d

Question 11

What is a unit matrix?

Answer 11

This is a matrix that when multiplied by another matrix has no effect on that matrix. Its values is essentially 1

Question 12

How is a matrix inverted?
e.g. x^-1 where
x = a b
c d

Answer 12

1/Determinent * | d -b |

| -c a |

Question 13

What is a singular matrix?

Answer 13

This is a matrix where the determinant is zero

Question 14

Can a singular matrix be inverted?

Answer 14

No

Question 15

What is matrix transportation?

Answer 15

This is when the matrix is reordered.
a b c ==> a d
d e f b e
c f

Question 16

What is the standard rotation matrix for 2 dimensions

Answer 16

R(θ) =
cosθ -sinθ
sinθ cosθ

Question 17

What is required for a matrix to be a rotation matrix?

Answer 17

It must have the form:
a -b
b  a
Where a^2 + b^2 = 1
and -1 ≤ a or b ≤ 1

Question 18

How can a rotation matrix be applied?

Answer 18

It must be multiplied to the matrix which is wanting to be rotated

Question 19

For a point [x,y] of distance r from the origin, how can this be modelled with 2 equations?

Answer 19

x = rcos(θ)
y = rsin(θ)

Question 20

For a vector [x,y] of length r (from the origin),what are the two equations for each dimension that can be made for the rotation of this vector?

Answer 20

x = rcos(θ+ϕ)
y = rsin(θ+ϕ)

Question 21

What is revolute?

Answer 21

This is the angle of the rotation link relative to the horizon plane.

Question 22

What is the revolute equation for the position of the end effector connected to a link of length L with a joint of angle θ relative to the horizontal?

Answer 22

Lcosθ

Lsinθ

Question 23

What does prismatic mean?

Answer 23

This is when the length of the link changes by a factor of T

Question 24

What is the prismatic equation for for the position of the end effector connected to a link of changing length L by a factor of T with a joint of angle θ relative to the horizontal?

Answer 24

L(T)cosθ

L(T)sinθ

Question 25

What equation can be formed for the position of the end effector connected to a link of changing length L2 by a factor of T2 with a joint connected to link L1 of angle ϕ (relative to the angle of the link L1) with a joint of angle θ relative to the horizontal?

Answer 25

L1cosθ + L2(T)cos(θ+ϕ) | L1sinθ + L2(T)sin(θ+ϕ)

Question 26

What equation can be formed for a 3 link robot each with length of link L with 3 joints with a relative angle θ to each other?

Answer 26

Lcosθ1 + Lcos(θ1+θ2) + Lcos(θ1+θ2+θ3) | Lsinθ1 + Lsin(θ1+θ2) + Lsin(θ1+θ2+θ3)

Question 27

How can equations be used to model the end effector location of a parallel robot?

Answer 27

Multiple equations can be formed for each parallel linkage and then the end effector location has to meet both of these equations for it to be a valid location

Question 28

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Answer 28

DO IT

Question 29

What is the Jacobean matrix?

Answer 29

This is a matrix which is used to calculate the change of the end effector based off the change of the angles of the joints.

Question 30

What are the applications of the Jacobean matrix?

Answer 30

If we know how the end effector location changes then we can calculate the speed of it. This is useful because if we want it to move at a constant speed then we need to know how fast to move the joint angles to maintain a constant end effector speed

Question 31

What is the equation for the change in the end effector location?

Answer 31

``` [dx/dy] = J.[dθ1/dθ2] dx = J.dq ```

Question 32

What is the equation for the rate of change of the end effector location?

Answer 32

``` [dx/dy]/dt = J.[dθ1/dθ2]/dt dx/dt = J.dq/dt v = J.q* ```

Question 33

How can we calculate what each joint is doing in the equation: v = J.q*

Answer 33

v = J1.∆θ1/t + J2.∆θ2/t Where: J = | J1 | | J2 |

Question 34

What is singularity?

Answer 34

This is when the motion the robot is required to do is not possible due to the nature of the jacobean matrix.

Question 35

What are 3 examples of singularity?

Answer 35

> When maintaining a constant end effector speed, the joint angles may have to move at infinite speed. This is impossible. > The motion of the joints is such that the end effector speed is 0. The joint motions cancel each other out. > When the robot cannot move the end effector to the desired position because it is out of the robots work space. In this case, it will cause the robot to get stuck and may not be able to fix itself.

Question 36

What is it called when the speed of the joints is calculated from the rate of change of the end effector location?

Answer 36

Inverse jacobean