Jacobians- velocities and static forces

Question 1

How can a(bVq) be represented as a rotation matrix?

Answer 1

a(bVq) = abR x Vq
where lower case letters are super/sub scripts

Question 2

How can bVq be represented in terms of differentiation?

Answer 2

lim∆t bQ(t+∆t) - bQ(t) / ∆t

Question 3

How dowe define the rotation of frame {b} relative to frame {a}?

Answer 3

aΩb

Question 4

how do we represent aΩb mapped to frame {c}?

Answer 4

c(aΩb)

Question 5

How else can we represent uΩc?
Where u is a universe frame

Answer 5

Wc (omega)

Question 6

Define aWb

Answer 6

the angular velocity of {b} relative to {universal} expressed in {a}

Question 7

If we have frames {a} and {b} which are not coincident, how do we represent the linear velocity of bVq in terms of {a}

Answer 7

aVq = a(bVq) + aVb.org

where aVb.org is the velocity of the position vector aPb.org

Question 8

If frames {a} and {B} are coincident, and we have an angular velocity aΩb and linear velocity bVq, how is aVq defined?

Answer 8

aVq = (aΩb x abRbQ) + abRbVq

Question 9

In the general case where origins are not coincident, we have aΩb and bVq, how do we find aVq?

Answer 9

aVq = (aΩb x abRbQ) + (abRbVq) + aVb.org

Question 10

What is the vector cross product of rotational velocity?

Answer 10

aVq = aΩb X aQ

Question 11

What is the purpose of Jacobians?

Answer 11

It can define cartesian velocities in terms of joint velocities (map from joint to cartesian)
v = J(ø)x ø

Question 12

How can we find the Jacobian using the partial differentiation method?

Answer 12

1. Conduct forward kinematics to get transformation matrix to end effector frame
2. Partially differentiate the position vector of the end effector
3. Find jacobian using ºv=ºJ(ø)ø

Question 13

When using partial differentiation how are terms differentiated in terms of øi ?

Answer 13

Terms that do not include øi are cancelled out and terms with øi are differentiated

Question 14

Identity for cos(ø1 + ø2)

Answer 14

cos(ø1+ø2) - sin(ø1 +ø2) = cos(ø1+ø2)

Question 15

Identity for sin(ø1 +ø2)

Answer 15

cosø1sinø2 + cosø2+sinø1