lecture 2: kinematics: rigid motions and homogeneous ......rigid motions and homogeneous...
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Lecture 2: Kinematics:
Rigid Motions and Homogeneous Transformations
• Frames, Points and Vectors
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 1/20
Lecture 2: Kinematics:
Rigid Motions and Homogeneous Transformations
• Frames, Points and Vectors
• Rotations:◦ In 2 and 3 Dimensions◦ Transformations by Rotation
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 1/20
Lecture 2: Kinematics:
Rigid Motions and Homogeneous Transformations
• Frames, Points and Vectors
• Rotations:◦ In 2 and 3 Dimensions◦ Transformations by Rotation
• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 1/20
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
x0
y0
V1
o0
P
A coordinate frame in R2: the point P = [x0p, y0
p] can be
associated with the vector ~V1. Here x0p denotes x-coordinate of
point P in (x0, o0, y0)-frame, i.e. in the 0-frame.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 2/20
0 0.5 1 1.5 2 2.5 3
0
0.5
1
1.5
2
2.5
3
x0
y0
V1
o0
P x1y
1
V2
o1
Two coordinate frames in R2: the point P = [x0p, y0
p] = [x1p, y1
p]
can be associated with vectors ~V1 and ~V2. Here x1p denotes
x-coordinate of point P in the 1-frame.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 2/20
Frames, Points and Vectors
• A point corresponds a particular location in the space.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 3/20
Frames, Points and Vectors
• A point corresponds a particular location in the space.
• A point has different representation (coordinates) in differentframes.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 3/20
Frames, Points and Vectors
• A point corresponds a particular location in the space.
• A point has different representation (coordinates) in differentframes.
• A vector is defined by direction and magnitude.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 3/20
Frames, Points and Vectors
• A point corresponds a particular location in the space.
• A point has different representation (coordinates) in differentframes.
• A vector is defined by direction and magnitude.
• Vectors with the same direction and magnitude are thesame.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 3/20
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
3.5
x0
y0
V10
o0
P x1y
1
o1
The coordinates of the vector ~V1 in the 0-frame [x0p, y0
p].
What would be coordinates of ~V1 in the 1-frame?
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 4/20
0 0.5 1 1.5 2 2.5 3 3.5
0
0.5
1
1.5
2
2.5
3
3.5
x0
y0
V10
o0
P x1y
1
o1
V11
We need to consider the vector ~V 11
of the same direction and
magnitude as ~V1 but with the origin in o1. Conclusion: we cansum vectors only if they are expressed in parallel frames.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 4/20
Lecture 2: Kinematics:
Rigid Motions and Homogeneous Transformations
• Frames, Points and Vectors
• Rotations:◦ In 2 and 3 Dimensions◦ Transformations by Rotation
• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 5/20
Rotations in 2-D
To find an appropriate way to parametrize rotations in 2-D, let ustrack vectors x0
1(·), y0
1(·) as θ varies, i.e.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 6/20
Rotations in 2-D
To find an appropriate way to parametrize rotations in 2-D, let ustrack vectors x0
1(·), y0
1(·) as θ varies, i.e.
x01(θ) =
[cos(θ)
sin(θ)
], y0
1(θ) = x0
1
(θ+
π
2
)=
[cos
(θ+ π
2
)
sin(θ+ π
2
)]
=
[− sin(θ)
cos(θ)
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 6/20
Rotations in 2-D
The matrix
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
is called the rotation matrix.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 7/20
Rotations in 2-D
The matrix
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
is called the rotation matrix.
It has a number of interesting properties:
• det R(θ) = cos2(θ) + sin2(θ) = 1
• R(θ)−1 =
[a b
c d
]−1
=1
det R(θ)
[d −b
−c a
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 7/20
Rotations in 2-D
The matrix
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
is called the rotation matrix.
It has a number of interesting properties:
• det R(θ) = cos2(θ) + sin2(θ) = 1
• R(θ)−1 =
[cos(θ) sin(θ)
− sin(θ) cos(θ)
]= RT (θ)
A n × n-matrix X that satisfies the property, X−1 = XT
⇒{XXT = In, det(XXT ) = 1 = det(X) det(XT ) = det(X)2
}
is called orthogonal, X ∈ O(n). If det X = 1 ⇒ X ∈ SO(n)
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 7/20
Rotations in 2-D
Let us consider another way for computing
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 8/20
Rotations in 2-D
Let us consider another way for computing
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
As known, scalar product between two vectors is
~a ·~b = a1b1 + a2b2 + a3b3 = |~a| · |~b| · cos
(~̂a,~b
)
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 8/20
Rotations in 2-D
Let us consider another way for computing
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
As known, scalar product between two vectors is
~a ·~b = a1b1 + a2b2 + a3b3 = |~a| · |~b| · cos
(~̂a,~b
)
All vectors of coordinate frames of magnitude 1, therefore
x01(θ) =
[x0
1(θ) · x0
x01(θ) · y0
], y0
1(θ) =
[y01(θ) · x0
y01(θ) · y0
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 8/20
Rotations in 2-D
Let us consider another way for computing
R(θ) =[x0
1(θ)| y0
1(θ)
]=
[cos(θ) − sin(θ)
sin(θ) cos(θ)
]
As known, scalar product between two vectors is
~a ·~b = a1b1 + a2b2 + a3b3 = |~a| · |~b| · cos
(~̂a,~b
)
All vectors of coordinate frames of magnitude 1, therefore
x01(θ) =
[x0
1(θ) · x0
x01(θ) · y0
], y0
1(θ) =
[y01(θ) · x0
y01(θ) · y0
]
⇒ R(θ) =[x0
1(θ)| y0
1(θ)
]=
[x0
1(θ) · x0 y0
1(θ) · x0
x01(θ) · y0 y0
1(θ) · y0
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 8/20
Rotations in 3-D
Rotation matrix for 3-dimensions is then
R01(θ) =
[x0
1(θ)| y0
1(θ)| z0
1(θ)
]
=
x01(θ) · x0 y0
1(θ) · x0 z0
1(θ) · x0
x01(θ) · y0 y0
1(θ) · y0 z0
1(θ) · y0
x01(θ) · z0 y0
1(θ) · z0 z0
1(θ) · z0
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 9/20
Rotations in 3-D
Rotation matrix for 3-dimensions is then
R01(θ) =
[x0
1(θ)| y0
1(θ)| z0
1(θ)
]
=
x01(θ) · x0 y0
1(θ) · x0 z0
1(θ) · x0
x01(θ) · y0 y0
1(θ) · y0 z0
1(θ) · y0
x01(θ) · z0 y0
1(θ) · z0 z0
1(θ) · z0
Properties to check:
• Columns of R01(·) are mutually orthogonal, for instance
[x0
1(θ)x0, x0
1(θ)y0, x0
1(θ)z0
] [y01(θ)x0, y0
1(θ)y0, y0
1(θ)z0
]T=
= x01(θ) · y0
1(θ)︸ ︷︷ ︸
= 0
·|x0|2 + x01(θ) · y0
1(θ)︸ ︷︷ ︸
= 0
·|y0|2 + x01(θ) · y0
1(θ)︸ ︷︷ ︸
= 0
·|z0|2
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 9/20
Rotations in 3-D
Rotation matrix for 3-dimensions is then
R01(θ) =
[x0
1(θ)| y0
1(θ)| z0
1(θ)
]
=
x01(θ) · x0 y0
1(θ) · x0 z0
1(θ) · x0
x01(θ) · y0 y0
1(θ) · y0 z0
1(θ) · y0
x01(θ) · z0 y0
1(θ) · z0 z0
1(θ) · z0
Properties to check:
• Columns of R01(·) are mutually orthogonal;
• R01(θ)
[R0
1(θ)
]T= I3
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 9/20
Rotations in 3-D
Rotation matrix for 3-dimensions is then
R01(θ) =
[x0
1(θ)| y0
1(θ)| z0
1(θ)
]
=
x01(θ) · x0 y0
1(θ) · x0 z0
1(θ) · x0
x01(θ) · y0 y0
1(θ) · y0 z0
1(θ) · y0
x01(θ) · z0 y0
1(θ) · z0 z0
1(θ) · z0
Properties to check:
• Columns of R01(·) are mutually orthogonal;
• R01(θ)
[R0
1(θ)
]T= I3
• det R01(θ) = 1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 9/20
Example of Rotation in 3-D
1-Frame is rotated through θ-angle around z0-axis.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 10/20
Example of Rotation in 3-D
1-Frame is rotated through θ-angle around z0-axis.
x01(θ)x0 = cos θ y0
1(θ)x0 = − sin θ z0
1(θ)x0 = 0
x01(θ)y0 = sin θ y0
1(θ)y0 = cos θ z0
1(θ)y0 = 0
x01(θ)z0 = 0 y0
1(θ)z0 = 0 z0
1(θ)z0 = 1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 10/20
Example of Rotation in 3-D
1-Frame is rotated through θ-angle around z0-axis.
R01(θ) =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
{=: Rz,θ
}
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 10/20
Example of Rotation in 3-D
1-Frame is rotated through θ-angle around z0-axis.
This basic rotation matrix clearly satisfies the properties
Rz,0 = I3, Rz,θRz,φ = Rz,θ+φ, [Rz,θ]−1 = Rz,−θ
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 10/20
Basic Rotations in 3-D
In the way we have introduced the basic rotation matrix
Rz,θ =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
we can introduce basic rotation matrices
Rx,θ, Ry,θ
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 11/20
Basic Rotations in 3-D
In the way we have introduced the basic rotation matrix
Rz,θ =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
we can introduce basic rotation matrices
Rx,θ, Ry,θ
They are
Rx,θ =
1 0 0
0 ∗ ∗
0 ∗ ∗
, Ry,θ =
∗ 0 ∗
0 1 0
∗ 0 ∗
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 11/20
Basic Rotations in 3-D
In the way we have introduced the basic rotation matrix
Rz,θ =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
we can introduce basic rotation matrices
Rx,θ, Ry,θ
They are
Rx,θ =
1 0 0
0 cos θ − sin θ
0 sin θ cos θ
, Ry,θ =
cos θ 0 sin θ
0 1 0
− sin θ 0 cos θ
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 11/20
Transformations by Rotations in 3-D
The 0-frame is our world, the 1-frame is fixed to a rigid body.
What will happen with points of body (let say p)if we rotate the body, i.e. the 1-frame?
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 12/20
Transformations by Rotations in 3-D
How to trace the change of position of the point in the 0-frame?
Let say, we are interested in coordinates of the point p, in the1-frame they are constant, but the 0-frame they are changed!
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 12/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
Clearly, with the rotation the basis of the 1-frame is changingand we know how it does that!
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
Clearly, with the rotation the basis of the 1-frame is changingand we know how it does that!
For instance,
x01
=
x1 · x0
x1 · y0
x1 · z0
=
x01
· x0 y01
· x0 z01
· x0
x01
· y0 y01
· y0 z01
· y0
x01
· z0 y01
· z0 z01
· z0
︸ ︷︷ ︸= R
1
0
0
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
For instance,
x01
= R
1
0
0
, y0
1= R
0
1
0
, z0
1= R
0
0
1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
For instance,
x01
= R
1
0
0
, y0
1= R
0
1
0
, z0
1= R
0
0
1
p0 = u · x01+ v · y0
1+ w · z0
1= R
u
0
0
+ R
0
v
0
+ R
0
0
w
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Transformations by Rotations in 3-D
The coordinates of point p in the 1-frame is p1 = [u, v, w]T , i.e.
p1 = u · ~x1 + v · ~y1 + w · ~z1
and the coordinates u, v, w do not change when the 1-frame isrotated. We need to find the coordinates of p in the 0-frame.
For instance,
x01
= R
1
0
0
, y0
1= R
0
1
0
, z0
1= R
0
0
1
p0 = u · x01+ v · y0
1+ w · z0
1= R
u
v
w
= R0
1p1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 13/20
Lecture 2: Kinematics:
Rigid Motions and Homogeneous Transformations
• Frames, Points and Vectors
• Rotations:◦ In 2 and 3 Dimensions◦ Transformations by Rotation
• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 14/20
Rotation with Respect to the Current Frame:
Suppose that we have 3 frames:
(o0, x0, y0, z0), (o1, x1, y1, z1), (o2, x2, y2, z2).
Any point p will have three representations:
p0 = [u0, v0, w0]T , p1 = [u1, v1, w1]
T , p2 = [u2, v2, w2]T
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 15/20
Rotation with Respect to the Current Frame:
Suppose that we have 3 frames:
(o0, x0, y0, z0), (o1, x1, y1, z1), (o2, x2, y2, z2).
Any point p will have three representations:
p0 = [u0, v0, w0]T , p1 = [u1, v1, w1]
T , p2 = [u2, v2, w2]T
We know that
p0 = R01
p1, p1 = R12
p2, p0 = R02
p2
How are the matrices R01, R1
2and R0
2related?
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 15/20
Rotation with Respect to the Current Frame:
Suppose that we have 3 frames:
(o0, x0, y0, z0), (o1, x1, y1, z1), (o2, x2, y2, z2).
Any point p will have three representations:
p0 = [u0, v0, w0]T , p1 = [u1, v1, w1]
T , p2 = [u2, v2, w2]T
We know that
p0 = R01
p1, p1 = R12
p2, p0 = R02
p2
How are the matrices R01, R1
2and R0
2related?
We can compute p0 in two different ways
p0 = R01
p1 = R01
R12
p2, p0 = R02
p2
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 15/20
Rotation with Respect to the Current Frame:
Suppose that we have 3 frames:
(o0, x0, y0, z0), (o1, x1, y1, z1), (o2, x2, y2, z2).
Any point p will have three representations:
p0 = [u0, v0, w0]T , p1 = [u1, v1, w1]
T , p2 = [u2, v2, w2]T
We know that
p0 = R01
p1, p1 = R12
p2, p0 = R02
p2
How are the matrices R01, R1
2and R0
2related?
We can compute p0 in two different ways
p0 = R01
p1 = R01
R12
p2, p0 = R02
p2
⇒ R01
R12
≡ R02
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 15/20
Example 2.5:
Suppose we rotate• first the frame by angle φ around current y-axis,• then rotate by angle θ around the current z-axis.
Find the combined rotation
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 16/20
Example 2.5:
The rotations around y- and z-axis are basic rotations
Ry,φ =
cos φ 0 sin φ
0 1 0
− sin φ 0 cos φ
, Rz,θ =
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 16/20
Example 2.5:
Therefore the overall rotation is
R = Ry,φRz,θ =
cos φ 0 sin φ
0 1 0
− sin φ 0 cos φ
cos θ − sin θ 0
sin θ cos θ 0
0 0 1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 16/20
Example 2.5:
Therefore the overall rotation is
R = Ry,φRz,θ =
cφcθ −cφsθ sφ
sθ cθ 0
−sφcθ sφsθ cφ
,
{⇒ p0 = R p2
}
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 16/20
Example 2.5:
Important Observation: Rotations do not commute
Ry,φRz,θ 6= Rz,θRy,φ
So that the order of rotations is important!
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 17/20
Example 2.5:
Important Observation: Rotations do not commute
Ry,φRz,θ 6= Rz,θRy,φ
So that the order of rotations is important!
Indeed
Ry,φRz,θ =
cφcθ −cφsθ sφ
sθ cθ 0
−sφcθ sφsθ cφ
and
Rz,θRy,φ =
cφcθ −sθ cθsφ
sθcφ cθ sφsθ
−sφ 0 cφ
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 17/20
Rotation with Respect to the Fixed Frame:
How to compute the rotation if the basic rotations are done withrespect to fixed frames?
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 18/20
Rotation with Respect to the Fixed Frame:
Given two frames and the rotation:
(o0, x0, y0, z0), (o1, x1, y1, z1), p0 = R01p1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 19/20
Rotation with Respect to the Fixed Frame:
Given two frames and the rotation:
(o0, x0, y0, z0), (o1, x1, y1, z1), p0 = R01p1
Given a linear transform A, for which we know how it acts onvectors of the 0-frame
~a0 = A~b0
How it acts on the vectors of the 1-frame?
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 19/20
Rotation with Respect to the Fixed Frame:
Given two frames and the rotation:
(o0, x0, y0, z0), (o1, x1, y1, z1), p0 = R01p1
Given a linear transform A, for which we know how it acts onvectors of the 0-frame
~a0 = A~b0
How it acts on the vectors of the 1-frame?
To compute its action, we need to observe that vectors in bothframes are in one-to-one correspondence, i.e
• given b0, then b1 =[R0
1
]−1
b0
• given b1, then b0 = R01b1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 19/20
Rotation with Respect to the Fixed Frame:
Given two frames and the rotation:
(o0, x0, y0, z0), (o1, x1, y1, z1), p0 = R01p1
To compute its action, we need to observe that vectors in bothframes are in one-to-one correspondence, i.e
• given b0, then b1 =[R0
1
]−1
b0
• given b1, then b0 = R01b1
To define A acting in the 1-frame, use its definition in 0-frame
[R0
1
]−1
a0
︸ ︷︷ ︸= a1
=[R0
1
]−1
Ab0 =[R0
1
]−1
AR01︸ ︷︷ ︸
B
b1
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 19/20
Rotation with Respect to the Fixed Frame:
We have two rotations• the basic rotation R0
1= Ry0,φ: by angle φ around y0-axis
• the rotation R12
defined as the rotation by angle θ aroundz0-axis (not z1-axis)
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 20/20
Rotation with Respect to the Fixed Frame:
We have two rotations• the basic rotation R0
1= Ry0,φ: by angle φ around y0-axis
• the rotation R12
defined as the rotation by angle θ aroundz0-axis (not z1-axis)
The combined rotation will be
R02
= R01R1
2= Ry0,φR1
2= Ry0,φ
[{Ry0,φ}−1 Rz0,θRy0,φ
]
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 20/20
Rotation with Respect to the Fixed Frame:
We have two rotations• the basic rotation R0
1= Ry0,φ: by angle φ around y0-axis
• the rotation R12
defined as the rotation by angle θ aroundz0-axis (not z1-axis)
The combined rotation will be
R02
= Ry0,φ
[{Ry0,φ}−1 Rz0,θRy0,φ
]= Rz0,θRy0,φ
c©Anton Shiriaev. 5EL158: Lecture 2 – p. 20/20