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Lecture 3: Kinematics: Rigid Motions and Homogeneous Transformations Composition of Rotations: Rotations with Respect to the Current Frame Rotations with Respect to the Fixed Frame Example c Anton Shiriaev. 5EL158: Lecture 3 – p. 1/15

Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

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Page 1: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Lecture 3: Kinematics:

Rigid Motions and Homogeneous Transformations

• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame◦ Example

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 1/15

Page 2: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Lecture 3: Kinematics:

Rigid Motions and Homogeneous Transformations

• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame◦ Example

• Parameterizations of Rotations:◦ Euler Angles◦ Roll, Pitch and Yaw Angles◦ Axis/Angle Representation

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 1/15

Page 3: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Lecture 3: Kinematics:

Rigid Motions and Homogeneous Transformations

• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame◦ Example

• Parameterizations of Rotations:◦ Euler Angles◦ Roll, Pitch and Yaw Angles◦ Axis/Angle Representation

• Rigid Motions and Homogeneous Transformations

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 1/15

Page 4: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 5: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

If we are given (N − 1)-rotation matrices

R01, R1

2, . . . , R(N−2)(N−1)

that represent consecutive rotation between the current frames

{(x0y0z0), (x1y1z1)

},

{(x1y1z1), (x2y2z2)

}, . . . ,

{(xN−2yN−2zN−2), (xN−1yN−1zN−1)

}

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 6: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

If we are given (N − 1)-rotation matrices

R01, R1

2, . . . , R(N−2)(N−1)

that represent consecutive rotations between the current frames

{(x0y0z0), (x1y1z1)

},

{(x1y1z1), (x2y2z2)

}, . . . ,

{(xN−2yN−2zN−2), (xN−1yN−1zN−1)

}

The formula to compute the position of the point in the 0-framehaving known its position in the 1-frame

p0 = R01p1

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 7: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

If we are given (N − 1)-rotation matrices

R01, R1

2, . . . , R(N−2)(N−1)

that represent consecutive rotations between the current frames

{(x0y0z0), (x1y1z1)

},

{(x1y1z1), (x2y2z2)

}, . . . ,

{(xN−2yN−2zN−2), (xN−1yN−1zN−1)

}

The formula to compute the position of the point in the 0-framehaving known its position in the 2-frame

p0 = R01p1, p1 = R1

2p2

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 8: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

If we are given (N − 1)-rotation matrices

R01, R1

2, . . . , R(N−2)(N−1)

that represent consecutive rotations between the current frames

{(x0y0z0), (x1y1z1)

},

{(x1y1z1), (x2y2z2)

}, . . . ,

{(xN−2yN−2zN−2), (xN−1yN−1zN−1)

}

The formula to compute the position of the point in the 0-framehaving known its position in the (N − 1)-frame

p0 = R01p1, p1 = R1

2p2, . . . , p(N−2) = R(N−2)(N−1)p

(N−1)

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 9: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

What Have We Learned:

Given N -frames in the 3-dimensional space

(o0, x0, y0, z0), (o1, x1, y1, z1), . . . (oN−1, xN−1, yN−1, zN−1)

If we are given (N − 1)-rotation matrices

R01, R1

2, . . . , R(N−2)(N−1)

that represent consecutive rotations between the current frames

{(x0y0z0), (x1y1z1)

},

{(x1y1z1), (x2y2z2)

}, . . . ,

{(xN−2yN−2zN−2), (xN−1yN−1zN−1)

}

The formula to compute the position of the point in the 0-framehaving known its position in the (N − 1)-frame is

p0 = R01 R1

2 R23 · · · R

(N−2)(N−1) p(N−1)

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 2/15

Page 10: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 3/15

Page 11: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

The total rotation will be then

R = Rx,θRz,φ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 3/15

Page 12: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

The total rotation will be then

R = Rx,θRz,φ

For any point of the 2-frame with coordinates p2 =[x2, y2, z2

]T

its coordinates in the 0-frame are computed simply as

p0 = R p2 =[Rx,θRz,φ

]p2

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 3/15

Page 13: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 4/15

Page 14: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

The total rotation will be then

R = Rx,θRz,φR3

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 4/15

Page 15: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

The total rotation will be then

R = Rx,θRz,φR3

We have computed this rotation as

R3 =[Rx,θRz,φ

]−1

· Rz,α ·[Rx,θRz,φ

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 4/15

Page 16: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

The total rotation will be then

R = Rx,θRz,φR3 = Rz,αRx,θRz,φ

We have computed this rotation as

R3 =[Rx,θRz,φ

]−1

· Rz,α ·[Rx,θRz,φ

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 4/15

Page 17: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 5/15

Page 18: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

The total rotation will be then

R = Rx,θRz,φR3 Ry,β =[Rz,αRx,θRz,φ

]Ry,β

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 5/15

Page 19: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

• 5: A rotation of δ about the fixed axis x

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 6/15

Page 20: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

• 5: A rotation of δ about the fixed axis x

The total rotation will be then

R = Rz,αRx,θRz,φRy,βR5

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 6/15

Page 21: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

• 5: A rotation of δ about the fixed axis x

The total rotation will be then

R = Rz,αRx,θRz,φRy,βR5

We have computed this rotation as

R5 =[Rz,αRx,θRz,φRy,β

]−1

· Rx,δ ·[Rz,αRx,θRz,φRy,β

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 6/15

Page 22: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Example 2.8:

Find the rotation R defined by the following basic rotations:

• 1: A rotation of θ about the current axis x;

• 2: A rotation of φ about the current axis z

• 3: A rotation of α about the fixed axis z

• 4: A rotation of β about the current axis y

• 5: A rotation of δ about the fixed axis x

The total rotation will be then

R = Rx,δ · Rz,α · Rx,θ · Rz,φ · Ry,β

We have computed this rotation as

R5 =[Rz,αRx,θRz,φRy,β

]−1

· Rx,δ ·[Rz,αRx,θRz,φRy,β

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 6/15

Page 23: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Lecture 3: Kinematics:

Rigid Motions and Homogeneous Transformations

• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame◦ Example

• Parameterizations of Rotations:◦ Euler Angles◦ Roll, Pitch and Yaw Angles◦ Axis/Angle Representation

• Rigid Motions and Homogeneous Transformations

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 7/15

Page 24: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Parameterizations of Rotations:

Any rotation matrix R is• of dimension 3 × 3, i.e. it is 9-numbers

• belongs to SO(3), i.e.◦ its 3 columns are vectors of length 1 (3 equations)

◦ its 3 columns are orthogonal to each other (3 equations)

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 8/15

Page 25: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Parameterizations of Rotations:

Any rotation matrix R is• of dimension 3 × 3, i.e. it is 9-numbers

• belongs to SO(3), i.e.◦ its 3 columns are vectors of length 1 (3 equations)

◦ its 3 columns are orthogonal to each other (3 equations)

Except the particular cases, only 3 of 9

numbers that parameterize the rotation matrix,

can be assigned freely!

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 8/15

Page 26: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Euler Angles:

Euler angles are angles of 3 rotations about current axes

RZYZ := Rz,φ · Ry,θ · Rz,ψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 9/15

Page 27: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Euler Angles:

Euler angles are angles of 3 rotations about current axes

RZYZ :=

cφ −sφ 0

sφ cφ 0

0 0 1

· Ry,θ · Rz,ψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 9/15

Page 28: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Euler Angles:

Euler angles are angles of 3 rotations about current axes

RZYZ :=

cφ −sφ 0

sφ cφ 0

0 0 1

·

cθ 0 sθ

0 1 0

−sθ 0 cθ

· Rz,ψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 9/15

Page 29: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Euler Angles:

Euler angles are angles of 3 rotations about current axes

RZYZ :=

cφ −sφ 0

sφ cφ 0

0 0 1

·

cθ 0 sθ

0 1 0

−sθ 0 cθ

·

cψ −sψ 0

sψ cψ 0

0 0 1

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 9/15

Page 30: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Roll, Pitch Yaw Angles:

Roll, pitch and yaw angles are angles of 3 rotations about thefixed axes x, y and z

Rxyz := Rz,φ · Ry,θ · Rx,ψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 10/15

Page 31: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Roll, Pitch Yaw Angles:

Roll, pitch and yaw angles are angles of 3 rotationsabout the fixed axes x, y and z

Rxyz := Rz,φ · Ry,θ ·

1 0 0

0 cψ −sψ

0 sψ cψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 10/15

Page 32: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Roll, Pitch Yaw Angles:

Roll, pitch and yaw angles are angles of 3 rotationsabout the fixed axes x, y and z

Rxyz := Rz,φ ·

cθ 0 sθ

0 1 0

−sθ 0 cθ

·

1 0 0

0 cψ −sψ

0 sψ cψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 10/15

Page 33: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Roll, Pitch Yaw Angles:

Roll, pitch and yaw angles are angles of 3 rotationsabout the fixed axes x, y and z

Rxyz :=

cφ −sφ 0

sφ cφ 0

0 0 1

·

cθ 0 sθ

0 1 0

−sθ 0 cθ

·

1 0 0

0 cψ −sψ

0 sψ cψ

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 10/15

Page 34: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Axis/Angle Representation for a Rotation Matrix:

Any rotational matrix can be expressed as a rotation of angle θ

about an axis k = [kx, ky, kz]T

R~k,θ = R · Rz,θ · R−1, R = Rz,α · Ry,β

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 11/15

Page 35: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Lecture 3: Kinematics:

Rigid Motions and Homogeneous Transformations

• Composition of Rotations:◦ Rotations with Respect to the Current Frame◦ Rotations with Respect to the Fixed Frame◦ Example

• Parameterizations of Rotations:◦ Euler Angles◦ Roll, Pitch and Yaw Angles◦ Axis/Angle Representation

• Rigid Motions and Homogeneous Transformations

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 12/15

Page 36: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Rigid Motions

A rigid motion is an ordered pair (R, d), where R ∈ SO(3) andd ∈ R3. The group of all rigid motions is known as SpecialEuclidean Group denoted by SE(3).

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 13/15

Page 37: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Rigid Motions

A rigid motion is an ordered pair (R, d), where R ∈ SO(3) andd ∈ R3. The group of all rigid motions is known as SpecialEuclidean Group denoted by SE(3).

It is the way to compute coordinates in different frames

p0 = R01p1 + d0

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 13/15

Page 38: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Rigid Motions

A rigid motion is an ordered pair (R, d), where R ∈ SO(3) andd ∈ R3. The group of all rigid motions is known as SpecialEuclidean Group denoted by SE(3).

It is the way to compute coordinates in different frames

p0 = R01p1 + d0

If there are 3 frames corresponding to 2 rigid motions

p1 = R12p2 + d1

2

p0 = R01p1 + d0

1

then the overall motion is

p0 = R01R1

2p2 + R01d1

2 + d01

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 13/15

Page 39: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

HT is just a convenient way to write the formula

p0 = R01R1

2 p2 + R01d1 + d0

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 14/15

Page 40: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

HT is just a convenient way to write the formula

p0 = R01R1

2 p2 + R01d1 + d0

Given two rigid motions (R01, d0

1) and (R12, d1

2), consider theproduct of two matrices

[

R01 d0

1

01×3 1

] [

R12 d1

2

01×3 1

]

=

[

R01R1

2 R01d1

2 + d01

01×3 1

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 14/15

Page 41: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

HT is just a convenient way to write the formula

p0 = R01R1

2 p2 + R01d1 + d0

Given two rigid motions (R01, d0

1) and (R12, d1

2), consider theproduct of two matrices

[

R01 d0

1

01×3 1

] [

R12 d1

2

01×3 1

]

=

[

R01R1

2 R01d1

2 + d01

01×3 1

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 14/15

Page 42: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

HT is just a convenient way to write the formula

p0 = R01R1

2 p2 + R01d1 + d0

Given two rigid motions (R01, d0

1) and (R12, d1

2), consider theproduct of two matrices

[

R01 d0

1

01×3 1

] [

R12 d1

2

01×3 1

]

=

[

R01R1

2 R01d1

2 + d01

01×3 1

]

Given a rigid motion (R, d) ∈ SE(3), the 4 × 4-matrix

H =

[

R d

01×3 1

]

is called homogeneous transformation associated with (R, d)

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 14/15

Page 43: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

To use HTs in computing coordinates of points, we need toextend the vectors p0 and p1 by one coordinate. Namely

P 0 =

[

p0

1

]

, P 1 =

[

p1

1

]

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 15/15

Page 44: Lecture 3: Kinematics: Rigid Motions and Homogeneous ......Rigid Motions and Homogeneous Transformations • Composition of Rotations: Rotations with Respect to the Current Frame Rotations

Concept of Homogeneous Transformation

To use HTs in computing coordinates of points, we need toextend the vectors p0 and p1 by one coordinate. Namely

P 0 =

[

p0

1

]

, P 1 =

[

p1

1

]

Then

P 0 =

[

p0

1

]

=

[

R01p1 + d0

1

]

=

[

R01 d0

01×3 1

]

︸ ︷︷ ︸

H01

[

p1

1

]

︸ ︷︷ ︸

P 1

that is in shortP 0 = H0

1P 1

c©Anton Shiriaev. 5EL158: Lecture 3 – p. 15/15