Basic Knowledge

(1)

HAL Id: cel-02130071

https://hal.inria.fr/cel-02130071

Submitted on 15 May 2019

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Basic Knowledge

Philippe Martinet

To cite this version:

Philippe Martinet. Basic Knowledge. Doctoral. GdR Robotics Winter School: Robotica Principia, Centre de recherche Inria Sophia Antipolis – Méditérranée, France. 2019. �cel-02130071�

(2)

Application fields History and Bibliography Definitions Basic Knowledge

Basic Knowledge

Classification

Homogeneous transformation

x

y

z

R

_i

x

y

z

R

f

Geometry

i

_T

f

i

_T

f

Homogeneous transformation matrix









P

i

R

i

R (A): Orientation

(3)

Basic Knowledge

Classification

Homogeneous transformation

Geometry

Consider a 3D point in space

f i R R

1 z

y

x

f

T

i

1 z

y

x

















⋅

=

















f R

1 z

y

x

















Then

f

P

i

z

y

x

f

R

i

z

y

x

f R i R

+













⋅

=













(4)

Basic Knowledge

Classification

Homogeneous transformation

Geometry

i

T

f

Homogeneous transformation matrix

R : Orientation

s, n, a Cosinus directors

RPY angles (Roll (z), Pitch(y), Yaw(x))

Briant angles (x,y,z)

Euler angles (z,x,z)

u.

θθθθ

, u.sin(

θθθθ

), u.sin(

θ/2

),

Quaternion

λλλλ

₁

,

λλλλ

₂

,

λλλλ

₃

,

λλλλ

₄

P : Position

Cartesian coordinates

Cylindrical coordinates

Spherical coordinates

Different representations

(5)

Basic Knowledge

Classification

Homogeneous transformation

Geometry

i

T

f

Homogeneous transformation matrix

R : Orientation

s, n, a Director Cosinus

P : Position

Cartesian coordinates

Different representations (i.e)













=

z z z y y y x x x

a

n

s

a

n

s

a

n

s

R













=

z y x

P

No rotation

No translation

P=(0,0,0)

T

(6)

Basic Knowledge

Classification

Homogeneous transformation

Geometry

i

T

f

Homogeneous transformation matrix

R : Orientation

s, n, a Director Cosinus

Main properties of the rotation matrix

1 a

n

s

=

0 n

a

0 a

s

0 n

s

=

⋅

=

⋅

=

⋅

n

s

a

s

a

n

a

n

s

=

×

=

×

=

×









=

_y _y _y x x x

a

n

s

a

n

s

R

(7)

Basic Knowledge

Classification

Homogeneous transformation: Rotation matrix

Geometry

i

R

f

Rotation matrix Rot(x,

θθθθ

x

)

Matrix to change the frame for one vector

(8)

Basic Knowledge

Classification

Homogeneous transformation: Rotation matrix

(9)

Basic Knowledge

Classification

Homogeneous transformation: Rotation matrix

(10)

Basic Knowledge

Classification

Homogeneous transformation properties

Geometry

i

T

f

Homogeneous transformation matrix

Prop. 1)

(11)

Basic Knowledge

Classification

Homogeneous transformation properties

Geometry

i

T

f

Homogeneous transformation matrix

Prop. 3)

(12)

Basic Knowledge

Classification

Homogeneous transformation properties

Geometry

i

T

f

Homogeneous transformation matrix

Prop. 5)

(13)

Basic Knowledge

Classification

Homogeneous transformation properties

Geometry

i

T

f

Homogeneous transformation matrix

Prop. 7)

T is defined in R

_i

(14)

Basic Knowledge

Classification

Homogeneous transformation properties

(15)

Basic Knowledge

Classification

Homogeneous transformation properties

Geometry

i

T

f

Homogeneous transformation matrix

(16)

Basic Knowledge

Cartesian coordinates

Classification

(17)

Basic Knowledge

Cylindrical coordinates

Classification

(18)

Basic Knowledge

Spherical coordinates

Classification

Rigid body pose parameterization: position

(19)

Basic Knowledge

Euler angles

Classification

Rigid body pose parameterization: orientation

(20)

Basic Knowledge

RPY angles (z,y,x)

Classification

Rigid body pose parameterization: orientation

A

_RPY

(21)

Basic Knowledge

Bryant angles (x,y,z)

Classification

Rigid body pose parameterization: orientation

(22)

Basic Knowledge

Orientation (u,

θθθθ

)

Classification

Rigid body pose parameterization: orientation

(23)

Basic Knowledge

Orientation (u.

θθθθ

)

Classification

Rigid body pose parameterization: orientation

Rodrigues formula

(24)

Basic Knowledge

Orientation (u.

θθθθ

)

Classification

Rigid body pose parameterization: orientation

( )

u

θ

I

u

s

θ

u

(

c

θ

)

A

,

T

=

₃

−

ˆ

.

+

ˆ

2

.

1 −

( ) ( )

u

θ

A

u

θ

u

s

θ

A

,

−

,

T

=

2 .

ˆ

.

[ ] [ ]

( ) ( )

2 ,

,

.

T

u

A

u

A

s

u

s

u

θ

×

=

θ

∧

=

θ

−

θ

( )

(

A

u

θ

)

Trace

(

I

u

s

θ

u

(

c

θ

)

Trace

,

=

₃

+

ˆ

.

+

ˆ

2

.

1 −

( )

(

)

(

)

( )

2

ˆ

.

1

3 ,

c

Trace

u

A

Trace

θ

=

+

−

θ

( )

(

A

u

,

θ

)

=

3 +

(

1 −

c

θ

)( )

.

−

2 Trace

( )

(

A

u

θ

)

c

θ

Trace

,

=

1 +

2 .

( )

(

A

u

θ

)

c

θ

Tr

,

=

1 +

2 .

( )

(

)

2

1 ,

cos

θ

=

Tr

A

u

θ

−

s

_y

- n

_x

= 2 u

_z

s

θθθθ

a

_x

- s

_z

= 2 u

_y

s

θθθθ

n

_z

- a

_y

= 2 u

_x

s

θθθθ

          − − − − − − =           −           0 0 0 y z x z z y x y z x y x z y x z y x z y x z z z y y y x x x a n a s n a n s s a s n a a a n n n s s s a n s a n s a n s

(

)

(

)

(

)

1 







₋

y z

a

n

1

(25)

Basic Knowledge

Quaternion (

λλλλ

₁

,

λλλλ

₂

,

λλλλ

₃

,

λλλλ

₄

)

Classification

(26)

x

y

z

R

₀

Case of serial manipulator

robot

x

y

z

R

₁

x

y

z

x

y

z

R

_k

R

k+1

C

_k

C

₀

C

_n

Consider a robot with n+1 rigid bodies C

_k

We associate n+1 frames

y

z

x

R

_e

C

₁

C

2 Basic Knowledge

Classification

Multi-Rigid bodies

(27)

Case of serial manipulator robot

The problem to solve is to obtain the position and orientation

of the end effector frame R

_e

in the fixed frame R

₀













=

+ + +

1

000 P

R

T

k 1 k 1 k k 1 k k

e

n

1 n

3

2

1

0 e

0

T

=

⋅

_L

−

⋅

Elementary frame transform

Basic Knowledge

Classification

Multi-Rigid bodies

(28)

Basic Knowledge

Classification

Rigid body kinematics

Circular motion

v: tangential velicity

(29)

Basic Knowledge

Classification

Rigid body kinematics

Rotating frame

R

_f

: fixed frame (origin O fixed)

R

_m

: mobile frame just in rotation w.r.t R

_f

(30)

R

_f

: fixed frame

R

_m

: Mobile frame

P : one point in

R

x

y

z

R

_f

x

y

z

R

m

D

d

x P

P

O

P

V

O

V

P

V

_m R R m R_f

=

(

)

_f

+

(

)

_m

+

ω

ωω

ω

×

)

(

d

dt

d

D

d

dt

d

m f R R

×

+

=

(

)

ω

ωω

ω

)

(

&

O

_m

O

_f

Basic Knowledge

Classification

(31)

Kinematic

P

O

P

V

P

V

_m R R_f

=

(

)

_m

+

ωω

ω

×

)

(

Remarks :

If D=0 then

If D=0 and then

V

P

0

m R

=

)

(

d

P

O

P

O

P

V

m m R_f

⋅

×

−

×

=

ω

ωω

ω

ωω

ω

ωω

ω

~

)

(

]

ˆ

[

]

[

]

[

0

0 ~

_ωω

_ω

_ωω

_ω

_ωω

_ω

_ωω

_ω

_ωω

_ω

ω

ωω

ω

ωω

ω

ωω

ω



=







−

=

∧

AS

x z y z

Using this relation we can established

The kinematic evolution of a multi-rigidbody robot

See next slide

P

O

P

V

O

V

P

V

_m R R m R_f

=

(

)

_f

+

(

)

_m

+

ω

ωω

ω

×

)

(

Basic Knowledge

Classification

(32)

Basic Knowledge

Classification

Multi-Rigid bodies kinematics

Angular velocity of R_iw.r.t R₀ expressed in R_i Velocity of O_i+1 w.r.t R_i expressed in R_i

(33)

Basic Knowledge

Classification

Multi-Rigid bodies kinematics

Angular velocity of R_i w.r.t R₀ expressed in R_i

Angular velocity of R_i w.r.t R_i-1 expressed in R_i

Angular velocity of R_i-1 w.r.t R₀ expressed in R_i-1

=

C

_i

C

i+1

(34)

Basic Knowledge

Classification

Multi-Rigid bodies kinematics

Angular velocity of R_i w.r.t R₀ expressed in R_i

Angular velocity of R_i w.r.t R_i-1 expressed in R_i

Angular velocity of R_i-1 w.r.t R₀ expressed in R_i-1

=

C

_i

C

i+1

(35)

Basic Knowledge

Classification

Multi-Rigid bodies kinematics

Considering two frames R

_a

and R

_b

rigidly linked

(case for R

_n

and R

_E

)

(36)

Basic Knowledge

Classification

Multi-Rigid bodies kinematics

Considering two frames R

_i

and R

_j

and a twist V

_i

=(v

_i

,

ω

_i

)

T

_{expressed in O}

i

Projection

with

(37)

Basic Knowledge

Classification

Differential translation and rotation of frames

Consider a differential translation vector dPi expressing the translation

of the origin of frame Ri, and of a differential rotation vector

δ

i, equal to

ui.d

θ

, representing the rotation of an angle d

θ

about an axis, with

unit vector ui, passing through the origin Oi.

(38)

R

₌

a

R

_b

R.R

T

_{= I3}

x

y

z

R

_bbbb

x

y

z

Ra

_a_a_a_a T TT T_b_bb_b ==== a a a a_R_R_R_R b bb b aaaattttbbbb 0 00 000000000 1111

Ω

_b/a

|

a

˙

R.R

T

+ R. ˙

R

T

= 0

R.R

˙

T

= −R. ˙

R

T

= −

R.R

˙

T

˙

R.R

T

= S(t)

R

˙

= S(t).R

˙p

_a

(t) = ˙

R

(t).p

_b

(t)

˙p

a

(t) = S(t).R.p

b

(t)

˙p

_a

(t) = Ω × p

_a

(t)

˙p

_a

(t) = Ω × (R.p

_b

(t) )

˙p

a

(t) = [Ω]

×

.R.p

b

(t)

S

(t) = [Ω]

×

p

_a

(t) = R(t).p

_b

(t)

Consider p

_b

(t) constant

Application fields History and Bibliography Definitions

Basic Knowledge

Classification

(39)

x

y

z

R

_bbbb

x

y

z

Ra

a a a a_T_T_T_T b bb b ==== a a a a_R_R_R_R b bb b aaaattttbbbb 0 00 000000000 1111 b bb b_T_T_T_T a a a a ==== b bb b_R_R_R_R a a a a bbbbttttaaaa 0 00 000000000 1111 b

_t

a

represents the position

b

_R

a

represents the orientation

Ω

_a/b

|

b

Ω

_b/a

|

_{a ×}

=

a

R

˙

_b

.

a

R

T

_b

Ω

_a/b

|

_{b ×}

=

b

R

˙

_a

.

b

R

T

_a

Ω

_b/a

|

a

Ω

_b/a

|

a

Ω

b/a

|

a

= −Ω

a/b

|

a

Ω

b/a

|

a

=

a

R

b

.

Ω

b/a

|

b

Ω

_a/b

|

b

Ω

a/b

|

b

= −Ω

b/a

|

b

Ω

_a/b

|

b

=

b

R

a

.

Ω

a/b

|

a

Basic Knowledge

Classification

(40)

Basic Knowledge

Classification

Representation of forces (wrench)

A collection of forces and moments acting on a body can be reduced to a wrench Fi

at point Oi, which is composed of a force fi at Oi and a moment mi about Oi:

Note that the vector field of the moments constitutes a screw where the vector of the