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151-0851-00 Vlecture: CAB G11 Tuesday 10:15 – 12:00, every weekexercise: HG G1 Wednesday 8:15 – 10:00, according to schedule (about every 2nd week)office hour: LEE H303 Friday 12.15 – 13.00Marco Hutter, Roland Siegwart, and Thomas Stastny

20.09.2016Robot Dynamics - Kinematics 1 1

Lecture «Robot Dynamics»: Kinematics 1

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Builds upon notation of other dynamics classes at ETH and IEEE standards

Vector: (often also ) Vector from point B to P: Reference coordinate system (calligraphic)

:= orthonormal basis of R3

Numerical representation of a vector: Addition of vectors: Use the same reference frame:

20.09.2016Robot Dynamics - Kinematics 1 2

Recapitulation: Vectors, Position, and Vector Calculus

r r

BPr{ }A

BPrA

AB

P

AP AB BP r r rABAP BP rr rA A A

BPrAPr

ABr

{ }A

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Cartesian coordinates

Position vector

Cylindrical coordinates

Position vector

Spherical coordinates

Position vector 20.09.2016Robot Dynamics - Kinematics 1 3

Parameterization of Vectors

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Only for Cartesian coordinates it holds that

NEVER do this for other representations (requires special algebra!!) => we will encounter similar problems for rotations

20.09.2016Robot Dynamics - Kinematics 1 4

Parameterization of VectorsExample

1 1 01 0 11 0 1

AP AB BP

r r rA A A

10 :0

Pc

Pz

Ps

χχχ

01 :1

Pc

Pz

Ps

χχχ

11 :1

Pc

Pz

Ps

χχχ

AP AB BP χ χ χ

1,0,0

1,0,0

1,0,0

T

T

T

0,1,1

1, ,12

2, ,2 4

T

T

T

1,1,1

2, ,14

13, ,arctan4 2

T

T

T

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The velocity of point P relative to point B, expressed in frame A is:

Question: What is the relationship between the velocityand the time derivative of the representation

20.09.2016Robot Dynamics - Kinematics 1 5

Differentiation of Representation Linear Velocity

BPrA

rχ

r r χrr χχ

1

P

P

r E χ χ

χ E χ r

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Cartesian coordinates:

Cylindrical coordinates:

20.09.2016Robot Dynamics - Kinematics 1 6

Differentiation of Representation Linear Velocity

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Position of P with respect to A expressed in A:

Position of P with respect to A expressed in B:

Write the unit vectors of B expressed in A as:

=>

20.09.2016Robot Dynamics - Kinematics 1 7

Rotations

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The rotation matrix transforms vectors expressed in B to A:

The matrix is orthogonal:

Belongs to special orthonormal group SO(3) (and not R3) This causes difficulties and requires special algebra

Consecutive rotations:

20.09.2016Robot Dynamics - Kinematics 1 8

Rotation Matrix

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Passive rotation = mapping of the same vector from frame {B} to {A}

Active rotation = rotating a vector in the same frame

20.09.2016Robot Dynamics - Kinematics 1 9

Passive and Active Rotation

R

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Find the elementary rotation matrixs.t

20.09.2016Robot Dynamics - Kinematics 1 10

Elementary Rotation

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Homogeneous TransformationCombined Translation and Rotation

Homogeneous transformation = translation and rotation

Inverse

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This allows to transform an arbitrary vector between different reference frames(classical example: mapping of features in camera frame to world frame)

20.09.2016Robot Dynamics - Kinematics 1 12

Homogeneous TransformationsConsecutive Transformation

{A}{B}

AB

P

{C}

CAP BP

BP CP

r rr r

TT T T

T

A AB B

AC AB BCB BC C

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Find the position vector Find the transformation matrix

Find the vector

20.09.2016Robot Dynamics - Kinematics 1 13

Homogeneous TransformationSimple Example

1 0 0 00 0 1 30 1 0 10 0 0 1

TAB

1 1

1 0 0 0 0 00 0 1 3 1 20 1 0 1 1 20 0 0 1 1 1

AP BPr r

T

A BAB

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Angular velocity describes the relative rotational velocity of B wrt. Aexpressed in frame A

The relative velocity of A wrt. B is: Given the rotation matrix between two frames, the angular velocity is

Transformation of angular velocity:

Addition of relative velocities: 20.09.2016Robot Dynamics - Kinematics 1 14

Angular Velocity

ω ωAB BA

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Given the rotation matrix

=>

20.09.2016Robot Dynamics - Kinematics 1 15

Angular VelocitySimple Example

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Rotation matrix: 3x3 = 9 parameters Orthogonality = 6 constraints

Euler Angles 3 parameters singularity problem

Angle Axis 4 parameters (angle and axis) unitary constraint

Quaternions 4 parameters no singularity constraints

20.09.2016Robot Dynamics - Kinematics 1 16

Outlook (next week)Rotation Parameterization