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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
27.09.2016Robot Dynamics - Kinematics 2 1
Lecture «Robot Dynamics»: Kinematics 2
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Topic Title20.09.2016 Intro and Outline L1 Course Introduction; Recapitulation Position, Linear Velocity, Transformation27.09.2016 Kinematics 1 L2 Rotation Representation; Introduction to Multi-body Kinematics 28.09.2016 Exercise 1a E1a Kinematics Modeling the ABB arm04.10.2016 Kinematics 2 L3 Kinematics of Systems of Bodies; Jacobians11.10.2016 Kinematics 3 L4 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation
Error; Multi-task Control12.10.2016 Exercise 1b E1b Kinematic Control of the ABB Arm18.10.2016 Dynamics L1 L5 Multi-body Dynamics19.10.2016 Exercise 2a E2a Dynamic Modeling of the ABB Arm25.10.2016 Dynamics L2 L6 Dynamic Model Based Control Methods26.10.2016 Exercise 2b E2b Dynamic Control Methods Applied to the ABB arm01.11.2016 Legged Robots L7 Case Study and Application of Control Methods08.11.2016 Rotorcraft 1 L8 Dynamic Modeling of Rotorcraft I15.11.2016 Rotorcraft 2 L9 Dynamic Modeling of Rotorcraft II & Control16.11.2016 Exercise 3 E3 Modeling and Control of Multicopter22.11.2016 Case Studies 2 L10 Rotor Craft Case Study29.11.2016 Fixed-wing 1 L11 Flight Dynamics; Basics of Aerodynamics; Modeling of Fixed-wing Aircraft30.11.2016 Exercise 4 E4 Aircraft Aerodynamics / Flight performance / Model derivation06.12.2016 Fixed-wing 2 L12 Stability, Control and Derivation of a Dynamic Model07.12.2016 Exercise 5 E5 Fixed-wing Control and Simulation13.12.2016 Case Studies 3 L13 Fixed-wing Case Study20.12.2016 Summery and Outlook L14 Summery; Wrap-up; Exam
27.09.2016Robot Dynamics - Kinematics 2 2
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Position vector: Parameterization:
Cartesian
Cylindrical coordinates
Spherical coordinates
Relation between linear velocity and parameter differentiation… with the parameterization specific matrix
27.09.2016Robot Dynamics - Kinematics 2 3
Last Time: Position Parameterization
3e e r r χ
3P χ
ee P P P
P
rr χ E χχ
3 3
P PE χ
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Rotation matrix: 3x3 = 9 parameters Orthonormality = 6 constraints
Euler Angles 3 parameters, singularity problem
Angle Axis 4 parameters, unitary constraint, singularity problem
Rotation vector 3 parameters, singularity problem
Quaternions 4 parameters no singularity
27.09.2016Robot Dynamics - Kinematics 2 4
Rotation Parameterization
,R rotvec χ φ n
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Last time: Elementary rotation
27.09.2016Robot Dynamics - Kinematics 2 5
Euler Angles Consecutive elementary rotations
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Three elementary rotations ZYZ and ZXZ: proper Euler angles ZYX: Tait-Bryan angles XYZ: Cardan angles
Example Z-Y-Z
27.09.2016Robot Dynamics - Kinematics 2 6
Euler Angles Consecutive elementary rotations
, 1 2R eulerZYZ z y z C C χ C C CAD AD AB BC CD
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From Euler Angles to Rotation Matrix ZYZ example
, 1 2R eulerZYZ z y z C C χ C C CAD AD AB BC CD
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A rotation matrix has the following form
As a function of ZYZ Euler Angles, we found
27.09.2016Robot Dynamics - Kinematics 2 8
From Rotation Matrix to Euler AnglesZYZ example
Atan2 function:uses sign of both arguments to determine the correct quadrant
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Rotation parameters
Rotation matrix from Euler Angles
27.09.2016Robot Dynamics - Kinematics 2 9
Euler Angles Rotation MatrixZYX example
Euler Angles from Rotation matrix
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What is the relation
Analog to linear velocity: Find , s.t.
27.09.2016Robot Dynamics - Kinematics 2 10
Time Derivatives and Rotational Velocity
ω χAD AD
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Time Derivatives and Rotational VelocityZYX example
ω ω ω ωA AD A AB A BC A CD
ω C ω C C ωA AB AB B BC AB BC C CD
z y xz y x e C e C C e A B CA AB B AB BC C
z y x
zyx
e C e C C e
A B CA AB B AB BC C
0 0 10 0 1 0 0
0 0 1 0 0
z z y y y z
x z z y z
y y y
c s c s c cs c c s
s c s
C C eCAB BC C
0 00 1
0 0 1 0 0
z z z
y z z z
c s ss c c
C eBAB B
001 0
z y z
z y z
y
s c cc c s
s
ω χ ,det cosR eulerZYX y E
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Angle axis parameterize the rotation by:
Rotation vector (aka Euler vectors)
Rotation matrix is given by:
27.09.2016Robot Dynamics - Kinematics 2 12
Angle Axis and Rotation Vector
Rotation angleRotation axis
Parameters from rotation matrix
3 3, cos sin 1 cos T C n I n nnAB
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Complex numbers in 4D
As vector
Real part
Imaginary part
Unitary constraint
Inverse
Identity 27.09.2016Robot Dynamics - Kinematics 2 13
Unit QuaternionsRotation parameterization w/o singularity problem
0 1 2 3i j k ξ2 2 2 1i j k ijk
Hamiltonian convention
1 0 0 0 Tξ
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Rotation matrixfrom unit quaternion
Unit quaternionsfrom rotation matrix
Algebra
27.09.2016Robot Dynamics - Kinematics 2 14
Unit Quaternions Rotation matrix
ξ ξ ξAC AB BC C C CAC AB BC
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Product of quaternions Given two quaternions q and p, the product is defined as
⊗⊗ ⊗ ⊗ ⊗ ⊗ ⊗⊗⊗
⊗
:
⊗
:
27.09.2016Robot Dynamics - Kinematics 2 15
Unit QuaternionsAlgebra
0 1 2 3i j k ξ2 2 2 1i j k ijk
Hamiltonian convention
2ij ji ijk kjk kj iki ik j
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The pure (imaginary) quaternion of a coordinate vector is given by
0
Given the unit quaternion representing the orientation of w.r.t. , one can show: ⊗ ⊗
Proof (see Quaternion Kinematics by Joan Solà on lecture homepage)
Decompose vector in parallel and orthogonal part to get vector rotation formula
Show that equation above does exactly the same
27.09.2016Robot Dynamics - Kinematics 2 16
Unit QuaternionsRotating a vector
r C rB BI I
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Derivation of rotation matrix ( ): ⊗ ⊗ 0
0
ζζ
ζζ
0
0
ζ ζ ζ
ζ ζ ζ 2ζ0
0
1 00 ζ 2ζ
0
ζ 2ζ
2ζ 1 2ζ27.09.2016Robot Dynamics - Kinematics 2 17
Unit QuaternionDerivation of rotation matrix
= - I
=
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Given the rotation matrix 2 1 2
Use this with the angle axis representation cos
sin
, 2 cos 1 2 cos sin sin
cos sin cos
27.09.2016Robot Dynamics - Kinematics 2 18
Unit QuaternionLink to angle axis
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Angle Axis
Rotation Vector
Quaternion with
27.09.2016Robot Dynamics - Kinematics 2 19
Derivative Angle Axis, Rotation Vector, Quaternions Angular Velocity
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Rotation matrix
EulerZYX
Angle Axis
Quaternions
27.09.2016Robot Dynamics - Kinematics 2 20
Quiz
60 CAB
0 00 0
60atan2( 3 / 2,1 / 2)
1 11 1 / 2 1 / 2 1cos cos 1 / 2 602
3 / 2 ( 3 / 2) 11 0 0 0
2sin 600 0 0
1 1 / 2 1 / 2 1 31 1 / 2 1 / 2 11 1 1
2 2 01 / 2 1 / 2 1 101 / 2 1 1 / 2 1
ξAB
cos 2
sin 2
1 0 0
0 1/ 2 3 / 2
0 3 / 2 1 / 2
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Given a vector in A frame
Rotate this to B frame using quaternions
27.09.2016Robot Dynamics - Kinematics 2 21
Quiz 2
60
010
rA
31 12 0
0
ξAB
31 12 0
0
ξBA
00 T Tl r
p r ξ p r ξ M ξ M ξrrB BA A BA BA BA
AB
3 1 0 0 3 1 0 0 0001 3 0 0 1 3 0 0 01 1
1 / 212 20 0 3 1 0 0 3 10 3 / 20 0 1 3 0 0 1 3
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Rotation matrix
Quaternion
27.09.2016Robot Dynamics - Kinematics 2 22
Quiz 360z 60y
1/ 2 3 / 2 0
3 / 2 1/ 2 00 0 1
CAB
1/ 2 0 3 / 20 1 0
3 / 2 0 1 / 2
CBC
31 02 0
1
ξAB
31 02 1
0
ξBC
3 31 0 1 02 20 1
1 0
ξ ξ ξAC AB BC
3 0 0 1 30 3 1 01 1 0
2 2 10 1 3 001 0 0 3
311 132 2
3
29 1 3 3 2 3 6 3 6 3 2 3
1 1 6 3 2 3 9 1 3 3 2 3 3 62 2
2 3 6 3 6 2 3 3 9 1 3 3
CAC
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