Lecture «Robot Dynamics»: Kinematics 2

Transcription

1 Lecture «Robot Dynamics»: Kinematics V lecture: CAB G11 Tuesday 1:15 12:, every week exercise: HG G1 Wednesday 8:15 1:, according to schedule (about every 2nd week) office hour: LEE H33 Friday Marco Hutter, Roland Siegwart, and Thomas Stastny Robot Dynamics - Kinematics

2 Topic Title Intro and Outline L1 Course Introduction; Recapitulation Position, Linear Velocity, Transformation Kinematics 1 L2 Rotation Representation; Introduction to Multi-body Kinematics Exercise 1a E1a Kinematics Modeling the ABB arm Kinematics 2 L3 Kinematics of Systems of Bodies; Jacobians Kinematics 3 L4 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation Error; Multi-task Control Exercise 1b E1b Kinematic Control of the ABB Arm Dynamics L1 L5 Multi-body Dynamics Exercise 2a E2a Dynamic Modeling of the ABB Arm Dynamics L2 L6 Dynamic Model Based Control Methods Exercise 2b E2b Dynamic Control Methods Applied to the ABB arm Legged Robots L7 Case Study and Application of Control Methods Rotorcraft 1 L8 Dynamic Modeling of Rotorcraft I Rotorcraft 2 L9 Dynamic Modeling of Rotorcraft II & Control Exercise 3 E3 Modeling and Control of Multicopter Case Studies 2 L1 Rotor Craft Case Study Fixed-wing 1 L11 Flight Dynamics; Basics of Aerodynamics; Modeling of Fixed-wing Aircraft Exercise 4 E4 Aircraft Aerodynamics / Flight performance / Model derivation Fixed-wing 2 L12 Stability, Control and Derivation of a Dynamic Model Exercise 5 E5 Fixed-wing Control and Simulation Case Studies 3 L13 Fixed-wing Case Study Summery and Outlook L14 Summery; Wrap-up; Exam Robot Dynamics - Kinematics

3 Last Time: Position Parameterization Position vector: Parameterization: e e 3 r r χ χ P 3 Cartesian Cylindrical coordinates Spherical coordinates Relation between linear velocity and parameter differentiation 33 with the parameterization specific matrix EPχ P r r χ E χ e e P P P χ P Robot Dynamics - Kinematics

4 Rotation Parameterization 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 χ φ n R, rotvec Robot Dynamics - Kinematics

5 Euler Angles Consecutive elementary rotations Last time: Elementary rotation Robot Dynamics - Kinematics

6 Euler Angles Consecutive elementary rotations Three elementary rotations ZYZ and ZXZ: proper Euler angles ZYX: Tait-Bryan angles XYZ: Cardan angles Example Z-Y-Z C C χ ReulerZYZ C z C y C z AD AD, AB 1 BC CD 2 Robot Dynamics - Kinematics

7 From Euler Angles to Rotation Matrix ZYZ example C C χ ReulerZYZ C z C y C z AD AD, AB 1 BC CD 2 Robot Dynamics - Kinematics

8 From Rotation Matrix to Euler Angles ZYZ example A rotation matrix has the following form As a function of ZYZ Euler Angles, we found Atan2 function: uses sign of both arguments to determine the correct quadrant Robot Dynamics - Kinematics

9 Euler Angles Rotation Matrix ZYX example Rotation parameters Rotation matrix from Euler Angles Euler Angles from Rotation matrix Robot Dynamics - Kinematics

10 Time Derivatives and Rotational Velocity What is the relation ω AD χ AD Analog to linear velocity: Find, s.t. Robot Dynamics - Kinematics

11 Time Derivatives and Rotational Velocity ZYX example AωAD AωAB AωBC AωCD C AB A ωab CAB B ωbc CAB CBC C ωcd A B C ez z C ey y C C ex x A AB B AB BC C z A B C z y x y e C e C C e A AB B AB BC C x c z sz sz e B y sz cz 1 c B z 1 cz sz cy sy1 cycz C C e C x sz cz 1 AB BC C cys z 1sy c y s y s c 1 c c z y z c s z y z s y ω χ det E cosy ReulerZYX, Robot Dynamics - Kinematics

12 Angle Axis and Rotation Vector Angle axis parameterize the rotation by: Rotation angle Rotation axis Rotation vector (aka Euler vectors) Rotation matrix is given by:, cos sin 1cos C n I n nn AB 33 T Parameters from rotation matrix Robot Dynamics - Kinematics

13 Unit Quaternions Rotation parameterization w/o singularity problem Complex numbers in 4D ξ 1i 2j 3k Hamiltonian convention i j k ijk 1 As vector Real part Imaginary part Unitary constraint Inverse Identity ξ 1 T Robot Dynamics - Kinematics 2 13

14 Unit Quaternions Rotation matrix Rotation matrix from unit quaternion Unit quaternions from rotation matrix Algebra ξac ξab ξbc C C C AC AB BC Robot Dynamics - Kinematics

15 Unit Quaternions Algebra Product of quaternions Given two quaternions q and p, the product is defined as : : Hamiltonian convention ξ i j k i j k ijk 1 2 ij ji ijk k jk kj i ki ik j Robot Dynamics - Kinematics

16 Unit Quaternions Rotating a vector The pure (imaginary) quaternion of a coordinate vector is given by Given the unit quaternion representing the orientation of w.r.t., one can show: r C r Proof (see Quaternion Kinematics by Joan Solà on lecture homepage) Decompose vector in parallel and orthogonal part to get vector rotation formula B BI I Show that equation above does exactly the same Robot Dynamics - Kinematics

17 Unit Quaternion Derivation of rotation matrix Derivation of rotation matrix ( ): ζ ζ ζ ζ ζ ζ ζ ζ ζ ζ 2ζ 1 ζ 2ζ = - I = ζ 2ζ 2ζ 1 2ζ Robot Dynamics - Kinematics

18 Unit Quaternion Link to angle axis Given the rotation matrix Use this with the angle axis representation cos sin, 2cos 1 2cos sin sin cos sin cos Robot Dynamics - Kinematics

19 Derivative Angle Axis, Rotation Vector, Quaternions Angular Velocity Angle Axis Rotation Vector Quaternion with Robot Dynamics - Kinematics

20 Quiz Rotation matrix C AB 1 1/2 3/2 3/2 1/2 6 EulerZYX Angle Axis Quaternions atan 2( 3 / 2,1/ 2) 6 111/21/21 1 cos cos 1/ /2 ( 3/2) 1 1 2sin6 ξ AB 11/21/ /21/ /2 1/ /2 1 1/2 1 cos 2 sin 2 Robot Dynamics - Kinematics

21 Quiz 2 Given a vector in A frame 1 Rotate this to B frame using quaternions A r 6 ξ AB ξ BA p r T T ξ p r ξ M l ξ M r ξ B BA A BA BA BA B r A r / /2 Robot Dynamics - Kinematics

22 Quiz 3 z 6 y 6 Rotation matrix C AC? 1/2 3/2 1/2 3/2 C AB 3/2 1/2 C BC 1 1 3/2 1/2 Quaternion ξ 1 AB 2 ξ 1 1 BC 2 1 ξ AC ξab ξbc C AC Robot Dynamics - Kinematics