Lecture «Robot Dynamics»: Kinematics 3 › ... › RobotDynamics2016 › 3-kinematics.pdf · 20.12.2016 Summery and Outlook L14 Summery; Wrap-up; Exam Robot Dynamics - Kinematics

||

151-0851-00 Vlecture: CAB G11 Tuesday 10:15 – 12:00, every weekexercise: HG E1.2 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

4.10.2016Robot Dynamics - Kinematics 3 1

Lecture «Robot Dynamics»: Kinematics 3

||

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; Jacobians05.10.2016 Exercise 1b L3 Differential Kinematics and Jacobians of the ABB Arm11.10.2016 Kinematics 3 L4 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation

Error; Multi-task Control12.10.2016 Exercise 1c 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 4.10.2016Robot Dynamics - Kinematics 3 2

||

Machines are built and controlled to achieve extremely accurate positions, independent of the load the robot carries

Very stiff structure Play-free gears and transmissions High-accurate joint sensors

End-effector accuracy +/- 0.02mm!

Large variety of robot arms


Multi-body KinematicsIntro

||

Base frame is rigidly connected to ground Often indicated as CS 0

Base frame is free floating Often indicated as CS B (base)

6 unactuated DOFs!


Fixed Base vs. Floating Base Robot

||

joints revolute (1DOF) prismatic (1DOF)

1 links moving links 1 fixed link


Classical Serial Kinematic LinkagesGeneralized robot arm

|| 4.10.2016Robot Dynamics - Kinematics 3 6

Configuration ParametersGeneralized coordinates

Generalized coordinatesA set of scalar parameters q that describe the robot’s configuration Must be complete (Must be independent)

=> minimal coordinates Is not unique

6 parameters• 3 positions• 3 orientations

5 constraints

n moving links: 6n parametersn 1DoF joints: 5n parameters6n – 5n = n DoFs

Degrees of Freedom Nr of minimal coordinates

||

End-effector configuration parameters A set of m parameters that completely specify

the end-effector position and orientation with respect to I

Operational space coordinates the m0 configuration parameters are independent

=> m0 number of degrees of freedom of end-effector


End-effector Configuration Parameters

{I}

{E}

||

Most general robot arm: q = = = = =

SCARA robot arm q = = = = =

ANYpulator: robot arm with 4 rotational joints q = = = = =


End-effector Configuration ParametersExample

1... jnq q

6 6

, , , , ,x y zx y z , , , , ,x y zx y z

, , ,

6 4

, , , , ,x y zx y z , , , zx y z

1 2 3 4, , ,q q q q

6 4

, , , , ,x y zx y z

||

Planar robot arm 3 revolute joints 1 end-effector (gripper) <= don’t consider this for the moment

What are the joint coordinates (generalized coordinates)?

What are the end-effector parameters?


End-effector Configuration ParametersSimple example

||

Joint Coordinates

Operational Coordinates 4.10.2016Robot Dynamics - Kinematics 3 10

Configuration Space Joint Space

||

Joint Coordinates => Joint Space

Operational Coordinates => Operational Space4.10.2016Robot Dynamics - Kinematics 3 11

Configuration Space Joint Space

1

2

3

x

z

1

2

3

q

e

xz

χ

obstacle

||

End-effector configuration as a function of generalized coordinates

For multi-body system, use transformation matrices

Note: depending on the selected end-effector parameterization, it is not possible to analytically write down end-effector parameters!


Forward Kinematics

en

||

What is the end-effector configuration as a function of generalized coordinates?


Forward KinematicsSimple example

0 01 12 23 3IE I E T T T T T T

1 1 2 2 3 3

1 1 0 2 2 1 3 3 2 3

1 0 0 0 c 0 0 c 0 0 c 0 0 1 0 0 00 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 00 0 1 0 0 0 0 0 0 10 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1

s s s

s c l s c l s c l l

123 123 1 1 2 12 3 123

123 123 0 1 1 2 12 3 123

c 00 1 0 0

...0

0 0 0 1

s l s l s l s

s c l l c l c l c

||

Forward Kinematics

Forward Differential Kinematics Analytic:


Forward Differential KinematicsAnalytical Jacobian

P

R

ee e

e

χχ χ q

χ

with

||

Given (from last example)

Determine the analytical Jacobian


Analytical JacobianPlanar robot arm

||

Analytic:

Geometric:

Algebra:


Forward Differential Kinematics

with

Depending on parameterization!!

Independent of parameterization

||

Definitions

Remember the difference:

Velocity

Time derivative of coordinates:4.10.2016Robot Dynamics - Kinematics 3 17

Velocity in Moving Bodies

||

For non-moving reference frames:

For moving reference frames:

Vector differentiation in moving frames (A = inertial/reference frame):


Vector Differentiation in Moving FrameEuler differentiation rule

||

Apply transformation rule as learned before

Differentiate with respect to time

Substitute

Rigid body formulation


Velocity in Moving BodiesRigid body formulation

||

Rigid body formulation at a single element

Apply this to all bodies

Angular velocity propagation

…get the end-effector velocity


Geometric Jacobian Derivation

with

||

Position Jacobian

Rotation Jacobian from


Geometric Jacobian Derivation

||

Preparation: determine the rotation matrices

Determine the rotation axes Locally Inertial frame

Determine the position vectors

Get the Jacobian


Geometric JacobianPlanar robot arm

…

…=…

||

Analytical Jacobian

Relates time-derivatives of config. parameters to generalized velocities

Depending on selected parameterization (mainly rotation) in 3D Note: there exist no “rotation angle”

Mainly used for numeric algorithms

Geometric (or basic) Jacobian

Relates end-effector velocity to generalized velocities

Unique for every robot

Used in most cases


RecapitulationAnalytical and Kinematic Jacobian

χ q

||

Kinematics (mapping of changes from joint to task space) Inverse kinematics control Resolve redundancy problems Express contact constraints

Statics (and later also dynamics) Principle of virtual work Variations in work must cancel for all virtual displacement Internal forces of ideal joint don’t contribute

Dual problem from principle of virtual work4.10.2016Robot Dynamics - Kinematics 3 24

Importance of Jacobian

0

TTi i E E

iTT

E

W

f x τ q F x

τ q F J q q

FE

1

23

T

qxFτ

JJ

||

151-0851-00 Vlecture: CAB G11 Tuesday 10:15 – 12:00, every weekexercise: HG E1.2 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


Floating Base Kinematics

||

Generalized coordinates

Generalized velocities and accelerations? Time derivatives depend on parameterization

Often

Linear mapping 4.10.2016Robot Dynamics - Kinematics 3 26

Floating Base SystemsKinematics

with

,q q

||

Position of an arbitrary point on the robot

Velocity of this point


Floating Base SystemsDifferential kinematics

BQ jr qB bC qIB IB br qI

QJ qI

TT

T

T

C C

C

ω

C CC

ω

CC BI BI

IB

I IB IB

I

B

IB I BIBB IB

with

||

A contact point is not allowed to move:

Constraint as a function of generalized coordinates:

Stack of constraints


Contact Constraints

||

Contact constraints

Point on wheel

Jacobian

Contact constraints

=> Rolling condition4.10.2016Robot Dynamics - Kinematics 3 29

Contact ConstraintWheeled vehicle simple example

x

qUn-actuated base

Actuated joints

sincos0

IP

x rr r

rI

0x r

10 00 0

IP P

rx

r J q 0

I I

1 cos0 sin0 0

P

rr

JI

||

Contact Jacobian tells us, how a system can move. Separate stacked Jacobian

Base is fully controllable if

Nr of kinematic constraints for joint actuators:

Generalized coordinates DON’T correspond to the degrees of freedom Contact constraints!

Minimal coordinates (= correspond to degrees of freedom) Require to switch the set of coordinates depending on contact state (=> never used)


Properties of Contact Jacobian

relation between base motion and constraints

-

c b jn n n

||

Floating base system with 12 actuated joint and 6 base coordinates (18DoF)


Quadrupedal Robot with Point Feet

Total constraints

Internal constraints

Uncontrollable DoFs

0

0

6

3

0

3

6

1

1

9

3

0

12

6

0

||

Exercise TOMORROW Differential Kinematics Use it as extended office hour!

Next Lecture 11.10. (Dario Bellicoso) Script Section 2.9 (Kinematic Control Methods) Inverse Kinematics Inverse Differential Kinematics


Outlook

{E}

Inverse Kinematics

Configurationend-effector

Joint angles

{I}

Documents

Lecture «Robot Dynamics»: Kinematics 3 › ... › RobotDynamics2016 › 3-kinematics.pdf · 20.12.2016 Summery and Outlook L14 Summery; Wrap-up; Exam Robot Dynamics - Kinematics