lecture«robot dynamics»: multi-body kinematics...machines are built and controlled to achieve...

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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)

Marco Hutter, Roland Siegwart, and Thomas Stastny

03.10.2017Robot Dynamics - Kinematics 3 1

Lecture «Robot Dynamics»: Multi-body Kinematics

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19.09.2017 Intro and Outline Course Introduction; Recapitulation Position, Linear Velocity26.09.2017 Kinematics 1 Rotation and Angular Velocity; Rigid Body Formulation, Transformation 26.09.2017 Exercise 1a Kinematics Modeling the ABB

arm

03.10.2017 Kinematics 2 Kinematics of Systems of Bodies; Jacobians 03.10.2017 Exercise 1b Differential Kinematics of the ABB arm

10.10.2017 Kinematics 3 Kinematic Control Methods: Inverse Differential Kinematics, Inverse Kinematics; Rotation Error; Multi-task Control

10.10.2017 Exercise 1c Kinematic Control of the ABB Arm

17.10.2017 Dynamics L1 Multi-body Dynamics 17.10.2017 Exercise 2a Dynamic Modeling of the ABB Arm

24.10.2017 Dynamics L2 Floating Base Dynamics 24.10.201731.10.2017 Dynamics L3 Dynamic Model Based Control Methods 31.10.2017 Exercise 2b Dynamic Control Methods

Applied to the ABB arm

07.11.2017 Legged Robot Dynamic Modeling of Legged Robots & Control 07.11.2017 Exercise 3 Legged robot14.11.2017 Case Studies 1 Legged Robotics Case Study 14.11.201721.11.2017 Rotorcraft Dynamic Modeling of Rotorcraft & Control 21.11.2017 Exercise 4 Modeling and Control of

Multicopter28.11.2017 Case Studies 2 Rotor Craft Case Study 28.11.201705.12.2017 Fixed-wing Dynamic Modeling of Fixed-wing & Control 05.12.2017 Exercise 5 Fixed-wing Control and

Simulation12.12.2017 Case Studies 3 Fixed-wing Case Study (Solar-powered UAVs - AtlantikSolar, Vertical

Take-off and Landing UAVs – Wingtra)19.12.2017 Summery and Outlook Summery; Wrap-up; Exam

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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

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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

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joints revolute (1DOF) prismatic (1DOF)

1 links moving links 1 fixed link


Classical Serial Kinematic LinkagesGeneralized robot arm

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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

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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}

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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

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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

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Joint Coordinates

Operational Coordinates 03.10.2017Robot Dynamics - Kinematics 3 10

Configuration Space Joint Space

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Joint Coordinates => Joint Space

Operational Coordinates => Operational Space03.10.2017Robot Dynamics - Kinematics 3 11

Configuration Space Joint Space

1

2

3

x

z

1

2

3

q

e

xz

χ

obstacle

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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

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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

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Forward Kinematics

Forward Differential Kinematics Analytic:


Forward Differential KinematicsAnalytical Jacobian

P

R

ee e

e

χχ χ q

χ

with

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Given (from last example)

Determine the analytical Jacobian


Analytical JacobianPlanar robot arm

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Analytic:

Geometric:

Algebra:


Forward Differential Kinematics

with

Depending on parameterization!!

Independent of parameterization

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Definitions

Remember the difference:

Velocity

Time derivative of coordinates:03.10.2017Robot Dynamics - Kinematics 3 17

Velocity in Moving Bodies

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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

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Apply transformation rule as learned before

Differentiate with respect to time

Substitute

Rigid body formulation


Velocity in Moving BodiesRigid body formulation

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Rigid body formulation at a single element

Apply this to all bodies

Angular velocity propagation

…get the end-effector velocity


Geometric Jacobian Derivation

with

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Position Jacobian

Rotation Jacobian from


Geometric Jacobian Derivation

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Preparation: determine the rotation matrices

Determine the rotation axes Locally Inertial frame

Determine the position vectors

Get the Jacobian


Geometric JacobianPlanar robot arm

…

…=…

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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

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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 work03.10.2017Robot 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

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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

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Generalized coordinates

Generalized velocities and accelerations? Time derivatives depend on parameterization

Often

Linear mapping 03.10.2017Robot Dynamics - Kinematics 3 26

Floating Base SystemsKinematics

with

,q q

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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

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A contact point is not allowed to move:

Constraint as a function of generalized coordinates:

Stack of constraints


Contact Constraints

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Contact constraints

Point on wheel

Jacobian

Contact constraints

=> Rolling condition03.10.2017Robot 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

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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

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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

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Exercise TOMORROW Differential Kinematics Use it as extended office hour!

Next Lecture Script Section 2.9 (Kinematic Control Methods) Inverse Kinematics Inverse Differential Kinematics


Outlook

{E}

Inverse Kinematics

Configurationend-effector

Joint angles

{I}

lecture«robot dynamics»: multi-body kinematics...machines are built and controlled to achieve...

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