Dynamik und Regelung von Robotern mit paralleler kinematischer ...

© Bodo Heimann

Dortmunder Regelungstechnisches Kolloquium, 25. Januar 2007

Dynamik und Regelung von Roboternmit

paralleler kinematischer Struktur

Bodo Heimann und Houssem AbdellatifInstitut für Robotik

Leibniz Universität Hannover

• Einführung und einige Beispiele

• Effektive Berechnung der Dynamik von PKM

• Identifikation der dynamischen Modellparameter

• Modellbasierte Steuerung und Regelung

• Iterative Learning Control für PKM

• Zusammenfassung

© Bodo Heimann

Institute of Robotics (IfR) and its Research Areas

Mechatronics in Cars

Mobil Service Robots

Production Engineering

Robotics / Identification

Medical Applications

© Bodo Heimann

Hannover Center of Mechatronics (MZH): Member Institutes

ELEKTROTECHNIK

AKTORIK/SENSORIK

PROZESS- RECHENTECHNIK

MECHATRONIKMASCHINENBAU ELEKTROTECHNIK

INFORMATIONSTECHNIK

AKTORIK/SENSORIK

PROZESS- RECHENTECHNIK MODELLIERUNG

MECHATRONIK

GEMInstitute of Electro- and

Measurement TechnologyInstitute of

Production Engineering

Institute of Transport and Automation

IRTInstitute of

Automatic Control

Center of ProductionEngineering Hannover

Institute of Material Sciences

Institute of Real-Time

SystemsInstitute of Robotics

© Bodo Heimann

Dortmunder Regelungstechnisches Kolloquium, 25.Januar 2007

• Introduction and Motivation

• Computationally Efficient Dynamics of Parallel Robots

• Identification of Dynamical Models

• Model-based Control Strategy

• Iterative Learning Control for Parallel Robots

• Conclusions

© Bodo Heimann

Serial – Parallel – Hybrid Kinematics

SerialStructure

=

Open kinematicchains

Hybrid Structure

=

Open and closed

kinematicchains

Parallel Structure =

Only closedkinematic chains

© Bodo Heimann

Serial (open chains) vs. Parallel (closed chains) Robots

Serial :

Kuka-R15

Hybrid:

Tricept, ABB – IRB 940

Parallel:

Delta-Roboter, ABB – Flexpicker

© Bodo Heimann

Comparison: Robots with Serial and Parallel Kinematics

Serial Robots Parallel Robots

Absolute Accuracy - - +

Accuracy + - + +

Stiffness - + +

Load/Mass Ratio - - +

End Effector Acceleration + - + +

Workspace(Size, Obstacles, …)

+ + -

Flexibility(Different Tasks)

+ + + -

Mathematics, Numerical Effort(Kinematics, Dynamics)

+ - - -

© Bodo Heimann

Historical Remarks on Development of Parallel Kinematic Machines

1890 First theoretical approaches (Maxwell, Clerk J. / Mannhein, A. 1894)

1956 Gough published first hexapod (for tire experiments )

1965 Stewart-platform (flight simulator)

1979 First assembly cell by McCallion und Pham

1989 First commercial system by Clavel (Delta)

1990 Pierrot and Toyoda develop the Hexa-Robot

1994 Giddings&Lewis and Ingersoll present the PK-grinding machine at IMTS

Since 1995 intensive research activities on parallel kinematic machines

© Bodo Heimann

linear actuatorsbetween the two platforms

linear actuatorswith movable base points

rotary actuatorsbetween the two platforms

Requirements of the actuators:•High acceleration and velocity •Simple integration•High accuracy•Sufficient stroke and force

Drive Principles for PKMs

Good ratio of workspace to machine size

© Bodo Heimann

Examples for Different Drive Systems

Stewart-Gough PlatformTricepts

PaLiDA

Hexaglide Paraplacer Linapods

Flexpicker Hexa Robot Planar Robot

© Bodo Heimann

Singularities

det (J)=0

„minus“ 1 DOF „plus“ 1 DOF

Givenvelocityat the endeffector

Neededvelocity:

∞

Givenforce at the endeffector

Neededforce:∞

High actuator forcesHigh actuator velocity

det (J-1)=0

© Bodo Heimann

Planar Parallel Kinematic Machines

2-DOF planar PKM

Example: IWF

(TU Braunschweig)

DFG SFB: Robotic Systems for Manipulation and Assemblying

© Bodo Heimann

Parallel Kinematic Structures at IfR

1. Hydraulically driven hexapod MS:• 6-axes motion simulator• Maximum force: ~50 kN• Maximum speed: ~0.8 m/s• Maximum accel.: ~250 m/s2

• Moved mass: ~100 kg• Workspace (10°): ~100 mm

2. Electrically driven hexapod PaLiDA:• Robot for pick-and-place applications• Maximum force: ~0.8 kN• Maximum speed: ~2 m/s• Maximum accel.: ~25 m/s2

• Moved mass: ~35 kg• Workspace (20°): ~400 mm

© Bodo Heimann

Parallel Kinematic Structures at IfR

PKM PaLiDA of the IFW• DFG-SPP „Tool Machinery withParallel Kinematic Structure“:

- Common project with IFW- Modeling and identification- Model-based control

Hydraulic simulator

• Cooperation with Witzenmann:- Control of a hydraulic manipulator

© Bodo Heimann

Actuator Design

2. PaLiDA: Development of a new linear directly driven strut

1. MS: Combination of commercial hydraulic actuator withself-developed cardan joints

© Bodo Heimann







• Conclusions

© Bodo Heimann

State of the Art: Modeling and Control of Robot Manipulators

Inverse dynamics

Actuation forcesInertia matrix

generalized coordinates = actuation variables q

Coriolis forces

Gravitation forces

Nonlinearities, e.g. friction

Direct and simple implementation of control

© Bodo Heimann

State of the Art: Modeling and Control of Robot Manipulators

Computed torque control (feedback linearization)

robotcontroller ++-

Direct and simple implementation of control

All needed quantities are given with respect to measurable control variables q

© Bodo Heimann

The Different Case of Parallel Robot Manipulators

generalized coordinates actuation variables q

• Dynamics have to be calculated with respect to the cartesiancoordinates

• High computational effort is necessary

• Only actuated joint variables can be sensed

• Presence of non-sensed passive joints

• Minimal coordinates and velocities are not measurable

• Calculation of minimal coordinates needs solution of direct kinematics

• Direct kinematics solution is ambiguous

• etc. …….

Alternative strategy for model-based control is needed

© Bodo Heimann

Proposed Control Structure for Parallel Manipulators

parallelrobot

inverse kinematics

inverse dynamics

+controller

Computed force (torque) feedforward control

-

Benefits

• Dynamics are calculated from undisturbed desired data

• Computational implicit dynamics models can be used

• Direct kinematics is not needed

• Measurement of actuator positions is sufficient

© Bodo Heimann

Proposed Control Structure for Parallel Manipulators

parallelrobot

inverse kinematics

inverse dynamics

+controller

Computed force (torque) feedforward control

-

Necessary steps

• Real-time calculation of complete inverse dynamics

• Consideration of passive-joint friction

• Experimental and accurate model identification

• Rejection of exogenous disturbances

© Bodo Heimann

Calculation of Rigid-Body Inverse Dynamics

Possible formulations

Lagrange formalism

Newton-Euler formalism

Virtual Work/Power

Parameter linear notation

Decreasing explicitness but increasing computational efficiency

Choice of methodLagrangian formalism: Analytic differentiation of kinetic energy inefficientNewton-Euler: Elimination of reaction forces too costly

JOURDAIN‘s Principle of Virtual Power

© Bodo Heimann

Jourdain’s Principle of Virtual Power

generalized velocities

generalized forces

actuation velocities

actuation forces

Application for rigid-body dynamics:

Application for friction dynamics:

Uniform formulation of both dynamics types

Simple integration of passive-joint friction

© Bodo Heimann

Deduction of Rigid-Body Dynamics in Parameter Linear Form

Usual formulation: Nonlinear with respect to ri,Ci

Transformation to body-fixed coordinate frames:

Parameter linear formulation:

© Bodo Heimann

Example: PKM PaliDA

© Bodo Heimann

Dynamic Model of PaLiDA

Friction: velocity dependent forces within actuator and within upper cardan joint

End-effector platform:3 moments of inertia distance to center of gravity and mass

Over-all model consisting of 19 bodies with 10 rigid-body-parameters and 14 friction coefficients

Struts (3 bodies):cardan ring, stator, slider

6 parameters

Model parameters obtained after identification

© Bodo Heimann

Deduction of Friction Dynamics in Parameter Linear Form

ex

ey

ez

-z

x

y

l

rpassive joints

active joint

Friction model for passive & active joints

Jourdain ‘s Principle of Virtual Power

Integration of friction in passive and active joints within the model

© Bodo Heimann

Implementation of Inverse Dynamics in Real-Time Control

Optimized C-code

• Rigid-body model723 additions1076 multiplications6 trigonometric functions

• Friction model36 additions54 multiplications

Execution time incl. path-planning, controllers etc.: < 0,15 msHighly efficient: real-time application without any problems

MAPLE®

optimizedC-Code

dSPACE® -control

parameters(kinematics,

inertia, friction)

© Bodo Heimann







• Conclusions

© Bodo Heimann

Identification of Dynamical Parameters: Linear Approach

LP-model structure

LP-estimation model

information matrix

where

measurement vector

,

noise vector

and

Combination of N measurements

Gauss-Markov estimation

© Bodo Heimann

Challenges by Trajectory Optimisation

-5000

500

-500

0

500-1000

-800

-600

-400

-200

xy

z

-5000

500

-500

0

500-1000

-800

-600

-400

-200

xyz

Inclination by 10°

Hard workspace constraints

Optimal excitation of dynamic parametersIndependent accelerations of bodies

Inclination of EE-platformReduction of available workspace

≠

© Bodo Heimann

Design of Experimental Identification

Excitation trajectory defined in the workspace

Optimal input-design• trajectory optimization optimization of trajectory parameter

• deterministic framework: noise-free information, only additive output-noise

• statistic framework: noise-corrupted information and output

minimization of condition number κ of the information-matrix

minimization of parameter covariance lower bound

© Bodo Heimann

Examples of Optimized Trajectories of Different Order

0 2 4 6-1

-0.5

0

0.5

x y z

0 2 4 6-20

-10

0

10

20

αβγ

-0.4-0.200.2-0.4-0.200.2

-0.9

-0.8

-0.7

-0.6

-0.5

x y

z

t [s] t [s]

[m]

[°]

Example of an optimized trajectory of the 5th order:

Example of an optimized trajectory of the 7th order:

0 3 6 9-1

-0.5

0

0.5

x y z

t [s]

[m]

0 3 6 9-20

-10

0

10

20αβγ

t [s]

[°]

-0.4-0.20 0.2

-0.4-0.200.2-0.9

-0.8

-0.7

-0.6

-0.5

xy z

© Bodo Heimann

Repeatability Experiments

How repeatable are the identification results?

© Bodo Heimann

Identification Results for Rigid-Body Model

Parameter GM 95% Conf. IntervallIzz1+Iyy1+Izz3 -0.033 [-0.035,-0.031]

Ixx2+Ixx3-Iyy2-Izz3 1.135 [1.132, 1.139]Izz2+Iyy3 1.055 [1.053, 1.058]

sy2 0.622 [0.620,0.623]sz3 -1.348 [-1.349, -1.344]

I*xx,E 0.246 [0.245,0.247]I*yy,E 0.242 [0.241,0.243]I*zz,E 0.133 [0.132,0.135]s*

zE 1.711 [1.710,1.711]mE + 6m3 16.509 [16.507,16.513]

Very good results with tight confidence intervals

© Bodo Heimann

Repeatability Experiments

1 1.02 1.04 1.060.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

1. 100x measurement of optimized trajectory

2. Evaluation and identification for each measurement

3. Investigation of distribution of the resulted parameters

data

prob

abilit

y

75% confidence interval

0.54 0.56 0.580.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

13.8 14 14.20.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

18 19 20 210.01 0.02 0.05 0.10

0.25

0.50

0.75

0.90 0.95 0.98 0.99

Very good repeatability!

[kgm2]

second minimal inertia parameter

[kgm]

first moment of the stators

dry friction coefficient in 2nd actuator

[N]

[Nsm-1]

damping coefficient in 5th actuator

© Bodo Heimann

Validation and Accuracy of Identified Models

0 1

-80-60-40-20

0

0 1

-150-100

-500

50

0 1

-500

50100

0 1

-100

-50

0

0 1

-100

0

100

0 1

-50

0

50

100

Act

uato

r for

ces

[N]

time [s]

Validation of identified models on circular motion

measured forces

validation with rigid-body model

validation with integral model

high accurate modeling and identification!

© Bodo Heimann

Model-Based Control Performance

Actuator tracking errors at highest velocity

0 0.5 1 1.5

-2

0

2

e 1[m

m]

0 0.5 1 1.5-10123

e 2[m

m]

0 0.5 1 1.5

-2

0

2

e 3[m

m]

time [s] time [s]time [s]

single-joint control

feedforwardcontrol

0 0.5 1 1.5

-10

-5

0

e 4[m

m]

0 0.5 1 1.5-2

0

2

4

e 5[m

m]

0 0.5 1 1.5

-2

0

2

e 6[m

m]

time [s] time [s] time [s]

• Tracking errors still exist

• Need of further disturbance reductionILC-concept

© Bodo Heimann







• Conclusions

© Bodo Heimann

Concept of Iterative Learning Control (ILC)

Controller+

Inverse dynamics

++

memoryoffline ILC

stable feedback controlled system

(time domain)

-

• The system is executing a repetitive task • Same perturbations over all repititions are assumed• Calculation of errors and a corresponding input-correction at each trial j

ILC: Iterative Learning Control

ILC is conventional control in the iteration domain

e.g. linear ILC:

Design of ILC-controller: design of learning-gain matrix L

© Bodo Heimann

Discrete Proportional Gain: P-ILC

Most elementary control law:

is constant (scalar or matrix) learning gain

Problem: Bad transient behavior

Guaranteed zero-tracking convergence

• Monotonous convergence is not guaranteed

• Divergent behavior before final convergence

• Typical problem for mechanical systems: high-frequency poles

Not adequate for complex mechanical systems

Alternative design

5 10 150

0.5

1

1.5

2

2.5

Iteration

e RM

S[m

m]

Simulation

© Bodo Heimann

ILC with Zero-Phase Filtering and Phase Lead Compensation

Idea: Filtering transient frequencies new control law

Parameterization with condition of monotonous error decay:

learning gainlow pass filter, cutoff ωc

phase-lead compensation

Choose Φ und l, for maximal cutoff ωc!

transfer function of

-1 0 1 2-1

0

1

Re

Im

←ω = 15 Hz←ω = 40 Hz

-1 0 1 2-1

0

1

Re

Im ←ω = 10 Hz

←ω = 120 Hz

© Bodo Heimann

Experimental Results: Control Accuracy

motion 2: quadratic

0 0.5 1-1

0

1

e 1[m

m]

0 0.5 1-1

0

1

e 2[m

m]

0 0.5 1-1

0

1

e 3[m

m]

0 0.5 1-1

0

1

e 4[m

m]

1 2-1

0

1

e 1[m

m]

1 2-1

0

1

e 2[m

m]

1 2-1

0

1

e 3[m

m]

1 2-1

0

1

e 4[m

m]

motion 1: circular

before learningwith feedforward control

after 10 ILC-iterations

Tracking errors for two motions with respect to four actuators

Successful application of ILC: Important improvement of control accuracy within few iterations

© Bodo Heimann

Experimental Results: TCP-Accuracy

0.12 0.14 0.16-0.16

-0.14

-0.12

x [m]y

[m]

0.12 0.14 0.16-0.16

-0.14

-0.12

x [m]

y [m

]

desired

realRemark: Errors were multiplied by 5 for better illustration

before learning

Dynamic positioning at corner

after 10 ILC-iterations

0 1 20

0.5

1

1.5

2

2.5

t [s]

|Δx|

[mm

]

circular motion

0 0.5 10

0.5

1

1.5

2

2.5

t [s]

|Δx|

[mm

]

quadratic motionposition accuracy

before learning

after 10 ILC-Iterations

© Bodo Heimann

Conclusions

• Model identification– Use of harmonic trajectories that fits in hard constrained workspace– Input-optimal design of experiments optimization of excitation trajectories– High repeatability of the identification results.– Excellent agreement of measurement and calculation.

• Feedforward computed-force control– Significant improvements by model-based control– Best results with experimentally identified models– Tracking errors still exist

• Improvement of tracking by Iterative Learning Control– Appropriate for repetitive robotic tasks– Simple implementation but highly satisfying results

• Modeling– Combination of Jourdain’s Principle of Virtual Power with automatic determination

of base parameter set highly efficient approach– Losses in passive and active joints

© Bodo Heimann

Hannover Center of Mechatronics (MZH)

• Founded in 2000

• Fusion of different Institutes of 2 Faculties

(Mech. Engineering and Electrotechnical Engng. and Informatics)

• Goal: Investigation of electro-mechanical systems and transfer of new

results to industrial applications

© Bodo Heimann

Institut für Robotik (IfR)

KontaktAppelstraße 11

30167 HannoverDeutschland

[email protected]. +49(0)511/762-4179

Arbeitsgebiete

▫ Autonome mobile Serviceroboter

▫ Fahrzeugmechatronik

▫ Produktionstechnik

▫ Medizintechnik

▫ Robotik

© Bodo Heimann

Antriebsprinzipien für Parallelkinematiken

61,9%53,8%

30,8%28,6%

7,7%4,8% 8,0%5,8%

0%

10%

20%

30%

40%

50%

60%

70%

Bearbeitungs-systeme

Roboter

sonstige

Variable Strebe Bewegter Fußpunkt Kniehebel

© Bodo Heimann

Application of Delta-Robots

General (Hitachi)

Manipulation (Sick Pack Systems)

Medical Application (SurgicalRobotics Lab, Humboldt-Universität Berlin )

© Bodo Heimann

Areas of Application of Different Structures

82,5%94,7%

58,3%

94,4%

17,5%5,3%

41,7%

5,6%

0%10%20%30%40%50%60%70%80%90%

100%

r itsytems

robots iti it

hybridparallel

overall

© Bodo Heimann

Stewart-Platform: MSI

Technical Datas• Force: 45 kN • Torque: 4 kNm• Acceleration: 25 g• Velocity: 1 m/s• Frequency: > 40 Hz

• Low weight of TCP -> high dynamics• Symmetric construction -> low cost

© Bodo Heimann

Hybrid Spatial Systems: Tricept

ABB – IRB20940

TR600 Neos Robotics

5 actuators

3 translaional and 2 rotational

Laser cutting and Laser welding

max. velocity: 1.5m/s,

max. acceleration: 2g

Control: Siemens SINUMERIK 840D

Accuracy: < ±10μm

6 actuators

3 translational and 3 rotational

max. velocity: 0.8m/s

max. acceleration: 1g

Accuracy: < ±0.02mm

© Bodo Heimann

Spatial PKM (4-DOF)

Flexpicker IRB340 (ABB)for high velocities

DELTA-Roboter

3 actuators

3 DOF (translation)+ 1 DOF (revolute)

Manipulator for „pick and place“

max. velocity: 10m/s, max. accel.: 10g

Control: PC-IO (Delta),

S4Cplus (ABB)

Accuracy: < ±0,2mm (Delta), ± 0,1mm (ABB)

Load: max. 1kg

RRPaR

© Bodo Heimann

Determination of Base (Minimal) Parameter Set

Analysis of Li and Ki leads to analytic reduction rules

• Elimination of parameters without influence trivial

• Elimination of all linear dependencies in H:elimination of parameters, of which corresponding columns are linearly dependentno relative motion in “corresponding direction” is possiblepossible combination of parameters of adjoining links

Analysis of the contributions Hi of adjoining links:- calculation of Hi as a function of Hi-1:

- definition of Li and Ki by MDH-parameters

© Bodo Heimann

Impact of Identification on Control Performance

0.4 0.8 1.2 1.60

0.01

0.02

0.03

0.04

feedforward with identified model

single-joint control

feedforward with nominal model

Evolution of error against end effector velocity

• Model-based control is better than single-joint control

• Nominal model is not sufficient for improving control

• Best performance is achieved with experimentally identified dynamics model

© Bodo Heimann

Experimental Results: Evolution of Error

Tracking errors for actuator 1: evolution with respect to time and iteration

Spectrum of errors for actuator 1: evolution with respect to error frequency and iteration

Dynamik und Regelung von Robotern mit paralleler kinematischer ...

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