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© 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
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
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
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
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
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
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
Vielen Dank für Ihre
Aufmerksamkeit !
Hannover
Neues Rathaus Altstadt
© 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