Parametric Identification of Mechanical systems

1
Parametric Identification of Mechanical systems

• Dr. Gentiane Venture
• ?????? ?????

gentiane_at_ynl.t.u-tokyo.ac.jp
2
What is SYSTEM IDENTIFICATION ?

• System

perturbation
system
input
output
3
What is SYSTEM IDENTIFICATION ?

• Describing the system by a relation between its
  INPUT and its OUTPUT
• System identification Finding the relation
  describing the system and its characteristics

4
How to describe a system ?

• White-box model
• The type of model is known and based on first
  principles
• Ex physical process from the Newton equations
• Gray-box model
• Part of the model is known only
• Black-box model
• 0 is known

5
Groups of identification method

• Parametric identification
• Estimates specific parameters, model structure,
  optimal observer
• Using LS, ARMAX, Kalman filtering
• Non-parametric identification
• Look for characteristic behavior
• Using transient response analysis, Fourier
  analysis

6
Example of parametric identification

• Find the stiffness k of the spring in the static
  case
• m is known and vertical static equilibrium
• By measuring l0 and lm

l0
lm
Fk(lm-l0)
m
mg
EXAMPLE 1
7
An other example

• Find the stiffness k of the spring in the dynamic
  case
• m is known, dynamic vertical movement
• By measuring l0 and lm(t)
• Why and How to solve such a problem ?
• ?IDENTIFICATION

l0
lm(t)
Fk(lm(t)-l0)
m
mg
EXAMPLE 2
8
Why doing identification ?

• Having the model of a system is very important
  for
• analysis,
• simulation,
• prediction,
• monitoring,
• diagnosis,
• control system design...

9
How doing?

• Modeling properly the system
• Finding input, output, mechanical description
• Defining the parameters to be estimated
• Measuring the input and output
• Using an appropriate optimization method
• Interpreting the obtained results

10
Modeling of mechanical systems

• Multi-body system n bodies

11
Modeling of mechanical systems

• Parameterization of each body Inertial
  parameters
• Mass Mj
• Inertia matrix Jj
• First moment of inertia MSj
• XSj the vector of standard parameters for body j
• XSj Mj IXX IXY IXZ IYX IYY IYZ
  IZX IZY IZZ MSX MSY MSZ
• XS the vector of standard parameters for the
  whole system
• XS XS1 XS2 XSj XSn

12
Modeling of mechanical systems

• Newtons laws to model a dynamic system
• q ?????????, qj ? j?????
• XS ?????????????????? Mj, Jj, MSj j 1n
• G ???????????
• Q ????????
• H ????????????????
• Ge, Gv, Gf ??????, ??, ????

13
Modeling of mechanical systems

• The inverse dynamic model is linear in the
  standard parameters XS
• Ge, Gv, Gf are linear in the stiffness kj, the
  viscosity hj, the friction fsj and the off-set oj

14
Parameters to estimate

• Vector of standard parameters for each body XSj
• In the case of elastic joint j
• stiffness kj, viscosity hj, friction fsj ,
  off-set oj
• They are concatenate in one vector
• XEj XSj kj hj fsj oj

15
Identification model

• Linear system
• Sampled along a movement ? over-determinate
  system (more equations than the number of
  parameters to estimate)
• ? LINEAR LEAST SQUARES METHOD

16
Identifiability Base parameters

• Can all the parameters in XE be estimated at the
  same time ?
• NO! It depends on the rank of DE
• Necessity of computing the vector of BASE
  PARAMETERS X the minimal set of parameters that
  can be estimated

17
Identifiability Base parameters

• An example of structural regrouping
• k k1 k2 only can be estimated but k1 and k2
  cannot be separated

k1
k2
k1
k2
lm(t)
l0
Fk(lm(t)-l0)
m
mg
EXAMPLE 2
18
Identifiability Base parameters

• An example of numerical suppression
• In the static case
• Only k can be estimated
• h cannot be estimated in the static case (or with
  small velocities)

k, h
EXAMPLE 1
19
Computation of the base parameters

• Symbolical (from the formal system)
• Only dependent on the structure
• Eliminate the column of DE that have not effect
  on the model (suppression)
• Determine the columns of DE that are linked
  (regrouping)
• Using dynamic model or energy model
• Numerical (from the sampled system)
• Dependent on both structure and sampling movement
• Use a QR decomposition of DE

20
Resolution of the identification model
Sampling along a movement
Idem for D and W but they are matrixes
21
Identification model - sampling

Sampling along a movement
EXAMPLE 2
22
Linear Least Squares ?????

• Algorithm to solve linear, multi variable,
  over-determinate systems
• W is on minimal rank (the base parameters have
  been computed)
• Powerful, robust , fast
• Implemented in common computation software
  Matlab, Scilab
• Example in Matlab gtgt XW\Y

23
Performing a good identificationand
Interpretation of the results

• Using good movements that excite the dynamics to
  estimate by avoiding numerical suppression or
  regrouping of parameters (EXAMPLE 1)
• Checking the condition number of W
• Computing the relative standard deviation

24
Using good movements

• Physical sense and common sense
• Knowledge of the system
• Experience

25
Condition number of W ???
where
and Si the singular value of W such as
26
Condition number of W ???

• W is well-conditioned if
• A well-conditioned system is less-sensitive in
  model and measurements error and leads to a
  better solution
• Example 3 sensitivity to perturbation

27
Condition number of W ???

• A classical mathematical example of sensitivity
  to perturbation
• The two following systems have the same solution
• If we perturb both systems we obtain

EXAMPLE 3
28
Condition number of W ???

• A well-conditioned system can be obtained with
• Good exciting movements
• A designed movement obtained by optimization of a
  criterion that gives a specific path to follow
• By sequential excitation of the concerned joints
  the other joints are blocked

29
Relative standard deviation ????

• Computed using classical and simple results from
  statistics
• W supposed to be deterministic
• r supposed to be a 0 mean noise

30
Relative standard deviation ????

• sXjlt10 good estimation
• sXj gt10 bad estimation
• BUT when Xjltlt1 impossible to conclude

31
Validation

• By comparison with a priori value
• By comparison of reconstructed joint torque and
  forces
• Using the same test as identification step
  direct
• Using different tests cross validation
  (preferred)

plot
Measured
Estimated
32
Validation

• Example

33
Practical identification sensors, noise and
filtering

• Measurements performed by sensors
• Access to all the variables not possible
  Necessity to reconstruct the missing data
• Integration, derivation,
• Eventual geometric considerations
• In the real world noise
• Effect of noise can bias the results
• ? Necessity of using filters

34
Practical identification sensors, noise and
filtering

• Butterworth filter 0-phase filter, fwd and rev

35
Practical identification sensors, noise and
filtering

• Associated with central derivative or trapezoidal
  integration

36
Applications

• Robots
• Cars and wheeled vehicles
• Human body

37
Robots

• Method widely applied to robotics and robot
  systems
• Manipulator robots
• Parallel robots
• Wide size robots with elasticity
• Extensions to mobile robots

38
Car and wheeled vehicles

• Modeling

39
Car and wheeled vehicles

• Sensor and measurements

40
Car and wheeled vehicles

• Designed movements and sequential movements quite
  impossible to perform Experience and good
  knowledge of the dynamic of the system

41
Car and wheeled vehicles
42
Human body Toward medical applications

• Most complex mechanical system
• But some models can be done
• Movements consider are passive muscles are not
  activated
• Aim identifying the joint stiffness of limbs for
  medical purposes

43
????????

• ??????????????
• ???????????????????????(??/????30 fps)

44
?? ????????

• ?????
• ??????????
• ??
• ?????
• ?????????

45
???????????

• ?????
• ????????
• ?????????????????
• ?????????????????????????????????????????

• 155 ???, 366 ??

46
?????

• ??????? (EMGs)
• ????????? 1 KHz

EMGs??????????????????????????????????
47
?????????

• ???? ???
• ???? ?????

??? ?????
48
??????????????????

• ????Q 0 ? ?????? G 0

H ????????????????????
49
?????
Y W.X r
50
?????
Y W.X r