Efficient dynamic closed-loop simulations of flexible manipulators - - PowerPoint PPT Presentation

Efficient dynamic closed-loop simulations

• f flexible manipulators
• Linearised equations of motion (for e.g. closed-loop simulations)
• Superposition of rigid link motion and small elastic deformations
• Perturbation method
• Mode-Acceleration Method / Adaptive Modal Integration
• Examples:

One-link manipulator with constrained motion Spatial two-link flexible manipulator with PID control

• Conclusions

FMSA4CP-LP / 1 Ronald Aarts

§ 12.6 or paper LP-1

Equations of motion

Flexible manipulator with em: large relative displacements and rotations, εm: flexible deformation parameters. Equations of motion, adapted from slide DvM/93:

¯

Mee ¯ Meε ¯ Mεe ¯ Mεε ¨ e m ¨ ε m

• +
• DemFT

DεmFT

• M(D2F ·( ˙

em , ˙ εm))·( ˙ em , ˙ εm) − f

• = −
• σem

σεm

• Components of the reduced mass matrix

¯ Mee = DemFT MDemF, ¯ Meε = DemFTMDεmF, ¯ Mεe = DεmFTMDemF, ¯ Mεε = DεmFT MDεmF.

FMSA4CP-LP / 2 Ronald Aarts

Equations of motion (2)

With generalised coordinates (D.O.F .) q =

• em

εm

• ¯

M(q)¨ q + C(q, ˙ q) ˙ q + g(q) + ¯ Kq = Bu,

• ¯

M(q) is the reduced mass matrix ¯ M = DFT MDF

• C(q, ˙

q) ˙ q represents the Coriolis and centrifugal forces

• g(q) is the vector of external nodal forces, including gravity,
• σem are the driving forces and torques, i.e. control input vector −u (note

the sign), and B =

• I
• σεm is the stress resultant vector of flexible elements, characterised by

Hooke’s law: Symmetric stiffness matrix Kεε with the elastic constants and ¯ K =

• 0 0

0 Kεε

• The direct solution of the non-linear equations of motion is rather time

consuming (both low and high frequent behaviour).

FMSA4CP-LP / 3 Ronald Aarts

Perturbation method

Model the vibrational motion of the manipulator as a first-order perturbation δq

• f the nominal rigid link motion q0

q = q0 + δq or

• em

εm

• =
• em
• +
• δem

δεm

• ; u = u0 + δu

The perturbation method involves two steps:

• 1. Compute nominal rigid link motion q0 from the non-linear equations of motion

with the rigidified model, i.e. all εm ≡ 0. ¯ Mee

0 ¨

em

0 + DemFT

• M0(D2F0·( ˙

em

0, 0))·( ˙

em

0, 0) − f

• = −σem

= u0, ¯ Mεe

0 ¨

em

0 + DεmFT

• M0(D2F0·( ˙

em

0, 0))·( ˙

em

0, 0) − f

• = −σεm

0 .

For a known nominal trajectory em

0 , ˙

em

0 , ¨

em

0 the generalised stress resultants

σem = −u0 and σεm are obtained.

FMSA4CP-LP / 4 Ronald Aarts

• 2. Compute the (small) vibrational motion δq from linearised equations of

motion: ¯ M0δ¨ q + C0δ ˙ q + ¯ K0δq =

• δu

σεm

• .

σεm are the generalised stress resultants applied as internal excitation forces. ¯ M0 is the system mass matrix, C0 is the velocity sensitivity matrix, ¯ K0 is the combined stiffness matrix defined as ¯ K0 =

• 0 0

0 Kεε

• + G0 + N0,

including the structural stiffness matrix Kεε

0 , the geometric stiffening matrix G0

and the dynamic stiffening matrix N0.

FMSA4CP-LP / 5 Ronald Aarts

Perturbation method: Applications

¯ M0

• δ¨

em ¨ εm

• + C0
• δ ˙

em ˙ εm

• + ¯

K0

• δem

εm

• =
• −δσem

σεm

• .
• 1. Constrained motion (§ 12.6 or paper LP-1): em = em

0 , so δem ≡ 0.

→ Solve differential equation for εm and compute δσem

0 .

→ Example of one-link manipulator.

• 2. Prescribed forces and torques (§ 12.6): σem = σem

0 , so δσem

≡ 0. → Solve differential equation for em and εm.

• 3. Controlled trajectory motion (paper LP-1): δσem

= −δu from control system. → Solve differential equation for em and εm. → Example of two-link manipulator.

FMSA4CP-LP / 6 Ronald Aarts

Constrained motion: One-link flexible manipulator

• ¯

mee ¯ Meε ¯ Mεe ¯ Mεε δ¨ em ¨ εm

• +
• 0 ¯

Kεε

0 + ¯

Nεε

0 + ¯

Gεε δem εm

• =
• −δσem

σεm

• Constrained motion: em = em(t) ; δem(t) = 0.

em(t) =

                

t < 0, Ω T

• 1

2t2 + T 2 4π2(cos 2πt T − 1)

• 0 ≤ t ≤ T,

Ω(t − 1 2T) t > T. Flexible motion: ¯ Mεε

0 ¨

εm +

• ¯

Kεε

0 + ¯

Nεε

0 + ¯

Gεε

• εm = σεm

FMSA4CP-LP / 7 Ronald Aarts

Tip deflection δy using different superposition approximations

5 10 15 20 25 30 −0.7 −0.6 −0.5 −0.4 −0.3 −0.2 −0.1 0.1 t [s] δy [m] 1) M0, K0, N0 and G0 2) K0, N0 and G0 3) M0, K0 4) K0 5) M0, K0, N0

1 2 3 4 5

FMSA4CP-LP / 8 Ronald Aarts

Comparison of the maximum tip deflection Model number of elements

• Max. deflection [m]

SPACAR, non-linear

4 0.536 Wu & Haug 4 substructures 0.556 idem 6 substructures 0.543 superposition, case 1 4 0.537 idem, case 2 4 0.531 idem, case 3 4 0.569 idem, case 4 4 0.569 idem, case 5 4 ∞

case 1 ¯ M εε

0 ¨

εm+

¯

Kεε

0 + ¯

Nεε

0 + ¯

Gεε

• εm = σεm

case 2

¯

Kεε

0 + ¯

Nεε

0 + ¯

Gεε

• εm = σεm

case 3 ¯ M εε

0 ¨

εm+

¯

Kεε

• εm = σεm

case 4

¯

Kεε

• εm = σεm

case 5 ¯ M εε

0 ¨

εm+

¯

Kεε

0 + ¯

Nεε

• εm = σεm

FMSA4CP-LP / 9 Ronald Aarts

Natural frequency of the first constrained bending mode

5 10 15 20 25 30 0.5 1 1.5 2 2.5 3 3.5

t [s] ω1 [rad/s]

1) M0, K0, N0 and G0 5) M0, K0, N0

Frequency equation: det(−ω2

i ¯

Mεε

0 + ¯

Kεε

0 + ¯

Nεε

0 + ¯

Gεε

0 ) = 0

FMSA4CP-LP / 10 Ronald Aarts

Perturbation method for controlled trajectory motion

As before, model the vibrational motion of the manipulator as a first-order perturbation δq of the nominal rigid link motion q0 q = q0 + δq, Apply the two steps of the perturbation method:

• 1. Compute nominal rigid link motion q0 from the non-linear equations of motion

with all εm ≡ 0, i.e. all vibrational modes set to zero.

• 2. Compute the vibrational motion δq from linearised equations of motion:

¯ M0δ¨ q + C0δ ˙ q + ¯ K0δq = σ0, σ0 =

• δud

σεm

• .

σεm are the generalized stress resultants applied as internal excitation forces. δud is the control input vector (minus u0) and ...

FMSA4CP-LP / 11 Ronald Aarts

δud is the control input vector δu = u − u0, possibly minus the proportional action of the controller represented by a matrix Kp δu = −Kpδem + δud. ¯ M0 is the system mass matrix, C0 is the velocity sensitivity matrix, ¯ K0 is the combined stiffness matrix defined as ¯ K0 =

• 0 0

0 Kεε

• + G0 + N0 +
• Kp 0

0 0

• .

It includes the structural stiffness matrix Kεε

0 , the geometric stiffening matrix

G0, the dynamic stiffening matrix N0 and the matrix Kp of the proportional control action. Note that one can also take δud = δu in which case Kp = 0 and the proportional control action is not included in the linearised equation of motion. Using a realistic Kp is particular beneficial for the modal analysis to be discussed next.

FMSA4CP-LP / 12 Ronald Aarts

Mode-superposition method

Equations of motion in n principal coordinates η δq = Φη Φ =

• φ1, φ2, . . . , φn
• δ ˙

q = Φ ˙ η + ˙ Φη ( ¯ KS

0 − ω2 i ¯

M0)φi = 0, i = 1, 2, ..., n, δ¨ q = Φ¨ η + 2 ˙ Φ ˙ η + ¨ Φη Symmetric ¯ KS

0 = 1 2( ¯

K0 + ¯ KT

0 )

¯ K0 includes Kp. ˆ M ¨ η + ˆ C ˙ η + ˆ Kη = ˆ σ, where ˆ M = ΦT ¯ M0Φ modal mass matrix, ˆ C = ΦT C0Φ + 2ΦT ¯ M0 ˙ Φ modal damping matrix, ˆ K = ΦT ¯ K0Φ + ΦT C0 ˙ Φ + ΦT ¯ M0 ¨ Φ modal stiffness matrix, ˆ σ = ΦT σ0 modal force vector. “Adaptive Modal Integration” (AMI): Time-varying nature of modal matrix Φ is taken into account.

FMSA4CP-LP / 13 Ronald Aarts

Mode-Displacement Method (MDM)

Solution δˆ q using only ˆ n < n modes δˆ q = ˆ Φˆ η, ˆ Φ =

• φ1, φ2, . . . , φˆ

n

• .

Mode-Acceleration Method (MAM)

Improved convergence after rewriting the equations of motion δq = ¯ K−1

0 (σ0 − ¯

M0δ¨ q − C0δ ˙ q), and substitution the MDM solution δˆ q in the right hand side δ˜ q = ¯ K−1

0 σ0 − ¯

K−1

0 ( ¯

M0δ¨ ˆ q + C0δ˙ ˆ q). First term in the expression of the MAM solution δ˜ q represents a pseudo static response of the system.

FMSA4CP-LP / 14 Ronald Aarts

Controlled trajectory motion

A (0.536, 0, 0) B (0, 1.300, 0) x y z

Motion trajectory

0.0 0.2 0.8 1.0 1.76 t [s] Tip velocity [m/s]

Velocity profile of the manipulator tip

0.0 0.5 1.0 −200 200 400 600 t [s] u0 [N/m]

σ1 σ2 σ3

Nominal torques u0 for the three actuators MIMO PID feedback control: δu = − ¯ Mee

0 H(s) δem.

¯ Mee

0 : mass matrix for decoupling between the actuators.

H(s): controller with three SISO PID controllers on the diagonal.

FMSA4CP-LP / 15 Ronald Aarts

Block diagram for simulations in Simulink

Non-linear simulation:

ltv y−Reference ltv times M0 dYtip ltv Setpoint U0 Selector Selector Ytipref Selector Selector Ytip Selector Selector Eref Selector Selector E spasim SPASIM

In1 Out1

MIMO lead

In1 Out1

MIMO PI

Output y = y0 + δy Nominal torques u0 Reference

• utput y0

SPASIM block for (non-linear) mechanism simulation. Nominal torques u0 (applied as feedforward) and reference output y0 are read from files.

FMSA4CP-LP / 16 Ronald Aarts

Perturbation method with modal analysis (“AMI”):

ltv times M0 dYtip ltv Setpoint Sigma Selector Selector Ytip Selector Selector E

In1 Out1

MIMO lead

In1 Out1

MIMO PI ltv LTV

• mega*omega

Kp

Output δy Reference is 0 LTV-block for simulation of Linear Time-Varying state space system ˙ xss = Ass xss + Bss uss yss = Css xss xss =

• δq

δ ˙ q

• r
• η

˙ η

• ,

uss =

• δud

σεm

• ,

yss: User defined outputs. Proportional controller part Kp is included in the LTV block and has to be excluded in the controller.

FMSA4CP-LP / 17 Ronald Aarts

Analysis 1: Natural frequencies along the trajectory

0.0 0.2 0.4 0.6 0.8 1.0 50 100 150 t [s]

• Nat. cons. freq. [rad/s]

0.0 0.2 0.4 0.6 0.8 1.0 10 12 14 t [s]

• Nat. sym. freq. [rad/s]

Four lowest natural frequencies ωc,i for a constrained manipulator. The bandwidth of the PID controllers is set to ωb = 12 rad/s. Three lowest closed-loop natural frequencies ωi during the controlled trajectory motion of the manipulator computed with a symmetric ¯ KS

0 .

FMSA4CP-LP / 18 Ronald Aarts

Analysis 2: Open loop Bode plots for actuator 1 + controller (SISO)

Frequency (rad/sec) Phase (deg); Magnitude (dB)

−40 −20 20 40 10 10

1

10

2

10

3

−270 −180 −90 90

Initial configuration

Frequency (rad/sec) Phase (deg); Magnitude (dB)

−40 −20 20 40 10 10

1

10

2

10

3

−270 −180 −90 90

Final configuration The solid lines are with the PID controller. The dashed lines are for a controller with an additional pole: Unstable behaviour is expected from the graph near 100 rad/s, which can be confirmed by simulations.

FMSA4CP-LP / 19 Ronald Aarts

Simulations 1a: Deviations from the nominal trajectory

0.5 1 1.5 2 −20 −10 10 t [s] δe [mrad]

δe1 δe2 δe3

Actuator rotations

non−linear method perturbation method AMI, 4 modes

0.5 1 1.5 2 −20 −15 −10 −5 5 t [s] δxyztip [mm]

δxtip δytip δztip

Tip co-ordinates Deviations from the nominal trajectory according to three simulation methods: No differences, so the AMI method performs well with only 4 modes (3 modified rigid link modes + 1 additional mode).

FMSA4CP-LP / 20 Ronald Aarts

Simulations 2a: Errors and CPU time

2 4 6 8 10 12 PM 0.001 0.01 0.1 1 10 Number of modes Maximum error [mm]

xtip ytip ztip

Maximum errors tip co-ordinates

2 4 6 8 10 12 PM NL 10

1

10

2

10

3

10

4

Number of modes

Number of time steps CPU time [s]

CPU time

• The maximum error in the tip co-ordinates found with the AMI method with
• nly 3 modes is comparable to the difference between the perturbation

method and the non-linear simulation (indicated with “PM”).

• With more modes the accuracy hardly improves at the expense of slower
• simulations. A significant reduction in CPU time is obtained in comparison

the perturbation method (“PM”) and the non-linear simulation (“NL ”).

FMSA4CP-LP / 21 Ronald Aarts

Simulations 1b: Deviations from the nominal trajectory

Second case: A stiffer manipulator with a controller bandwidth of approximately 60 rad/s = 9.5 Hz.

0.5 1 1.5 2 −0.1 0.1 t [s] δe [mrad]

δe1 δe2 δe3

Actuator rotations

non−linear method perturbation method AMI, 4 modes

0.5 1 1.5 2 −0.8 −0.6 −0.4 −0.2 t [s] δxyztip [mm]

δxtip δytip δztip

Tip co-ordinates

• Conclusions as before.

FMSA4CP-LP / 22 Ronald Aarts

Simulations 2b: Errors and CPU time

2 4 6 8 10 12 PM 10

−6

10

−5

10

−4

10

−3

10

−2

10

−1

Number of modes Maximum error [mm]

xtip ytip ztip

Maximum errors tip co-ordinates

2 4 6 8 10 12 PM NL 10

1

10

2

10

3

10

4

Number of modes

Number of time steps CPU time [s]

CPU time (faster PC)

• For the stiffer manipulator the approach works as well. As before the AMI

method with only 3 modes gives accurate results for e.g. the tip position.

FMSA4CP-LP / 23 Ronald Aarts

Conclusions

• The presented perturbation method allows an efficient numerical simulation
• f the controlled trajectory motion of a flexible manipulator as well as a

straightforward vibration control formulation.

• A further reduction of the simulation time was obtained by applying a modal

reduction technique, which we refer to as the Adaptive Modal Integration (AMI) method.

• For the spatial flexible two-link manipulator, results of both the perturbation

method and the AMI method agree well with the results obtained from a full non-linear analysis. In the AMI method only three (modified rigid link) or four degrees of freedom are needed to reach a satisfying accuracy.

• Crucial elements in the AMI method are the availability of accurately

linearized equations and a careful modal analysis in which the time-varying nature of the mode shape functions and the proportional feedback gains are taken into account.

FMSA4CP-LP / 24 Ronald Aarts