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Research Papers

Modeling and Parameter Estimation of Robot Manipulators Using Extended Flexible Joint Models

[+] Author and Article Information
Stig Moberg

Division of Automatic Control,
Department of Electrical Engineering,
Linköping University,
Linköping SE-581 83, Sweden;
ABB AB—Robotics
Västerås SE-721 68, Sweden
e-mail: stig.moberg@se.abb.com

Erik Wernholt

Division of Automatic Control,
Department of Electrical Engineering,
Linköping University,
Linköping SE-581 83, Sweden
e-mail: erik@wernholt.se

Sven Hanssen

ABB AB—Robotics
Västerås SE-721 68, Sweden;
Department of Solid Mechanics,
Royal Institute of Technology,
Stockholm SE-100 44, Sweden
e-mail: sven.hanssen@se.abb.com

Torgny Brogårdh

ABB AB—Robotics,
Västerås SE-721 68, Sweden
e-mail: torgny.brogardh@se.abb.com

The same modeling principle can of course be applied to parallel kinematic structures and serial structures with parallelogram linkages.

Due to the symmetrical inertia tensor, only six components of Ji need to be defined.

The electrical drive system is not included in this model and is assumed to be ideal for the frequency range considered here. The drive system can be taken into account as described in Sec. 3.4.

The friction of the gearbox is actually split into two components and distributed between fm and fg.

In this paper, the number of DOF refers to the number of independent coordinates necessary to specify the (internal) configuration of a system.

Bearing elasticity is called joint flexibility in Ref. [18], while the transmission elasticity is called drive flexibility.

These nonlinearities can also be regarded as deterministic disturbances.

The path contains straight lines, circular segments, corner zones, and has a geometry that is expected to excite the manipulator over a wide frequency range.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received November 24, 2010; final manuscript received December 17, 2013; published online February 10, 2014. Assoc. Editor: Rama K. Yedavalli.

J. Dyn. Sys., Meas., Control 136(3), 031005 (Feb 10, 2014) (13 pages) Paper No: DS-10-1346; doi: 10.1115/1.4026300 History: Received November 24, 2010; Revised December 17, 2013

This paper considers the problem of dynamic modeling and identification of robot manipulators with respect to their elasticities. The so-called flexible joint model, modeling only the torsional gearbox elasticity, is shown to be insufficient for modeling a modern industrial manipulator accurately. Another lumped parameter model, called the extended flexible joint model, is therefore used to improve the model accuracy. In this model, nonactuated joints are added to model the elasticity of the links and bearings. The unknown elasticity parameters are estimated using a frequency-domain gray-box identification method. The conclusion is that the obtained model describes the movements of the motors and the tool mounted on the robot with significantly higher accuracy than the flexible joint model. Similar elasticity model parameters are obtained when using two different output variables for the identification, the motor position and the tool acceleration.

Copyright © 2014 by ASME
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References

Figures

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Fig. 1

The IRB4600 robot from ABB

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Fig. 2

Examples of elastic elements modeling torsional deflection in three directions: three nonactuated elastic joints (left), two nonactuated elastic joints together with one elastic joint actuated by a motor (right). Dampers are not shown.

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Fig. 3

Three rigid bodies with fixed coordinate systems A, B, and C (solid). The joint vectors li connect the dashed coordinate systems, starting in N. The transformation (rotation) from the one dashed to the next solid coordinate system is given by the kinematic joint angles.

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Fig. 4

A nine DOF extended flexible joint model with two links, two motors (M), three elastic elements (EE), and three rigid bodies (RB)

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Fig. 5

Closed-loop measurement setup

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Fig. 6

Partly linearized system

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Fig. 7

System description including nonlinearities and unmodeled dynamics

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Fig. 8

Stiffening spring characteristics

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Fig. 9

Nonlinear friction characteristics

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Fig. 11

Normalized stiffness for all 12 springs in case 7. True parameter values are 1. The uncertainty is indicated by two standard deviations.

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Fig. 12

One diagonal FRF element in configuration 1 of the simulation study. The magnitude of FRF from motor torque to motor acceleration is shown.

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Fig. 13

An example of measured signals for one axis in one experiment. The steady-state response from 4 s to 14 s is used for FRF estimation.

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Fig. 14

Illustration of all configurations used for identification (maximum payload) and validation (maximum payload and without payload). The robot joints 2, 3, and 5 are illustrated with a dot, the payload with a circle. Note that validation configurations 4–7 are not identical due to different positions of joint 1, 4, and 6.

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Fig. 15

Cost of all models evaluated for the identification FRF set and in the first validation FRF set, both with maximum payload. Same cost function as used in the identification.

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Fig. 16

Cost of all models evaluated for the identification FRF set and in the first validation FRF set, both with maximum payload, and for the second validation FRF set with no payload. Same cost function as used in the identification, but with all elements including motors 5 and 6 removed (unexcited when no payload).

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Fig. 17

All FRF elements in configuration 14 of identification set, showing the measured FRF (thick) and the 18 DOF model FRF (thin). The shaded area indicates the measured FRF uncertainty (1 std). The magnitude of the 6 × 6 FRF is shown, with input motor torque and output motor acceleration.

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Fig. 18

One diagonal FRF element in configuration 6 of identification set

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Fig. 19

The same FRF element as in Fig. 18 but in configuration 9 of validation set 1

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Fig. 20

One nondiagonal FRF element in configuration 3 of identification set

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Fig. 21

Normalized stiffness for all 12 springs in the 18 DOF model. The uncertainty is indicated by two standard deviations.

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Fig. 22

One diagonal FRF element in configuration 12 of validation set 2. The model with stiffening transmission springs is denoted NLS.

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Fig. 23

Illustration of all configurations used for identification using motor sensor and arm accelerometer (maximum payload)

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Fig. 24

The cost for models obtained by using tool acceleration or motor position as output variables. Cost for both output variables is shown.

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Fig. 25

Stiffness parameters for 15 DOF model. The parameters obtained by using motor position are normalized to one and indicated by a circle. The parameters obtained by tool acceleration are indicated by a square. Two standard deviations are indicated.

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Fig. 26

One FRF element (motor torque to motor acceleration)

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Fig. 27

One FRF element (motor torque to tool acceleration)

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Fig. 28

Real (solid) and simulated (dotted) tool speed in all directions. Eighteen DOF extended flexible joint model used in simulation.

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Fig. 29

Real (solid) and simulated (dotted) motor torque for axes 1, 2, and 3. Eighteen DOF extended flexible joint model used in simulation.

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Fig. 30

Real (solid) and simulated path. Eighteen DOF extended flexible joint model (dotted) and flexible joint model (dashed) used in simulation.

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Fig. 31

Error of simulated model when compared with the real robot path. Eighteen DOF extended flexible joint model (thick solid) and flexible joint model (thin solid).

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