PARAMETRIC MULTIBODY MODELING OF … · Stefano Baglioni University of Perugia Dep. of Industrial...
Transcript of PARAMETRIC MULTIBODY MODELING OF … · Stefano Baglioni University of Perugia Dep. of Industrial...
1 Copyright © 2013 by ASME
Proceedings of the ASME 2013 International Mechanical Engineering Congress & Exposition IMECE2013
November 13-21, 2013, San Diego, California, USA
IMECE201364815
PARAMETRIC MULTIBODY MODELING OF ANTROPOMORPHIC ROBOT TO PREDICT JOINT COMPLIANCE INFLUENCE ON END EFFECTOR POSITIONING
Stefano Baglioni University of Perugia
Dep. of Industrial Engineering Via G. Duranti 1
06125 Perugia, Italy
Filippo Cianetti University of Perugia
Dep. of Industrial Engineering Via G. Duranti 1
06125 Perugia, Italy
Claudio Braccesi University of Perugia
Dep. of Industrial Engineering Via G. Duranti 1
06125 Perugia, Italy
Luca Landi University of Perugia
Dep. of Industrial Engineering Via G. Duranti 1
06125 Perugia, Italy
ABSTRACT Nowadays, in the field of robotic, one of the most important
objectives is to reduce robot error positioning and improve its
dynamic behaviour. One of the main source of error in end
effector positioning is due to the joint compliance: robot joint
components under operating conditions can be deformed as a
function of their stiffness/damping properties. Generally, for
industrial robots, harmonic drive gearings are used, their
principal characteristics are high transmission ratio and law
weight, on the other hand, to realize high transmission ratio,
harmonic drive gearings work on inner gear elastic
deformation, conferring to the robot joints an excessive
compliance that, in some robot applications, cannot be
neglected. In this research activity multibody modelling and
simulation approach has been used to analyse joint compliance
influence on robot position accuracy. The principal aim of this
work was the formulation of a modelling procedure that
starting from classical robots modelling approach (i.e. Denavit
Hartenberg) defines an universal database and a parametric
modelling procedure that allows the designer to use any
multibody commercial codes to analyse anthropomorphic
robots considering or not the compliance effect. All the
procedure was developed and managed into a numerical code
environment (Matlab/Simulink).
An example of commercial anthropomorphic robot was
considered by assuming its principal kinematic and dynamic
characteristics. Parametric models of the robot have been
developed in two different multibody modelling environments
(Simmechanics, Adams/View). Moreover the models structure
has been built in order to control the robot movements both in
motion (open loop) or in force (closed loop). In this case they
are interfaced with Simulink code in a so called co-simulation
approach that allows to developed a generic control system
and test it by using one or more models, less or more refined.
1. INTRODUCTION The objective of this study is the development of a procedure
able to carry out dynamic analysis for generic industrial
anthropomorphic robot manipulators in order to reduce end
effector error positioning and improve robot dynamic
behavior.
The mean by it has been realized is the multibody approach
which represents a very flexible and direct way to analyse
dynamic systems.
To reach this aim, the authors built a fully parametric robot
modeling procedure which, starting from the robot
characteristic properties evaluation, allows an easy multibody
modeling that, if opportunely expressed, can be used in any
multibody code available.
In detail, through a compiled procedure, an “universal”
database of all necessary parameters describing the robot has
been developed and made available for the multibody model
construction.
Thus, the first fundamental step of the developed procedure is
the robot geometry identification and representation in a
simulation environment. It has been done through a systematic
procedure of reference coordinates system definition,
according to the well known Denavit-Hartenberg convention,
subsequently reappraised by Veitschegger-Wu.
Another important step realized is the identification and
treatment of all technical information which defines the robot
dynamic behaviour. For example, in the subsequently analysed
test case these parameters have been identified both through
CAD 3D or by web catalogues examination.
2 Copyright © 2013 by ASME
Finally, once the robot multibody model has been generated
both kinematic and dynamic analysis can be carried out.
In addition, the developed procedure has been realized in order
to obtain more or less accurate robot representation. In this
way it is given to the user the possibility to focus the analysis
on determined components evaluating their properties or their
influence on the global robot dynamic behaviour.
A practical example is represented by the joint compliance
evaluation. Joints compliance influence on end effector error
positioning has been evaluated through dynamic analysis
comparison between a model made up of flexible joint and
another of rigid joint.
Furthermore, to test the control system behaviour, another
kind of analysis has been made available: the so called co-
simulation. Since Simulink is a suitable software for control
systems development, improvement and validation, a
procedure to accomplish dynamic simulation of the Simulink
control system coupled to the robot multibody model has been
developed.
Finally, the afore explained procedure has been validate
analysing a specific robot test case in two different multibody
environments (Adams and Simmechanics) and comparing the
results.
2. DIRECT KINEMTATIC: DENAVIT-HARTENBERG VERSUS VEITSCHEGGER-WU CONVENTION The robot direct kinematic analysis allow the knowledge of
the movement of each element constituting a robot structure
while it is solving an assigned operation.
An anthropomorphic robot can be schematized through a
number i of arms connected each other through joints in which
an in-built fixed reference coordinate system is placed.
In this way it is possible to represent any robot in a generic
pose that generally corresponds to the so called “zero state” or
“reference position” of the robot.
Practically, the robot geometry, defined by the position and
orientation of each coordinates system, can be described by a
series of homogenous transformation between triads.
Generally those coordinate system can be arbitrary placed, but
a systematic procedure of their definition is provided by the
Denavit-Hartenberg convention [1].
With reference to Figure 1, assuming the i-th axis as the
motion axis of the joint which connect the (i-1)-th Arm to the
i-th Arm, according to Denavit-Hartenberg convention it is
possible to define the i-th triad as follows:
Align zi with the axis of motion (rotary or sliding) of (i+1)-th
joint.
Locate the origin Oi of the i-th triad at the intersection of zi
and the common normal between zi and zi-1 axis, and Oi’ on
the intersection between the normal common and zi-1 axis
Establish xi axis along the common normal between zi and zi-
1 axis assuming positive the direction from joint i to (i+1)-th
joint.
Assign yi to complete the right-handed coordinate system.
Once the coordinate systems in-built to the arms has been
placed it is possible to define the i-th homogenous
transformation through the four parameters of Denavit-
Hartenberg which define position and orientation of the i-th
triad relative to (i-1)-th triad.
ai distance between Oi and Oi’,
di displacement of Oi' along zi-1,
αi angle, about xi, axis between zi-1 axis and zi axis,
considering positive counterclockwise rotations,
θi angle, about zi-1, axis between xi-1 axis and xi axis,
considering positive counterclockwise rotations,
About those parameters, ai and αi are always constant and
dependent from consecutive joint geometry defined by the
presence of i-th arm; instead one between di and θi is a
variable as a function of the kind of joint to be represented
(translational di or rotational θi ) while the other is a constant.
Now it is possible to express the homogenous transformation
which connects the (i-1)-th triad to the i-th triad as product of
transformation matrices as in equation 1. Where c and s refer
to the trigonometric functions sine and cosine.
The aforementioned robot kinematic identification has been
subsequently modified by Veitschegger-Wu [2] through the
introduction of an additional kinematic parameter βi in order to
1000
00
00
000
1000
100
00
00
'1
'
1
ii
ii
i
i
ii
ii
i
i
i
i
i
ics
sc
a
d
cs
sc
AAA
1000
0 iii
iiiiiii
iiiiiii
dcs
sascccs
casscsc
(1)
),(11
i
i
i
i
iVW yRotAA
1000
iiiiii
iiiiiiiiiiiiii
iiiiiiiiiiiiii
dccssc
sacscssccssccs
cacsssccsssscc
(2)
3 Copyright © 2013 by ASME
allow the evaluation of small position errors even in case of
two parallel consecutive joint axes [3].
According to Figure 2, it is possible to notice how the new
introduced parameter describes a rotation of the (i+1)th- joint
motion axis about yi axis.
Analytically the new transformation is obtained
postmultiplying the Ai Denavit-Hartenberg matrix by a matrix
which describes the additional matrix rotation Rot(y, βi) ,
shown in equation 3, in this way small position and orientation
variations can be always described through small variations of
the five kinematic parameters: ai , αi , di , θi e βi (Equation 2).
In equation 2 it is assumed:
1000
00
0010
00
),(ii
ii
ics
sc
yRot
(3)
Figure 2: Veitschegger-Wu kinematic parameters [2]. Figure 1: Denavit-Hartenberg kinematic parameters [4].
Figure 3: 3D kinematic Veitschegger-Wu representation of a
6 DOF anthropomorphic robot arm.
Figure 4: 3D representation of anthropomorphic robot arm
with geometry correction.
4 Copyright © 2013 by ASME
In the developed multibody modeling procedure this
additional parameter βi allow to realize, for the degrees of
freedom of each joint, the axis misalignment from the ideal
axis. Thus it is possible to simulate the small errors that can be
done in the robot assembly process or due to the deterioration
of the joint component during the robot life. Anyway, for βi
equal to 0 Rot(y, βi) becomes an identity matrix and the
Veitschegger-Wu convention degenerate into Denavit-
Hartenberg convention.
3. ROBOT GEOMETRY IDENTIFICATION The explained Denavit-Hartenberg or Veitschegger-Wu
convention provides a “minimal” representation of the robot
focused on motion axis identification. Substantially it
summarize the robot kinematic behavior adequate only for
kinematic analysis solution.
Instead, in case of dynamic analysis through multibody
modeling the minimal kinematic representation is not
sufficient as it not always provide complete information on the
effective joint positions as required by this approach.
As example, in case of two consecutive joints with orthogonal
motion axis the cited convention detects only the shorter
distance between the two motion axis, i.e. it doesn’t provide
information on the effective robot joint spatial configuration.
As we will show, for anthropomorphic robot manipulator that
happens during the transition from the elbow triad to the first
spherical wrist triad.
To leap over this problem additional information are required
to complete a correct robot multibody modeling. Thus, in the
modeling procedure developed by the authors other five
parameters aGCi , αGCi , dGCi , θGCi e βGCi , with the same
meaning of ai , αi , di , θi e βi , have been introduced. They
describe a new homogenous transformation, called 1i
iGCA
(similar to equation 2) from the i-th coordinate system
detected by Veitschegger-Wu convention to another coordinate
system opportunely oriented and located in the exact place
where the “real” joint is designed. Those parameters which
allow to identify the effective robot joint spatial configuration
have been called “Geometry Correction Parameters”.
The overall homogenous transformation which starting from
the (i-1)th- triad brings to the coordinate system placed on the
effective i-th joint position is analytically described by
equation 4.
111
i
iGC
i
iVW
i
iTot AAA (4)
Comparing Figure 3 and Figure 4 it is possible to better
understand the differences between the “Veitschegger-Wu”
convention, useful for kinematic analysis, and the developed
procedure with geometry correction useful for dynamic
analysis in multibody simulations.
In these schematizations the arms are represented by black
lines while the joints are represented by triads where the red
line represents the x axis, the green line represents the y axis
and the blue represents the z axis (motion axis).
In detail it is possible to notice the different position of the
fourth coordinate system which represents the first triad of the
wrist. According to Veitschegger-Wu convention it is placed at
the minimum distance along the normal common between the
motion axis of the third and fourth joint. Instead, after
adopting the geometry corrections, it is placed in the effective
spatial location of the robot wrist.
The identification of the correct geometry is crucial for the
robot modeling and dynamic simulation in multibody
environment so as to analyze the effective robot element
behaviors such as flexible joint.
4. MULTIBODY MODELING According to aforesaid to accomplish dynamic analysis of an
anthropomorphic robot manipulator the tool chosen by the
authors is the multibody modelling environment.
The authors’ ideas is to realize a procedure that starting from
an universal database definition allow to obtain parametric
robot models in any multibody code.
The developed procedure, in which the user has to provide the
necessary inputs as ascii universal files, is explained in the
following steps.
Preliminarily, in Matlab environment, the inputs defining the
robot characteristics are processed to make available a
common database, also this constituted by ascii universal files,
where the adopted multibody code can find all the needed
information.
Then, the multibody modelling begins with the definition of a
base reference coordinate system (absolute reference) and the
set-up of the working environment (units, gravity,…).
Subsequently, the robot structure is developed iteratively
implementing the aforementioned Veitschegger-Wu plus
“Geometry Corrections” convention so as to identify the robot
characteristic points which define the geometry: arms, joints
and degrees of freedom.
Figure 5: ith- Robot stage for iterative multibody modelling.
5 Copyright © 2013 by ASME
Substantially, through this iterative procedure, in order to
obtain a multibody model of a given n-DOF (degrees of
freedom) anthropomorphic manipulator, the whole robot
structure has been conceived as a sequence of n fundamental
stage each one representing the principal robot elements: i-th
arm (body plus joint), i-th actuator (which provides the
specific motion law (angular displacement or torque)) and i-th
sensor monitoring the variables of interest.
In Figure 5, for the developed iterative multibody modelling
procedure, a flowchart representing the i-th stage of a generic
robot model is shown.
The generated robot multibody model is of parametric kind
since the element properties are described by variables whose
values are defined in the universal database.
In so doing, for the user is easy and quick to realize robot
models with different characteristics (geometry, constraints,
mass and inertia properties, motion laws,…). As we will see,
for the different multibody environments this step is assisted
by specific tools developed by authors.
Another fundamental aspect to mention is: this procedure
allows to generate different kind of models in order to realize
more or less accurate representations of the robot test case.
Thus, the feasible robot multibody models differ each other
about the robot modeling complexity level defined by the user:
starting from a simplified representation of the robot with rigid
ideal joint and motion controlled in open loop (useful to
evaluate geometry and kinematic behavior) up to robot model
with joint compliance and motion controlled in closed loop
(useful to verify motor and control system performance or the
robot dynamic behavior) [5].
Anyway, it will be always given the possibility to generate
models with a desired complexity level considering only some
of the aforementioned robot element behavior.
Future development will increase the model complexity level
introducing the possibility of adding more robot component
behaviors such as friction in joints, servo motor rotational
inertia influence on robot dynamic performance, joint
clearances, flexible components,…
Then, in order to validate the developed procedure, multibody
models of a test case anthropomorphic robot have been
realized in two commercial multibody codes: Simmechanics
[6] and MSC. Adams [7].
Adams is a reliable tool that is widely accepted in industry, it
offers a 3D based graphical interface and a simple command
language supporting the user in modelling, set-up simulations
and postprocessing. Moreover it is able to interfaces to several
other commercial tools and software. On other hand
Simmechanics, being a Matlab toolbox, is suitable for very
fast model set-up and debugging. Both software allow a
control system validation through Simulink Co-simulation.
This step will be deeply analysed in further paragraph.
In Adams environment the parametric multibody model
construction has been done using a command language
iterative procedure (macro files) coupled to the generation of a
dedicated toolbar useful for modifying generated models or for
the desired complexity level set-up.
An overview of the implemented toolbar capabilities is shown
in Figure 6.
While, in Simmechanics environment the parametric
multibody modelling take place interactively: the user must
build the robot structure adopting the subsystems, each one
with the desired complexity level, developed by the authors.
Finally, in both case, before running dynamic simulation, it is
necessary to choose the solver and to set its properties. Solver
choice and properties settings can widely vary according to the
simulation type, results accuracy and simulation length.
Anyway, since the solver setting is not the main target of this
work, it is just important to mention that both multibody code
are able to solve the dynamic equation through a wide range of
integrators that can lead to full agreement results.
Figure 6: Snap-shot of “Robotic” utilities ADAMS toolbox (developed by the authors).
6 Copyright © 2013 by ASME
5. COMPLIANCE DUE TO JOINT ELASTICITY One of the main source of error in end effort positioning is due
to the joint compliance based on the following assumption:
“joint components under operating conditions can be
elastically deformed as a function of their stiffness/damping
properties”.
Generally for industrial robots harmonic drive gearings are
used, their principal characteristics are high transmission ratio
and law weight. On other hand to realize high transmission
ratio an harmonic drive gearing works on inner gear elastic
deformation, by the way its stiffness along motion direction is
not satisfactory and it confers an excessive compliance to the
robot joints that in some applications cannot be neglected.
According to aforesaid, joint elasticity is predominant if
compared to arms deflection under load, thus the robot
structure can be seen as a number n of rigid arms connected
each other through elastic joints [8,9,10].
A complete review compliance approach state of art is
presented in [11].
In this study case only revolute joint and related compliance,
along motion direction, has been considered.
To model the compliance, that is a deflection or a rotational
error respect to the desired imposed motion, it is necessary to
define a kinematic chain that allow to split the imposed
relative rotation from that the robot arm shows due to joint
flexibility.
So, the joint compliance has been conceived as the
composition of a so called rigid joint, which allow to define
the desired motion law by imposing a motion (open loop) or
by applying a controlled torque (closed loop), and of a so
called elastic joint added to introduce the joint compliance
behaviour.
In detail the i-th kinematic chain related to the i-th joint,
between (i-1)-th and i-th arm, is composed by a revolute joint
between (i-1)-th arm and a i-th dummy part (rigid joint), a
revolute joint between the i-th dummy part and the i-th arm
(elastic joint) and a i-th torsional spring damper force in
parallel with the elastic joint (Figure 7).
In this study we considered and modelled only torsional spring
damper in order to evaluate compliances simply along motion
direction. Further development will consider the
implementation of a six components spring damper so as to
evaluate objects deformations along all space directions.
In Figure 7 and Figure 8 the multibody modeling of a joint
with compliance respectively through a flowchart for
Simmechanics and through graphic representation for Adams
environment is represented.
Figure 7: Compliance modelling in Simmechanics environment.
Figure 8: Compliance modelling in Adams environment.
7 Copyright © 2013 by ASME
It is important to highlight that the introduced dummy part has
been defined with mass and inertia properties that can be
considered negligible if compared with the other robot model
components.
The i-th torsional spring-damper force is defined by the
following equation:
iiiii bKT (5)
in which Ki is the stiffness, bi is the damping, i and i
respectively represent the relative rotation and angular
velocity.
6. CONTROL SYSTEM DEVELOPMENT: “CO-SIMULATION” Since Simulink is a suitable software for control systems
development, improvement and validation, a procedure which
allows to set up a dynamic analysis of the Simulink control
system coupled to the multibody model has been developed.
This kind of simulation is the so called co-simulation [12].
The co-simulation consist of a dynamical simulation where the
equation describing the control system are solved by the
Simulink integrator while the motion equation of the robot are
solved by the adopted multibody integrator.
This is possible when the user opportunely defines the
necessary input and output variables for both control system
and multibody model. Moreover the user has to properly set-
up the communication parameters by which the two software
exchange the information relative to the input/output variables.
Substantially the target of a co-simulation is to make a
connection so that any change in one of the models affects the
behaviour of the other.
In detail, in case of Simmechanics multibody model there are
not particular devices to be taken in to account since both
Simulink and Simmechanics are both Matlab toolboxes. In this
case the co-simulation can be done directly connecting the
Simmechanics multibody model of the robot to the Simulink
model of the control system.
Instead, in case of Simulink-Adams dynamic co-simulation an
Adams dedicated procedure has been developed by the
authors. This procedure brings the user to create a Simulink
block representing the multibody model which can be
connected to the Simulink model of the control system.
In Figure 9 a flowchart representing the co-simulation in
Simulink environment is shown. From the figure it is possible
to see the connection between the multibody and the control
system and the typical variables in the running of a control
system modelling. In detail from the multibody model are
carried-out the variables describing the robot motion in the
joint space ( i rotation, i angular velocity and i
angular
acceleration) while the input variables are the motion low that
can be given as T torque, I current intensity, i rotation.
Anyway, if previously defined, the two software can exchange
information about any kind of variable.
7. ROBOT TEST CASE DESCRIPTION In this paragraph, the main properties of the robot test case,
selected to validate the developed multibody modelling
procedure, are presented.
The analysed robot is a 6 DOF anthropomorphic robot
manipulator made-up of six arm and six revolute joints
connected to the servo motors through harmonic drive
gearings.
In Figure 10 the robot in the zero position posture is
represented as its top, lateral, front and isometric view.
Figure 9: Control system development. Co-simulation.
8 Copyright © 2013 by ASME
In Table 1 and Table 2 respectively the five Veitschegger-Wu
parameters which define the robot kinematic and the five
“Geometry Correction” parameters which define the exact
geometry position of each joint are shown.
In Tables 3, Table 4 and Table 5 the main parameters which
describe the robot mass and inertia properties are shown. For
each i-th Am those parameters have been evaluated referring
to the i-th triad defined by the i-th homogeneous
transformation according to Veitschegger-Wu convention.
After harmonic drive catalogue investigation, for each joint,
the value of the stiffness K has been identified. Then the
damping constants have been evaluated in order to obtain a
mechanical system with an adequate level of damping. In
Table 7 the elastic properties of stiffens K and damping
constant b for each joint connecting the (i-1)-th Arm to the ith-
Arm, are presented
Finally, in order to make the robot end effector able to follow
the desired trajectory, called “Greca”, the imposed “Motion
Laws” have been defined as function of joint rotation θ(t).
This trajectory, shown in Figure 11, is a typical path for
anthropomorphic robot manipulators assigned to milling or
painting manufacturing.
The Motion Laws are given to the multibody software as an
input function described by spline (Akima method).
The desired 3D end effector trajectory presented in Figure 11
is obtained with the less complex model of the robot: motion
controlled through joint angular rotation i.e. no control system
influence and rigid joint i.e. no compliance introduced.
In the following paragraph the effect of the joint compliance
and of the control system will be analyzed to determine the
robot dynamic performance and to validate the developed
multibody modeling procedure.
Table 1: Robot Veitschegger-Wu parameters
Component α [°] a [m] θ [°] d [m] 𝛽 [°]
Arm 1 90 0.26 0 0.27 0
Arm 2 0 0.68 90 0.0 0
Arm 3 90 0.0 0 0.0 0
Arm 4 -90 0.0 0 0.67 0
Arm 5 90 0.0 0 0.0 0
Arm 6 0 0.0 0 0.155 0
Table 2: Robot “Geometry Corrections” parameters
Component α [°] a [m] θ [°] d [m] 𝛽 [°]
Arm 1 0 0 0 0.072 0
Arm 2 0 0 0 0.072 0
Arm 3 0 0 0 0.432 0
Arm 4 0 0 0 0 0
Arm 5 0 0 0 0.132 0
Arm 6 0 0 0 0 0
Table 3: Center of mass position
Component X [m] Y [m] Z [m]
Arm 1 -0.1504 -0.1227 -0.0175
Arm 2 -0.361 -0.0003 0.1595
Arm 3 0.0187 -0.006 -0.0276
Arm 4 0.0 0.164 -0.0042
Arm 5 -0.001 -0.0016 0.0539
Arm 6 -0.003 0.0 -0.022
Table 4: Robot Mass and Inertia
Component Mass [Kg]
Inertia Ixx [Kg·m2]
Inertia Iyy [Kg·m2]
Inertia Izz [Kg·m2]
Arm 1 75.4 1.92 2.63 2.9
Arm 2 38.3 1.6·10-1 2.15 2.15
Arm 3 40.5 1.36 1.31 3.9·10-1
Arm 4 10.9 1.14·10-1 3.38·10-2 1.12·10-1
Arm 5 5.53 4.02·10-2 2.25·10-2 2.87·10-2
Arm 6 1.02 4.5·10-4 4.5·10-4 5.3·10-4
Table 5: Robot Inertia
Component Inertia Ixy [Kg·m2]
Inertia Ixz [Kg·mm2]
Inertia Izy [Kg·m2]
Arm 1 9.21·10-1 -1.87·10-1 -1.60·10-1
Arm 2 -1.07·10-3 5.30·10-2 5.50·10-4
Arm 3 -3.04·10-3 2.19·10-2 2.19·10-2
Arm 4 2.26·10-5 -3.91·10-5 7.64·10-3
Arm 5 -1.10·10-5 -2.37·10-4 2.29·10-3
Arm 6 0.0 0.0 0.0
Table 6: Joint Stiffness and damping constant
Component K [N·m/rad] b [N·m·sec/rad]
Joint 1 1.8·105 1.3·103
Joint 2 6.7·104 5.5·102
Joint 3 1.8·105 4·102
Joint 4 6.7·104 1.1·102
Joint 5 3.1·104 3.9·101
Joint 6 3.1·104 4.3
9 Copyright © 2013 by ASME
8. ROBOT DYNAMIC BEHAVIOUR AND MULTIBODY MODELS VALIDATION In Figure 12 a comparison between the desired and effector
trajectory and the trajectories evaluated through multibody
dynamic simulation is presented. It refers to a robot model in
which the Motion Laws are given as rotation in the joint space
(open loop) and revolute joints are considered with
compliance.
In detail the upper part of the figure shows the whole 3D path
while the bottom part put in evidence a detail of the path in
Figure 11: Desired trajectory “Greca”.
Figure 10: Adams Model of tested anthropomorphic robot.
10 Copyright © 2013 by ASME
which the joint compliance effect is more evident. The blue
line is related to the desired trajectory, the red line is related to
Adams multibody simulation while the black line refers to the
Simmechanics simulation.
From the upper part of this figure it is possible to notice a shift
along z axis between the desired trajectory and the one done
by a robot with elastic joint as expected.
From the bottom part of the figure it is possible to see the
differences between desired and simulated trajectories on the
x-y plane, so as to better understand the compliance influence.
The small differences in end effector path between Adams and
Simmechanics analysis are due to different evaluation of the
angular velocity and acceleration laws. The two codes have
different kind of derivative operator.
To step over this problem, it is recommended to provide to
both multibody all three functions describing the motion law:
rotation, angular velocity and acceleration.
According to aforesaid it is possible to say the two software
provide an equal estimation of the joint compliance behavior.
This results comparison highlights the goodness of both
models realized to predict the joint compliance dynamic
behavior.
In Figure 13 it is possible to notice a detail of the same end
effector trajectory in case of models made-up with elastic joint
and motion controlled in closed loop (imposed torque). The
blue line is related to the desired trajectory, the red line is
related to Adams multibody simulation while the black line
refers to the Simmechanics simulation.
From this figure it is possible to notice that the implemented
control system increase the oscillations of the end effector
trajectory making both developed multibody simulation
macroscopically similar to the real dynamic robot behaviour
(experimental tests and numerical comparison with multibody
simulation are still in progress).
Anyway, it is possible to notice how both models are in full
agreement about the oscillation peak entity evaluation.
Still, it is possible to see the Simmechanics model presents a
slight anticipation of the trajectory oscillation behavior. This
phenomenon is due to the added Simmechanics block called
“Memory” recommended when the Simmechanics multibody
modeling contains closed loops (an example is the
implemented control system realized through a feedback
control on joint angular rotation).
This results comparison highlights the goodness of both
models realized to predict the joint compliance plus control
system dynamic behavior.
CONCLUSIONS In conclusion it is possible to assert that the following
objectives has been reached:
A fully parametric procedure to generate any robot geometry
in multibody simulation environment based on the
Veitschegger-Wu convention has been developed. All the
necessary inputs, needed to accomplish a multibody analysis
are provided by the user as simple file.dat.
The developed parametric multibody modeling procedure
Figure 13: Robot compliances dynamic analysis. One
corner detail of trajectories. Closed loop simulation.
Figure 12: Robot compliances dynamic analysis. Full and
one corner detail of trajectories. Open loop simulation.
11 Copyright © 2013 by ASME
allow to generate different multibody models in order to
realize more or less accurate representations of the robot.
They differ each other by the number and kind of robot
elements modeled (multibody model complexity level). The
definition of the complexity level is chosen by the user that
in this way can control the influence of each modeled
component in robot dynamic performance.
A dedicated procedure, called Co-simulation, which allows
to set-up dynamic analysis of a control system developed in
Simulink environment coupled to a generic multibody code
has been developed.
For each complexity level the models realized in the two
cited multibody environments bring to fully agreement
results so as to validate the multibody modeling procedure
developed by the authors.
Future development will increase the model complexity level
introducing the evaluation of different robot component
behavior such as friction in joints, servo motor rotational
inertia influence on robot dynamic performance, joint
clearances, flexible components,…
Then, the model made-up by the higher complexity level will
be compared to experimental results in order to validate the
multibody robot end effector trajectory prediction.
REFERENCES [1] Denavit J. and Hartenberg R.S., "A kinematic notation
for lawer-pair mechanism based on matrices", Trans.
ASME Ser. E, J. App1. Mech., Vol. 22, p. 215-221, 1955.
[2] Veitschegger W. K., Wu C., "Robot accuracy analysis
based on kinematics", IEEE Journal of Robotics and
Automation, Vol. 2, No. 3 p. 171-179, 1986.
[3] Hayati S.A., “Robot geometric link calibration” 22nd
IEEE Conf. on Decision and Control, pp1477-1483, Dec
1983.
[4] Siciliano B., Sciavicco L., Villani L., Oriolo G.,
"Robotics: Modeling, Planning and Control", Springer-
Verlag, 1st Edition, 2009.
[5] Corke P., "Robotics, Vision and Control – Fundamental
Algorithm in Matlab", Springer-Verlag, 2011.
[6] The Math Works Inc., “Simmechanics User’s Guide”,
July 2002.
[7] MSC Software, “Adams/View User’s Guide”, 2010.
[8] Abele E., Bauer J. et al., "Comparison and validation of
implementations of a flexible joint multibody dynamics
system model for an industrial robot", CIRP Journal of
Manufacturing Science and Technology, Vol.4, p.38-43,
2011.
[9] Zollo L., Siciliano B., et al., "Compliance control for an
anthropomorphic robot with elastic joints: theory and
experiments", Journal of Dynamic Systems,
Measurement and Control, Vol.127, p.321-328,
September 2005.
[10] Abele E., Rothenbücher S., Weigold M., "Cartesian
compliance model for industrial robots using virtual
joints", Production Engineering Research and
Development, Vol. 2, p. 339-343, 22 July 2008.
[11] Erkaya S., "Investigation of joint clearance effects on
welding robot manipulators", Robotics and Computer
Integrated Manufacturing, Vol.28, p. 449-457, February
2012.
[12] Cheraghpour F., Vaezi M. et al., "Dynamic modeling and
Kinematic Simulation of Stäubli TX40 Robot Using
MATLAB/ADAMS Co-Simulation", International
conference of mechatronics, April 2011, Istanbul,
Turkey.