Elsevier Editorial System(tm) for Robotics and Computer Integrated Manufacturing Manuscript Draft Manuscript Number: Title: Active Preload Control of a Redundantly Actuated Stewart Platform for Backlash Prevention Article Type: Research Paper Keywords: Backlash prevention; Stewart platform; redundant actuation; active preload control; online optimization Corresponding Author: Mr. Boyin Ding, Ph.D student Corresponding Author's Institution: University of Adelaide First Author: Boyin Ding, Ph.D student Order of Authors: Boyin Ding, Ph.D student; Benjamin S Cazzolato, Ph.D; Steven Grainger, Ph.D; Richard M Stanley, Engineer; John J Costi, Ph.D Abstract: There is an increasing trend to use Stewart platforms to implement ultra-high precision tasks under large interactive loads (e.g. machining, material testing) mainly due to their high stiffness, and high load carrying capacity. However, the backlash or joint clearance in the system can significantly degrade the accuracy and bandwidth. This work studied the application of actuation redundancy in a general Stewart platform to regulate the preloads on its active joints for the purpose of backlash prevention. A novel active preload control method was proposed to achieve a real-time approach that is robust to large six degree of freedom interactive loads. The proposed preload method applies an inverse-dynamics based online optimization algorithm to calculate the desired force trajectory of the redundant actuator, and uses a force control scheme to achieve the required force. Simulation and experimental results demonstrate that this method is able to eliminate backlash inaccuracies during application of large interactive loads and therefore ensure the precision of the system.
MECHANICAL ENGINEERING FACULTY OF ENGINEERING, COMPUTER AND MATHEMATICAL SCIENCES
BOYIN DING MECHANICAL ENGINEERING THE UNIVERSITY OF ADELAIDE SA 5005 AUSTRALIA
TELEPHONE +61 8 8313 2579 FACSIMILE +61 8 8313 4367 [email protected] CRICOS Provider Number 00123M
10th Oct 2013
Re: Research paper submission
Dear Editor
We would like you consider the attached paper entitled “Active preload control of a redundantly
actuated Stewart platform for backlash prevention” for submission to the Journal of Robotics
and Computer Integrated Manufacturing.
We certify that this article is original, that it is not under consideration by another journal or been
previously published. All named authors were involved in the conception of the idea, data
collection, data analysis and drafting of the final manuscript.
Yours sincerely,
Boyin Ding
School of Mechanical Engineering
Cover Letter
An online active preload control method with actuation redundancy was
proposed to prevent backlash on a Stewart platform.
The arrangement of the redundantly actuated manipulator demonstrated effective
active preload distribution efficiency, particularly when placing the redundant
leg into the robot inner space.
The proposed preload control method significantly mitigated backlash limit
cycles and consequently higher bandwidth control can be achieved on the robot
with higher accuracy.
The proposed method was robust to large six degrees of freedom interactive
loads.
*Highlights (for review)
Manuscript for Robotics and Computer Integrated Manufacturing
1
Active Preload Control of a Redundantly Actuated Stewart Platform for Backlash
Prevention
Boyin Dinga, Benjamin S. Cazzolato
a, Steven Grainger
a, Richard M. Stanley
b, and John
J. Costib
aSchool of Mechanical Engineering, University of Adelaide, Adelaide, SA 5005,
Australia
bBiomechanics & Implants Research Group, Medical Device Research Institute and
School of Computer Science, Engineering and Mathematics, Flinders University,
Bedford Park, SA 5042, Australia
Correspondence author: Boyin Ding
Permanent address: School of Mechanical Engineering, University of Adelaide,
Adelaide, SA 5005, Australia
Email address: [email protected]
Telephone number: +61 (08) 8313 2579
Fax number: +61 (08) 8313 4367
*ManuscriptClick here to view linked References
Manuscript for Robotics and Computer Integrated Manufacturing
2
Active preload control of a redundantly actuated Stewart platform for backlash
prevention
Abstract
There is an increasing trend to use Stewart platforms to implement ultra-high precision
tasks under large interactive loads (e.g. machining, material testing) mainly due to their
high stiffness and high load carrying capacity. However, the backlash or joint clearance
in the system can significantly degrade the accuracy and bandwidth. This work studied
the application of actuation redundancy in a general Stewart platform to regulate the
preloads on its active joints for the purpose of backlash prevention. A novel active
preload control method was proposed to achieve a real-time approach that is robust to
large six degree of freedom interactive loads. The proposed preload method applies an
inverse-dynamics based online optimization algorithm to calculate the desired force
trajectory of the redundant actuator, and uses a force control scheme to achieve the
required force. Simulation and experimental results demonstrate that this method is able
to eliminate backlash inaccuracies during application of large interactive loads and
therefore ensure the precision of the system.
Keywords: Backlash prevention, Stewart platform, redundant actuation, active preload
control, online optimization
Manuscript for Robotics and Computer Integrated Manufacturing
3
1. Introduction
Parallel robots are well known for their advantages in providing higher rigidity and
stiffness, being more compact in structure, and having greater payload capacity than their
serial counterparts. As a result, they are often used in applications where precision of the
order of micrometres is required from the robot during interaction (e.g. manufacturing
and assembling). However, joint clearances or backlash can largely degrade the accuracy
of parallel robots in these applications [1,2] as well as severely limiting bandwidth.
Many linear and non-linear control methods have been proposed to mitigate backlash
inaccuracies on a single actuated joint [3]. These methods often require a highly accurate
backlash model which is difficult to approximate in practice. Flexure joints have been
developed to remove backlash at the expense of limited range of motion [4,5].
Recent research found it was possible to achieve backlash prevention for parallel robots
by controlling the preloads on their actuated joints. Preload control can be further
divided into two categories: the active method and the passive method. The active
method uses actuation redundancy while considering dynamic effects for the purpose of
backlash prevention along a specified path [6,7]. This approach requires an offline
optimization process and its performance is highly sensitive to model error, which
prevents the proposed approach working in many real-time applications. The passive
method on the other hand uses preloaded passive joints in order to eliminate backlash
throughout a desired workspace when given norm-bounded external loads [8]. Although
much simpler than the active method, the passive method is hard to realize on a parallel
robot with more than three degrees of freedom and is not feasible with large external
loads of the order of 100N or greater.
Manuscript for Robotics and Computer Integrated Manufacturing
4
In this paper the authors investigate combining the benefits of both active and passive
preload methods using actuation redundancy to prevent backlash on a six degree of
freedom Stewart platform. Rather than using offline optimization based on feed-forward
dynamics, an online optimization algorithm is developed combined with a feedback
force control scheme to achieve a real-time method which is robust to both model
inaccuracy and load disturbance. The proposed approach is ideal for applications where
the Stewart platform is required to implement a high-precision task under large external
loads, e.g. materials testing, machining, assembling, etc. Section 2 presents the backlash
free condition, which is the essential goal for preload control. Based on the backlash free
condition, the overall solution is formulated, followed by four main problems to be
further treated in Section 3: the configuration of the redundant manipulator (Subsection
3.1), the inverse dynamics equation (Subsection 3.2), the preload optimization algorithm
(Subsection 3.3), and the force control scheme on the redundant actuator (Subsection
3.4). Section 3.5 presents the simulation results on a custom-built Stewart platform-based
manipulator developed for testing biological materials [9], followed by Section 3.6
which uses physical experiment to further verify the results.
2. Problem Statement
A general Stewart platform mechanism consists of six linear actuators, which are
connected via universal joints to a fixed base below and via spherical joints to a moving
platform above. Ballscrews driven by rotary motors are often used as the linear
actuators. The backlash in the ballscrew actuators is dominant compared to all other
sources of backlash. The backlash-free condition for a linear actuator is shown in Fig. 1,
Manuscript for Robotics and Computer Integrated Manufacturing
5
where represents the actuator control forces, represents the backlash-free threshold,
and represents the actuator payload limit. The backlash-free condition physically
means the magnitude of the actuator control force must remain above a certain level and
its sign must remain fixed during the period of the task for backlash prevention (Muller,
2005, Wei and Simman, 2010). Its mathematical expression is:
, , . (1)
In order to prevent backlash on a Stewart platform, all of its six ballscrew actuators
must satisfy the backlash-free condition. For achieving this, a new preload control
method is proposed with a redundant linear actuator attached to the moving platform
(Fig. 2). Overall, the concept is to use the redundant actuator to regulate the preloads on
the original position-controlled ballscrews for the purpose of ensuring the control forces
remain in the backlash-free region. As the solution is not unique, this forms an optimal
force control problem in which the redundant actuator is required to generate minimum
internal preloads to satisfy the backlash-free condition with lowest cost. Therefore, a
preload optimization algorithm is used to search for the desired preload on the
redundant actuator ( ) based on the backlash-free condition, the inverse dynamics
equation, and the varying parameters (e.g. external forces and moments, and the end-
effector trajectory). In series with the optimization algorithm, a feedback force control
scheme is used to drive the redundant actuator to achieve the desired preload. A
kinematics-based dual loop PID control scheme [9] is used to control the original six
ballscrews for accurately positioning the robot end-effector. This requires six linear
encoders mounted in parallel with the six ballscrews to measure their absolute lengths.
The use of dual loop PID control not only ensures the accuracy of the positioning when
Manuscript for Robotics and Computer Integrated Manufacturing
6
backlash is eliminated by preload control but also guarantees the stability of the plant in
the case backlash is not eliminated effectively.
3. Theoretical Analysis
Four main problems are treated in this section for achieving the proposed preload
control method in practice. Firstly, the configuration of the redundant actuator in the
Stewart platform is analysed for ease of control. Secondly, a simplified inverse
dynamics equation is derived for the redundant manipulator configuration. Thirdly, an
online optimization algorithm is proposed to determine the preload requirement on the
redundant actuator in real-time. Finally, the force control on the redundant actuator is
investigated for the purpose of accurate tracking and disturbance rejection.
3.1 Redundant Manipulator Configuration
The redundant actuation of parallel robots has been widely studied due to the
advantages of eliminating singularities, increasing manipulator stiffness, payload and
acceleration, and reducing power consumption [10,11]. These aspects are not the focus
of this study. Instead, the redundant actuation of the Stewart platform is used to regulate
the preloads assigned on the ballscrews for the purpose of backlash prevention. From a
controllability point of view, this is difficult as all six ballscrews must satisfy the
backlash-free condition with the regulation from only one actuator. There is no doubt
that the configuration of the redundant manipulator is fundamental to the success of the
proposed method.
Manuscript for Robotics and Computer Integrated Manufacturing
7
As a Stewart platform is symmetrical in its nominal configuration, an external preload
along the centre axis of the manipulator (z-axis of the global coordinate system) can
effectively create preloads on all six legs. Therefore, the redundant actuator is
configured to align with the centre axis (Fig. 3). The top end of the actuator is connected
via a spherical joint to a rigid support frame whose mounting point is on the centre axis
while the bottom end is attached via a spherical joint to the centre of the moving
platform. Although misalignment between the redundant actuator and the centre axis
occurs during motion of the moving platform, effective preloads can still be achieved on
all six legs within the envelope of motion of a typical Stewart platform. With a
sufficiently long redundant actuator assembly, it is possible to apply all compressions or
all tensions on the six legs, and therefore largely decrease the overall control difficulty.
Moreover, a passive element (mass-spring-damper system) is introduced into the
redundant actuator assembly to achieve a moderately compliant coupling with the
Stewart platform. This inherently increases the disturbance rejection and force
resolution of the system [12], and thus makes the implementation of force control much
easier. A force sensor is attached to the redundant actuator to measure its preload.
Details of parameter selection for the passive element will be discussed in Section 3.3.4.
In this study, the redundant leg is placed at the upper space of the robot assuming that
the inner space is used for implementing tasks which is often the case in applications
involving large interaction forces (e.g. material testing and machining). If the upper
space of the robot is required for task implementation, the redundant leg can be placed
inside the inner space.
3.2 Inverse Dynamics Equation
Manuscript for Robotics and Computer Integrated Manufacturing
8
The inverse dynamics model of a general Stewart platform has been presented in detail
in Do and Shahimpoor [13], Dasgupta and Mruthyunjaya [14], and Harib and
Srinivasan [15]. In the proposed preload control method, the inverse dynamics is used to
predict the actuator control forces for optimizing the preload on the redundant actuator.
To ease the computational expense of optimization, a simplified inverse dynamics
model is derived for the redundant manipulator with the following assumptions: 1) As
the motion range of the Stewart platform is limited around its nominal pose in high
precision and high load applications, the centre of gravity of each leg is always fixed at
the point which is the equivalent centre of gravity when the leg is at its nominal length
(mid stroke). 2) A universal joint is used at the stationary end of each leg (includes the
redundant leg) and therefore there is no rotational movement about the longitudinal axis
of the leg. 3) Friction is not considered. 4) Motor dynamics and actuator transmission
system dynamics are not considered.
Figure 4 shows the free-body diagram of one leg and the moving platform. Each leg
(actuator) consists of a cylinder and a piston. As the moving platform and each leg are
connected via a frictionless spherical joint, there is no moment but a single force exerted
at which can be decomposed as a force along the longitudinal axis of the leg ( )
and a force normal to the longitudinal axis ( ). results from the actuator control
force and is caused by the rotational dynamics of the leg. In order to solve ,
must be solved first. Considering the moments acting on ith leg about the rotation centre
of the leg , Euler’s equation gives:
(2)
where is the unit vector along the leg, is the leg length, is the distance between
the leg rotation centre and the leg centre of gravity, is the leg mass, represents the
Manuscript for Robotics and Computer Integrated Manufacturing
9
inertia tensor of leg, and are the angular velocity and the angular acceleration of
the leg respectively, is the acceleration of the gravity centre of the leg, and is the
gravitational vector. The global basis can be obtained via the following equation
(3)
where is the inertia tensor of the leg relative to the leg inertia coordinate system
and remains as a constant, is the rotation matrix describing the orientation of
relative to the global coordinate system O.
By assuming there is no rotation moment about the longitudinal axis of the leg (i.e.
and ), the kinematics of the leg can be written as Eqs. (4)-(8)
[14],
,
(4)
where represents the end-effector position, represents the end-effector orientation,
represents the position of the ith spherical joint in the end-effector coordinate
system o, and represents the position of the ith universal joint.
(5)
where represents the elongation speed of leg i, and and represents the angular
velocity and the linear velocity of the end-effector respectively. The angular velocity
and acceleration of the leg respectively are given by:
, (6)
(7)
where and represents the angular and linear acceleration of the end-effector
respectively. The acceleration of the centre of gravity of leg i can be written as
Manuscript for Robotics and Computer Integrated Manufacturing
10
. (8)
By substituting Eqs. (3)-(8) into Eq. (2), can be solved. Then, considering the
dynamics of the moving platform, Newton’s equation for the moving platform gives
,
(9)
where represents the moving platform mass, represents the external forces
acting on the platform in the end-effector coordinate system, and represents the
position vector of the gravity centre of the moving platform in the end-effector
coordinate system. Considering the moments acting on the moving platform about ,
Euler’s equation gives
,
(10)
where represents the external moments acting on the platform in the end-effector
coordinate system, represents the position vector of the external force exerting point
in the end-effector coordinate system, and represents the inertia tensor of the moving
platform and can be obtained via
(11)
where is the inertia tensor of the moving platform relative to end-effector coordinate
system and remains as a constant. The dynamics equation of the moving platform
can be written in matrix form:
Manuscript for Robotics and Computer Integrated Manufacturing
11
(12)
with
,
,
,
,
,
,
,
,
where represents the kinematics Jacobian matrix of a general Stewart platform,
represents the preloads on six original legs , represents the
inertia matrix of the moving platform, represents the centrifugal and Coriolis terms
of the moving platform, represents the gravity vector of the moving platform,
represents the terms generated from the dynamics of the legs, represents the statics
vector of the redundant leg, represents the axial preload on the redundant leg
, and represents the external loads. In Eq. (12), if , the external
loads, and the trajectory of the end-effector are known, then can be calculated.
Finally, considering the dynamics of the actuator piston, the actuator control forces can
be derived:
, (13)
Manuscript for Robotics and Computer Integrated Manufacturing
12
where is the mass of the actuator piston, and is the elongation acceleration of leg
i which can be written as
. (14)
Although simplified, the inverse dynamics model of the redundant manipulator is still
difficult to solve in real-time. The model can be further simplified by eliminating all the
Coriolis and centrifugal terms. In applications where the motion of the manipulator is
slow and external loads are large, the dynamics of the legs and pistons can be ignored
and therefore the inverse dynamics model can be finally simplified as a closed form:
(15)
with
,
,
where represents the control forces for the original six actuators, represents the
gravity matrix of the pistons, and represents the gravity vector of all seven legs.
3.3 Preload Optimisation
3.3.1 Optimisation Problem Formulation
Even if assuming the trajectories of the end-effector and the external loads are known,
the solution for in Eq. (15) is not unique in satisfying the backlash-free condition as
described in Eq. (1). This problem can be solved by minimizing the total internal
Manuscript for Robotics and Computer Integrated Manufacturing
13
preloads acting on the seven actuators . The lower the total internal preloads
means the lower the energy consumption of the system and smaller the redundant
actuator. Therefore, the backlash prevention optimal control problem can be formed as:
, (16)
where
,
,
where L represents the 2-norm of the total internal preloads at time t, represents the
maximum allowed preload on the redundant leg. As is the only unknown in L and all
six need to satisfy the backlash prevention condition, Eq. (16) is a one-dimensional
quadratic optimization problem subject to seven inequality constraints. Furthermore,
there are possible combinations of signs of , each of which has to be
considered independently in Eq. (16) and therefore the required computational time for
solving Eq. (16) is enlarged 64 times. As mentioned in Section 3.1, with the redundant
actuator configuration, it is easy to add all compression (positive preloads) or all tension
(negative preloads) on the six original legs and thus it is far more feasible to cause all
positive or all negative rather than the other cases. This reduces the possible
combinations of signs from 64 to 2 and the optimal control problem is simplified as
only two possible cases:
, or
, (17)
Manuscript for Robotics and Computer Integrated Manufacturing
14
3.3.2 Optimisation Algorithm
Problem (17) can be solved by offline optimization methods if a prescribed trajectory of
the end-effector and a prescribed trajectory of the external loads are both given.
However, in real-time applications, the external loads are caused by the interaction
between the robot and environment, and thus the prescribed trajectory of external loads
is generally unpredictable. This prevents the offline optimization methods from working
in applications where the external loads are dominant. In order to address this issue, the
authors developed an online optimization algorithm. This approach requires the external
loads to be measured by a 6-DOF load sensor which normally exists in high interactive
force applications. With the external load feedback, can be observed and predicted
for determining at each discrete time . As the proposed optimization algorithm is
based on online feedback measurement rather than offline processing, may slip into
the backlash-free condition before a control decision is made. Furthermore, when
tracking the determined on the redundant leg under force control, force tracking
errors must appear and cause preload errors on the six original legs. This can also lead
to stray into the backlash problem region. Therefore, the backlash free-condition in
Eq. (17) is redefined for compensating the delays in measuring external loads and
controlling :
, or , (18)
where represents a safety margin which narrows the original backlash-free condition.
If Eq. (18) is satisfied, are in the safe zone, where not only satisfy the
backlash-free condition but also are away from the backlash problem region. If Eq. (18)
Manuscript for Robotics and Computer Integrated Manufacturing
15
is not satisfied, are in the danger zone, and are either very close to or
already in the backlash problem region. With the definition of Eq. (18), a decision can
be made before the backlash problem actually occurs.
The flow chart of the proposed algorithm is shown in Fig. 5. At each discrete time ,
the external forces ( ) and moments
are measured from the 6-DOF load
cell. and represent the current desired end-effector pose and acceleration
respectively. represents the preload requirement for the redundant leg calculated
at the last discrete time . Using the inverse dynamics Eq. (15), the current control
forces of the six position-controlled actuators can be approximated as well as the
current total internal preloads index . Then are checked in Eq. (18). If are in
the safe zone, is regulated within its range to minimize the total internal preloads
index in the range of the safe zone. In order to guarantee the smoothness of and
decrease the computation burden, only the two points ( ) around with a
small increment d are considered. The total internal preloads index and for
these two points are calculated and compared. The smaller one is then compared with
the current index . If is smaller, the preload requirement at remains the
same as . Otherwise, the control forces under the new preload ( ) are
calculated via Eq. (15) and checked in Eq. (18). If are in the safe zone,
is equal to or . If not, remains the same as . In the case when are
in the danger zone, is regulated in its range to quickly move the control forces into
the safe zone. This is achieved by iteratively searching along both positive and
negative directions simultaneously from . At each iteration j,
and
are
increased in their directions with an increment of d. Then the corresponding control
Manuscript for Robotics and Computer Integrated Manufacturing
16
forces
and
are calculated via Eq. (15) and checked in Eq. (18). If none of
and
are in the safe zone, the next iteration starts. Otherwise, the iteration ends and
the rest of the code simply ensures the discrete increment between and is
below a maximum allowed value .
3.4 Force Control
In order to achieve the optimised preload trajectory on the redundant leg, an accurate
force control is required. This subsection investigates the redundant leg dynamics as
well as the control algorithm for preload tracking.
3.4.1 Dynamics Model of the Redundant Leg
Figure 6 shows a simplified schematic of the redundant leg, where the system is
modelled as a linear three-mass system under two assumptions. Firstly, we assume the
connection between motor and ballscrew piston is infinitely rigid compared to the mass-
spring-damper (MSD) system. Secondly, by assuming the ballscrew backlash is
infinitely small compared to the MSD system displacement, backlash non-linearity is
ignored. Analysing the torque balance on the motor, we have:
(19)
where is the motor moment of inertia, is the viscous motor friction, is the
motor rotational angle, is the motor driving torque, is the control force for driving
the ballscrew piston, and is the torque to force ratio of the ballscrew. As the ballscrew
piston and MSD cylinder are bolted together, the differential equation describing their
dynamics can be written in the form of a collective mass,
Manuscript for Robotics and Computer Integrated Manufacturing
17
,
, (20)
where is the total mass of the ballscrew piston and MSD cylinder, is the
viscous friction of the ballscrew piston, is the displacement of the ballscrew piston,
the displacement of the MSD piston is equal to the displacement of the redundant leg
length , and are the spring stiffness and damping of the MSD system
respectively, and is the angle to displacement ratio (also known as lead) of the
ballscrew. Analysing the force balance on the MSD piston, we have:
(21)
where is the mass of the MSD piston, is the preload on the redundant leg. With
Eqs. (19) to (21), the block diagram of the redundant actuator dynamics can be found in
Fig. 7. Clearly, the preload to be controlled is subject to the acceleration term
and gravity term of the MSD piston, and the stiffness term
and damping term of the MSD system. The
acceleration term and gravity term of the MSD piston are the disturbances in as they
are not controllable via . Therefore, the MSD piston mass is ideally made as
small as possible to minimize such disturbances. The stiffness term of the MSD system
is the major term in which can be controlled by regulating using a position
control loop of the redundant actuator. The selection of an appropriate MSD system
stiffness is critical. Very high can lead to low disturbance rejection. This
physically means any disturbance or error in spring movement can lead to large force
errors in . Conversely, a very low can decrease the bandwidth of force control if
the actuator slew rate is limited. The stability of force control is subject to the MSD
Manuscript for Robotics and Computer Integrated Manufacturing
18
damping term. There is a trade-off in selecting the MSD damping . A low can
cause control instability, while a high can lead to a large time constant and therefore
decrease the bandwidth.
3.4.2 Force Control Algorithm
As is mainly governed by the relative displacement between the ballscrew piston and
MSD piston, a position-based explicit control algorithm [12] is applied to control .
Figure 8 shows the algorithm in the form of a block diagram, where superscript d
represents the desired value and superscript s represents the real value. An outer force
control loop is placed around an inner position control loop. The force loop calculates
the desired relative displacement between the ballscrew piston and MSD piston
for minimizing the force error between the desired force and the measured
force .
is derived from the preload optimization algorithm while is measured
from the sensor. The absolute displacement of the leg length is calculated via
inverse kinematics from the end-effector desired pose and is used to compensate the
impact of the displacement of the MSD piston on . The sum of and
gives
the total desired displacement of the ballscrew piston , while the real displacement
of the ballscrew piston is obtained from the motor rotary encoder. The internal
position loop calculates the motor torque based on the displacement error
between and
for driving the ballscrew piston to achieve the desired
displacement. A PID controller is applied to the position control loop. The force
controller consists of a pure integral term and a low pass filter:
Manuscript for Robotics and Computer Integrated Manufacturing
19
(22)
where is an approximation of the spring stiffness, is the integral gain, and is
the low pass filter time constant. Integral control is commonly used in position-based
force control and yields good accuracy. is normally set as half of the position-loop
bandwidth with the resulting bandwidth of the force loop half the position-loop
bandwidth. With integral control, the force error is proportional to the desired velocity
of the actuator and therefore any discontinuity in force error can result in discontinuity
in actuator motion. In order to ensure smooth actuator motion, a low pass filter is used
in series.
4. Numerical Simulation
This section uses a custom-built Stewart platform-based manipulator as an example to
verify the preload control method with the assumption that an additional leg consisting
of a ballscrew (same as the original leg ballscrews) and a mass-spring-damper system is
mounted at the top of the manipulator. Simulations are implemented on a high fidelity
model of this system in the aspects of the redundant manipulator configuration, the
preload optimization algorithm, and the force control strategy.
4.1 Preload Distribution Efficiency of the Redundant Manipulator
This subsection assesses the proposed redundant manipulator configuration in terms of
its efficiency to distribute active preloads on the six position-controlled legs. A
comparison was undertaken between locating the redundant leg at the robot upper space
and placing the redundant leg into the robot inner space. The geometrical parameters of
Manuscript for Robotics and Computer Integrated Manufacturing
20
the original manipulator together with the assumed geometries of the redundant leg are
shown in Tables 1 and 2, which respectively present the coordinates of the fixed
universal joints and the coordinates of the moveable spherical joints of the seven legs in
the global coordinate system and in the end-effector coordinate system. Since the
manipulator used for simulation implements tasks in its inner space, the redundant leg
was firstly assumed to be located at the robot upper space (as illusrated in Fig. 3) and its
joint coordinate locations were selected considering the dimensions of the robot moving
platform, the length of the ballscrew actuator and the length of the MDS system. Then
the dimensions of the other case, where the redundant leg is placed into the robot inner
space, were obtained by simply mirroring the upper space leg dimensions about the XY
plane of the end-effector coordinate system when the robot is at its nominal central
pose [0mm 0mm 490.7mm 0° 0° 0°]. With the mirrored leg dimensions, a more
comparable result can be obtained between the two cases.
Given the geometrical parameters shown in Tables 1 and 2, the active preloads
distributed on the six original legs arising from the unit compressive preload of the
redundant leg can be calculated. In order to obtain the overall distribution efficiency of
the redundant manipulator within the workspace of the robot, such a relationship is
quantified during the movement of the robot along each of the three translational axes
and the three rotational axes about a virtual centre of rotation (the interaction point
between the robot and the environment). For the case that the redundant leg is located at
the upper space, the virtual centre of rotation is defined as [0mm 0mm -100mm] in the
end-effector coordinate system. For the other case, the virtual centre of rotation is
defined as [0mm 0mm 100mm].
Manuscript for Robotics and Computer Integrated Manufacturing
21
As a typical example, the preload distribution efficiency on the x-axis translation is
shown in Fig. 9, where the left subfigure shows the case when the redundant leg is
located at the upper space while the right subfigure shows the case when the redundant
leg is in the inner space. The solid lines represent the preload ratio of forces between the
th leg and the redundant leg and the pink dashed lines represent the boundaries of the
low efficiency zone [-0.05 0.05]. Preload ratios outside this zone means that effective
preload can be distributed to the corresponding leg, whilst ratios within the zone leads
to low distribution efficiency, in which circumstance the preload on the corresponding
leg is difficult to control since very small active preload can be assigned on the axial
direction of the leg. The worst case scenario is when the six preload ratios do not have
the same sign. When this occurs, the hypothesis in Eq. 17 that the redundant leg can
cause all tensions or all compressions on the original six legs is no longer valid, which
can consequently cause null solution in the preload control algorithm. From Fig. 9a
(upper case), we can see that the preload ratios are outside the low effciency zone and
have the same signs only when the robot motion on the x translation is restricted to
mm, which is about its half range of motion on this axis ( mm). By contrast,
Fig. 9b (inner case) shows that effective preloads can be assigned on all six position-
controlled legs (magnitude of the ratios > 0.1) over the full motion range. The results on
the other five degrees of freedom are listed in Table 3. Clearly, when the redundant leg
is placed at the robot upper space, the motion range of the robot must be restricted to
ensure an acceptable overall preload distribution efficiency. By contrast, placing the
redundant leg into the robot inner space leads to a satisfied overall preload distribution
efficiency over the full robot motion range. In the later sections, the redundant leg is
placed at the robot upper space to verify the proposed preload control method in both
Manuscript for Robotics and Computer Integrated Manufacturing
22
simulation and experiment since the original manipulator was designed to implement
tasks in its inner space.
4.2 Preload Optimisation Algorithm
The proposed preload optimization algorithm is assessed in this subsection under the
following two assumptions. 1) The robot interacts with a stiff environment within its
inner space and therefore undergoes large 6-DOF external loads. For simplicity, the
environment is assumed to have a linear stiffness matrix with diagonal terms only. 2) As
the simulated motion is slow and the resulting external loads are large, the end-effector
acceleration term and the leg dynamics terms are negligible and thus are ignored in Eq.
(15) during optimization. The geometrical and physical parameters required for preload
optimization are listed in Table 4. The backlash-free condition ( , , and ) for the
robot ballscrews are estimated from experiments. The loop running the preload
optimization algorithm has a loop rate of 100Hz. The initial preload on the redundant
leg is defined as 100N (a positive value means compression) for initially moving the
control forces of all the original ballscrews into the positive backlash-free region, such
that all six control forces are positive. Simulations are implemented by deforming the
environment about its centre of rotation (interaction point) in shear (x and y axis
translation), axial loading (z axis translation), bending (x and y axis rotation), and
torsion (z axis rotation). A sinusoidal waveform with +/-3mm (+/-10 degrees for
rotation) amplitude and 0.1Hz is applied on each of the above six degrees of freedom
sequentially for three cycles. In addition to the major movement, a sinusoidal waveform
with +/-0.1mm amplitude and 0.1Hz is superposed to all three translational axes for
Manuscript for Robotics and Computer Integrated Manufacturing
23
simulating the coupled forces arising from the movement of the environment centre of
rotation.
Figures 10 and 11 show the simulation results under x-axis shear and x-axis bending
respectively, where subplot (a) shows the optimized control forces on the six position-
controlled legs (solid lines) and the backlash-free threshold (pink dashed line) and
subplot (b) shows the required preload on the redundant leg. The results demonstrate
that the proposed algorithm is able to restrict the control forces on the six position-
controlled legs to the backlash-free region by generating a consistent desired preload
trajectory on the redundant leg. As we can further see from the plots, as soon as the
control force on any of the six legs approaches the margin of the danger zone (defined
as 80N in simulation), the algorithm enlarges the desired preload on the redundant leg in
order to move the control forces on the six legs away from the backlash problem region.
When the control forces on the six legs are in the safe zone, the algorithm gradually
decreases the total internal preloads on all seven legs. Results also demonstrate that the
proposed method is robust to large dynamic external loads. The cases on the other four
degrees of freedom have similar results but are not shown here due to redundancy.
4.3 Force Control Algorithm
The force control of the redundant leg was simulated in Matlab Simulink 7.6.1. The
redundant leg dynamics were modelled as the simplified system shown in Fig. 7, where
the numerical values used for simulation are listed in Table 5. The parameters
corresponding to the actuator dynamics ( , , , , and ) were obtained from
the Aerotech BM250 motor and EDRIVE VT209-07 actuator manuals. The parameters
Manuscript for Robotics and Computer Integrated Manufacturing
24
corresponding to the MSD system dynamics ( , and ) were selected to achieve
a high bandwidth force control as well as good disturbance rejection. For example, the
maximum backlash in the actuator is about 0.05mm which can only result in 5N
disturbance with the selected spring stiffness. Therefore, ballscrew backlash of the
redundant leg is negligible in simulation, as is the tracking error of the robot end-
effector, which is normally within 0.05mm when the control forces on the position-
controlled legs remain in the backlash-free region. The integral based force control
algorithm shown in Fig. 8 runs in a 100Hz loop. The parameters of the force controller
and the position controller in the equivalent continuous time domain are listed in Table
5.
The simulation has been undertaken on the steepest preload trajectory (Fig. 10(b))
obtained above. As the desired preload on the redundant leg is periodic, only the section
between the 11th second and 20th second which corresponds to the 2nd preload
optimization cycle is displayed. Fig. 12 shows the simulated force tracking results. The
maximum tracking error is approximately 45N. This physically means the maximum
force error assigned on each of the six position-controlled legs is about 8N which is
lower than the safety margin defined . Thus, a backlash-free condition is ensured
even with a delay in the control so long as the safety margin is sufficiently large to
tolerate the force control error.
5. Physical Experiment
Manuscript for Robotics and Computer Integrated Manufacturing
25
The assembly of the custom-built redundant manipulator [9, 16] for experiment is
shown in Fig. 13. The original manipulator consists of six EDRIVE VT209-07 actuators
driven by Aerotech BM250 motors. An AMTI MC31-6-1000 load cell is mounted on
the top platform to measure the 6-DOF loads reacted from deforming the testing
sample—a stiff polymer specimen for this experiment. A pyramid shape support frame
was designed to mount the redundant leg at the upper space. The framework was
manufactured from powder coated RHS steel. Static and vibration analyses were
implemented on the framework in ANASYS Workbench during the design process. The
final design has a stiffness of about 80000N/mm on the compression/tension axis and a
stiffness of about 14000N/mm on the shear axes. The first natural frequency of the
framework is about 87Hz. A seventh EDRIVE VT209-07 actuator is used to drive the
redundant leg. The piston of the actuator is coupled to a NET motorbike shock absorber,
which acts as the MDS system. The motorbike shock absorber was chosen due to its
availability, compact size and stiffness. The static performance of the shock absorber
(Fig. 14) was directly measured using an Instron model 8511 material testing machine.
The shock absorber exhibits a desired linear performance with a stiffness of about
30N/mm and a damping constant of about 8N/(mm/s) only under a compressive force
between 150N and 1000N. This is not ideal in real applications but is sufficient for
verifying the proposed concept. The shock absorber is then coupled to a Novatech F214
load stud. The location of the load stud allows direct measurement of the preload
exerted on the manipulator top platform. Spherical joints are used on both ends of the
redundant leg to couple the leg to the manipulator and the frame.
Figure 15 shows the overall control hardware configuration for the preload control
experiment. The control system of the original manipulator runs a host-target structure
Manuscript for Robotics and Computer Integrated Manufacturing
26
[9]. A host computer runs Windows and LabVIEW graphical user interface for
operating the system. Connected with the host computer via Ethernet, a target real-time
controller (NI PXI 8106) is used to handle upper level control of the manipulator. At the
lowest level, two FPGA boards (NI PXI 7852R) connect with the real-time controller
via DMA and run the dual loop PID controllers for the six robot legs. The control
signals are then sent to six Aerotech Soloist amplifiers (CP20) which drive the leg
motors. An AMTI MSA-6 strain gauge amplifier converts the AMTI load-cell analog
signal to a digital form and is sent serially over RS232 in order to minimize the noise
arising from the motor servo amplifiers. The converted RS232 signal is then input into
the real-time controller via a serial port on the controller and is decoded using the built-
in NI VISA. In this way, the measured loads are obtained at a 200Hz sampling rate and
the noise in the obtained signal is about ±6N and ±0.3Nm. For the same reason, a
custom-built strain gauge amplifer is used to digitize the Novatech load stud signal and
it is sent via a custom-written RS232 protocol on the FPGA. Unfortunately the obtained
preload signal contains noise as high as ±70N which mainly arises from the large
measurement range of the load stud (±15000N). This figure is far beyond the acceptable
range for the experiment. To reduce the noise, a smaller force sensor with a lower
capacity is required, however this would reduce the stiffness of the redundant leg and
consequently degrade its dynamic performance. An alternative solution—estimating the
preload from the deformed displacement (travel) of the shock absorber—is applied to
avoid directly measuring the preload. The deformed displacement of the shock absorber
can be obtained by comparing the difference between the robot travel pose and 7th leg
travel pose. Then the linear function between the travel of the shock absorber and the
Manuscript for Robotics and Computer Integrated Manufacturing
27
force response as shown in Fig. 14 can be used to estimate the preload on the redundant
leg. The preload optimization and the force control algorithm shown in Figs. 5 and 8 run
at 100Hz on the real-time controller. A Maxon EPOS2 70/10 position controller is used
to run the inner position loop (as a form of velocity control at 10kHz loop rate) on the
redundant leg shown in Fig. 8. As LabVIEW real-time controller does not support the
Maxon LabVIEW driver (which only works under LabVIEW Windows), the velocity
command of the motor on the 7th leg, which is calculated from the force control loop,
has to be sent to the Maxon controller indirectly via the host PC at 50Hz sampling rate.
A high density polymer specimen was mounted on the redundantly actuated
manipulator to emulate the robot interacting with a stiff environment which undergoes
large reactive external loads. Most of the control parameters for the experiment were
defined the same as the values for the simulation, and where they differ are stated in
Table 6. The backlash-free threshold and the safety margin were increased to 80N
and 20N respectively to compensate the delay and error arising from the limitations of
the control hardware set-up. The payload generated by the redundant leg was restricted
between 150N and 1000N to ensure that the shock absorber remains within its linear
range. For the redundant leg, the force control gains were selected by trial and error and
position control gains were tuned by the Maxon EPOS2 controller auto-tuning system.
The robot was commanded to deform the polymer specimen along each of the 6-DOF
under two circumstances. In the first circumstance, the robot was controlled without the
redundant leg but with a dead mass preload (180N) on top of the robot. Under the
second circumstance, the robot was controlled with the redundant leg using the
proposed active preload control method. To obtain comparable results, all the common
parameters (e.g. control gains of the position-controlled legs) and testing protocols were
Manuscript for Robotics and Computer Integrated Manufacturing
28
defined as the same for both circumstances. The testing protocols included shearing the
specimen by 1mm along the x and y axis, compressing the specimen by 0.4mm along
the z axis, bending the specimen by 6 degrees about the x and y axis, and twisting the
specimen by 6 degrees about the z axis. For shearing, compression and torsion testing,
the displacements were applied as a form of haver-sine waveform at 0.02Hz for three
cycles. For bending testing, the displacements were applied as a form of haver-sine
waveform at 0.01Hz for three cycles. These protocols were chosen for the following
reasons. Firstly, the displacements were selected to ensure the resulting external loads
on the robot were within the allowable range, which can be addressed by the force
capacity (150N to 1000N) of the shock absorber. Secondly, the testing speed was
defined in a very slow manner to minimize the error from preload estimation, where
only the static force was considered and to also tolerate the delay in the control
hardware set-up. Finally, backlash instabilities normally occurred at slow test speeds
which meant that the actuators spent considerable time in the backlash region during
zero crossings of actuator load. Furthermore a slow motion allowed the limit cycles
arising from backlash instabilities to become dominant and obvious within the overall
robot dynamic tracking inaccuracies.
Figure 16 shows the experimental results under x axis shear, where subfigures (a) and (b)
represent the three translational and three rotational errors of the robot respectively, and
subfigure (c) represents the preloads on the six position-controlled legs. The plots on the
left hand side illustrate the results under the dead mass preload method, while the plots
on the right hand side illustrate the results under the active preload control method using
the redundant leg. A maximum of 100N reactive shear resulted from the x axis shear
testing. Under the dead mass preload method, the preloads on the position-controlled
Manuscript for Robotics and Computer Integrated Manufacturing
29
legs were inevitably moved into the backlash-problem region as shown in subfigure (c).
As soon as this happened, the stability margin of the corresponding leg was narrowed
and consequently limit cycles arose from backlash instabilities as shown in subfigures
(a) and (b). Such high frequency limit cycles can be harmful to the ballscrews and other
mechanical components of the robot. By contrast, under the active preload control
method, the preloads on all six position-controlled legs were consistently kept in the
backlash-free region as shown in subfigure (c). Under such a condition, the non-linear
dynamics of the backlash was eliminated in the leg dynamics and consequently
backlash instabilities disappeared as shown in subfigure (a) and (b). As a result, the
robot tracking abilities were significantly improved. The experiments on the other five
degrees of freedom have similar results. The RMS tracking errors of the robot for the
dead mass preload (DMP) method and for the active preload control (APC) method on
each of the six degrees of freedom testings were computed and compared in Table 7.
The RMS errors arising from APC were within 5µm on translational axes and 5 arc-
second on rotational axes which are about 2 to 15 times smaller than the counterparts
arising from DMP. This proved the efficacy of the proposed active preload control
method.
6. Discussion and Conclusion
This paper studied the use of actuation redundancy to eliminate backlash inaccuracy for
a general 6-DOF Stewart platform. A novel redundancy arrangement with a refined
active preload control method was proposed for real-time control applications.
Simulation results demonstrated that placing the redundant leg into the robot inner
space results in a more effective preload distribution efficiency of the redundant
Manuscript for Robotics and Computer Integrated Manufacturing
30
manipulator within its workspace compared to placing the redundant leg at the robot
upper space, particularly along the horizontal (shear) axes of the robot. Thus, it is
suggested to apply the inner space case in applications which require use of the robot's
full range of motion (e.g. machining, assembling). Simulation results also demonstrated
that the proposed real-time preload control algorithm can effectively achieve backlash-
free conditions of the robot under large dynamically varying external loads. Because of
the hardware limitations, the experiment was restricted to low speed tests, however,
based on simulation results, it is expected that using improved hardware, the bandwidth
of testing could increase. The experimental results further demonstrated that the
proposed method can significantly mitigate (or even completely eliminate with an
improved design) backlash instabilities from control and consequently higher bandwidth
control can be achieved on the robot with higher accuracy compared to the same system
without the redundant leg.
In order to make the proposed active preload method fully applicable in industry, further
design and research work is required. Firstly, the design of the redundant leg assembly
is critical. A bicycle shock absorber is not ideal, not only because of the unsatisfactory
dynamic performance on its longitudinal axis but also due to the unexpected dynamic
behaviour on its transversal axes. Thus, a more sophisticated mass-damper-spring
system needs to be designed to allow a single degree of freedom linear compliant
motion along its longitudinal axis only. As the redundant leg actively controls the
preloads on all six position-controlled legs, the load capacity of the redundant leg is
required to be approximately 4 times higher than the position-controlled leg to ensure
the controllability of the system.
Manuscript for Robotics and Computer Integrated Manufacturing
31
References
[1] Khalil, I.S.M., Golubovic, E., and Sabanovic, A., 2011, “High precision motion
control of parallel robots with imperfections and manufacturing tolerances,” In Proc.
2011 International Conference on Mechatronics ( ICM), Istanbul, Turkey, pp. 39-44.
[2] Briot, S., and Bonev, I.A., 2008, “Accuracy analysis of 3-dof planar parallel robots,”
Mechanism and Machine Theory 43, 445-458.
[3] Nordin, M., and Gutman, P.-O., 2002, “Controlling mechanical systems with
backlash—a survey,” Automatica 38, 1633-1649.
[4] McInroy, J.E., 2002, “Modeling and design of flexure jointed Stewart platforms for
control purposes,” IEEE Transactions on Mechatronics 7(1), 95-99.
[5] Kang, B.H., Wen, J.T.-Y., Dagalakis, N.G., and Gorman, J.J., 2005, “Analysis and
design of parallel mechanisms with flexure joints,” IEEE Transaction on Robotics 21(6),
1179-1184.
[6] Muller, A., 2005, “Internal preload control of redundantly actuated parallel
manipulators—Its application to backlash avoiding control,” IEEE Transactions on
Robotics 21(4), 668-677.
[7] Boudreau, R., Mao, X., and Podhorodeski, R., 2011, “Backlash elimination in
parallel manipulators using actuation redundancy,” Robotica 30, 379-388.
[8] Wei, W. and Simaan, N., 2010, “Design of planar parallel robots with preloaded
flexures for guaranteed backlash prevention,” ASME Journal of Mechanisms and
Robotics 2, 011012 (1-10).
Manuscript for Robotics and Computer Integrated Manufacturing
32
[9] Ding, B., Stanley, R.M., Cazzolato, B.S., and Costi, J.J., 2011, “Real-time FPGA
control of a hexapod robot for 6-DOF biomechanical testing,” In Proc. 37th
Conference
of the IEEE Industrial Electronics Society (IECON), Melbourne, Australia, pp. 211-216.
[10] Wang, H., Zhang, B.J., Liu, X.Z., Luo, D.Z., and Zhong, S.B., 2011, “Singularity
elimination of Stewart parallel manipulator based on redundant actuation,” Advanced
Materials Research 143-144, 308-312.
[11] Nahon, M.A., and Angles, J., 1989, “Force optimization in redundantly-actuated
closed kinematics chains,” in Proc. IEEE International Conference of Robotics
Automation (ICRA), Scottsdale, AZ, USA, pp. 951-956.
[12] De Schutter, J., and Brussel, H.V., 1988, “Compliant robot motion II. A control
approach based on external control loops,” International Journal of Robotics Research
7(44), 18-33.
[13] Do, W.Q.D., and Shahimpoor, M., 1998, “Inverse dynamics analysis and
simulation of a platform type of robot,” Journal of Robotic Systems 5(3), 209-227.
[14] Dasgupta, B., and Mruthyunjaya, T.S., 1998, “The Stewart platform manipulator: a
review,” Mechanism Machine Theory 35, 15-40.
[15] Harib, K. and Srinivasan, K., 2003, “Kinematic and dynamic analysis of Stewart
platform-based machine tool structures,” Robotica 21(5), 541-554.
[16] Ding. B., Cazzolato, B.S., Grainger, S., Stanley, R.M., and Costi, J.J., 2013,
“Active preload control of a redundantly actuated Stewart platform for backlash
prevention,” In Proc. 2013 IEEE Conference on Robotics and Automation (ICRA),
Karlscrhe, Germany.
Manuscript for Robotics and Computer Integrated Manufacturing
33
Figure Captions
Fig. 1 Backlash-free condition for a linear actuator
Fig. 2 Schematics showing the preload control method where and represents
the external forces and moments, , , and represents the end-effector trajectory,
velocity, and acceleration, represents the desired preload on the redundant actuator,
and and represent the control forces for driving the redundant actuator and the
original six ballscrews respectively.
Fig. 3 Configuration of the redundant manipulator where BSP represents the ballscrew
piston, M1 and M2 represent the upper mass and bottom mass of the mass-spring-
damper system respectively, FS represents the force sensor, and SJ represents the lower
spherical joint.
Fig. 4 Free-body diagram of one leg and the moving platform, where represents the
fixed joint centre of leg i, represents the moving joint centre of leg i, represents the
gravity centre of leg i, represents the end-effector, represents the gravity centre of
the moving platform, and represents the point where the external loads are exerted on
the moving platform. The end-effector coordinate system frame o is attached to o, a
leg inertia coordinate system frame is attached to and rotates in coincidence with
leg i, and a global coordinate system frame O is fixed for reference. (i=1:7).
Fig. 5 Online optimization algorithm at discrete time .
Manuscript for Robotics and Computer Integrated Manufacturing
34
Fig. 6 Simplified schematic diagram of the redundant actuator.
Fig. 7 Block diagram of the redundant actuator dynamics.
Fig. 8 Block diagram of position-based force control.
Fig. 9 Preload distribution efficiency on x-axis translation.
Fig. 10 Optimized control forces and desired preload under N x-axis shear.
Fig. 11 Optimized control forces and desired preload under Nm x-axis bending.
Fig. 12 Simulated force control performance on the redundant leg.
Fig. 13 Assembly of the redundantly actuated manipulator—experimental rig.
Fig. 14 Measured static response of the shock absorber (tested under displacement
control with a haversine waveform of a -35mm amplitude at 0.01Hz for three cycles).
Fig. 15 Schematics showing the control hardware configuration for the preload control
experiment and the communication between the hardware elements.
Fig. 16 Comparison between dead mass preload (left figures) and active preload control
using the redundant leg (right figures). The robot was commanded to shear the
Manuscript for Robotics and Computer Integrated Manufacturing
35
specimen by 1mm along x-axis using a haver-sine waveform at 0.02Hz for three cycles.
Maximum shear force reached 100N.
Manuscript for Robotics and Computer Integrated Manufacturing
36
Table Captions
Table 1 Coordinates of the fixed universal joints in the global coordinate system O.
Table 2 Coordinates of the movable spherical joints in the end-effector coordinate
system o.
Table 3 Preload distribution efficiency of the redundant manipulator on each of the six
degrees of freedom.
Table 4 Geometrical and physical parameters for preload optimization simulation.
Table 5 Model parameters for force control simulation.
Table 6 Control parameters for physical experiment (same as the simulation parameter
if not listed).
Table 7 A comparison between the RMS tracking errors of the robot under dead mass
preload and under active preload control.
Table 1. Coordinates of the fixed universal joints in the global coordinate system O
(upper space)
(inner space)
X (mm) 46.6 341.0 294.4 -294.4 -341.0 -46.6 0.0 0.0
Y (mm 366.8 -143.1 -233.8 -233.8 -143.1 366.8 0.0 0.0
Z (mm) 0.0 0.0 0.0 0.0 0.0 0.0 1573.1 -591.7
Table 1
Table 2. Coordinates of the movable spherical joints in the end-effector coordinate system o
(upper space)
(inner space)
X (mm) 166.3 196.3 30.0 -30.0 -196.3 -166.3 0.0 0.0
Y (mm 130.6 78.7 -209.3 -209.3 78.7 130.6 0.0 0.0
Z (mm) 0.0 0.0 0.0 0.0 0.0 0.0 136.5 -136.5
Table 2
Table 3. Preload distribution efficiency of the redundant manipulator on each of the six degrees of
freedom
Full Range of Motion Restricted Motion Range Lowest Preload Ratio (absolute)
Upper Case Inner Case Upper Case Inner Case
X axis translation mm mm N/A 0.05 0.1
Y axis translation mm mm N/A 0.05 0.1
Z axis translation mm N/A N/A 0.18 0.18
X axis rotation ° ° N/A 0.05 0.08
Y axis rotation ° ° N/A 0.05 0.08
Z axis rotation ° N/A N/A 0.12 0.12
Table 3
Table 4. Geometrical and physical parameters for preload optimization simulation
Parameters Values Description (units)
Linear stiffness of the environment (N/mm, Nm/degree)
Position of the platform centre of gravity in o (mm)
Position of the interaction point in o (mm)
240 Length between leg rotation centre and gravity centre
(mm)
20 Platform mass (kg)
2 Actuator piston mass (kg)
5 Leg mass (kg)
70 Backlash-free threshold (N)
4000 Actuator payload limit (N)
4000 Redundant actuator payload limit (N)
10 Safety margin (N)
2 Preload searching resolution (N)
20 Preload discrete increment limit (N)
100 Initial preload on the additional leg (N)
Table 4
Table 5. Model parameters for force control simulation
Parameters Values Description (units)
0.404 Lead of the ballscrew actuator (mm/rad)
2215 Torque to force ratio of the ballscrew actuator (N/Nm)
0.0001 Motor moment of inertia ( )
0.005 Motor viscous friction ( )
5 Viscous friction of the ballscrew piston ( )
2 Total mass of the ballscrew piston and MSD cylinder (kg)
1 Mass of the MSD piston (kg)
5 Damping coefficient of the MSD system ( )
100 Spring stiffness of the MSD system ( )
1.2 Proportional gain of the position PID controller (
2.4 Integral gain of the position PID controller ( )
0.004 Derivative gain of the position PID controller ( )
10 Integral gain of the force controller (1/s)
0.01 Low pass filter time constant of the force controller
Table 5
Table 6. Control parameters for physical experiment (same as the simulation parameter if not listed)
Parameters Values Description (units)
Position of the specimen centre of rotation in o (mm)
80 Backlash-free threshold (N)
20 Safety margin (N)
1000 Redundant actuator payload upper limit (N)
150 Redundant actuator payload lower limit (N)
30 Spring stiffness of the shock absorber ( )
2 Integral gain of the force controller ( )
0.016 Low pass filter time constant of the force controller
Table 6
Table 7. A comparison between the RMS tracking errors of the robot under dead mass preload and under
active preload control
Shear
(x-axis)
Shear
(y-axis)
Compression
(z-axis)
Bending
(x-axis)
Bending
(y-axis)
Torsion
(z-axis)
ffffMethod
Axis ffffff DMP APC DMP APC DMP APC DMP APC DMP APC DMP APC
Tx ( m) 6.40 2.04 2.51 1.90 18.68 1.97 6.34 4.70 9.66 3.14 10.39 4.06
Ty ( m) 4.92 1.19 6.40 2.61 18.28 2.05 14.16 3.79 7.53 4.14 7.63 4.00
Tz ( m) 1.87 0.63 1.32 0.84 9.87 0.73 3.01 1.09 3.28 1.22 3.03 1.38
Rx ( ) 6.02 1.27 4.17 2.03 18.64 1.69 10.50 2.95 7.37 3.19 10.47 2.77
Ry ( ) 4.08 0.89 3.44 2.14 27.44 1.80 7.81 2.91 8.25 2.96 13.32 3.27
Rz ( ) 6.23 1.70 3.52 1.85 17.20 1.86 10.53 3.85 8.00 3.40 7.42 4.31
Maximum
Load 100N 120N 500N 45Nm 45Nm 27Nm
Table 7
With backlash Backlash free Backlash free
Figure 1
Feedback force
control
Preload optimization
based on backlash- free
condition and inverse
dynamics
Kinematics based dual-
loop PID control
Figure 2
Support
Frame
Redundant
Actuator
x y
z
M1
M2
FS
BSP
SJ
Figure 3
o
O
o
Figure 4
Y
Y
Y Y
Read loads Trajectory
generation Initialization
Inverse dynamics
Y
N Y
N
Calculate
Check if in
the safe zone
Calculate
Check if <
Calculate Calculate
Check if in
the safe zone
Check if in
the safe zone
Y
N N
Check if < Check if < N N
END
Calculate
Calculate
Check if
in
the safe zone
Check if
in
the safe zone
N
Y
N
Check if
Y Y
N N
Minimize Move into safe zone
BEGIN
Check if
Figure 5
MSD piston
Ballscrew piston and
MSD cylinder
Motor
Figure 6
+
+
+
+
Figure 7
Motor encoder
+
+
+
Fig. 7
Force sensor
Motor
Position
controller
Inverse
kinematics
Force
controller
Figure 8
-150 -100 -50 0 50 100 150-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
Robot displacement along x-axis translation (mm)
Pre
load r
atio (
fui/fu
7)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
LE line
-150 -100 -50 0 50 100 150-0.3
-0.2
-0.1
0
0.1
Robot displacement along x-axis translation (mm)
Pre
load r
atio (
fui/fu
7)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
LE line
a) Redundant leg at the upper space b) Redundant leg at the inner space
Figure 9 (color on web)
0 5 10 15 20 25 300
100
200
300
400
500
Time (seconds)
Actu
ato
r contr
ol fo
rce (
N)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
BFT
0 5 10 15 20 25 300
200
400
600
800
1000
1200
Time (seconds)
Pre
load (
N)
7th leg
(a) Optimized actuator control forces on the six legs (b) Desired preload on the redundant leg
Figure 10 (color on web)
0 5 10 15 20 25 3050
100
150
200
250
Time (seconds)
Actu
ato
r contr
ol fo
rce (
N)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
BFT
0 5 10 15 20 25 30100
200
300
400
500
600
700
Time (seconds)
Pre
load (
N)
7th leg
(a) Optimized actuator control forces on the six legs (b) Desired preload on the redundant leg
Figure 11 (color on web)
12 14 16 18 20400
500
600
700
800
900
1000
1100
Time (seconds)
Forc
e (
N)
Actual force
Desired force
16.2 16.3 16.4 16.5
760
780
800
820
840
Time (seconds)
Actual force
Desired force
(a) Force tracking performance in a cycle (b) Zoomed-in section showing the maximum error
Figure 12 (color on web)
Support
Frame Shock
Absorber
Force
Sensor
Polymer
Specimen
Ballscrew
Actuator
& Motor
6-DOF
Load-cell
Figure 13 (color on web)
-40 -30 -20 -10 0
-1200
-1000
-800
-600
-400
-200
0
200
Displacement (mm)
Forc
e (
N)
Figure 14
LabVIEW GUI
Maxon LabVIEW
driver (50Hz)
Kinematics based
control (1kHz)
Preload optimization
and force control
(100 Hz)
Dual loop PID
control (10kHz)
FPGA RS232
protocol (40MHz)
Current control
(20kHz)
Ethernet DMA Analog Host PC RT controller FPGA
Servo amp (Soloist)
for six motion legs
Velocity control
(10kHz)
Servo amp (Maxon)
for the redundant leg
RS232
RS232 RS232
AMTI load-cell
signal A/D (200Hz)
Novatech load stud
signal A/D (1kHz)
Strain gauge amp (AMTI) Strain gauge amp (Custom)
Figure 15
0 50 100 150-0.06
-0.04
-0.02
0
0.02
0.04
Time (second)
Err
or
(mm
)
Tx
Ty
Tz
0 50 100 150-0.06
-0.04
-0.02
0
0.02
0.04
Time (second)
Err
or
(mm
)
Tx
Ty
Tz
0 50 100 150-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Time (second)
Err
or
(degre
e)
Rx
Ry
Rz
0 50 100 150-0.015
-0.01
-0.005
0
0.005
0.01
0.015
Time (second)
Err
or
(degre
e)
Rx
Ry
Rz
0 50 100 1500
50
100
150
200
250
300
Time (second)
Forc
e (
N)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
0 50 100 1500
50
100
150
200
250
300
Time (second)
Forc
e (
N)
1st leg
2nd leg
3rd leg
4th leg
5th leg
6th leg
(a) Three translational tracking errors of the robot
(b) Three rotational tracking errors of the robot
(c) Preload assigned on the six position-controlled legs
Figure 16 (color on web)