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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

http://ees.elsevier.com/rcim/viewRCResults.aspx?pdf=1&docID=2055&rev=0&fileID=49072&msid=28FB1DC6-0706-498D-B935-EFFD5910F3BE


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


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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.


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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,


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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


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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.


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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


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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


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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


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. (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:


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)


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


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)


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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)


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


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,


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


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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:


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


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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].


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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


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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


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


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


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


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


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


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


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


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.


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).


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.


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 .


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


35

specimen by 1mm along x-axis using a haver-sine waveform at 0.02Hz for three cycles.

Maximum shear force reached 100N.


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


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)


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


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


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


Support

Frame Shock

Absorber

Force

Sensor

Polymer

Specimen

Ballscrew

Actuator

& Motor

6-DOF

Load-cell


-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


manuscript draft article type: research...

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