Abstract - An analytical-numerical method based on solution

of inverse dynamics is presented for determining the load

workspace of generally configured parallel motion systems with

up to 6 degrees of freedom. The kinematic workspace is also

obtained by the method, since the solution of inverse kinematics

is embedded in the procedure. The method is first developed for

3DoF motion systems to take the advantages of illustrating the

workspace by use of 3D plots. Then it is extended to higher

degrees of freedom. For such types of 3DoF motion systems, a

generalized trajectory is defined for the manipulator in the

kinematic workspace that consists of straight paths connecting

26 selected points of workspace's boundary. The platform

performs piecewise sinusoidal movements on the trajectory such

that it stops at each of the selected point and goes ahead toward

the other with maximum allowable speed. As the platform

moves on this trajectory, legs are exposed to any possible static

and dynamic forces. The forces obtained in this way are used to

determine the force workspace which is defined as the subspace

of kinematic workspace in which the structural forces do not

exceed their critical values. In this paper, the criterion of

structural failure is considered to be the buckling of legs. The

generalization of the method to higher degrees of freedom is

straightforward. As case studies, a 3DoF heave-roll-pitch

motion platform and a hexapod are chosen to apply the method

to. The results of the method can be used in the design of both

the control system and structure of parallel manipulators. The

method is also advantageous in structural design of large sizes

motion systems in flight simulator application.

I. INTRODUCTION

PATIAL Parallel mechanisms come in a big variety of

designs that vary depending on their degrees of freedom,

their structure and type that are determined in best conformity

with performance requirements considering cost and

complexity [1]. Different combinations of degrees of freedom

of platform are used in motion systems of simulators,

manipulating, surgery, general machining tasks such as

drilling, grinding, milling, welding, etc [2, 3].

In all of the applications mentioned above, determination of

workspace of the end effecter is essential. During the last two

decade, several researchers have performed studies regarding

the workspace of 6DoF parallel manipulators [4]. Study on

Ali Mahmoodi, Phd. Student, Aerospace Engineering Department,

Amirkabir university of technology, Tehran, Iran. (corresponding author: fax:

+98-21-66462210; e-mail: alimahmoodi@yahoo.com).

Masoumeh Aminzadeh, Msc. stusent, Aerospace Engineering Department,

Amirkabir University of Technology (e-mail: mas.aminzadeh @gmail.com)

Mohammad Bagher Menhaj, professor, Electrical Engineering

Department, Amirkabir University of Technology, Tehran, Iran (e-mail:

Menhaj@aut.ac.ir ).

Mehdi SabzehParvar, Associate Professor, Aerospace Engineering

Department, Amirkabir university of technology, Tehran, Iran (e-mail:

sabzehpar@cic.aut.ac.ir).

force workspace is essential since performing a big variety of

motion is an integral part of parallel manipulators' task.

However, little work has been done on determining load

workspace caused by dynamic forces. This is due to the fact

that in comparison with kinematic workspace, the force

workspace seems to be insignificant since the designers can

easily enlarge it by reinforcing the structure. But

determination of the force workspace is essential from two

aspects. First it helps the designers to optimize the size and

materials of the motion system. Second it is very helpful in

designing the large size motion systems. Structural

reinforcement will be difficult if the structure is built in larger

sizes. As the size of the structure increases, the structural

tensions increase extensively. Hence enlarging the motion

system's size will have an upper restriction. This explanation

shows the significance of structural forces in large scale

motion systems. Thus, the force workspace becomes

dominant in such systems. Large scale motion systems are

very important in flight simulation application since the

fidelity of simulation increases when the linear workspace

extend [5].

The present study, offers a method for determining the force

or load workspace based on solution of inverse dynamics on a

generalized trajectory. It is explained for the case of 3DoF

motion system to show the results in 3D plots. Then it is

extended to a 6DoF motion system. The method is applicable

for all manipulators of parallel type.

II. METHODOLOGY

The load or force workspace stands for the space within the

kinematic workspace in which the structural forces remain in

the allowable region, while the manipulator moves on every

possible trajectory. Since the determination of structural

forces in general are too complex, the buckling of legs, the

most crucial structural phenomenon, is determined to be the

criterion of structure failure. This assumption yields simple

form of mathematical formulations for structural resistance.

The structural forces are affected by static and dynamic loads

which in turn are functions of platform’s position/orientation

and linear/angular velocities and accelerations. Thus a variety

of platform’s movements has to be studied to make sure that

each leg is exposed to any possible force.

In this paper a generalized trajectory is defined in kinematic

workspace on which the legs are assured to experience any

possible force. Without loss of generality, the trajectory is

defined as follows for a 3DoF heave-roll-pitch platform for

flight simulator applications, due to getting the advantage of

demonstrating the results in 3D plots. The procedure is

extendable to any other types of parallel manipulators with up

to 6 degrees of freedom. The considered 3Dof motion system

A Generalized Method for Determining Load Workspace of

Spatial Parallel Mechanisms

A. Mahmoodi, M. Aminzadeh, M.B. Menhaj and M. SabzehParvar

S

2010 8th IEEE International Conference onControl and AutomationXiamen, China, June 9-11, 2010

ThCP4.5

978-1-4244-5196-8/10/$26.00 ©2010 IEEE 1360

as shown in Fig.1 consists of three electrical actuators which

are connected to a platform by universal joints and to the base

by ball joints. A vertically mounted pneumatic actuator with

ball joints at its both ends is to support the static weight of the

platform. It is located underneath the platform center with an

almost constant pressure supply. The center point of the

platform is constrained to move vertically by utilization of

three pairs of scissors which are synchronized by a gear

mechanism. The scissors also prevent the platform from

yawing. The platform is connected by a universal joint to the

top plate that connects the scissors’ upper ends. Thus with the

aid of the three electrical actuators, the platform can

accomplish heave, roll and pitch motion.

Fig.1. the prototype 3DoF heave-roll-pitch motion system

The kinematic workspace is determined by geometric

constraints which are the limits of legs’ length and the

allowable bending of the joints. Referring to Fig.2 these

constraints are formulated for a parallel motion system as

below:

, i = 1..3 (1)

in which ℜℜℜℜ is the Euler transformation matrix and is the

ith

upper joint vector with respect to the body coordinate

frame (see Fig. 2). The leg vector , can be stated as:

, i = 1..3 (2)

in which is the ith lower joint vector with respect to the

fixed coordinate frame and is the displacement vector of

the platform. The length of each leg can be obtained from:

(3)

The leg length constraints can be stated as:

maxmin LLLi⟨⟨

(4)

The joint angles can be obtained as:

, i = 1 … 3 (5)

in which is the unit vector of and is the reference unit

vector that is perpendicular to the platform for calculating

upper joint angles and to the base for calculating lower joint

angles. The joint angle constraint is as follows:

maxθθ ⟨i

(6)

Note that the magnitudes of θmax for the lower and upper

joints are different.

Fig. 2. Geometry of a leg in relation with fixed a body

coordinate frames

For the 3DoF sample motion system shown in Fig.1 with

characteristics listed in appendix A, we apply inequalities (4)

and (6) numerically to a plenty of points containing the

workspace, and gather those satisfy (4) and (6). The

workspace obtained in this way is plotted in Fig. 3.

Fig.3. Kinematic workspace of the sample 3DoF parallel

mechanism

The workspace is stated in terms of φ, θ and z coordinates

which represent roll, pitch and heave respectively.

To produce the generalized trajectory, we define 26 vectors

initiating at the center of the workspace i.e. the platform’s

center in neutral position, pointing toward its border in the

directions depicted in Fig.4. In this figure consider a cube

centered at the origin of the φ-θ-z coordinate frame with its

edges parallel to the axes; then plot 8 vectors from the origin

ThCP4.5

1361

to the vertices of the cube, 12 to the midpoint of the edges and

6 to the center of its faces.

Fig.4 26 auxiliary vectors for constructing the

generalized trajectory

For a cube with an edge length of 2, the vectors may be stated

as:

which are every possible triple permutation of the digits -1, 0

and 1, deducting the zero vector {0 0 0}T. Then draw lines

along these vectors till they intersect the boundary of the

workspace. This can be found through the solution of the

inverse kinematics iteratively while the platform moves along

the lines. In each step the satisfaction of the inequalities (4)

and (6) is investigated and the motion is stopped at the first

coming step in which at least one of the inequalities is no

longer held. The vectors obtained in this way for the sample

3DoF motion system are as:

The step sizes for angular and linear motions are chosen to be

0.01 radian and 0.01 meter respectively.

The 26 boundary points selected in this way are the main

corners of the workspace. Now, we would like to define a

trajectory in the workspace that involves a vast variety of

forces to be exerted to the legs. To do that, a sinusoidal

movement from each of these boundary points to another is

considered in such a way that the platform stops at each point

while it experiences its maximum allowable speed in each

portion of the whole path. The maximum allowable speed is

the maximum speed for a leg that can be provided by the

actuator. Fig. 5 shows the path and Fig. 6 depicts the

trajectory components of the path. It can be seen that the time

span between two adjacent points is variable depending on

speed limit of legs. To obtain these time spans, first arbitrary

amounts are chosen; the inverse kinematics is then solved in

all points of each time span and the maximum prismatic speed

among the legs is determined. Then this maximum speed is

compared to the allowable speed of a leg. The ratio of

maximum speed to the allowable speed will be the scale that

the arbitrarily-chosen time span should be multiplied by.

Thus the new time span can be determined in each piece of

path shown in Fig 5. The time spans obtained in this way,

with the data of the sample motion system presented in

appendix A are shown on the horizontal axis of the plot in Fig

6.

Fig. 5. The generalized trajectory in the workspace

Fig. 6. The continuous sinusoidal heave, roll and pitch

movements, connecting the 26 selected corners of the

workspace. The travel time between two adjacent points is

variable and is determined in such a way that at least one leg

reaches its speed limit.

Now the leg forces can be calculated by solution of inverse

dynamics of the mechanism which is extracted from

ThCP4.5

1362

reference [6], for the sample 3DoF heave-roll-pitch motion

system.

We can also compare the leg forces with the critical forces

that cause buckling in the legs. The buckling forces of

actuators will be variable since their lengths are changing. A

schematic picture of an actuator is depicted in Fig. 7. It

mainly consists of two parts: a cylinder and a piston. The

cylinder is supposed to be stiff enough to resist against

buckling. Thus it can be neglected and the portion of the

piston rod that remains out of the cylinder is considered to be

a cantilever beam under buckling. (See Fig. 7).

Fig. 7. Simplified model for leg buckling

The critical load causing initiation of the leg buckling can be

stated as [7]:

(7)

in which "E" is the Yang module of elasticity, "I" is the

bending moment of the rod, (which is for a circular

section rod with radius "r") and "l" is the length of that portion

of the piston rod that remains out of the cylinder. Referring to

Fig. 7, "l" can easily be obtained as: 4 in which

"L" is the total length of the leg (between two joint centers),

and "LS" is the length of the cylinder.

Implementing the above formulations with the system

specifications shown in appendix A, in a MATLAB program,

the legs' longitudinal forces corresponding to the generalized

trajectory are obtained. These forces together with the legs'

critical forces are depicted in Fig. 8.

For the sample system the actual forces are far below the

amount of critical forces i.e. the legs are fully much

overdesigned in structural viewpoint. Hence the load

workspace coincides with the kinematic workspace according

to the definition presented in the beginning of this section.

The procedure can be generalized to 6DoF parallel

manipulators in 6 dimensional space. However the 3D plots

would no longer be available. Alike Fig.4 consider a cube in a

six dimensional space. The 26 vectors there are converted to

36-1=728 vectors which are composed of independent

coordinates φ, θ, ψ, x, y and z which are associating with

Roll, pitch, yaw, surge, sway and heave motions respectively.

The vectors takes every possible hexed permutation of the

digits -1, 0 and 1, deducting the zero vector {0 0 0 0 0 0}T.

Fig. 8. The longitudinal forces in comparison with critical

forces of the sample 3DoF system corresponding to the

trajectory shown in Figs 5 and 6

The procedure is applied to a sample Stewart platform shown

in Fig.9 with characteristics listed in appendix B.

ThCP4.5

1363

Fig. 9. Sample 6-DoF Stewart platform for Determining its

load workspace

For the Stewart platform, the procedure requires a much

larger amount of calculations in comparison with the 3DoF

sample platform, for both large number of vectors and high

computational load due to solving inverse dynamics. Thus the

inverse dynamics solution is achieved by the method

presented in reference [8] which is much faster than the other

available methods.

A selected part of the platform's trajectory is shown below:

Fig. 8. A selected part of the Stewart platform's

generalized trajectory

the corresponding actuator forces and the actuator forces in

comparison with the legs' critical forces are demonstrated in

figures 9 and 10 respectively. Like the 3DoF motion system,

for the Stewart platform, the leg forces are far below the

critical forces i.e. it is much over designed for the payload.

Fig.9. actuator forces associated with the trajectory shown in

Fig.8

Fig.10. actuator forces in comparison with legs' critical forces

III. CONCLUSION

In this paper an important concept in design of robot

manipulators was addressed. Load workspace, as a space in

the kinematic workspace of the system where the structural

constraints permit the system to move within, was defined.

The workspace of interest is not only governed by the

kinematics of the system but also affected by dynamics since

the system is exposed to accelerations that impose forces to

the structure. Therefore, for a point to be within this

workspace, both the inverse kinematics of the system must be

solved for kinematical constraints not to be exceeded, and the

inverse dynamics must be solved so that the resulted forces do

not exceed critical forces imposed by the structure. To do that,

a generalized trajectory is defined for the manipulator in the

kinematic workspace that leads to a vast variety of forces

exerted to the legs. Thus the simulation of the platform's

dynamics while it tracks the trajectory, leads to maximum

amount of structural forces. The trajectory consists of straight

paths connecting selected points that are all over the

workspace boundary. Each path must be tracked by

sinusoidal movements, such that it stops at each selected

point and goes ahead towards the other, reaching its

ThCP4.5

1364

maximum allowable speed. Moving on such a trajectory, the

platform is exposed to high accelerations in various points of

the workspace. Hence the highest amount of static forces

together with the dynamic ones, are exerted to legs and

structure. The forces obtained in this way are used to

determine the force workspace. The buckling of legs, is

determined to be the criterion of structural failure. The critical

forces causing the initiation of legs' buckling vary as leg

lengths change. The leg forces obtained by tracking the

generalized trajectory, in each computational iteration are

compared to these critical forces. Every portion of the

trajectory in which the leg forces exceed the critical values,

must be excluded from the kinematic workspace. The

workspace obtained in this way will be the safe zone from the

structural view point.

The procedure is applied to a 3DoF heave-roll-pitch motion

platform and the workspace is illustrated in 3D plots. Then it

is applied to a 6DoF hexapod system as well. The

generalization of the method to parallel type manipulators is

straight forward. The method comes very effective in the

structural design of spatial motion systems and other

applications of parallel manipulators specially in large scale

ones to be used in flight simulators

REFERENCES

[1]. E. Flint and E. Anderson, “Multi-degree of freedom

parallel actuation system architectures for motion control”,

AIAA 2001-4750

[2]. D. Zhang, F. Xi, and C. Mechefske, “Kinematic analysis

of a spatial 3-DoF parallel manipulator," CCToMM

Symposium on Mechanisms, Machines, and Mechatronics

(2001)

[3]. T. Ukiana, M. Morita, K. Koseki, Y. Ohba, K. Fujii,

K. Arai, T., "Design of 3-DOF parallel mechanism with thin

plate for micro finger module in micro manipulation",

Tanikawa, IEEE/RSJ International Conference on Intelligent

Robots and System, 2002.

[4]. B.Dasgupta, T.S.Mruthyunjaya., “The Stewart Platform

Manipulator: A Review”, Mechanism and Machine Theory 35

(1), 15-40 (2000)

[5]. Telban, R. J., and Cardullo, F. M., Motion Cueing

Algorithm Development: Human-Centered Linear and

Nonlinear Algorithms. 2005, NASA CR-2005-213747,

NASA Langley Research Center, Hampton, VA.

[6]. M. Aminzadeh, A. Mahmoodi, and M.

Sabzehparavar, ”Dynamic analysis of a 3DoF Motion

Platform”. International Journal of Robotics: Theory and

Application, Vol. 1, No.1, IJR08, 2009.

[7]. Ansel. C.Ugural “Mechanics of Materials” Copyright,

2008, John Wiley & Sons

[8]. A. Mahmoodi, M.B. Menhaj, M. Sabzehparvar, " An

efficient method for solution of inverse dynamics of Stewart

platform"Aircraft Engineering and Aerospace Technology"

Volume: 81, Issue: 5, PP: 398 – 406, 2009

APPENDIX A

The characteristics of the sample 3DoF mechanism

Mass of total payload (platform and its payload)

Equivalent mass of scissors and gear set

Total payload matrix of moment inertia

Displacement vector of total payload center of mass

from pneumatic actuator upper joint

Position vector of the upper joint of ith leg relative to

the platform-fixed frame’s origin

The unit vector of ith leg directed from the lower

joint to the upper joint

Constant upward force produced by pneumatic

balancer

Prismatic viscous friction of pneumatic cylinder

,

APPENDIX B

The characteristics of the sample Stewart platform

parameter value

Platform and payload mass (kg) 100

Upper leg mass (kg) 1

Lower leg mass (kg) 6

Top radius (m) 0.45

Offset angle from 120 degree spacing on top (deg) Base radius (m) 0.4

Offset angle from 120 degree spacing on top (deg) Offset angle between top and bottom points (deg)

ThCP4.5

1365