Force isotropy of three limb spatial parallel manipulator

Force Isotropy of Three-Limb Spatial Parallel Manipulator, A. Chandrashekhar, G. Satish
Babu, Journal Impact Factor (2015): 8.8293 Calculated by GISI (www.jifactor.com)
www.iaeme.com/ijmet.asp 1 editor@iaeme.com
1
Asst. Prof., Mechanical Engineering Department,
The ICFAI Foundation for Higher Education, Hyderabad, India
2
Professor, Department of Mechanical Engineering,
JNTUH College of Engineering Hyderabad, INDIA
ABSTRACT
One of the criteria in optimal robot design is that the robot can achieve isotropic
configurations i.e. configurations where the condition number of its Jacobian matrix equals one. At
these configurations, the likelihood of error is equal in all directions and equal forces may be exerted
in all directions. A three-limb spatial parallel manipulator with two identical limbs is considered in
the present paper. Nearly isotropic configurations are identified by using the condition number
concept. A MATLAB code is developed and used for the analysis. The results are graphically
presented.
Keywords: Spatial Robotic Manipulator, Condition Number, Force Isotropy, Performance
Measures.
I. INTRODUCTION
J. Kenneth Salisbury and John J. Craig (1) are the first to apply the condition number concept
to mechanisms. They have used the condition number of the Jacobian Matrix as an optimization
criterion to obtain ideal dimensions for the mechanisms with the two revolute joints and Stanford
JPL Articulated hand. For the anticipated tasks of robot manipulators, it is important to locate its
workspace in the optimum location. Several measures of workspace are used for this purpose. The
size of reachable volume is an important performance measure. Another measure of workspace
quality is the accuracy with which forces can be exerted. It has been found by Salisbury and Craig
that at certain interior points in the workspace forces may be exerted with maximum accuracy. They
have shown that condition number of the force transformation JT it is possible to compare error
propagation with different manipulator configurations. Point in the workspace that minimize the
condition number of the Jacobian matrix are the best conditioned to minimize error propagation from
input torques to output forces. J P. Merlet (2) had reviewed the papers defining the various accuracy
indices and discussed the suitability of these indices for parallel robots. They have also examined the
concept of Jacobin and inverse Jacobin matrices which are essential for finding the positioning
accuracy of the end effector. They have proved that the condition number is dependent on the choice
FORCE ISOTROPY OF THREE-LIMB SPATIAL PARALLEL
MANIPULATOR
A. Chandrashekhar1
, G. Satish Babu2
Volume 6, Issue 6, June (2015), pp. 01-08
Article ID: 30120150606001
International Journal of Mechanical Engineering and Technology
© IAEME: http://www.iaeme.com/IJMET.asp
ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
IJMET
© I A E M E

of the matrix norm. The Two-norm and Frobenius-norm are used for this purpose. In order to
examine the consistency of a condition number and the corresponding maximal positioning errors a
set of poses of Gough platform are considered. Hollerbach and Suh (3) examined the methods for
resolving kinematic redundancies of manipulators by the effect on joint torques. Kazuhiro Kosuge
and Katsuhisa Furuta (7) used the inverse of the condition number of Jacobian Matrix as a measure
of kinematic controllability. For dynamic controllability another measure based on condition number
concept in the Jacobian matrix and the matrix concerned with the inertia term is considered by them.
Klein and Blaho (4) proposed several dexterity measures for an optimizing posture for a given end-
effector position and for designing the optimum link lengths of an arm. The four measures
Determinant, Condition number, Minimum singular value and Joint range availability are determined
for the entire reach of the planar three link revolute jointed manipulator.
Stephen L.chiu (5) has proposed various kinematic performance measures to quantify the
optimality of manipulator configurations. They have shown that the condition number is not a true
measure of accuracy, and that accuracy is inversely related to manipulability. They have explained
that manipulability must be obtained at the expense of accuracy and vice versa. Using the concept of
the velocity and force ellipsoids, they have shown the relationships between the transmission ratios
and manipulator performance characteristics. The transmission ratio has been viewed as a measure of
manipulability. They have shown that transmission ratio is also a direct measure of error
amplification. Kircanski (6) determined all the isotropic configurations of planar manipulators with
two, three and four degrees of freedom and 3 DOF spatial manipulator. The solutions are obtained in
the form of polynomials. The condition numbers are obtained as explicit analytic functions of joint
coordinates and link length ratios. Tanev and Stoyanov (7) applied the performance indices, the
dexterity index, manipulability, condition number and minimum singular value to a SCARA type
robot and compare the indices. Gosselin and Angeles (8) have proposed a new performance index for
the kinematic optimization of robotic manipulators. The index termed by them as global conditioning
index, (GCI) can be used to assess the distribution of condition number over the whole workspace.
The concept of global index is applicable to other local kinematic or dynamic indices. The GCI is
applied to the serial two-link manipulator, to a spherical three degree of freedom serial wrist, and to
three-degree-of-freedom parallel planar and spherical manipulators. Gosselin (9) proposed new
dexterity indices that can be applied to planar and spatial manipulators. These indices are based on
the condition number of the Jacobian matrix of the manipulators. As the existing indices are affected
by a scaling of the manipulator, a new formulation of these equations is proposed by them to avoid
the problem of dimensional dependence. Two dexterity indices, based on a redundant formulation of
the velocity equations and minimum number of parameters are presented by them.
II. ERROR PROPAGATION AND INDEX FOR MINIMIZATION OF ERROR
PROPAGATION
To optimize the workspace of a manipulator in an optimum location for an anticipated tasks,
plays an important role in designing a manipulator. The size of reachable volume is an important
performance measure. It has been found that only at certain points in workspace attain maximum
accuracy, with which the forces or torques can be exerted. These are the points where the condition
number of the Jacobian matrix is minimum and are best conditioned to minimize the error
propagation from input torques to output forces. In general, best conditioning occurs when the
condition number is equal to one. Such best conditioning points are called isotropic points. Hence
condition number is taken as a measure to minimize the error propagation.
A large dimension along a given axis of the manipulability polyhedron indicates a large
amplification error. It is therefore necessary to quantify this amplification factor. Let us consider the
linear system

J-1
DX = DΘ
Where J-1
is a n x n inverse kinematic Jacobian matrix. The error amplification factor in this
system expresses how a relative error in Θ gets multiplied and leads to a relative error in X. It will be
used as a performance index. We use a norm such that
||J-1
DX|| ≤ ||J-1
||||DX||
And obtain
||∆ ||
|| ||
≤ || |||| ||
||∆ ||
|| ||
The error amplification factor, called the condition number k, is therefore defined as
K (J) = k (J-1
) = ||J-1
||||J||
The condition number is thus dependent on the choice of the matrix norm. The norms which
are widely used are:
The Two-Norm: the square root of the largest eigenvalue of matrix A –T
A-1
: the condition
number of A-1
is thus the square root of the ratio between the largest and the smallest eigenvalues of
A –T
A-1
.
The Euclidean norm: which is also called as Frobenius norm defined by the m x n matrix A
as ||A||= ∑ ∑ | | , if li denotes the Eigen values of A –T
A-1
, then the condition number is the
ratio between ∑ and Πλi. For a weighted Frobenius norm AT
A is substituted with AT
WA. Where
W is the weight of matrix.
III. THREE LIMB SPATIAL PARALLEL MANIPULATOR WITH TWO IDENTICAL
LIMBS
Inverse Kinematic Analysis of Spatial Parallel Manipulator
The inverse kinematics problem involves mapping a known pose of the output platform of
the mechanism to a set of input joint variables that will achieve that pose. The inverse kinematics
problem of the parallel manipulator can be described in closed form.
Fig.1. Kinematic model of spatial parallel manipulator

A kinematic model of the manipulator is shown in Fig.1 Vertices of the output platform are
denoted as platform joints Pi (i = 1, 2, 3), and vertices of the base platform are denoted as bi, (i = 1,
2, 3). A fixed global reference system R: O - xyz is located at the center of the side with the z
axis normal to tile base plate and the y axis directed along . Another reference frame, called the
top frame ′
∶ ′
− ′ ′ ′
, is located at the center of the side P1, P2. The z' axis is perpendicular to the
output platform and y’ axis directed alongP1, P2. The length of link for each leg is denoted as L,
where Pi Bi, = L, i = 1, 2, 3. What we should note that, in some case, the length of the link P3B3 can
be different from that ofP1B1, andP2B2. For this analysis, the pose of the moving platform is
considered known, and the position is given by the position vector ℜ
′)(O and the orientation is given
by a matrix Q.
Then there are
( ′
) = ( ) (1)
Where x = 0 and
! = "
cos φ 0 sin φ
0 1 0
− sin φ 0 cos φ
* (2)
Where the angle φ is the rotational degree of freedom of the output platform with respect to y axis.
The coordinate of the point Pi in the frame R’ can be described by the vector (+ ) ′ (i = 1, 2. 3), and
(+ ) ′ = ,
0
−-
0
. , (+ ) ′ = ,
0
-
0
. , (+0) ′ = 1
−-
0
0
2 (3)
Vectors ( ) ′ (i = I. 2,3) will be defined as the position vectors of base joints in frame R, and
( ) ′ = ,
0
−
3
. , ( ) ′ = ,
0
3
. , ( 0) ′ = ,
−
0
30
. (4)
The vector (i = 1, 2, 3) in frame 0 - xyz can be written as
(+ ) = !(+ ) ′ + ( ′
) (5)
Then the inverse kinematics of the parallel manipulator can be solved by writing following
constraint equation:
‖[7 − ] ‖ = 9 (6)
(Where i=1, 2 ,3)
Hence, for a given manipulator and for prescribed values of the position and orientation of
the platform, the required actuator inputs can be directly computed from (6). that is
= ± :L − (−r + y + R) + z (7)
= ± :L − (r + y − R) + z (8)
0 = ± :L − (−r cos φ + R) − y + r sin φ + z (9)

From (7)-(9), we can see that there are eight inverse kinematics solutions for a given pose of
the parallel manipulator. To obtain the inverse configuration as shown in Fig. 4.1, each one of the
signs "±" in (7)-(9) should be "+."
IV. JACOBIAN ANALYSIS OF PARALLEL MANIPULATORS
The Jacobian analysis of parallel manipulators is a much more difficult problem than that of
serial manipulators because there are many links that form a number of closed loops. An important
limitation of parallel manipulator is that singular configurations may exist within its work space
where the manipulator gains one or more degrees of freedom and therefore loses its stiffness
completely. The Jacobian matrix is converted into two matrices: one associated with the direct
kinematics and the other with the inverse kinematics. Depending on which matrix is singular, a
closed-loop mechanism may be at a direct kinematic singular configuration, an inverse kinematic
singular configuration, or both.
A parallel manipulator is one which consists of a moving platform and a fixed base connected
by several limbs. The moving platform serves as the end effector. Because of the closed-loop
construction, some of the joints can be controlled independently and the other joints are passive.
Generally, the number of actuated joints should be equal to the number of degrees of freedom of the
manipulator.
Let the actuated joint variables be denoted by a vector q and the location of the moving
platform be described by a vector x. Then the kinematic constraints imposed by the limbs can be
written in the general form [20].
F(x, q) = 0, (10)
Differentiating Eq. (7) with respect to time, the following relationship between the input joint
rates and end-effector output velocity can be written as
@ABC = @DDC (11)
Where
@B =
∂∂∂∂E
∂∂∂∂B
and @D = −
∂∂∂∂E
∂∂∂∂D
The derivation above leads to two separate Jacobian matrices. Hence the overall Jacobian
matrix, J, can be written as
DC = BC (12)
Where J = Jq
-1
Jx. We note that the Jacobian matrix defined in Eq. (3) for a parallel
manipulator corresponds to the inverse Jacobian of a serial manipulator.
Jacobian matrix of 2URP and 1 RRP manipulator is obtained by differentiating Eq. (6).
= FG FH = I J (13)
J = "
− 0 0
0 − 0
0 0 - KLMφ + − 0
* (14)
I = "
−- + + − 0
- + − − 0
- KLMφ + − 0 - KLMφ + -( − 0)NOKφ
* (15)

V. RESULTS
A MATLAB code is generated for the inverse kinematic analysis, Jacobian matrix and
Condition number of the three-limb spatial parallel manipulator with two identical limbs. After
verifying the results of various kinematic structures and different configurations in each structure,
the optimal structure is identified.
Figure.2. Flow chart for calculating force isotropy index
Figure.3. Error propagation index vs Orientation angle

Figure.4. Error Propagation index Vs Horizontal reach
VI. CONCLUSION
The three-limb spatial parallel manipulator has been analyzed for its force isotropy, using the
condition number concept. The objective of this paper is to identify the isotropic configuration of the
manipulator. After verifying several configurations in different structures, we have obtained nearly-
isotropic configurations.
Our future work extends to find the exact isotropic configurations, by using optimization
techniques such as Genetic Algorithm.
REFERENCES
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5. Stephen L.Chiu. “Task Compatibility of Manipulator Postures,” The international Journal of
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Robotic Manipulators,” ASME Journal of Mechanical Design, Vol. 113, pp 220-226,
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9. Clement M. Gosselin. “The optimum design of robotic manipulators using dexterity indices,”
Robotics and Autonomous Systems 9, pp 213-226 (1992).

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Force isotropy of three limb spatial parallel manipulator

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