DOI : 10.17577/IJERTV2IS2295
- Open Access
- Total Downloads : 1031
- Authors : Kassa Zelalem Zenebe
- Paper ID : IJERTV2IS2295
- Volume & Issue : Volume 02, Issue 02 (February 2013)
- Published (First Online): 28-02-2013
- ISSN (Online) : 2278-0181
- Publisher Name : IJERT
- License:
This work is licensed under a Creative Commons Attribution 4.0 International License
Formulation of a Generalized Jacobian and Inverse Acceleration Analysis of a Linear Delta Manipulator Based on Reciprocal Screws Method
Kassa Zelalem Zenebe
School of Mechanical Engineering, Tianjin University of Technology and Education, Tianjin, 300222, China
Abstract
This paper presents the inverse velocity and acceleration analyses of a three DOF Delta linear parallel manipulators mainly focusing on screw algebra. First the generalized Jacobian of the Delta parallel manipulator is derived using reciprocal screws technique. Geometrical observation method is employed to determine the reciprocal screw systems. This generalized Jacobian will be used to formulate the acceleration of the platform. For the acceleration analysis a new Hessian matrix of serial kinematic chains (or limbs) is developed in an explicit and compact form using Lie brackets. This idea is then extended to cover parallel manipulators.
-
Introduction
A parallel manipulator is a mechanical system that uses several computer-controlled serial chains to support a single platform, or end-effector. Perhaps, the best known parallel manipulator is formed from six linear actuators that support a movable base for devices such as flight simulators. This device is called a Stewart platform or the Gough-Stewart platform in recognition of the engineers who first designed and used them [1].
Parallel manipulators have a kinematic structure that has some advantages over serial manipulators such as rigidity, precision, velocity and acceleration. Nowadays, these characteristics are highly demanded in the manufacturing of aeronautical components and for the automotive industry. Parallel manipulators have found a place in applications such as flight simulators or pointing devices, but they have not jumped into industrial production yet. one of the basic reason is the kinematic, stiffness, singularity and dynamics analysis of a parallel manipulators is not an easy task.
Velocity, accuracy stiffness and dynamics are the important performance factors that should essentially be considered in design of parallel manipulators, particularly in the many circumstances where high speed, high precision and high rigidity are basic requirements. The
common manipulation required for velocity, accuracy, stiffness modelling is to formulate a specific linear map between two vector spaces at a given configuration. Huang [2] has recently developed a mathematical framework to unify all these three. He propose a general approach for Jacobian analysis of lower mobility manipulators by simultaneously taking into account both instantaneous permitted and restricted motions of the end-effector. This Jacobian is known as generalized Jacobian, unlike an overall Jacobian it incorporate restricted motions of the end-effector.
Jacobian analysis has been a focus of research area for many years as far as parallel manipulators are concerned. For this reason there are so many research works available on the subject. Jacobian analysis for different parallel manipulators has been conducted using different approaches. Among these approaches are direct differentiation method, the kinematic influence coefficient method and screw- based method. A Jacobian analysis of 6-DOF parallel manipulator was presented using a screw- based method [1] [3]. Screw-based method is more powerful and easily gives deep insight into the topological structure of robotic system and this approach is well outlined by Tian Huang [2]. A paper on inverse, direct, and intermediate Jacobians of a delta robot was published by Lopez using a direct differentiation method [4].
In relation to acceleration analysis, screw theory based approaches could potentially be the most powerful method for acceleration analysis. In order to overcome the difficulty of expressing the twist derivatives in a screw form, a novel term named the accelerator or acceleration motor was proposed and employed for the acceleration analysis of serial and parallel kinematic chains [5] . However, the terms associated with the second derivatives in the acceleration equations can only be written in a lengthy form of Lie brackets rather than in a compact form of the Hessian matrix [5] [6].
-
Delta robot
The delta robot is a parallel robot, i.e. it consists of multiple kinematic chains interconnecting the base with the end-effector. The key concept of the delta
robot is the use of parallelograms. These parallelograms restrict the movement of the end
since the
Ai Bi is the vector along the actuated joint,
platform to pure translation. mostly the robot's base
then
is mounted above the workspace and all
Ai Bi
Oo ' o 'Ci OAi BiCi
(2)
the actuators are located on it. From the base, three middle jointed arms extend. The ends of these arms are connected to a small triangular platform. This type of configuration is widely applicable as a sorting and collating in various types of packaging and food industry [7]. Actuation of the input links will move the triangular platform along the X, Y or Z direction. Actuation can be done with linear or rotational actuators, with or without reductions.
Nowadays linearly actuated delta robots are used in the manufacturing industry for high-speed machining centres.
-
Position Equation
A delta robot employs three parallelograms to completely restrain the orientation of the moving platform with only three purely translational degrees of freedom. The description of the kinematics chain is shown in figure 1. A reference coordinate system O xyz is attached to the centre
of the base platform with X and Y axes lying on the fixed plane and Z axis pointing up vertically.
Where all the vectors in Eq. (2) are expressed in the reference coordinate system O xyz .
-
Screw and reciprocal screw
The concept of screw and reciprocal screws was first studied by Ball [8]. The instantaneous motion of a rigid body can be represented by a six- component vector called twist, $t . Similarly, all the forces acting on a rigid body can be described by another six-component vector called wrench,
$w . In this regard, the first three components of a twist represent the angular velocity and the last three components represent the linear velocity of a point in the rigid body which is instantaneously coincident with the origin of the reference frame. On the other hand, the first three components of a wrench represent the resultant force and the last three components represent the resultant moment about the origin of the reference frame.
Although twist and wrench have different physical meanings, they have the same mathematical representation called screws. A unit
screw $
is defined by a pair of vectors:
$ =
s0
s
s s
(3)
T
T
S1, S2 , S3 , S4 , S5 , S6
where s is a unit vector along the direction of the screw axis, s0 is the position vector of any point on the screw axis with respect to a reference frame, and is called the pitch [1]. A screw of intensity
can be written as $
$ . Two screws $1 and
$2 are said to be reciprocal to each other if their orthogonal product equals to zero.
$1 $2 0
(4)
Figure 1. Description of the kinematic chain
Another coordinate system Ai xi yi zi is attached to the fixed base at point A , such that x axis is in
If one screw is twist and the other is wrench, the physical meaning for reciprocity between these two screws is tht the instantaneous work for the wrench along the twist is zero.
A screw system is a span of ki ( ki 6 )
i i
linearly independent screws, and is often called a
line with line OA and zi axis is parallel to z axis.
ki system. For this screw system, any ki
linearly
the constant angle i is measured from the x axis to xi . some variables associated with the ith limb are defined in Figure 1.
Therefore the loop-closure equation of each limb
can be written as;
independent screws can be a basis of the system. The reciprocal screw system for the given one is a
6 ki system in which any one screw is reciprocal to all the screws in the given system. Similarly, any
6 ki linearly independent screws form a basis of the reciprocal system. Hence we only need to find
OAi
Ai Bi
BiCi
Oo '
o 'Ci
(1)
6 ki
convenient linearly independent screws to
achieved by mechanical joints.
represent the reciprocal system.
As described above the twist space form a 6- dimensional vector space, which is composed of a pair of complementary subspaces called The twist subspace of permissions and The twist subspace of restrictions. The twist subspace of permissions is a
ki -dimensional subspace of the twist space
spanned by the twists permitted by the system
constraints with
$ta
being its element. The twist
subspace of restrictions is a 6
ki dimensional
subspace of the twist space spanned by the twists restricted by the system constraints with $tc being its element [4].
The wrenche space also form a 6-dimensional
vector space, which is composed of a pair of complementary subspaces called The wrenche subspace of actuations and The wrenche subspace of constraints. The wrenche subspace of actuations is a ki -dimensional subspace of the wrenche space spanned by the wrenches of actuations generated
by ki
independent actuation forces and/or
moments with $wa
being its element. The wrench
subspace of constraints is a 6 ki dimensional subspace of the wrenche space spanned by the wrenches of constraints generated by 6 ki independent joint reaction forces and/or moments with $wc being its element.
From the above definitions we can see that:
Figure.2 Schematic representation of a single limb of linearly actuated delta robot.
For example, rotation and translation can be obtained by revolute and prismatic joints respectively. In robot manipulator design, revolute and prismatic joints are widely used. In fact, most
$ta $tc
$wc
$wa
0 and
0
(5)
of the other joints can be conceived as the combination of several revolute and prismatic joints. For example, a universal joint can be
Hence if a screw system of twists represents the degrees of freedom (DOF) of a rigid body, the reciprocal system of wrenches is the constraint forces acting on it. The dual proposition also holds: When a screw system of wrenches act on a rigid body, then the reciprocal system of twists is the DOF of the body. Therefore, the reciprocity property makes it possible to obtain motion information from the corresponding constraint counterpart and vice versa.
-
Determination of the basis Elements of the Four Subspaces
This section employ geometrical observation method to determine the reciprocal screw systems of linearly actuated delta robot. we only needs to implement three simple observations to fully determine the basis of a reciprocal screws of a screw system [9]. Simply by inspecting the joint axes and applying these observations, one can easily obtain a basis of the reciprocal screw system. Relative motion between two rigid bodies can be
substituted by two revolute joints whose axis intersects with each other perpendicularly and a spherical joint can be represented by three mutually perpendicular revolute joints with the axes of the joints passing through the center of the sphere.
At the same time, common constraints in statics are pure force or moment. For both the motion and constraint, the screw associated with rotation or a pure force is a zero pitch screw( =0) which we call a line vector, while the screw associated with translation or a pure moment is an infinite pitch screw which we call a couple vector.
-
The three basic observations
The following three basic observations will help us to find the basis of the reciprocal screws of a screw system. The proofs for the observations are discussed by Jianguo [9].
Observation 1: Two line vectors are reciprocal to each other if and only if they are coplanar.
Observation 2: Two couple vectors are always reciprocal to each other.
Observation 3: A line vector is reciprocal to a couple vector if and only if they are perpendicular to each other.
$ ci
ta,Gi
ts2,i si (12)
si
where si is coincident with the axis of the revolute
-
The basis Elements of the Four Subspaces
A single limb of linearly actuated Delta robot with 3-DOF translational motion capabilities is
joint, and is also the normal vector of parallelogram
DEFG i . Within the scope of velocity analysis,
the instantaneous twist of the platform due only to the twist of the parallelogram joint can be given as
shown in figure 2. It consists of a base, a platform
$ta,3,i
3,i $ta,D
3,i $ta,G
and three identical limbs made of prismatic- i i
revolute-parallelogram-revolute(PRPaR) joints. In
$
$
(13)
each limb, the axis of the prismatic joint is parallel to the z-axis of the base plate, the axes of two
3,i ta,Ei
3,i ta,Fi
revolute joints are parallel to each other, and the
where
3,i
represents the angular velocity of the
axes of the revolute joints in the parallelogram joint are normal to the parallelogram DEFG i , Since
revolute joint. Substituting Eq.(9) through Eq.(12) into Eq. (13) gives
the parallelogram joint can be visualized as a 1- DOF compound joint [2]. We have 4-screw system
$ta,3,i
3,ib$ ta,3,i
3,ib i
(14)
i
i
e
e
single limb i.e. ki 4 . The joint axes are arranged 0
such that, s2,i
si , and si
b .
Hence the unit screw becomes,
Hence, the unit screws of permissions, e
$ta, ja,i ( ja
1, 2, 3, 4)
in the ith limb( i
1, 2,3 ) can
$ta,3,i i
b
b
s
s
0
(15)
be generated as follows:
Since a prismatic joint is a couple vector
where ei
i . Therefore,
$ ta,3,i
can be
0
0
$ta,1,i s1i
c
c
a
a
i
i
(6)
considered to be the unit screw of permissions associated with the 1-DOF parallelogram joint of limb i.
Note that except the actuated joint the rest three joints are revolute joints (i.e. line vector) therefore the unit screws about a reference point o' due to the two vectors s2,i will give us
Next the two unit wrench of constraints, $ wc ,k ,i ( kc 1, 2 ), which are orthogonal to all $ ta , j ,i , can then be identified using the above observations:
$ta,2,i
(ci
bb )
i
i
s2,i
s2,i
(7)
Since the unit screw of permissions are composed of one couple vector and three line vectors we can take two linearly independent
and
$ta,4,i
ci s2,i
. (8)
couple vectors which are perpendicular to
s2i .
$ ta,3,i
s2,i
is a unit screw of permissions of a 1-DOF
Note that by observation 2 any couple vector is a reciprocal of any other couple vector. So
parallelogram joint. it will be the resultant unit screw of permissions associated with the revolute
joints of the ith limb at Di, Ei , Fi , and Gi
$ wc,1,i
0
, (16)
si
0
$ wc,2,i s s
(17)
$ ci
ts2,i
bbi si
, (9)
i 2i
ta,Di s
Locking sequentially the
ka th
( ka
1, 2,, 4 )
i
c ts
bb s
joint, the unit wrench $
wa,ka ,i
, which is orthogonal
$ta,Ei
i 2,i i i
a
a
s
, (10)
to $ ta , j ,i ( ja
1, 2,, 4 ,
ja ka ) and dual to
a
a
$ta,F
i
ci ts2,i si
, (11)
$ ta ,k ,i , can be identified as by the three basic observations mentioned above, take a line vector
i si
along a vector
b , from observation 3 this line
i
i
vector is not perpendicular to the couple vector s1i
that means it will be a dual to s1i . from observation 2 it will be a reciprocal to any coplanar line vector. From observation 3 this same line vector is
ki
ja 1
a, ja ,i $ ta, ja ,i
6 ki
jc 1
c, jc ,i $ tc, jc ,i
(22)
perpendicular to ei . Hence
$ wa,1,i
b
b
i will
where
$ ta, ja ,i
and
$ tc, jc ,i
are unit screw of
ja th
b
b
c
c
i
i
i elements of the twist subspace of permissions and
be a dual to $ ta,1,i and orthogonal to $ta,2,i , $ta,3,i ,
jc th elements of the twist subspace of restrictions
c
c
and $ta,4,i . Like wise
respectively.
a, ja ,i
and
c, j ,i
are their
respective intensity within the ith limb.
$ e3,i
, (18)
since
$ wa,1,i
is the unit wrench of actuations
wa,2,i
ci e3,i
associated with the actuated joint, labelled
ja 1 , in the
ith ( i
1, 2,3 ) limb. Note that
$ wa,3,i
s2,i
ci s2,i
, (19)
$ wa,1,i
is dual to $
ta,1,i
but orthogonal to $
ta, ja ,i
( ja
1, 2,, 4 , ja
1 ) and
$ tc, jc ,i
( jc
1, 2 ).
$ wa,4,i
e4,i
ci bbi
e4,i
(20)
Similarly, let
$ wc,kc ,i
be the
kc th
unit wrench
of constraints in the ith limb. Also, note that
where e3,i
is a unit vector which is perpendicular
$ wc,kc ,i
is orthogonal to
$ ta, ja ,i
( ja
1, 2,, 4 )
to both s1,i and ei passing through point Ci and
and $
( j 1, 2 , j k ) and dual to
e is a unit vector which is perpendicular to both
tc, jc ,i c c c
4,i
s1,i
and
ei passing through point
Bi . Hence
$ tc,kc ,i
. Thus, Taking inner products on both sides
e3,i e4,i s1,i ei .
of equation (22) with
$ wa,1,i
and
$ wc,kc,i ,
Finally, with the
kc th
( kc
1, 2 ) constraints
respectively gives
provided by
$ wc,k ,i
being sequentially released,
$ T $
$ T $
, i 1, 2,3
(23)
c wa,1,i t a,1,i wa,1,i ta,1,i
the unit screw of restrictions, $ tc, j ,i
( jc
1, 2 ),
$ T
$
$ T $
, k 1, 2 , (24)
c
a
a
which is orthogonal to $ wa,k ,i ( ka 1, 2,, 4 ) and
wc,kc ,i t c,kc ,i wc,kc ,i tc,kc ,i c
$ ( k 1, 2 , k j ) except for $
, can
Therefore Eq.(21) and Eq.(22) can be written as
wc ,kc ,i c c c
wc, jc ,i
J x $t J (25)
be identified as
$ tc,1,i
ci si si
, $
tc,2,i
ci bi bi
J x J xa , J
J
J
xc
J a 0
0 J c
-
-
Jacobian Analyses
Jacobian analysis relates the linear map between actuator rates and velocity twist of the end-
The overall Jacobian matrix will be
J J-1J x
(26)
effector.. The Jacobian matrix ( J ) relates the two by:
Hence
J$t q
J$t q (21)
Where, q
(qT 0)T q q q
q T
where $t is the twist of the platform and q is the
a a a1,1
a1,2
a1,3
actuated joints rate. According Huang [2] for parallel manipulators the twist, $t , of the platform to can be expressed as a linear combination of the basis elements of the twist space associated with
Therefore substituting all the screws components of the generalized Jacobian for a linearly delta robot will be;
the ith limb, for i 1, 2,3
c b T
T
T
$t $ta $tc $ta,i
$tc,i
J xa
T
b
b
1
2 2
2 2
bT
1 1 b1 s1,1
T
T
c2 b , J a
bT
0
2
2
3
3
s1,2
b
b
T T
b
b
3
3
3 c3
0 bT s1,3
Jxc
03 1 03 1 03 1 03 1 03 1 03 1
$Li
a,1,i $ ta,1,i
a,2,i $ ta,2,i
c,6 n ,i $ tc,6 n ,i
T
T
s1 s1
s21 s2
s2 s22
s3 s3
s23
$
$
$
i i
i i
1 0 a,2,i ta,2,i a,3,i ta,3,i c,6 ni ,i tc,6 ni ,i
i i i i
i i i i
s1 s21
1
c,5 n ,i $ tc,5 n ,i c,6 n ,i $ tc,6 n ,i
a
a
c
c
J c s2
s22
1
2r , 2 , 2 a, j ,i , and
2 c, j ,i denote,
a
a
0 s3
s23
respectively, the variation of r , , a, j ,i , and
From Eq. (26) we can see that the generalized
c, j ,i . The bracket in $Li
denotes the Lie
c
c
Jacobian matrix of a linearly actuated 3-DOF delta parallel manipulator will be a 9×6 matrix.
product. From the properties of the Lie product,
$Li can also be written as
-
Acceleration Analysis
-
Acceleration analysis of a limb
where
$Li
$a,i
$ac,i
$c,i
(28)
The formulae for expressing the angular acceleration, and the acceleration of an arbitrary
$a,i a,1,i a,2,i
$ta,1,i
$ta,2,i
…
point of the end effector of a serial chain have been developed by Rico and Duffy [6] [8]. Those formulae contained the direction and moment parts of the screws representing the kinematic pairs that
a,1,i a,ki ,i
a,2,i a,3,i
a,2,i a,ki ,i
$ta,1,i
$ta,2,i
$ta,2,i
$ta,k ,i
i
i
$ta,3,i
i
i
$ta,k ,i
…
…
join the end effector with the body, or link, employed as reference system. Here, the coordinate system must have the arbitrary point chosen as the
a,k 1,i a,k ,i $ta,ki 1,i $ta,ki ,i
i i
i i
origin. That same paper shows that if the angular velocity of the end effector, with respect to the
$ac,i a,1,i c,1,i
$ta,1,i
$tc,1,i
reference system, vanishes, then the acceleration of
…
a,1,i c,6 ki ,i
$ta,1,i
$tc,6
ki ,i
an arbitrary point in the end effector can be
expressed in a simpler form. In fact, the
a,2,i c,1,i $ta,2,i
$tc,1,i
…
acceleration state of the rigid body can be written
a,2,i c,6
ki ,i
$ta,2,i
$tc,6
ki ,i
…
in terms of the screws, representing the kinematic
pairs which join the end effector to the reference
i
i
a,ki ,i
c,1,i
$ta,k ,i $tc,1,i
…
i
i
frame, together with their Lie products.
a,k ,i c,6
k ,i
$ta,k ,i $tc,6
ki ,i
As outlined in a recent work by Liu and Huang [5] acceleration analysis of a parallel manipulator can be best performed by the concept of a Lie bracket.
i i
$c,i c,1,i c,2,i $
tc,1,i $tc,2,i
…
Taking the variation ofEq. (22) and expressing the derivatives of screws in the form of Lie brackets
c,1,i c,6 ki ,i
$tc,1,i
$tc,6
ki ,i
yields
c,2,i c,3,i
$tc,2,i
$tc,3,i
…
i
i
A Ji 2 i
$Li , i
1, 2,3
(27)
c,2,i c,6 ki ,i
$tc,2,i $tc,6 k ,i …
c,5 k ,i c,6 k ,i $tc,5 k ,i $tc,6 k ,i
i i
i i
Where A 2r r T
T
2 T ,
i i
Then equation (27) can be written as
T
T
2 2 T 2 T
A Ji 2 i
T i Hi i , i
1, 2,3
(29)
i a,i c,i
Where Hi
Ha,i Hac,i ,
i
i
T
T
2 a,i 2 a,1,i 2 a,2,i 2 a,n ,i
0 Hc,i
T
T
2 2 2 2
c,i c,1,i c,2,i c,6 ni ,i
0 $ta,1,i $ta,2,i
$ta,1,i $ta,3,i
$ta,1,i $ta,k ,i
taking inner products on both sides of Eq. (30) with
0 0 $ta,2,i $ta,3,i
i
i
i
$ta,2,i $ta,k ,i
$ wa,1,i
and $
wc,kc ,i
respectively [3], to give
Ha,i
T
0 0 0 $ta,k 1,i $ta,k ,i
JA q
q Hq
(31)
i i
0 0 0 0
a
a
Where q = qT
T
0 , qa
qa,1,1
qa,1,2
T
T
qa,1,3 ,
qa,1,i
is the acceleration of the actuated joint
i
i
0 $tc,1,i $tc,2,i
$tc,1,i $tc,3,i
$tc,1,i $tc,6 k ,i
numbered 1 in the
ith
limb. H
RN N N (
i
i
0 0 $tc,2,i $tc,3,i $tc,2,i $tc,6 k ,i f
Hc,i
(N f
i 1 6
ki )
is known as the Hessian
0 0 0
0 0 0
$tc,5 k ,i $tc,6 k ,i
i i
i i
i 1
i 1
0
matrix of an f-DOF parallel manipulator. It is a three dimensional matrix composed of
a(c)
a(c)
Ha R
f N N
and
Hc R
f 6 ki
N N . The
a
a
$ $
$ $
expression of U th
layer of
Ha(c) is given as
Hac,i
ta,1,i ta,2,i ta,1,i tc,6 ki ,i
follow: the U th
layer of Ha (Ua
1, 2,… f ) ) and
$ $
$ $
the
U th
layer of
H ( U
1, 2,,6 k ,
ta,ki ,i tc,1,i ta,ki ,i tc,6 ki ,i
Hi is known as the Hessian matrix of the ith limb. It is a three-dimensional matrix having six layers,
,
Hn,i
c c c i
6
6
f
f
i 1 ki )
1 *
each containing a 6 6 upper triangular matrix
where [* *] represent the Lie product between two elements of the Lie algebra e(3). The constituent
0 m1 $ta,1,i $ta,2,i
m1 $ta,1,i $ta,3,i
m1 $ta,1,i $ta,6 ki,i
parts of Hi , Ha,i R6 ki
ki , Hac,i R6 ki
6 ki
0 0 m1 $ta,2,i $ta,3,i m1 $ta,2,i $ta,6 ki,i
6 6 k 6 k
2
and
Hc,i R
i i , are also three-
0 0 0
m1
$ta,5 ki,i $ta,6 ki,i
dimensional matrices having six layers.
0 0 0 0
In acceleration analysis where only ideal motions of the platform are considered, replacements can be made in Eq. (29) such that
where
n a, m1 m2
m
$ wa,1,i
$ ta,1,i
$ wc,k ,i
1
Ji LP
T
T
m T m
2 =Ji LP
T T T T
0T T ,
n c, 1 c 1 2
i a,i c,i i a,i
m2 $tc,kc ,i
2 2 T
T T
T
T
L
L
2 T
i
a,i
c,i
T
i a,i 0
T
where,
Ji LP
Ji 1J
Ja,i J ,
J L
A
Thus
2r
T
r
2 T
A v
v T T
T T
c,i
L
L
Ja,i
$ wa,1,i
$ wa,1,i $ta,1,i
i i
i i
T
T
,
i
i
A Jii
THii (i
1, 2, 3)
(30)
T
$ wa,ki ,i
$ wa,k ,i $ta,k ,i
and A becomes the accelerator of an ki
DOF limb
T T
with v and being the linear acceleration of the
$ wc,1,i
J L
$ wc,1,i $tc,1,i
reference point and the angular acceleration of the
c,i T T
$ wc,6 k ,i $ wc,6 ni,i $tc,6 k ,i
platform respectively.
-
Acceleration analysis of a parallel manipulator
Following the same approach, the acceleration equation of a parallel manipulator is obtained by
i i
Where, J+ is the pseudo-inverse of J and the superscript L simply identifies that the matrix applies to a single limb.
Hence the inverse acceleration equation of a parallel manipulator will be derived from Eq. (31)
Lets take the following numeric values for the dimensions of a linearly actuated delta robot, where c=0.2m, b=0.5m, t=0.05m, a=0.4m
q
JA $T JTHJ$
(32)
1 0, 2 2 3,
3 4 3 , and the
t t
t t
where q
T
q
q
a,i
T
0 q
qa1,1
qa1,2
T
qa1,3
platform is assumed to have the following velocity and acceleration behaviour;
The Hessian matrix, H , of the manipulator can be found by substituting the expressions just found
ax =amaxcos( 2 t T )
ay =amaxsin( 2 t T ) ,
for
$ ta , j ,i
( ja
1, 2,, 4 ),
$ tc ,1,i and $ tc ,2,i
and
amax
1m s , vz =1m s ,
az =0 , T 5s ,
a
a
J
J
i
i
Jacobians LP and J into the expressions given
vx =Tamaxsin( 2 t T )/2
above and then into Eq.(32) to give layers of H as follows
JLP T L JLP
vy =Tamaxcos( 2 t T )/2 .
Where
Ha,i
i a,i i
b Ts
, i 1, 2,3
When the platform of the manipulator moves according to the preceding rules, in Fig. 3 and Fig.
i 1,i
6 6
6 6
Ma1,i 0
4 the velocity and the acceleration of the actuated joints versus time can be evaluated using the
La,i
0 Ma2,i ,
proposed approach. These results are shown in Fig. 5 and Fig. 6.
0 T s s
0 T s s 3
bi 1,i
M
2,i
bi 1,i
2,i
a1,i
0 0 0
T
b bi bi 2
T
T
2 4
velocity,m/s
velocity,m/s
1
bi ei si 0 0
Ma2,i
H JLP
0 0
0 0
0 0 4 2
T
T
T
T
-
JLP , i
-1
-2
1, 2,3
c1,i i
c1,i i -3
vx vy vz
vx vy vz
J LP L J LP
0 1 2 3 4 5
Hc2,i
i c2,i i
s s T b
i 1, 2,3
time(s)
1 21 i
Figure 3. Velocity of the moving platform
Lc1,i
0 0 0 0 0 Mc1,i 1
Lc2,i
0 0 0 0 Mc2,i 0
0.5
-
0 T s
b 0
T s
T
acc,m/s2
acc,m/s2
b 0 0
c1,i bi
2,i i
bi 2,i i 0
Mc2,i
0 T 0
T 0 0 T
-0.5
where, si
s2,i
ax ay az
ax ay az
H a,i ( Hc1,i &Hc2,i ) represents the layer of H ( H ).
Ka th
( Kc th )
-1
0 1 2 3 4 5
time(s)
a c
Figure 4. Acceleration of the moving
Numerical Example
platform
i=1 i=2 i=3
i=1 i=2 i=3
2.5
2
qi,m/s2
qi,m/s2
1.5
1
0.5
0
-0.5
0 1 2 3 4 5
time(s)
Figure 5. Velocity of the three actuated joints
i=1 i=2 i=3
i=1 i=2 i=3
30
20
ai,m/s2
ai,m/s2
10
0
-10
-20
0 1 2 3 4 5
time(s)
Figure 6. Acceleration of the three actuated joints
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Conclusions
This paper presents the inverse velocity and acceleration analyses of a linear delta parallel manipulators using a general and systematic approach based on screw theory. With this approach, the process of acceleration modelling is much simplified with help of the generalized Jacobian. It results in a new Hessian matrix for a linearly actuated delta parallel manipulator being developed in a general and compact form.
Acknowledgment
This work is supported by the Scientific Research Foundation of Tianjin University of Technology and Education (Grant No. KYQD09005).