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13th World Congress in Mechanismand Machine Science, Guanajuato, México, 19-25 J une, 2011 IMD-123
1
Workspace of a 3-RRPS Parallel Robot Leg with a Constant Orientation
C. K. Qi
*
X. C. Zhao
†
Z. L. J in
‡
Shanghai J iao Tong University Shanghai J iao Tong University Yanshan University
Shanghai, China Shanghai, China Qinhuangdao, Chi

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13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 1
Workspace of a 3-RRPS Parallel Robot Leg with a Constant Orientation
C. K. Qi
*
X. C. Zhao
†
Z. L. Jin
‡
Shanghai Jiao Tong University Shanghai Jiao Tong University Yanshan University Shanghai, China Shanghai, China Qinhuangdao, China X. X. Zhang
§
C. X. Ma
**
F. Gao
††
Shanghai University Shanghai Jiao Tong University Shanghai Jiao Tong University Shanghai, China Shanghai, China Shanghai, China
Abstract
—
The leg design is very important for a multi-legged rescue robot which can carry heavy payload and walk in complex unstructured terrain. For parallel mechanism, the allowed payload is usually larger than that of serial mechanism. However, the workspace and dexterity of parallel mechanism (e.g., 6-limb SPS Stewart mechanism) may be reduced due to the limb interference. In this study, to balance the payload and the workspace, the less-limb parallel mechanism 3-R RPS will be used for the leg design of a six-legged rescue robot. After the kinematic model is established, the workspace of the 3- R RPS parallel leg will be analyzed with a constant orientation. The actuated and passive joint constraints and the limb interference are considered in the analysis. This leg workspace analysis will be useful for gait planning and motion control of the rescue robot.
Keywords: 3-RRPS parallel mechanism, limb interference, multi-legged robot, rescue robot, workspace analysis
I Introduction
Rescue robots are fast developing fields in the robotics because of the rescue demanding for the disasters in the world. However, rescue robots are still not mature to handle all tasks. Many theoretical and technical aspects need to solve or improve, e.g., better mechanical design, better sensing system, better modeling, control and planning, better system integration, more robustness and intelligence. The mechanical design is a first step for a rescue robot development. Considering various rescue requirements, we will focus on the rescue robot that can carry heavy payload and perform the rescue tasks in complex terrain
. Compared with the wheeled and tracked robots, the legged robots are more suitable for complex terrain though it could be more difficult to design and control. The allowed payload of legged robots is usually smaller than that of wheeled and tracked robots. To improve the payload capability, more powerful actuators are often
*
chenkqi@sjtu.edu.cn
†
xczhao@sjtu.edu.cn
‡
zljin@ysu.edu.cn
§
xianxia_zh@shu.edu.cn
**
cxma@sjtu.edu.cn
††
fengg@sjtu.edu.cn
required if the leg is designed with the serial mechanism. However, it is not always possible due to the limited power of the actuator. Therefore, with the reasonable power of the actuator, we will use the parallel mechanism for the leg design in order to carry the heavy payload. To achieve the flexible adaption to complex terrain, the leg is expected to have 6 degree-of-freedom (6-DOF). However, if the leg is designed with the parallel mechanism, its workspace may become small due to the mechanical constraints and the limb interference. For example, though a 6-SPS Stewart parallel mechanism has 6 DOF, it may result in a reduced workspace due to the 6-limb interference. There are some studies on the dexterity improvement using better mechanical design [1]. In this study, to balance the payload and workspace, we will use a less-limb parallel 6-DOF mechanism for the leg design. Various 6-DOF parallel mechanisms with 3 limbs have been studied. For example, 3-PPSP [2][3], 3-PRPS [4-6], 3-U
r
RS [7][8], 3-RRRS [9], 3-PPSR [6][10-12], 3-RRR [13] and 3-limb mechanism with five-bar linkages [14], where the number denotes the number of kinematic chains linking the moving platform to the base, and the set of letters defines the sequence of joints used in each kinematic chain, the letters P, R and S denote the prismatic, revolute and spherical joints. Specially, there are a few studies on the 3-UPS mechanism. The direct kinematics of 3-UPS is solved in [15], and the singularity analysis is performed in [16]. The 3-UPS parallel machine tool is established and analysized in [17][18]. In [19-21], the U pair is a spatial five-bar 2-DOF mechanism, which is chosen as the actuation input. However, in [22][23], the U pair is decomposed into one active R joint and one passive R joint, the P pair is active and the S joint is passive. The inverse kinematics, operational capability and workspace analysis of this type of 3-UPS are also studied. Until now, there are few studies reported on the applications of parallel mechanism to the leg design of the multi-legged robot. Though some studies on the kinematics and workspace of the 3-UPS mechanism have been conducted, there are still some problems due to the complexity of the 3-UPS leg for a practical multi-legged robot. Specially, the workspace analysis should consider the various constraints. To make it more clear, here the 3-
13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 2
UPS will be named as the 3-RRPS, where the actuated joints are indicated by underlining. The workspace of various parallel robots has been studied [24]. Several kinds of workspaces can be defined, e.g., orientation workspace and positional workspace. With different orientation requirements, the positional workspace can be further classified into: constant orientation workspace, reachable workspace, inclusive workspace, total orientation workspace, and dextrous workspace [25]. Positional workspace with a constant orientation has been widely studied. For example, a geometrical method is proposed to determine the positional workspace of 6-SPS and 6-RTS mechanisms [26]. However, the limb interference is not considered. There are also some studies on the orientation workspace [27-30] and the 6-dimensional positional/orientation workspace [31]. For simplicity, we only study the workspace for a constant orientation, but the similar procedure can also be applicable for the case of varying orientation. In the workspace analysis, there are mainly three types of kinematic constraints, i.e., the actuator constraint, the passive joint limitations, and the link interference conditions [27][32][33]. The actuator constraints and passive joint limitations can be easily handled while the link interference constraints are complicated. The interference of two limbs can be identified by comparing the nearest distance of two limbs and the diameter of the limbs [33]. For example, based on this idea, a geometrical method is proposed to identify the limbs interference of 6-SPS Stewart platform [34]. The numerical method is used to investigate the workspace of 6-DOF 3-PPSR parallel manipulator [12] by considering the limbs interference, revolute and spherical joint limitations. The leg length limits, angle limits, leg interference are considered in [35]. Guo, Shan, Chen & Chen [36] studied the workspace of 3/6-SPS Stewart platform. The interference checking over a given workspace or trajectory is studied in [37]. In this study, the constant orientation workspace will be obtained by considering all these three constraints. The workspace can be determined by at least four methods: geometrical method, continuation method, discretization method and optimization method [38]. In this study, we will follow the discretization method because it is an easy and stable method, though the computational expense may increase with the higher accuracy required. Using a numerical search algorithm, we can obtain the reachable workspace of the 3-R RPS parallel mechanism for a given orientation, from which the movement capability of foots of the rescue robot can be obtained. The results of this paper will lay the foundation for the successful applications of the rescue robot. Especially it will play an important role for the gait planning and control of the rescue robot. The rest of the paper is organized as follows. The 3-RRPS parallel leg is introduced in Section 2. In Section 3, the kinematics and workspace analysis are presented. Section 4 contains numerical simulations. Finally, a few conclusions are presented in Section 5.
II. The 3-RRPS parallel leg
The parallel mechanism is structurally more rigid than the serial mechanism, so it is expected to take more payload. Currently, we are developing a six-legged rescue robot whose leg consists of a 6-DOF 3-R RPS parallel mechanism as shown in Fig. 1. The underlined “R” and “P” joints are actuated joints and others are passive joints. With the limited power of actuators, this rescue robot can still carry heavy payload due to this kind of leg design.
Foot Passive R joint Actuated R joint Passive S joint Actuated P joint BaseLimb 1 Limb 2 Limb 3
Fig. 1. Prototype of the 3-RRPS parallel leg
III. Workspace analysis of the 3-RRPS parallel leg
A. Coordinate system
P
L
1
L
2
L
3
A
1
A
2
A
3
B
1
B
2
X Y
Z
X’
Y’
Z’
OF
G
3
B
3
G
1
G
2
P S R R
Fig. 2. Coordinate system of the 3-RRPS parallel leg
As shown in Fig. 2, the base and moving platform of the 3-RRPS parallel mechanism are connected by three RRPS chains. Each chain is connected to the base
13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 3
through a U pair. This U pair is further decomposed into an actuated R joint and a passive R joint. The motion centers
1
B
,
2
B
and
3
B
of three U pairs form an equilateral triangle
123
BBB
(
122331
BBBBBB
). On the other hand, each chain is also connected to the moving platform through an S joint. However, the motion centers
1
A
,
2
A
and
3
A
of three S joints is designed to form an isosceles triangle
123
AAA
(
121323
AAAAAA
). Obviously, these two triangles are not similar. The universe coordinate system
OX
is defined on the base, where the srcin
O
is located at the center of the equilateral triangle
YZ
123
BBB
. The axis
OY
is along with the direction . The axis is perpendicular to the base plane and the upward direction is positive. The axis
OX
is then determined by the right-hand screw rule. The coordinate system of the moving platform is defined as
1
OB
OZ
'''
PXY
Z
, however due to a special design the srcin
P
is not located at the center of the isosceles triangle
123
AAA
''
. Similarly, the axis
PY
is along with the direction . The axis is perpendicular to the plane of the moving platform and the upward direction is positive. The axis is then determined by the right-hand screw rule. When the leg is in the initial states, the axis directions of the coordinate systems
OX
and
'
'
1
PA
'
PZ
'
PX
YZ
PXYZ
are same (see Fig. 2).
B. Inverse kinematics
The distance between the S joint and the center of U pair is also the length of the P pair, which is denoted as
1
L
,
2
L
and
3
L
respectively. The coordinates of the vertices
1
A
,
2
A
and
3
A
of the triangle
123
AAA
with respect to the moving coordinate system are denoted as
1
P
A
,
2
P
A
and
3
P
A
. Similarly, the coordinates of the vertices
1
B
,
2
B
and
3
B
of the triangle
123
BBB
with respect to the base coordinate system are denoted as
1
O
B
,
2
O
B
and
3
O
B
. The coordinate of the srcin of the moving platform with respect to the base coordinate system is expressed as .
O
P
Given the position
O
F
and the orientation
,
nd a
-Y-X Euler angles) of the foot (Z
F
with respect to the coordinate system, the problem of inverse kinematics is to obtain all actuated inputs: the length of three P
OXYZ
pairs
1
L
,
2
L
,
3
L
and the angle of three active R joints
1
,
2
,
3
. First, using the Z-Y-X Euler angles the rotation matrix of the moving frame relative with respect to the base frame can be expressed by
ccccc
O p
ssscscssTssssccssccsscscc
Then, the coordinates of
P
with respect to the
OXYZ
can be obtained as
OOOP p
PFTF
, (2) and the coordinates of the vertices
1
A
,
2
A
,
3
A
with respect to the can be calculated by
OXYZ
1
OOOP p
1
APTA
, (3)
2
OOOP p
2
APTA
, (4)
3
OOOP p
3
APTA
. (5) For each chain, the distance between the S joint and the U pair can give the length of the P pair, which can be determined by
11111
||||||||
OO
LBAAB
, (6)
22222
||||||||
OO
LBAAB
, (7)
33333
||||||||
OO
LBAAB
. (8) If the position and the orientation of the moving frame are given, the length of the P pair can be calculated by (6)-(8). On the other hand, we can calculate the direction vector
ii
BA
of the P pair with respect to the
OXYZ
coordinate system
OOiiii
BAAB
. (9) Considering the relationship between the P pair and the U
pair, the angle between direction vectors
ii
BA
and
i
BO
will give the passive angle
i
of the R joint in the U pair. This angle can be easily calculated using the triangle functions which are ingored here. To calculate the angle
i
of the actuated R joint in the U
pair, the normal direction vector
ii
BG
of the plane will be used
ii
OBA
iiiii
BGOBBA
, (10) where the “
” is the cross product. Suppose
(,,)
iiiii
BGiljmkn
, then the angle
i
between
ii
BG
and the base plane
123
BBB
can be obtained by
222
arcsin
iiiii
nlmn
. (11) The equations (6)-(8) and (11) are the solutions of
inverse kinematics of the 3-RRPS parallel leg.
C. Kinematic constraints C.1 Actuated R joint and P pair constraints
For each leg, the length of three actuated P pairs should satisfy
minmax
iii
LLL
, ,
1,2,3
i
. (1) where
min
i
L
and
max
i
L
are the minimal and maximal length of the
i
th
-link respectively. The angle of actuated R joints in three U
pairs (see Fig. 3) should satisfy
minmax
iii
, ,
1,2,3
i
13th World Congress in Mechanism and Machine Science, Guanajuato, México, 19-25 June, 2011 IMD-123 4
where and
max
are the minimal and maximal angles of the actuated R
min
i
i
joints respectively.
C.2 Passive R joint constraints
As shown in Fig. 3, the passive R joint angle in the U
pair is the angle
i
between
ii
BA
and
i
BO
. The constraints of three passive R joints can be expressed by
minmax
iii
, ,
1,2,3
i
where
min
i
and are the minimal and maximal values respectively.
max
i
L
1
L
2
L
3
A
1
A
2
A
3
B
1
B
2
O
G
3
B
3
G
1
G
2
1
2
3
3
2
1
Fig. 3. Actuated R joint and passive R joint constraints
C.3 Passive S joint constraints
For easy implementation, the passive S joint is designed as a serial RRR mechanism (see Fig. 4a). The middle R joint angle
i
, i.e., the angle between lines
ii
AB
and (see Fig. 4b), will have the following constraints:
1
PA
minmax
iii
, ,
1,2,3
i
where and are the minimal and maximal values respectively.
min
i
max
i
R R R
(a)
P
L
1
L
2
L
3
A
1
A
2
A
3
B
1
B
3
2
1
3
B
2
(b) Fig. 4. Passive S joint constraints
C.4 Link interference constraints
. As shown in Fig. 5, each link consists of two cylinders and
ii
EA
ii
BE
, and their diameters are denoted by and
i
d
i
D
respectively (
i
dD
i
). The shortest distance between the links
i
L
and
j
L
over two sublinks and
ii
EA
jj
EA
are . Similarly, the shortest distance over another two sublinks
ij
dis
ii
BE
and
jj
BE
are
ij
DIS
. Then the conditions to avoid the link interference can be expressed by
()/2
iji
dddis
j j
,
()/2
iji
DDDI
S
. The shortest distance between the center-lines of two sublinks and
ij
dis
ii
EA
jj
EA
can be calculated as below
the length of the common perpendicular
ij
n
, if their common perpendicular has intersection points and
i
C
j
C
within two links
ii
EA
and
jj
EA
respectively (see Fig. 5 for an example).
the shortest distance between the endpoint
i
A
of the link
ii
EA
and the link
jj
EA
, if only the intersection point
i
C
of the link
ii
EA
and the common perpendicular of the two links is beyond the link
ii
EA
.
the distance between the two endpoints
i
A
and
j
A
, if two intersection points
i
C
and
j
C
are both beyond the links
ii
EA
and
jj
EA
. The shortest distance
ij
DIS
between the center-lines of two sublinks
ii
BE
and
jj
BE
can also be obtained in a same way.

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