A wide family of parallel manipulators (PMs) is the one that groups all PMs with three legs where the legs become kinematic chains constituted of a passive spherical pair (S) in series with either a passive prismatic pair (P) or a passive revolute pair (R) when the actuators are locked. The topologies of the structures generated by these manipulators, when the actuators are locked, are ten. Two out of these topologies are the SR-2PS topology (one SR leg and two PS legs) and the SP-2RS topology (one SP leg and two RS legs). This paper presents two algorithms. The first one determines all the assembly modes of the SR-2PS structures. The second one determines all the assembly modes of the SP-2RS structures. The presented algorithms can be applied without changes to solve, in analytical form, the direct position analysis (DPA) of all the parallel manipulators that generate a SR-2PS structure or a SP-2RS structure when the actuators are locked. In particular, the closure equations of two generic structures, one of type SR-2PS and the other of type SP-2RS, are written. The eliminants of the two systems of equations are determined and the solution procedures are presented. Finally, the proposed procedures are applied to real cases. This work demonstrates that (i) the DPA solutions of any PM that becomes a SR-2PS structure are at most eight, and (ii) the DPA solutions of any PM that becomes a SP-2RS structure are at most sixteen.

1.
Tsai
,
L.-W.
, 1999,
Robot Analysis: The Mechanics of Serial and Parallel Manipulators
,
Wiley
, New York.
2.
Innocenti
,
C.
, and
Parenti-Castelli
,
V.
, 1994, “
Exhaustive Enumeration of Fully-Parallel Kinematic Chains
,”
Proc. of ASME 1994 Int. Annual Winter Meeting
,
Chicago
, ASME, New York, DSC-Vol.
55-2
, pp.
1135
1141
.
3.
Wampler
,
C. W.
, 1996, “
Forward Displacement Analysis of General Six-in-Parallel SPS (Stewart) Platform Manipulators Using Soma Coordinates
,”
Mech. Mach. Theory
0094-114X,
31
(
3
), pp.
331
337
.
4.
Husty
,
M. L.
, 1996, “
An Algorithm for Solving the Direct Kinematics of General Stewart—Gough Platforms
,”
Mech. Mach. Theory
0094-114X,
31
(
4
), pp.
365
380
.
5.
Innocenti
,
C.
, 2001, “
Forward Kinematics in Polynomial Form of the General Stewart Platform
,”
J. Mech. Des.
1050-0472,
123
(
2
), pp.
254
260
.
6.
Lee
,
T.-Y.
, and
Shim
,
J.-K.
, 2001, “
Forward Kinematics of the General 6-6 Stewart Platform Using Algebraic Elimination
,”
Mech. Mach. Theory
0094-114X,
36
(
10
), pp.
1073
1085
.
7.
Innocenti
,
C.
, and
Parenti-Castelli
,
V.
, 1990, “
Direct Position Analysis of the Stewart Platform Mechanism
,”
Mech. Mach. Theory
0094-114X,
25
(
6
), pp.
611
621
.
8.
Innocenti
,
C.
, and
Parenti-Castelli
,
V.
, 1991, “
Direct Kinematics of the Reverse Stewart Platform Mechanism
,”
Proc. of 1991 IFAC Symposium on Robot Control
, Siroco’91,
Vienna
, IFAC, Laxenburg (Austria), pp.
75
80
.
9.
Parenti-Castelli
,
V.
, and
Innocenti
,
C.
, 1990, “
Direct Displacement Analysis for Some Classes of Spatial Parallel Mechanisms
,”
Proc. of 8th CISM-IFToMM Symp. on Theory and Practice of Robots and Manipulators
,
Cracow
, CISM, Udine (Italy), pp.
126
133
.
10.
Parenti-Castelli
,
V.
, and
Innocenti
,
C.
, 1992, “
Forward Displacement Analysis of Parallel Mechanisms: Closed Form Solution of PRR-3S and PPR-3S Structures
,”
J. Mech. Des.
1050-0472,
114
(
1
), pp.
68
73
.
11.
Di Gregorio
,
R.
, 2004, “
Direct Position Analysis of Parallel Manipulators Which Generate SP-2PS Structures
,”
Robotica
(in press).
You do not currently have access to this content.