A unified approach to solving driving forces in spatial parallel manipulators with less than six DOFs is proposed. First, the geometric constrained equations of the parallel manipulators are derived, and some independent pose parameters and a common transformation matrix with three translations and three Euler rotations are determined. Second, the common formulas for solving inverse kinematics, the Jacobian matrix, and velocity are derived. Third, a common virtual serial mechanism with three virtual prismatic joints and three virtual revolute joints corresponding to three Euler rotations is constructed. Fourth, a common analytic formula for solving driving forces in spatial parallel manipulators with less than six DOFs is derived by using the principle of virtual work and the virtual serial mechanism. Finally, a 3-SPR parallel manipulator with three DOFs and a 4SPS+SPR parallel manipulator with five DOFs are presented to illustrate the use of the unified approach to solve their inverse kinematics and the driving forces. The solving results are verified by the simulation mechanisms.

Issue Section:
Research Papers
Keywords:
force control, geometry, Jacobian matrices, manipulator kinematics, parallel manipulator, kinematics, driving force, serial mechanism
Topics:
Kinematics, Manipulators
1.
Niku
,
S. B.
, 2004,
Introduction to Robotics Analysis, Systems, Applications
,
Prentice-Hall
,
Beijing
.
2.
Huang
,
Z.
,
Kong
,
L. F.
, and
Fang
,
Y. F.
, 1997,
Theory on Parallel Robotics and Control
,
Machinery Industry Press
,
Beijing
.
3.
Zhang
,
D.
,
Wang
,
L.
, and
Lang
,
S. Y. T.
, 2005, “
Parallel Kinematic Machines: Design, Analysis and Simulation in an Integrated Virtual Environment
,”
ASME J. Mech. Des.
1050-0472,
127
(
4
), pp.
580
–
588
.
4.
Dasgupta
,
B.
, and
Mruthyunjaya
,
T. S.
, 1998, “
A. Newton-Euler Formulation for the Inverse Dynamics of the Stewart Platform Manipulator
,”
Mech. Mach. Theory
0094-114X,
33
(
8
), pp.
1135
–
1152
.
5.
Tsai
,
L. W.
, 2000, “
Solving the Inverse Dynamics of a Stewart-Gough Manipulator by the Principle of Virtual Work
,”
ASME J. Mech. Des.
1050-0472,
122
(
1
), pp.
3
–
9
.
6.
Gallardo
,
J.
,
Rico
,
J. M.
,
Frisoli
,
A.
,
Checcacci
,
D.
, and
Bergamasco
,
M.
, 2003, “
Dynamics of Parallel Manipulators by Means of Screw Theory
,”
Mech. Mach. Theory
0094-114X,
38
(
11
), pp.
1113
–
1131
.
7.
Miller
,
K.
, 2004, “
Optimal Design and Modelling of Spatial Parallel Manipulators
,”
Int. J. Robot. Res.
0278-3649,
23
(
2
), pp.
127
–
140
.
8.
Stock
,
M.
, and
Miller
,
K.
, 2003, “
Optimal Kinematic Design of Spatial Parallel Manipulators: Application to Linear Delta Robot
,”
ASME J. Mech. Des.
1050-0472,
125
(
4
), pp.
292
–
301
.
9.
Lu
,
Y.
, 2007, “
Using Virtual Work Theory and CAD Functionalities for Solving Driving Force and Driven Force of Spatial Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
42
, pp.
839
–
858
.
10.
Lu
,
Y.
, and
Hu
,
B.
, 2006, “
Solving Driving Forces of 2(3-SPR) Serial-Parallel Manipulator by CAD Variation Geometry Approach
,”
ASME J. Mech. Des.
1050-0472,
128
(
6
), pp.
1349
–
1351
.
11.
Russo
,
A.
,
Sinatra
,
R.
, and
Xi
,
F.
, 2005, “
Static Balancing of Parallel Robots
,”
Mech. Mach. Theory
0094-114X,
40
(
2
), pp.
191
–
202
.
12.
Ider
,
S. K.
, 2005, “
Inverse Dynamics of Parallel Manipulators in the Presence of Drive Singularities
,”
Mech. Mach. Theory
0094-114X,
40
(
5
), pp.
578
–
599
.
13.
Li
,
M.
,
Huang
,
T.
,
Mei
,
J.
,
Zhao
,
X.
,
Chetwynd
,
D. G.
, and
Hu
,
S. J.
, 2005, “
Dynamic Formulation and Performance Comparison of the 3-DOF Modules of Two Reconfigurable PKM—The Tricept and the TriVariant
,”
ASME J. Mech. Des.
1050-0472,
127
(
5
), pp.
1129
–
1136
.
14.
Bhaskar
,
D.
, and
Mruthyunjaya
,
T. S.
, 1998, “
Force Redundancy in Parallel Manipulators: Theoretical and Practical Issues
,”
Mech. Mach. Theory
0094-114X,
33
(
6
), pp.
727
–
742
.
15.
Nokleby
,
S. B.
,
Fisher
,
R.
,
Podhorodeski
,
R. P.
, and
Firmani
,
F.
, 2005, “
Force Capabilities of Redundantly-Actuated Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
40
(
5
), pp.
578
–
599
.
16.
Zhang
,
D.
, and
Gosselin
,
C. M.
, 2001, “
Kinetostatic Modeling of N-DoF Parallel Mechanisms With Passive Constraining Leg and Prismatic Actuators
,”
ASME J. Mech. Des.
1050-0472,
123
(
3
), pp.
375
–
381
.
17.
Zhang
,
D.
, and
Gosselin
,
C. M.
, 2002, “
Kinetostatic Modelling of Parallel Mechanisms with a Passive Constraining Leg and Revolute Actuators
,”
Mech. Mach. Theory
0094-114X,
37
(
6
), pp.
599
–
617
.
18.
Lu
,
Y.
, 2005, “
Position and Workspace Analyses of 3-SPR and 3-RPS Parallel Manipulators DETC2005–84117
,”
Proceedings of IDETC/CIE
, Sept., pp.
957
–
962
.
19.
Chakarov
,
D.
, 1999, “
Study of the Passive Compliance of Parallel Manipulators
,”
Mech. Mach. Theory
0094-114X,
34
(
3
), pp.
373
–
389
.
20.
Tsai
,
L. W.
, and
Joshi
,
S.
, 2000, “
Kinematics and Optimization of a Spatial 3-UPU Parallel Manipulator
,”
ASME J. Mech. Des.
1050-0472,
122
(
4
), pp.
439
–
446
.
21.
Gosselin
,
C. M.
, 1993, “
On the Direct Kinematics of Spherical 3-DOF Parallel Manipulators
,”
Int. J. Robot. Res.
0278-3649,
12
(
4
), pp.
394
–
402
.
22.
Carretero
,
J. A.
,
Podhorodeski
,
R. P.
,
Nahon
,
M. A.
, and
Gosselin
,
C. M.
, 2000, “
Kinematic Analysis and Optimization of a New Three Degree-of-Freedom Spacial Parallel Manipulator
,”
ASME J. Mech. Des.
1050-0472,
122
(
1
), pp.
17
–
24
.
23.
Lu
,
Y.
, 2006, “
CAD Variation Geometry and Analytic Approach for Solving Kinematics of a Novel 3-SPU/3-SPU Parallel Manipulator
,”
ASME J. Mech. Des.
1050-0472,
128
(
3
), pp.
574
–
580
.
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 by American Society of Mechanical Engineers
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