This paper presents a new approach to the velocity and acceleration analyses of lower mobility parallel manipulators. Building on the definition of the “acceleration motor,” the forward and inverse velocity and acceleration equations are formulated such that the relevant analyses can be integrated under a unified framework that is based on the generalized Jacobian. 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 by considering the loop closure constraints. A 3-PRS parallel manipulator with coupled translational and rotational motion capabilities is analyzed to illustrate the generality and effectiveness of this approach.

1.
Zhu
,
S. J.
,
Huang
,
Z.
, and
Ding
,
H. F.
, 2007, “
Forward/Reverse Velocity and Acceleration Analysis for a Class of Lower-Mobility Parallel Mechanism
,”
ASME J. Mech. Des.
0161-8458,
129
(
4
), pp.
390
396
.
2.
Joshi
,
S.
, and
Tsai
,
L. W.
, 2002, “
Jacobian Analysis of Limited-DOF Parallel Manipulators
,”
ASME J. Mech. Des.
0161-8458,
124
(
2
), pp.
254
258
.
3.
Huang
,
T.
,
Liu
,
H.
, and
Chetwynd
,
D. G.
, “
Generalized Jacobian Analysis of Lower Mobility Manipulators
,”
Mech. Mach. Theory
0094-114X (to be published).
4.
Tsai
,
L. -W.
, 2000, “
Solving the Inverse Dynamics of a Stewart-Gough Manipulator by the Principle of Virtual Work
,”
ASME J. Mech. Des.
0161-8458,
122
(
3
), pp.
3
9
.
5.
Khalil
,
W.
, and
Guegan
,
S.
, 2004, “
Inverse and Direct Dynamic Modeling of Gough–Stewart Robots
,”
IEEE Transactions on Robotics
,
20
(
4
), pp.
754
761
.
6.
Li
,
M.
,
Huang
,
T.
,
Mei
,
J. P.
,
Zhao
,
X. M.
,
Chetwynd
,
D. G.
, and
Hu
,
S. J.
, 2005, “
Dynamic Formulation and Performance Comparison of the 3-DOF Modules of Two Reconfigurable PKMs—The Tricept and the TriVariant
,”
ASME J. Mech. Des.
0161-8458,
127
(
6
), pp.
1129
1136
.
7.
Callegari
,
M.
,
Palpacelli
,
M. -C.
, and
Principi
,
M.
, 2006, “
Dynamics Modeling and Control of the 3-RCC Translational Platform
,”
Mechatronics
0957-4158,
16
, pp.
589
605
.
8.
Staicu
,
S.
, and
Zhang
,
D.
, 2008, “
A Novel Dynamic Modelling Approach for Parallel Mechanisms Analysis
,”
Rob. Comput.-Integr. Manufact.
0736-5845,
24
, pp.
167
172
.
9.
Staicu
,
S.
, 2009, “
Inverse Dynamics of the 3-PRR Planar Parallel Robot
,”
Robot. Auton. Syst.
,
57
, pp.
556
563
.
10.
Staicu
,
S.
,
Liu
,
X. -J.
, and
Li
,
J.
, 2009, “
Explicit Dynamics Equations of the Constrained Robotic Systems
,”
Nonlinear Dyn.
0924-090X,
58
(
1–2
), pp.
217
235
.
11.
Thomas
,
M.
, and
Twsar
,
D.
, 1982, “
Dynamic Modeling of Serial Manipulator Arms
,”
ASME J. Mech. Des.
0161-8458,
104
(
9
), pp.
218
228
.
12.
Huang
,
Z.
, 1985, “
Modeling Formulation of 6-DOF Multi-Loop Parallel Mechanisms
,”
Proceedings of the Fourth IFToMM International Symposium on Linkage and Computer Aided Design Methods, II-1
, pp.
155
162
.
13.
Huang
,
Z.
, 1985, “
Modeling Formulation of 6-DOF Multi-Loop Parallel Mechanisms
,”
Proceedings of the 4th IFToMM International Symposium on Linkage and Computer Aided Design Methods, II-1
, pp.
163
170
.
14.
Zhu
,
S. J.
,
Huang
,
Z.
, and
Guo
,
X. J.
, 2005, “
Forward/Reverse Velocity and Acceleration Analyses for a Class of Lower-Mobility Parallel Mechanisms
,”
ASME
, pp.
949
955
.
15.
Huang
,
Z.
,
Zhao
,
Y. S.
, and
Zhao
,
T. S.
, 2006,
The Advanced Spatial Mechanism
,
The High Education
,
Beijing
.
16.
Fang
,
Y.
, and
Huang
,
Z.
, 1997, “
Kinematics of a Three-Degree-Of-Freedom In-Parallel Actuated Manipulator Mechanism
,”
Mech. Mach. Theory
0094-114X,
32
(
7
), pp.
789
796
.
17.
Lu
,
Y.
, and
Shi
,
Y.
, 2009, “
Kinematic Analysis of Limited-DOF Parallel Manipulators Based on Translational/Rotational Jacobian and Hessian Matrices
,”
Robotica
0263-5747,
27
(
7
), pp.
971
980
.
18.
Lu
,
Y.
,
Shi
,
Y.
, and
Hu
,
B.
, 2008, “
Kinematic Analysis of Two Novel 3UPPU I and 3UPU II PKMs
,”
Robot. Auton. Syst.
,
56
, pp.
296
305
.
19.
Lu
,
Y.
, and
Hu
,
B.
, 2007, “
Analyzing Kinematics and Solving Active/Constrained Forces of a 3SPU+UPR Parallel Manipulator
,”
Mech. Mach. Theory
0094-114X,
42
(
10
), pp.
1298
1313
.
20.
Lu
,
Y.
, and
Hu
,
B.
, 2007, “
Unified Solving Jacobian/Hessian Matrices of Some Parallel Manipulators With n SPS Active Legs and a Passive Constrained Leg
,”
ASME J. Mech. Des.
0161-8458,
129
(
11
), pp.
1161
1169
.
21.
Lu
,
Y.
, and
Hu
,
B.
, 2008, “
Unification and Simplification of Velocity/Acceleration of Limited-DOF Parallel Manipulators With Linear Active Legs
,”
Mech. Mach. Theory
0094-114X,
43
(
9
), pp.
1112
1128
.
22.
Hunt
,
K. H.
, 1978,
Kinematic Geometry of Mechanisms
,
Oxford University Press
,
Oxford
.
23.
Mohamed
,
M. G.
, and
Duffy
,
J.
, 1985, “
A Direct Determination of the Instantaneous Kinematics of Fully Parallel Robot Manipulators
,”
ASME J. Mech., Transm., Autom. Des.
0738-0666,
107
(
2
), pp.
226
229
.
24.
Kumar
,
V.
, 1992, “
Instantaneous Kinematics of Parallel-Chain Robotic Mechanisms
,”
ASME J. Mech. Des.
0161-8458,
114
(
9
), pp.
349
358
.
25.
Ling
,
S. -H.
, and
Huang
,
M. Z.
, 1995, “
Kinestatic Analysis of General Parallel Manipulators
,”
ASME J. Mech. Des.
0161-8458,
117
(
12
), pp.
601
606
.
26.
Bonev
,
I. A.
,
Zlatanov
,
D.
, and
Gosselin
,
C. M.
, 2003, “
Singularity Analysis of 3-DOF Planar Parallel Mechanisms via Screw Theory
,”
ASME J. Mech. Des.
0161-8458,
125
(
3
), pp.
573
581
.
27.
Fang
,
Y.
, and
Tsai
,
L. -W.
, 2003, “
Inverse Velocity and Singularity Analysis of Low-DOF Serial Manipulators
,”
J. Robotic Syst.
,
20
(
4
), pp.
177
188
.
28.
Zoppi
,
M.
,
Zlatanov
,
D.
, and
Molfino
,
R.
, 2006, “
On the Velocity Analysis of Interconnected Chains Mechanisms
,”
Mech. Mach. Theory
0094-114X,
41
(
11
), pp.
1346
1358
.
29.
Sugimoto
,
K.
, 1990, “
Existence Criteria for Overconstrained Mechanisms: An Extension of Motor Algebra
,”
ASME J. Mech. Des.
0161-8458,
112
(
3
), pp.
295
298
.
30.
Brand
,
L.
, 1947,
Vector and Tensor Analysis
,
Wiley
,
New York
.
31.
Rico
,
J. M.
, and
Duffy
,
J.
, 1996, “
An Application of Screw Algebra to the Acceleration Analysis of Serial Chains
,”
Mech. Mach. Theory
0094-114X,
31
(
4
), pp.
445
457
.
32.
Rico
,
J. M.
, and
Duffy
,
J.
, 2000, “
Forward and Inverse Acceleration Analysis of In-Parallel Manipulator
,”
ASME J. Mech. Des.
0161-8458,
122
(
9
), pp.
1161
1169
.
33.
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
.
34.
Gallardo
,
J.
,
Rico
,
J. M.
, and
Alici
,
G.
, 2006, “
Kinematics and Singularity Analyses of a 4-DOF Parallel Manipulator Using Screw Theory
,”
Mech. Mach. Theory
0094-114X,
41
(
9
), pp.
1113
1131
.
35.
Crane
,
C. D.
, III
, and
Duffy
,
J.
, 2003, “
A Dynamic Analysis of a Spatial Manipulator to Determine the Payload Weight
,”
J. Robotic Syst.
,
90
(
7
), pp.
355
371
.
36.
Murray
,
R.
,
Li
,
Z. X.
, and
Sastry
,
S.
, 1994,
A Mathematical Introduction to Robotic Manipulation
,
CRC
,
Boca Raton, FL
.
37.
Wahl
,
J.
, 2002, “
Articulated Tool Head
,” U.S. Patent No. 6,431,802.
38.
Angeles
,
J.
, 2003,
Fundamentals of Robotic Mechanical Systems: Theory, Methods, and Algorithms
,
3rd ed.
,
Springer-Verlag
,
New York
.
You do not currently have access to this content.