This paper proposes a trajectory generation technique for three degree-of-freedom (3-dof) planar cable-suspended parallel robots. Based on the kinematic and dynamic modeling of the robot, positive constant ratios between cable tensions and cable lengths are assumed. This assumption allows the transformation of the dynamic equations into linear differential equations with constant coefficients for the positioning part, while the orientation equation becomes a pendulum-like differential equation for which accurate solutions can be found in the literature. The integration of the differential equations is shown to yield families of translational trajectories and associated special frequencies. This result generalizes the special cases previously identified in the literature. Combining the results obtained with translational trajectories and rotational trajectories, more general combined motions are analyzed. Examples are given in order to demonstrate the results. Because of the initial assumption on which the proposed method is based, the ratio between cable forces and cable lengths is constant and hence always positive, which ensures that all cables remain under tension. Therefore, the acceleration vector remains in the column space of the Jacobian matrix, which means that the mechanism can smoothly pass through kinematic singularities. The proposed trajectory planning approach can be used to plan dynamic trajectories that extend beyond the static workspace of the mechanism.

References

1.
Lim
,
W. B.
,
Yang
,
G.
,
Yeo
,
S. H.
,
Mustafa
,
S. K.
, and
Chen
,
I.-M.
,
2009
, “
A Generic Tension-Closure Analysis Method for Fully-Constrained Cable-Driven Parallel Manipulators
,”
IEEE International Conference on Robotics and Automation
(
ICRA '09
), Kobe, Japan, May 12–17, pp.
2187
2192
.
2.
Gouttefarde
,
M.
,
Krut
,
S.
,
Company
,
O.
,
Pierrot
,
F.
, and
Ramdani
,
N.
,
2008
, “
On the Design of Fully Constrained Parallel Cable-Driven Robots
,”
Advances in Robot Kinematics: Analysis and Design
,
Springer
,
Dordrecht
, pp.
71
78
.
3.
Gosselin
,
C.
,
2014
, “
Cable-Driven Parallel Mechanisms: State of the Art and Perspectives
,”
Bull. Jpn. Soc. Mech. Eng.: Mech. Eng. Rev.
,
1
(
1
), pp.
1
17
.
4.
Bosscher
,
P.
,
Riechel
,
A. T.
, and
Ebert-Uphoff
,
I.
,
2006
, “
Wrench-Feasible Workspace Generation for Cable-Driven Robots
,”
IEEE Trans. Rob.
,
22
(
5
), pp.
890
902
.
5.
Alp
,
A. B.
, and
Agrawal
,
S. U.
,
2002
, “
Cable Suspended Robots: Design, Planning and Control
,”
IEEE International Conference on Robotics and Automation
(
ICRA '02
), Washington, DC, May 11–15, pp.
4275
4280
.
6.
Pusey
,
J.
,
Fattah
,
A.
,
Agrawal
,
S.
, and
Messina
,
E.
,
2004
, “
Design and Workspace Analysis of a 6-6 Cable-Suspended Parallel Robot
,”
Mech. Mach. Theory
,
39
(
7
), pp.
761
778
.
7.
Fattah
,
A.
, and
Agrawal
,
S. K.
,
2002
, “
Workspace and Design Analysis of Cable-Suspended Planar Parallel Robots
,”
ASME
Paper No. DETC2002/MECH-34330.
8.
Barrette
,
G.
, and
Gosselin
,
C. M.
,
2005
, “
Determination of the Dynamic Workspace of Cable-Driven Planar Parallel Mechanisms
,”
ASME J. Mech. Des.
,
127
(
2
), pp.
242
248
.
9.
Cunningham
,
D.
, and
Asada
,
H. H.
,
2009
, “
The Winch-Bot: A Cable-Suspended, Under-Actuated Robot Utilizing Parametric Self-Excitation
,”
IEEE International Conference on Robotics and Automation
(
ICRA '09
), Kobe, Japan, May 12–17, pp.
1844
1850
.
10.
Lefrançois
,
S.
, and
Gosselin
,
C.
,
2010
, “
Point-to-Point Motion Control of a Pendulum-Like 3-Dof Underactuated Cable-Driven Robot
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Anchorage, AK, May 3–7, pp.
5187
5193
.
11.
Zanotto
,
D.
,
Rosati
,
G.
, and
Agrawal
,
S. K.
,
2011
, “
Modeling and Control of a 3-Dof Pendulum-Like Manipulator
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), Shanghai, China, May 9–13, pp.
3964
3969
.
12.
Zoso
,
N.
, and
Gosselin
,
C.
,
2012
, “
Point-to-Point Motion Planning of a Parallel 3-Dof Underactuated Cable-Suspended Robot
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), St Paul, MN, May 14–18, pp.
2325
2330
.
13.
Fantoni
,
I.
, and
Lozano
,
R.
,
2001
,
Non Linear Control for Underactuated Mechanical Systems
,
Springer, London
.
14.
Gosselin
,
C.
,
Ren
,
P.
, and
Foucault
,
S.
,
2012
, “
Dynamic Trajectory Planning of a Two-Dof Cable-Suspended Parallel Robot
,”
IEEE International Conference on Robotics and Automation
(
ICRA
), St Paul, MN, May 14–18, pp.
1476
1481
.
15.
Gosselin
,
C.
,
2012
, “
Global Planning of Dynamically Feasible Trajectories for Three-DoF Spatial Cable-Suspended Parallel Robots
,”
First International Conference on Cable-Driven Parallel Robots
, Stuttgart, Germany, Sept. 2–4, pp.
3
22
.
16.
Jiang
,
X.
, and
Gosselin
,
C.
,
2014
, “
Dynamically Feasible Trajectories for Three-DoF Planar Cable-Suspended Parallel Robots
,”
ASME
Paper No. DETC2014-34419.
17.
Davis
,
H. T.
,
1962
,
Introduction to Nonlinear Differential and Integral Equations
,
Dover Publications
,
New York
.
18.
Beléndez
,
A.
,
Pascal
,
C.
,
Méndez
,
D. I.
,
Beléndez
,
T.
, and
Neipp
,
C.
,
2007
, “
Exact Solution for the Nonlinear Pendulum
,”
Rev. Bras. Ensino Fís.
,
29
(
4
), pp.
645
648
.
19.
Marion
,
J. B.
,
1970
,
Classical Dynamics of Particles and Systems
,
Academic Press
,
New York
.
You do not currently have access to this content.