IEEE Robotics & Automation Magazine - September 2022 - 92

the forward kinematics (FK) for parallel robots is more
complex to solve. It is even more difficult in the case of CDPRs
because of the unilateral nature of cables, which requires them
to always be in tension. Inverse kinematics (IK) is easier than
the FK to solve for parallel robots and CDPRs with ideal cables
with no mass and elasticity.
Furthermore, the hefty
shape of the cables makes
the IK problem quite
complex. In this case,
both the position parameters
and cable tensions
are unknowns, and some
of the equations are not
algebraic. Therefore,
approaches that are efficient
with ideal cables
cannot be used for IK
analysis. By assuming a
hefty elastic model for the
cables, solving the IK
problem becomes even
more complex as the cable configuration considers both hefty
shape and elongation of the tensioning cables [58]. The details
about elastic and hefty cable models and their effect on the
kinematics and dynamics are explained in the sections " Cables
with Mass, " " Research Works Considering Elastic Cables, " and
" Research Works Considering Cables With Both Mass
and Elasticity. "
Research Works Assuming
Massless Inelastic Cables
The Newton-Euler method was widely used for deriving the
dynamics equations of both planar and spatial CDPRs [5],
[26], [59], [60]. In this approach, the position vector of each
cable and applied forces and wrenches on the MP were calculated
for deriving the dynamics equations. In [61], a new
Jacobian matrix was constructed with a chosen set of variables
arising from dynamics analysis using the Lagrangian
method. The dynamics equation presented in [59] has six
variables corresponding to the position and orientation
coordinates for the six DoF of the MP. However, the new
Jacobian matrix employed Cartesian coordinates of the vertices
of the MP consisting of nine variables, which reduces
the computation time for workspace definition. In [62], the
FK was studied using the multilayer perception type neural
network approach, which is faster than numerical approaches.
A backpropagation procedure was utilized for training
the network. In [63], an approach for finding the lowest stable
equilibrium pose of suspended CDPRs with an arbitrary
number of cables was studied. In this approach, the potential
energy of MP is minimized using the branch-andbound
algorithm.
The kinematics of the collaborative transport of cablesuspended
payloads by four mobile cranes was examined in
[64]. The estimation of kinematic errors caused by
92 * IEEE ROBOTICS & AUTOMATION MAGAZINE * SEPTEMBER 2022
A positive effect of the
five-bar mechanism on
decreasing the drive
torques and increasing
the load-carrying
capacity of the robot
was demonstrated.
machining, assembly, and operation was studied to improve
the exact positioning of the MP. Moreover, in [65], an IK
analysis of three quadrotors carrying a payload was presented.
The IK problem has infinitely many solutions for this
configuration. However, when the tensions of the cables are
also specified, the IK problem is shown to have a finite
number of solutions.
Different approaches are used in the literature to account
for the effect of the winding mechanism for improving the
dynamic performance of the CDPRs [66]. In [67], reflective
pulleys were integrated into the MP to improve the kinematics
and dynamics of the robot. These reflective pulleys must
have the same radius as the ones at the BP to compensate for
their impacts. However, according to [2], pulleys have a significant
effect only on the kinematics of small CDPRs and
can otherwise be neglected. In [68], a five-bar mechanism
was used to move the MP of a spatial IRPM. A positive effect
of the five-bar mechanism on decreasing the drive torques
and increasing the load-carrying capacity of the robot was
demonstrated. In [28] and [66], the dynamics of pulleys and
winches were considered for deriving kinematics and
dynamics equations of a spatial and a planar robot using geometrical
approaches.
Cables With Mass
Two approaches have been proposed in the literature for analyzing
the kinematics and dynamics of CDPRs, considering
the mass of cables. The first approach assumes cables as
straight elements, and the second one accounts for their hefty
or catenary shape.
In terms of the first approach, in [14], the cables were
assumed as straight lines with varying mass and velocity.
Dynamics equations of a 4-3 planar IRPM were derived
using the Lagrangian method. In [15], the virtual work and
Newton-Euler methods were used to analyze the dynamics
of the MPs and cables of the FAST, respectively. In [13],
cables were modeled as cylinder elements. The motion analysis
of a 7-6 CRPM was studied using the " bushing " joint of
ADAMS software.
The second approach was applied in [19], where the
cable configuration (Figure 3) is represented with a hyperbolic
function,
yx k cosh`
h
^ =+ + ,
k
x cc
12
j
and c2
(1)
where k Hq= /, q is the distributed mass of the cable, H is the
horizontal tension force, and c1
are two constants that
can be evaluated from the boundary conditions.
Dynamic equations were derived for a hefty configuration
of the cables considering aerodynamic forces acting
on the robot by discretizing the cable into a series of N
elastic segments joined at nodes [9]. The lumped-mass
method was used to develop partial differential equations,
which were solved using the adaptive Runge-Kutta algorithm.
In [10], besides considering the mass of cables, the
dynamics of pulleys were integrated for kinetostatic
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