Special Issue Paper
Received 23 June 2016 Published online 21 October 2016 in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.4210
MOS subject classiﬁcation: 97N50; 93A30; 37N15
Smooth trajectory generation for rotating
Communicated by J. Vigo-Aguiar
In this study, the generation of smooth trajectories of the end effector of a rotating extensible manipulator arm is consid-
ered. Possible trajectories are modelled using Cartesian and polar piecewise cubic interpolants expressed as polynomial
Hermite-type functions. The use of polar piecewise cubic interpolants devises continuous ﬁrst-order and – in some cases
– second-order derivatives and allows easy calculation of kinematics variables such as velocity and acceleration. More-
over, the manipulator equations of motion can be easily handled, and the constrained trajectory of the non-active end
of the manipulator derived directly from the position of the end-effector. To verify the proposed approach, numerical
simulations are conducted for two different conﬁgurations. Copyright © 2016 John Wiley & Sons, Ltd.
Keywords: mathematical modelling; smooth trajectories; manipulator; robotic arm
One of the important problems in robotics is the generation of smooth trajectories that minimise system energy, jerk, and deﬂection
caused by ﬂexibility, clearance, and components deformation. The problem can be approached by ‘interpolating’ or ‘approximating’
the desired path by piecewise interpolating curves through a set of a priori deﬁned points that control the manipulator trajectory.
Various approaches to generate trajectories have been considered so far. Among these, piecewise interpolating curves with slope
continuity, geometrically continuous Catmull–Rom splines , parametric and/or geometric continuous splines [2, 3], or uniform cubic
B-spline with parametric and geometric continuity, proved to be adequate in generating a smooth motion, especially when manipula-
tor dynamics  is considered. Lately, Cartesian univariate and bivariate splines have been successfully used in [3, 5] for preserving the
shape of multiscale data.
Interpolation in curvilinear or polar coordinate systems, which have received a considerable attention lately, may represent an excel-
lent alternative to smooth trajectory generation, especially when rotating robotic systems are involved. A general approach to polar
coordinate interpolation has been considered in [6, 7]. Polar coordinate interpolation including polynomial splines approximation has
been disused in , while single values splines and splines focales in [9, 10]. A survey on trigonometric splines has been presented in
, while trigonometric B-splines have been considered in .
In this paper, a mathematical approach for generating the smooth trajectory of the end effector of a rotating extensible robotic arm
is presented. Cartesian and polar piecewise polynomial interpolating curves are considered for the generation of the geometric path
of the end effector. To verify the proposed approach, the trajectory and the velocity proﬁle for the end effector and non-active end of
the constrained trajectory are computed for two different conﬁgurations.
2. Mathematical modelling of manipulator trajectory
2.1. System model
The rotating extensible robotic arm shown in Figure 1 is composed of a rigid guide OE and a sliding rod SP, which is constrained to a
curved trajectory by the end S. The non-active end of the sliding part SP is denoted by S, and its active end-effector is denoted by P.
The rigid guide of the robotic arm has length d
; the sliding part has the length d
. The total length of the robotic arm denoted by
r D d
(distance between the manipulator base location O and its end-effector P/ varies because of the rotation of the rigid guide of
Department of Design and Engineering, Faculty of Science and Technology, Bournemouth University, Poole, UK
Correspondence to: Mihai Dupac, Department of Design and Engineering, Faculty of Science and Technology, Bournemouth University, Poole, UK..
Copyright © 2016 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2018, 41 2281–2286