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is of importance to account for the requirements of application, such as high precision, speed, and payload.
Henceforth, highernaturalfrequenciesandlowelasticdisplacementsofaroboticmanipulatorwillallowhigher
[6]
operational speeds and working cycles for efficient productivity . Natural frequencies indicate the condition
in which a mechanism tends to vibrate [7,8] . Differing from a structure or element, the dynamic behavior of
[9]
a mechanism usually heavily depends on its architecture and configurations ; thus, it is not a trivial task to
characterize the robot dynamics throughout the workspace, which calls for the kineto-elastodynamic analysis
to provide the fundamentals of the modeling, design and control.

The elastodynamic modeling and analysis of a robotic manipulator have been reported previously [10,11] , and
they are roughly grouped into two categories: lumped modeling [12–15] and distributed-flexibilities model-
ing [9,16–19] . In general, with lumped modeling it is simpler to model the elastodynamic equation with accept-
able computational accuracy, while the latter provides a more accurate model but with the high-dimensional
generalized coordinate space and more complex procedure [20] . The commonly used method to study the
elasticity of the robotic manipulators is the virtual joint method (VJM) as it can provide acceptable compu-
tation accuracy that is close to that of finite element analysis (FEA) [21] . Besides, VJM can be time efficient.
VJM is based on pseudo-rigid body models with “virtual joints” [22–25] . Generally, the link flexibilities and
linear/torsional springs take into account the bending contributions to the mechanism [26–29] . The stiffness
formulated in the above approaches is limited to a subspace defined by the degrees of freedom (dofs) of the
manipulator end-effector. Pashkevich et al. [30] overcame this issue by introducing a full-mobility lumped-
parameter model by localizing 6-dof virtual springs to the links’ ends and/or joints. In these models, the
stiffness matrix is calculated in an unloaded equilibrium configuration of a robotic manipulator. On the other
hand, the external loads directly influence the manipulator equilibrium configuration and, consequently, may
modify the static properties. The lightweight design of the robotics accordingly decreases the link structural
stiffness; thus, the robot geometry change due to external loads should be considered [31–33] . Consequently,
elastodynamics of the robotic manipulators is an important concern in their design and applications. Based
on the matrix structural analysis, Cammarata et al. [9,34] proposed an algorithm to assemble the stiffness matrix
to investigate the manipulators with lower kinematic pairs. In this manner, the overall robotic manipulator in-
parallel architecture can be split into substructures for modeling the elastodynamics [35] . Wu et al. [36] analyzed
and compared the stiffness and natural frequencies of a 3-dof parallel manipulator with/without a redundant
leg, where the joint deformations are ignored in the stiffness modeling. The small-amplitude deformations
of the active joints can be considered as parameter uncertainties in terms of small variations to be integrated
into the dynamic model [37] . Briot and Khalil [14] used the Newton–Euler recursive approach to develop a gen-
eral symbolic elastodynamic calculation model for flexible parallel robots. Taghvaeipour et al. [15] derived the
posture-dependent stiffness matrix in the elastodynamic modeling by resorting to the generalized spring con-
cept. The previous models were established in the nominal configurations; hence, the geometry changes of the
manipulator in this work are considered in the elastodynamic modeling and analysis.

In this paper, the elastodynamic characteristics of a lightweight robotic arm are investigated. The arm grav-
ity and external load are taken into account to derive the stiffness matrix. Isocontours of natural frequencies
over the dexterous workspace are formulated and sensitivity analysis is conducted. The frequencies and dis-
placement responses of the robotics with payload are analyzed and compared with the dynamic behaviors of
the unloaded mode. The main contribution of this work lies in that a systematic approach of elastodynamic
analysis for serial robotic manipulators is formulated, where the arm gravity and external load are taken into
account to investigate the dynamic behaviors of the robotic arms, i.e., frequencies, sensitivity analysis, and
displacement responses, under the loaded mode.