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Wu. Intell Robot 2021;1(2):99-115 I http://dx.doi.org/10.20517/ir.2021.11 Page 109

4.2. Sensitivity analysis
Sensitivity analysis can be used to evaluate the influence of the geometric parameters and design variables to
the manipulator performances. Based on the elastodynamic equation, there exists

2
(− M + K)e = 0 (49)

UpondifferentiationofEquation(49), thederivativeequationwithrespecttoavariable isobtainedas follows:

2 M K 2 e
(−2 M − + )e + (− M + K) = 0 (50)

Taking the dot-product on both sides of Equation (50) yields

2 M K 2 e (51)
e (−2 M − + )e + e (− M + K) = 0

From
( )

2
2
e Me = 1; e (− M + K) = (− M + K)e = 0 (52)

we have
2 M K
− 2 − e e + e e = 0 (53)

or ( )
1 2 M K
= − − e e + e e (54)

2
Figure 6 illustrates the sensitivity of the first-order natural frequency to the first two active joints with constant
orientation [0, /2, 0]. It is found that the first-order natural frequency is much more sensitive to the second
joint, particularly in the upper and lower workspace regions, which implies that the robot’s dynamic perfor-
mance can be improved by replacing the second joint with a stiffer actuator. It is noted that the distributions
of sensitivity coefficients are not symmetric, which is because the robot configurations are not axisymmetric
about the vertical direction when the robot end-effector moves with some constant orientations, since the
robot under study is a 5-dof robotic arm. Moreover, if a payload with more mass were exerted to the robot, it
could be predicted that the sensitivity coefficients will be increased with very tiny varying trends, compared
to the present results.

4.3. Dynamic analysis of loaded system
With the payload 5 kg applied to the end-effector of the robotic arm, they constitute a new dynamic system
and the solved frequencies with constant-orientation [0, /2, 0] are illustrated in Figure 7, from which it is
observed that the frequencies of the loaded robotic system decrease about 20% compared to Figure 5. Table 4
lists the average frequencies [46] within the constant-orientation workspace defined by
∫
dΩ
¯ (55)
= ∫
dΩ
where Ωstandsfortheworkspacevolume. Differentfromthetraditionalindustrialrobotswithlowfrequencies,
thehighorderfrequencieshavelargevaluestomakethemanipulatorachievehigh-speedmotion. Comparedto
theaveragenaturalfrequencies,thefrequenciesoftheroboticswithpayloadreduce 10%–40%forthesixorders
of frequencies. From the view of kineto-elastodynamic characteristics, the difference between the frequency
of the loaded system and its natural frequency could be a consideration in the design of the mechanical system,
where the smaller difference implies higher rigidity and higher payload capability.