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Page 104 Wu. Intell Robot 2021;1(2):99-115 I http://dx.doi.org/10.20517/ir.2021.11

with
[ ]
K = diag act,1 act,2 K act,3 K act,4 act,5 (15)
where act, is the actuation stiffness and K and K are the upper and lower link stiffness matrices, respectively.

To calculate the stiffness matrix of the loaded mode, a neighborhood in the loaded configuration in which the
external loads and the joint location are supposed to be incremented by small values F and , which can
still satisfy the equilibrium conditions, is considered, leading to

0
(J + J ) G + (J + J ) (F + F) = K ( − + ) (16)
and the linearized kinematic constraint
t = J (17)

Based on Equation (14), expanding Equation (16) yields

H ⊗ G + J F + H ⊗ F = K (18)

where the symbol ⊗ represents the Kronecker product between matrices and H = J / , H = J / .
Combining Equations (17) and (18), the stiffness model of the robotic manipulator is reduced to
[ ] [ ] [ ]
0 J F t (19)
J K − K = 0

with

K = H ⊗ G + H ⊗ F (20)

From F = K t, the Cartesian stiffness matrix K of the robotic arm is calculated as
( ) −1
−1
K = J (K − K ) J (21)
3.2. Mass matrix
The mass matrix can be derived from the expression of the system’s kinetic energy, consisting of energies of
the revolute joints, links, and end-effector. The energy of the five active joints are
( 5 5 )
1 ∑ 2 ∑
¤
= , + , v v , (22)
,
2
=1 =3
with
¤
¤
v ,3 = E 3 ; v , = E 45 , = 4, 5 (23)
and
[ ]
E 3 = z 0 × q 2 z 1 × (q 2 − q 1 ) 0 3 (24a)
[ ]
E 45 = z 0 × q 4 z 1 × (q 4 − q 1 ) z 2 × (q 4 − q 2 ) 0 3×2 (24b)
where , is the moment of inertia of the th joint, , is the mass, and v , is the velocity in the Cartesian
space. Let I = diag[ ,1 , ,2 , ..., ,5 ]; then, Equation (22) can be written in a compact form, namely,

1 ¤
= M ¤ (25)
2
with
(26)
M = I + ,3 E E 3 + ( ,4 + ,5 )E E 45
3 45