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Wu. Intell Robot 2021;1(2):99-115 I http://dx.doi.org/10.20517/ir.2021.11 Page 105

The kinetic energy of the upper/lower links and the wrist link can be expressed as

1 ( )
= v M v + v M v + v M v (27)

2
with
[ ]
R 1 I R 0
M = 1 (28a)
0 1 3
[ ]
R 3 I R 0
M = 3 (28b)
0 1 3
[ ]
R 4 I R 0
M = 4 (28c)
0 1 3
where the subscripted I, , and v stand for the moment of inertia, mass, and velocities in the Cartesian space,
respectively, and
v = E ; v = E ; v = E ¤ (29)
¤
¤
with
[ ]
z 0 z 1 0 3
E = (30a)
z 0 × q z 1 × (q − q 1 ) 0 3
[ ]
z 0 z 1 z 2 0 3×2
E = (30b)
z 0 × q z 1 × (q − q 1 ) z 2 × (q − q 2 ) 0 3×2
[ ]
z 0 z 1 z 2 z 3 0 3×1
E = (30c)
z 0 × q 4 z 1 × (q 4 − q 1 ) z 2 × (q 4 − q 2 ) z 3 × (q 4 − q 2 ) 0 3×1
where q and q are the position vector of the centers of the mass of the upper and lower links, respectively.
Equation (27) can be cast in a matrix form as follows:

1
¤
= M ¤ (31)
2
with
(32)
M = E M E + E M E + E M E

Similarly, the kinetic energy of the end-effector can be obtained as
[ ]
1 RI R 0 3
= v M v ; M = (33)

2 0 3 1 3
where I is the moment of inertia of the end-effector and is the mass.

From the total kinetic energy of the robotic arm = + + , the mass matrix M for the robotic arm can
be expressed as
− −1
M = M + J (M + M )J (34)

3.3. Dynamic equation and analysis
The dynamic equation of the robotic arm can be formulated as

M¥ u + C¤ u + Ku = f − M¤ v = F (35)
where C is the damping matrix, F is the resultant force, and u and ¥ u are the elastic displacement and accelera-
tion, respectively. Since damping can only slightly influence the natural frequency and mode of free vibrations,