Dynamic Performance of a SCARA Robot Manipulator With ...

IEEE TRANSACTIONS ON ROBOTICS, VOL. 25, NO. 1, FEBRUARY 2009 207
TABLE I
SCARA PARAMETERS FOR NUMERICAL STUDY
Fig. 1. Schematic of the SCARA robot showing the geometric parameters and
coordinate frames.
where M is the mass matrix, C is a vector of Coriolis and centrifugal
terms, G is the gravitational terms, and τ is a vector of generalized
forces. These terms, in general, contain polynomial and trigonometric
nonlinearities. Polynomial nonlinearities cause no problems to the formulation.
Trigonometric functions can be solved easily using a Taylor
series expansion with a small-angle approximation, as will be shown
later in this paper.
B. PCT in Control Analysis
The addition of control laws to a serial manipulator simply changes
the right-hand side of (6). For example, using independent potential
difference (PD) control on the joints causes no issues when applying
PCT. Nonlinear control laws can be more vexing if the nonlinearity
is not polynomial or trigonometric based. Care must be exercised to
ensure that additional nonlinearities are not introduced by division as
well.
III. PCT APPLIED TO SCARA ROBOT
In this paper, we will assume a 4-DOF SCARA-type manipulator
having variation in the mass (and subsequently, the inertia) of both the
first two links as well as payload, as shown in Fig. 1 and Table I. Variation
in the link lengths and centers of mass could also be incorporated
into the model but are left out as their effects are assumed to be small
compared to the mass variation. The other mechanical parameters are
set to reasonable values, as given in Table I. The variation is assumed
to be uniformly distributed across the range. 2
A. Dynamics of the SCARA Robot
An open kinematic chain’s dynamics can be derived utilizing either
a Newton–Euler or Lagrangian formulation. Utilizing a Lagrangian
formulation, the equations of motion for the SCARA robot as shown
in Fig. 1 are found to be [21], [22]
⎡
⎤ ⎡ ⎤
p 1 + p 2 c 2 p 3 +0.5p 2 c 2 0 −p 5
¨θ 1
p 3 +0.5p 2 c 2 p 3 0 −p 5
¨θ 2
⎢
⎣ 0 0 p 4 0
⎥ ⎢
⎦ ⎣
¨θ
⎥
3 ⎦
−p 5 −p 5 0 p 5
¨θ 4
} {{ }
M
⎡ ⎤ ⎡
⎤ ⎡ ⎤
τ 1 −p 2 s 2 ˙θ2 −0.5p 2 s 2 ˙θ2 0 0 ˙θ
⎡ ⎤
1 0
τ
=
2
⎢
⎣ τ
⎥
3 ⎦ − 0.5 ∗ p 2 s 2 ˙θ1 0 0 0
˙θ 2
⎢
⎣ 0 0 0 0
⎥ ⎢
⎦ ⎣
˙θ
⎥
3 ⎦ + 0
⎢ ⎥
⎣ p 4 g ⎦
τ 4 0 0 0 0 ˙θ 4
0
} {{ }
F
(7)
where c 2 and s 2 are cos(θ 2 ) and sin(θ 2 ), respectively, and
p 1 =
4∑
I i + m 1 x 2 1 + m 2 (x 2 2 + a 2 1 )+(m 3 + m 4 )(a 2 1 + a 2 2 )
i=1
p 2 =2[a 1 x 2 m 2 + a 1 a 2 (m 3 + m 4 )]
4∑
p 3 = I i + m 2 x 2 2 + a 2 2 (m 3 + m 4 )
i=2
p 4 = m 3 + m 4
p 5 = I 4 (8)
where τ i is the input torque (or force), I i is the moments of inertia
around the centroid, m i is the mass, x i is the mass center, and a i is the
length for link i. In order to include the affect of mass on the inertia,
the inertia terms are further defined as
I i = m i κ 2 i , i =1,...,4 (9)
where κ i is the radius of gyration of the links. Assuming a PD control
law on each joint, the torques are
τ i = K pi (θ ides − θ i )+K di ( ˙θ ides − ˙ θ i ), i =1,...,4 (10)
2 It should be noted that virtually any type of distribution and even mixed
distributions (one variable being normally distributed and one being uniform)
can be incorporated into the framework.
where des denotes the desired values.
In order to use a general nonlinear ordinary differential equation
solver, the equations are expressed in state space form [23]. For this
particular problem, the governing equations can be put into eight
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