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386 C. Feng et al.
Through the redundant joint sensor, extra information can be obtained without employing any
external measuring devices (Hollerbach and Wampler 1996), and the calibration procedure is
usually implemented by minimizing a function of the closed-loop constraint errors. Obviously,
the self-calibration methods are more convenient to implement than the first class of calibration
methods, especially for the parallel manipulator with inherent redundant joint sensors. On the
other hand, it is usually difficult to minimize the closed-loop constraint error function, and the
calibration results are usually dependent on the adopted error functions and the optimization
methods.
There have been a couple of research articles on self-calibration of parallel manipulators.
A general framework was introduced by Zhuang (1997) for the self-calibration of parallel manip-
ulators, and the Levenberg–Marquardt algorithm was applied to optimize the error function. A
new error function based on projected tracking was proposed (Zhang et al. 2007). Self-calibration
of 2-DoF redundantly parallel manipulators was studied and error functions were proposed based
on the relationship of constraint torque to torsional deflection at actuated joints, geometric con-
straint between torsional deflections of joints, and constraint torque equilibrium (Jeong et al.
2004). In other research, on the basis of the closed-loop constraint and the geometric constraint,
an error function for self-calibration was proposed, and a Matlab
®
local optimization function
‘fmincon’ was used for optimization (Yiu et al. 2003). Although the mechanisms and the error
functions were studied in detail, the optimization algorithms applied in these articles were all
local optimization methods, and the solutions obtained by these methods would inevitably be
local solutions of the error functions, because of their multimodal properties. As a result, either
the calibration results are not highly accurate or the parameters involved have been calibrated
only partly. One cannot calibrate all of the parameters of the manipulator using local optimization
methods.
To address the problem of convergence to local optima, evolutionary algorithms were proposed
and have been playing a rapidly growing role in the past few decades. The evolutionary algorithms
deliberately introduce randomness into the search process and inherently accept weak candidate
solutions, thus the search process can escape from local optima to locate the global optimum.
Moreover, the algorithms are less sensitive to noise and model errors. These algorithms mainly
include genetic algorithms (GAs; Holland 1975), differential evolution (DE; Storn and Price 1995)
and particle swarm optimization (PSO; Kennedy and Eberhart 1995), and the like. Most of these
algorithms are also inherently parallel, which makes the algorithms more efficient in searching
for global solutions.
In this article, the self-calibration problem of a planar 2-DoF parallel manipulator with a
redundant joint sensor is studied. On the basis of the closed-loop constrained equations, a new
error function is proposed. Compared with the error function in the study of Yiu et al. (2003), the
presented error function eliminates the involvement of the passive joint positions. Furthermore,
the product items in the error function are decoupled, making it easier for the search process.
As a real-world optimization problem, the corresponding error function is very complex both
in the number of variables and its multimodal features; thus, it is very hard to converge to a
global minimum using local optimization methods. To address such a problem, evolutionary
optimization algorithms, including GA, PSO, and DE, are applied to search for the global optimum
of the error functions. First, simulations are run to compare the performance of the algorithms
under different error functions. The results show that DE produced the best solution under the
decoupled error function. Then, the calibration on real-world manipulator is carried out, on the
basis of DE with the decoupled error function, and all 12 kinematic parameters involved are
calibrated with high accuracy.
The rest of this article is organized as follows. In section 2, the mechanisms and kinematics
of the parallel manipulator to be calibrated are described and the error functions are constructed.
Section 3 covers the self-calibration designs using GA, PSO, and DE. In section 4, simulations
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