# A calibration procedure for reconfigurable Gough-Stewart manipulators

TL;DR: A calibration procedure for the identification of the geometrical parameters of a reconfigurable Gough-Stewart parallel manipulator is introduced and a numeric algorithm for an efficient solution to the problem is proposed.

Abstract: This paper introduces a calibration procedure for the identification of the geometrical parameters of a reconfigurable Gough-Stewart parallel manipulator. By using the proposed method, the geometry of a general Gough-Stewart platform can be evaluated through the measurement of the distance between couples of points on the base and mobile platform, repeated for a given set of different poses of the manipulator. The mathematical modelling of the problem is described and a numeric algorithm for an efficient solution to the problem is proposed. Furthermore, an application of the proposed method is discussed with a numerical example, and the behaviour of the calibration procedure is analysed as a function of the number of acquisitions and the number of poses.

## Summary (2 min read)

### 1 Introduction

- Parallel robots are closed-loop mechanisms that are characterized by high stiffness, payload capability and repeatability [1].
- Furthermore, the estimation of some parameters might not be available at all.
- In his book, Merlet [1] identifies three main calibration methods for parallel kinematic machines: external calibrations, which are based on measurements with external devices; constrained calibrations, which analyse the motion of the robot in a constrained configuration; auto-calibrations, that only rely on the internal sensors of the robot.
- While most of the works of the 1990s are focused on practical calibration methods, in the early 2000s several papers on calibration modelling were published.
- The new decade was also characterized by the rise of new technologies, such as vision-based metrology.

### 2 Mechanism description

- The Gough-Stewart mechanism, often called hexapod, is based on a 6-UPS parallel architecture with six identical limbs of varying length, which are controlled by linear motors.
- The position of each joint on the base platform is expressed by position vector fi, while the relative position of each joint on the moving platform with respect to centre point H is expressed by position vector mi.
- With reference to Fig. 1, the following parameters are used to define the geometry of the calibration system: .
- The location of the jth distance sensor on the moving platform is defined by point Sj.
- A simple iterative procedure based on the Newton-Raphson method with the steps in Fig. 2 is used to solve forward kinematics.

### 3 Calibration procedure

- This section presents the mathematical modelling of a calibration procedure that identifies the geometry of a reconfigurable Gough-Stewart platform, which is characterized by a variable position of the joints of the fixed and mobile platform, defined by vectors fi and mi.
- By assuming perfect passive joints, a general Gough-Stewart platform is characterized by 42 identifiable parameters, namely the xyz coordinates of the mobile joints (18) and fixed joints (18) and the limb offsets (6).
- A priori estimates are available for the full set of parameters.
- To compensate errors due to sensor positioning and assembly, the xyz coordinates of sensors (3nr) and of measurement targets (3nr) can be calibrated, for a total of 6nr additional parameters.

### 4 Calibration in unknown environments

- The previous section assumes a known coordinate system for the identification of the position of the joints of the base platform.
- This reference system can then be used to calibrate and identify the geometry of the fixed base and the position of the measuring targets.

### 5 Experimental validation

- The proposed calibration procedure is applied to the Free-Hex robot, a reconfigurable Gough-Stewart machining tool, in order to identify the position of its passive joints.
- Free-Hex, as explained in [33], is a parallel machine tool that is characterized by a mobile platform with fixed geometry and a reconfigurable base platform, with loose magnetic feet at the end of each limb.
- A second partial calibration has been performed by including the location of all the passive joints as parameters.
- When compared to the reference geometry of Table 1, the average correction is equal to 1.94 mm, with an average relative correction of 1.03% and a maximum relative correction of 1.63%.
- A smaller number of poses does not increase the number of iterations to convergence, with 30 to 90 iterations needed for convergence with different subsets.

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...Because its design and research involve a series of high and new technology fields such as machinery [1], hydraulic [2–4], control [5], computer, signal [6], and sensor [7], it has integrated the knowledge of multiple disciplines such as electromechanical and hydraulic and has been attached great importance by the academic circles [8]....

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##### Frequently Asked Questions (2)

###### Q2. What future works have the authors mentioned in the paper "A calibration procedure for reconfigurable gough-stewart manipulators" ?

Since the procedure follows a linear approximation with the assumption of small parameter variation, it is possible to study the dependency of limb length on position and geometry independently. A direct derivation of the total differential of Eq. ( 2. 2 ) yields the same result without decoupling the system and can be obtained by expanding Eq. ( A. 5 ) without applying conditions ( A. 6 ) or ( A. 9 ).