1. Introduction

A manipulator type 3D coordinate measurementsystem (CMS)is used in a wide range of fields such as product engineering, surface shape measurement, motion measurement and so on[1-5]. Also, the space where it is used is various; for example, a relatively large space in a measurement for surface shape of vehicle, a narrow space in a jaw motion measurement etc. Most 3D CMSsare composed of a multi-link arm and coordinates identifier. Joint angles information in case the probe at the tip of a link is on a measuring point is sent to anidentifier.Then, the estimate of coordinates is calculated by substituting them to the identification modelcorresponding to the forward kinematic model in Cartesian coordinates system. The accuracy improvement of 3D measurement has been needed increasingly according to the quality improvement demand of a product.

However, an estimate of coordinates does not always coincide with the actual coordinatessince links of actual ma-nipulator for 3D CMS cannot rotate like an ideal manipulator. Such an uncertainty depends upon dimension error of components, a change of parts size by temperature orhumidity, a secular change resulting from vibration and so on. This may also depend upon the fact that normal vectors of rotation have displacements from ideal directions. Considerable cost will be required for reducing such an uncertainty. Therefore, it is important to make a mathematical model which matches the actual multi-link manipulator in order to obtain the coordinates estimate as accurate as possible.

A many conventional kinematics model for a CMS manipulator is built under the assumption that the normal vector which is the axis of link rotation has an ideal direction. Therefore, the gap between the actual system and the mathematical model is an obstacle to coordinates measurement if a demand performance becomes higher.

Several estimation methods for obtaining uncertainties in the 3D measuring instrument model have been proposed [1-5]. However, researches in which uncertainties mentioned above have been taken into account cannot be found. In addition, conventional estimation methods of uncertainties in 3D measuring instrument model were almost linear estimation methods like a least squared method utilizing known teaching coordinates. Therefore, the number of estimates increases explosively when the number of links increases.As a result, the realization of the measurement system becomes difficult because large quantities of teaching coordinates are necessary.

Therefore, this paper proposes firstly a new forward kinematics model ofa manipulator for3D coordinate measurementsystem which takes into account inclinations of rotational normal vectors from ideal directions.Then, parameters with respect to those inclinations are estimated by utilizing the particle swarm optimization (PSO) technique and the linkangledata corresponding to the known coordinates for calibration.

The rest of the paper is organized as follows.Statement of problem and a new kinematics model are presented in the next section. In section 3, the conventional linear parameter estimation method for a kinematic model and its problem are introduced. The PSO scheme for a 3D CMS is proposedin the section 4. Some numerical simulation results for verifying the effectiveness of proposed scheme are shown in the section 5. The last section concludes the paper.

Figure 1. An example of manipulator for 3D coordinates measurement

2. New Kinematics Model and Statement of Problem

The following simple example gives the motivation of the study.

Example:Now consider a two link manipulator. The link 1 rotates around the -axis of Cartesian coordinates. The 2nd link which is attached to the end of the 1st link rotates on the plane orthogonal to the 1st link. Then, the 3D coordinates at the tip of 2nd link can be represented as

(1)

where

(2)

(3)

(4)

means the measured link angle which can be obtained from link an angle sensor like a rotary encoder, is the known link length for and are matrices which represent coordinate rotation of and around -axis and -axis respectively. Hence, the initial coordinates of the tip of link 1 is . The rotation axis of the link 1 is the -axis of Cartesian coordinates. The angle between the -axis and the link 1 which rotates counterclockwise is defined as a positive . Similarly, the initial coordinates of the tip of link 2 is

The rotation axis of the link 2 is the link . The angle between the -planeof Cartesian coordinatesand the link 2 which rotates so as to satisfy

(5)

is defined as a positive where

(6)

Figure 2. Definitions of normal vectors and rotation angles

Figure 3. Link rotation when the normal vector is deviated from the ideal direction

Then, the accurate coordinate of the 2nd link tip can be obtained by substituting link angle sensor data to the model(1) when the arm tip is placed to the measurement pointif physical arm equivalent to the model (1) can be produced.

However, it is unfortunately difficult even if any expert tries to challenge.Especially, the fact that real rotational normal vectors of arm are different from ideal normal directions cannot be avoided.Thus, the combination of the model (1) and the actual arm may reduce the accuracy of three-dimensional coordinate measurement. A simple solution is to give up the manufacture of an ideal arm and further to construct a precise model corresponding to the actual arm.

Then, there exist some positive constants and such that the following new model generates the exact coordinate when real rotational normal vectors of arm are different from ideal normal directions;

(7)

where is the tip position coordinates of the 2nd link. Parameters and represent deviations of normal vectors from ideal directions.

Therefore, an accurate measurement can be achieved by this model and the corresponding arm if these parameters are known a priori.

Also, precisionmodel corresponding to themore complexmulti-link arm like a 6 DOF PUMA manipulator (in which adjacent links are orthogonal to each other. See Fig. 4) for a practical 3D CMS can be constructedsimilarly as

(8)

where

(9)

(10)

(11)

(12)

Figure 4. The definition of rotation direction of each link

means the tip position coordinates of the 6th link. Parameters , and represent deviations of the normal vectors from ideal directions.

Therefore, an accurate measurement can be achieved by the model (6) and the corresponding 6 DOF PUMA manipulator for a practical 3D CMS if parameter vector is known a priori.

So that, the problem to be considered here is how we identify the unknown vector beforehand.

3. Linear Estimation

In order to estimate unknown parameters by using a steepest descent method or a least squares method, it is necessary that the 3D coordinates can be represented as an affine (or a linear) model on unknown parameters. However, (8) does not satisfy such a property, i.e. the coordinates is nonlinear function on . A solution for this problem is an overparameterization technique. For example, when the manipulator consists of 2 links, 3D coordinates of (7) can be overparameterized as follows;

(13)

where . Defi - nitions of these elements are listed in appendixes.

Therefore, if the tip of link 2 can be placed to multiple known points whose coordinates are the estimate of can be obtained by the following weighted least square on-line method;

(14)

(15)

where represents estimate of in the iteration, the design parameter satisfies is a small constant. is the -th known point. Elements of the matrix are calculated by using which is the link angle vector when the link tip is placed at . (14) minimizes the cost

(16)

Fig.5 shows simulation results when (14) is used. Parameters for the simulation are set as; .

Figure 5. Response of and using overparametrization and linear estimation

but avilable values for estimation include measurement noise (infinitesimal random number).

From Fig. 3, it can be seen that decreases monotonically by updating . However, known several hundred coordinates are needed in order to obtain small enough.It may lead to divergence of the number of in order to obtain good estimates which make small enough.Hence, the combination of an overparametrization and a linear estimation is far apart from a practical use in a 3D coordinates measurement.

Therefore, in order to solve the problem mentioned above, we will propose a new systemconfiguration forthe3D measurement which uses a multi-link manipulator, the new kinematic model proposed in (7) and a Particle Swarm Optimization (PSO) technique for direct estimation of .

4. Parameter Estimation Based on PSO for New Kinematic Model

PSO is a computational method that optimizes a problem by iteratively trying to improve a candidate solution with regard to a given measure of quality. PSO optimizes a problem by having a population of candidate solutions, here dubbed particles, and moving these particles around in the search-space according to simple mathematical formulae over the particle's position and velocity. Each particle's movement is influenced by its local best known position and is also guided toward the best known positions in the search-space, which are updated as better positions are found by other particles. This is expected to move the swarm toward the best solutions[9].

Therefore, it is suitable for estimating parameters of a precision kinematics model.

Let be the estimate of . Now, consider the following cost function which evaluates an average distance between and in the sense of root mean square.

(17)

where means a vector norm which may be the 1-norm, the Euclidean norm or any other norm.

Then, is a nonlinear function on and alsotakes multiple local minimum points on since is an argument of trigonometric functions though obtainingan optimal estimateto minimize is expected to contribute tothe precise3D coordinates measurement. Therefore, we propose to use the following PSO technique as a parameter estimation method for the new kinematic model (7).

Step1 Prepare known coordinates; Place the manipulator tip to each and storage the corresponding link angle sensor data .

Step2 Determine the particle number and set initial particle for . Set the scale , the particle update gain , the inertia gain , correction factors and the tolerance

Step3 For all particle , evaluate where Then, determineeach particle best and the group best which satisfy

(18)

(19)

If then go to Step5 else go to step4.

Step4 Update all particle and particle’s velocity by

(20)

where and are random number[0, 1]. Go to Step3.

Step5 Adopt as the optimal estimate of , and end the algorithm.

We adopted the update law (18) which is a typical PSO algorithm[9] though many modified PSO algorithm[6, 7] are proposed.

5. Numerical Simulation

In order to verify the effectiveness of PSO. Simple numerical simulations were carried out.

Fig. 6 shows all particles behavior when the manipulator consists of 2 links where simulation settings are as follows;

and in the range of were produced by utilizing pseud random numbers of PC. Then, calculated by (7) were used as respectively. The proposed algorithm was performed till without setting .

Elements of all particle vectors converged to true values in about 80 times iteration despite is only 4.Similar trends have been confirmed by multiple times simulations though the same result cannot be obtained since random numbers are used. Hence, sufficient result was obtained by setting in case of 6 links manipulator.

Figure 6. Simulation example of elements behaviour of all particles in the case of 2 link manipulator

6. Conclusions

We proposed ahybrid measurement system of three dimensional coordinates by combination of a multi-link manipulator and a new precision kinematics model in which unknown parameters are estimated a priori by a particle swarm optimization techniques.In other words, wehave proposed a method torealize aprecisethree-dimensional measurementsystembythe certain softwareto complement theinaccuraciesof thehardware.The methodproposed here isvery practicalbecausethe parameterscan be estimated precisely with only small number ofcalibration points(the number of required known coordinate is only twice thenumber of links). The effectiveness is verified by simple numerical simulation results.

This concept seems to be able to apply to the construction of precise inverse kinematics model of the robot arm for achieving a high precision position control or trajectory control. This is our future study.

Appendix

Basis functions and parameters in theoverparameterization kinematics model (13) for 2 link manipulator are defined as follows;