## Abstract

This chapter focuses on path tracking of a wheeled mobile manipulator designed for manufacturing processes such as drilling, riveting, or line drawing, which demand high accuracy. This problem can be solved by combining two approaches: improved localization and improved calibration. In the first approach, a full-scale kinematic equation is derived for calibration of each individual wheel’s geometrical parameters, as opposed to traditionally treating them identical for all wheels. To avoid the singularity problem in computation, a predefined square path is used to quantify the errors used for calibration considering the movement in different directions. Both statistical method and interval analysis method are adopted and compared for estimation of the calibration parameters. In the second approach, a vision-based deviation rectification solution is presented to localize the system in the global frame through a number of artificial reflectors that are identified by an onboard laser scanner. An improved tracking and localization algorithm is developed to meet the high positional accuracy requirement, improve the system’s repeatability in the traditional trilateral algorithm, and solve the problem of pose loss in path following. The developed methods have been verified and implemented on the mobile manipulators developed by Shanghai University.

### Keywords

- mobile manipulator
- localization
- tracking
- path following

## 1. Introduction

Recently, mobile manipulators have been used in various industries including aerospace or indoor decoration engineering, which requires a large workspace [1, 2, 3, 4]. The said mobile manipulator consists of an industrial manipulator mounted on a mobile platform to perform various manufacturing tasks such as drilling/riveting in aerospace industry or baseline drawing in decoration engineering. Wheeled mobile platforms with Mecanum wheels that can easily navigate through crowded spaces due to their omnidirectionality with a zero turning radius are commonly used. Path tracking is one of the important issues for mobile manipulators, in particular for performing manufacturing tasks. This chapter addresses this issue from the aspect of localization and calibration.

Localization is a key functionality of mobile manipulator in order to track and determine its position around the environment [5]. Many methods are proposed to address this issue [6]. They can be divided into two categories: absolute localization and relative localization.

Absolute localization relies on detecting and recognizing different features in the environment to obtain the position and posture. The features can be normally divided into two types: artificial landmarks and natural landmarks. Compared with the natural landmarks, artificial landmarks have advantages of high recognition, which will lead to high accuracy. There is no cumulative error problem when a localization method based on artificial landmarks is used. The key challenge is to identify and extract the needed information from the raw data of landmarks. For the relative localization, dead reckoning and inertial navigation are commonly carried out to obtain the systems’ position. It does not have to perceive the external environment, but the drift error accumulates over time.

Researchers have proposed several solutions, such as fuzzy reflector-based localization [7] and color reflector-based self-localization [8]. The main drawbacks of these two methods are that the anti-interference ability is poor and the computation is huge. To solve these problems pertaining to the localization method based on artificial landmarks, Madsen and Andersen [9] proposed a method using three reflectors and the triangulation principle. Betke and Gurvits [10] proposed a multichannel localization method with the three-dimensional localization principle and the least squares method. Because of the unavoidable errors in position and angle measurement of reflectors, the use of only a trilateral or triangular method will not achieve high accuracy [11]. Nevertheless, there are still many challenges for mobile manipulator working in industrial environments such as aerospace manufacturing or decoration engineering, which requires high maneuverability and high accuracy at the same time. The stationary industrial manipulator has high repeated localization accuracy, which the mobile manipulator cannot achieve. This chapter focuses on the improvement of path tracking of a mobile manipulator through enhanced localization combined with calibration. Calibration is required for the system kinematic model established based on nominal geometry parameter to improve motion accuracy. Muir and Neuman [12] proposed a kinematic error model for Mecanum wheeled platform and applied actuated inverse and sensed forward solutions to the kinematic control. Wang and Chang [13] carried out error analysis in terms of distribution among Mecanum wheels. Shimada et al. [14] presented a position-corrective feedback control method with vision system on Mecanum wheeled mobile platform. Qian et al. [15] conducted a more detailed analysis on the installation angle of rollers. An improved calibration method is presented in this chapter to improve the tracking accuracy of a mobile manipulator.

## 2. System modeling

The wheeled mobile manipulator, as shown in Figure 1, is built with a manipulator onto a wheeled mobile platform with four Mecanum wheels. This system aims to carry out fuselage drilling/riveting tasks at assembly stations in an adaptive and flexible way in the aerospace industry.

This system needs to localize in real time during machining process. The global frame

The position and posture of the system in the global frame can be defined as

where

## 3. Accuracy analysis of a wheeled mobile manipulator

### 3.1. Problem formulation

Figure 3 shows the kinematic model of the mobile platform with Mecanum wheels. According to kinematic analysis [16], the motion equation can be obtained as

where

The Mecanum wheel and its roller are shown in Figure 4. The roller is fitted on the edge of the wheel at a certain angle (generally 45°) with its central axis. Each roller can rotate freely around the central axis. The wheel relies on the friction between the roller and the ground to move. The material of roller’s outer rim is usually rubber, which will deform under pressure between ground and the wheel.

Figure 5 shows the distribution of roller deformation. F and T are the force and driving torque on the roller, respectively. The radiuses of wheels reduce differently under different pressures. The deformation zone and the position of the wheel center change with the action of driving torque and the shifting [17]. Furthermore, with such deformation,

As shown in Figure 6, based on the kinematic analysis and consideration on deformation of the roller, Eq. (2) can be revised as follows: [18].

where

### 3.2. Error modeling

By multiplying time

where

leading to

where

With the least squares method, the geometric errors can be solved as

where

The rank of matrix

The system is moved in the

According to Eq. (2),

Substituting Eq. (11) in to Eq. (7),

The Monte Carlo analysis is applied to deal with 50 samples. While

The results are given in Figure 8. These data are averaged to form a new result:

There are two kinds of situations for which the process capability index has to be solved. First,

Second,

The values of

The interval analysis is also carried out. In Eq. (6),

The same method is used to solve the process capability index of

The result is verified as well.

By setting two sets of four wheel speeds

The experiment with four movements is shown in Figure 12(a), while the actual movement is shown in Figure 12(b). It can be found that

## 4. Localization

### 4.1. System configuration

The localization component of mobile manipulator includes two laser range finders and a number of reflectors that are of cylindrical shape placed in the environment. Each reflector is covered by a reflective tape with high reflectivity (Figure 13).

### 4.2. Dynamic tracking

To achieve accurate localization, the system should have the ability to perceive external information through extracting the reflector features. In this research, as shown in Figure 13, there are n reflectors, and the feature of each reflector

The first step is filtering and clustering. The raw data obtained by each finder are a set of discrete data sequences

The density of the collected data points is proportional to the distance from the target point to the laser range finder. To improve the efficiency of the feature extraction process, an adaptive clustering method is adopted as given by Eq. (12). Unless the distance between two data points is less than the threshold δ, these data points are clustered for one reflector.

where

The second step is identification. After clustering, the data set can be correlated to each observable reflector. Each reflector data set is then used to calculate the position of the reflector in the laser range finder frame [19]. Let

then the set

The third step is feature extraction. The feature extraction algorithm of a reflector extracts its central position

First, the value of

As shown in Figure 14,

Furthermore, as mentioned before, the position of the finder in the platform frame is

The optimal triangulation localization algorithm based on angle measurement is carried out. A number of experiments were carried out on the reflector feature extraction algorithm before proposing a localization algorithm for the system. A single reflector was placed in the measuring range of the finder. The finder performed repeated scanning when it was placed at difference places. Two different feature extraction algorithms were tested to calculate the position of the reflector, and the results are shown in Figure 15. Algorithm A is the feature extraction algorithm proposed in this chapter, while algorithm B is the least square circle fitting method without a radius constraint. Apparently, the former one yields a better result.

After extracting the reflector center, the positions of all the reflectors

where

It can be seen from Figure 16 that the angle measurement accuracy is better than the distance measurement accuracy. Therefore, based on these results, this chapter proposes an optimal trilateral localization algorithm based on angle measurement. The idea is to use the azimuth angle

First, it is assumed that the finder can cover at least three reflectors at each position. After feature extraction, the positions of three reflectors

According to the cosine theorem, the relations between the above variables can be expressed by the following equations:

In Eq. (21), the known parameters

The position

where

The posture of the system also includes an azimuth angle

The azimuth angle

The dynamic tracking algorithm is then carried out. Localization of the system is based on landmarks. The system needs to meet the following two conditions to complete real-time localization:

The system requires real-time localization of the reflector in the environment.

The localization system correctly matches the real-time observed reflectors.

During the matching process, the reflectors observed in real time are made to correspond to all reflectors in the last moment in the environment one by one to extract the effective reflectors [20]. The localization can be achieved.

During localization, owing to the fact that some of the reflectors are obscured by obstacles or confused with a highly reflective metal surface object, the loss of position of the system is observed. To address the above problems, a dynamic tracking algorithm is proposed as shown in Figure 18.

After placing n reflectors in the global environment, the system actually observes q reflectors at the tth moment, and the coordinate value of the ith reflector in the platform frame is

If the difference between the theoretical value

Therefore, a set ** A**of effective reflectors at the tth moment can be obtained, and

### 4.3. Experimental verification

The experimental data are obtained by the LMS 100 laser range finder with a scanning range of 270° and an angular resolution of 0.25°. The experimental platform is shown in Figure 19. The outside of the reflector is wrapped by reflective tape. In the experimental environment, five reflectors are placed around the system. Their global coordinate values are (0,995); (0, 0); (0, 1774); (2905, −2449); and (3956, −2032), and the unit is mm, as shown in Figure 20.

The optimal triangulation method based on angle measurement is used for validation by the repeatability of the system. In the stationary state of the system, the environment is scanned by finders. Each result is indicated by a red dot in Figure 21. The repeatability obtained by the trilateral method is nearly 18 mm, while the repeatability of the optimal method is only 9 mm. It can be shown that the optimal method is better than the traditional method.

The mobile manipulator moves in the direction of the arrow in Figure 19, and each time the system moves a certain distance, the localization system will perform an experiment, i.e., it will use the left rear finder to calculate the current position. An average of 30 samples is taken for each experiment.

Figure 22 shows the results of static localization accuracy. The maximum distance error is 18 mm and the maximum angle error is 2°, which satisfies the localization requirement of the system.

The mobile manipulator moves in the designated route, and it needs to constantly calculate and record its own position in the moving process. As shown in Figure 23, the trajectory of the moving system based on the localization method is smoother.

This chapter demonstrates the feasibility of a tracking and localization algorithm for mobile manipulators. The following conclusions can be drawn from this study: (i) In the detection of a reflector in the laser range finder frame, the angle repeatability of the reflector is better than that of the distance repeatability based on the feature extraction algorithm; (ii) The repeatability localization accuracy using the optimal triangulation method based on the angle measurement is nearly 9 mm, which is better than that of the trilateral method; (iii) The localization error of the system is 18 mm, which satisfies the localization requirement of system. Improvements in the location method based on reflectors, such as optimizing the layout of reflectors and the map of reflectors selection strategy for localization, are still needed.

## 5. Summary

In this chapter, through analyzing the roller deformation of the Mecanum wheel, the changed parameters of the motion equation of mobile system are found. The relative variation of the parameters in the motion equation of the Mecanum motion platform is solved by Monte Carlo analysis and interval analysis. Using the relative variation of the parameters to revise the motion equation, the displacement errors in different spaces in theory are solved for and compared with the measured displacement errors. From the comparison, both the methods are found to satisfy the system’s requirement. Then, the feasibility of a tracking and locating algorithm for mobile manipulator is demonstrated. The following conclusions can be drawn from this study: (i) In the detection of a reflector in the laser range finder frame, the angle repeatability of the reflector is better than that of the distance repeatability based on the feature extraction algorithm; (ii) The repeatability localization accuracy using the optimal triangulation method based on the angle measurement is nearly 9 mm, which is better than that of the trilateral method; (iii) The localization error of the system is 18 mm, which satisfies the localization requirement of system. Improvements in the localization method based on reflectors, such as optimizing the layout of reflectors and the map of reflectors selection strategy for localization, are still needed.

The method in this chapter is also used in the research of MoMaCo (Mobile manipulator in Construction), which can draw baseline for architectural decoration engineering as shown in Figure 24. The application result also verified the effectiveness of the method.