Open access peer-reviewed chapter

Translational and Rotational Motion Control Considering Width for Autonomous Mobile Robots Using Fuzzy Inference

By Takafumi Suzuki and Masaki Takahashi

Submitted: December 19th 2010Reviewed: April 30th 2011Published: September 9th 2011

DOI: 10.5772/25000

Downloaded: 2245

1. Introduction

Obstacle avoidance methods for mobile robots have proposed in a broad range of studies and the availabilities have been discussed. Most of these studies regard the robots as points or circles and control methods of the translational movements are discussed. In these studies, it is pointed out that a non-circle robot can be transformed into a point robot by expanding the obstacles by the largest radius or maximum size of the robot. The effectiveness of avoiding obstacles by these approaches have been confirmed, however, according to the shape of the robot, these approaches reduce and waste the available free-space and can decrease the likelihood of getting to the goal. If wide-robots, which are horizontally long, are regarded as circles according as conventional approaches, they have possibilities not to go through between two divided objects due to the largest radius of the robot, even if they ought to be able to go through by using their shortest radius. This suggests necessity of suitable orientation angle at the moment of avoidance. Consequently, to enable wide-robots to avoid obstacles safely and efficiently, it is necessary to control not only the translational movement but also the rotational movement. In our current research, wide-robots with omni-directional platform have been employed, as shown in Fig.1. In situations like Fig.1, both wide-robots can go through only by changing the orientation angle in real time.

Some researches focus attention on the orientation angle of the robot (Kavraki, 1995)(Wang & Chirikjian, 2000). In these studies, by convolving the robot and the obstacle at every orientation and constructing the C-space, the suitable orientation angles of the robot for path planning are decided. However, these methods need environmental map and do not show the effectiveness for autonomous mobile robots about avoidance of unknown obstacles in these studies. Therefore, in order to avoid unknown obstacles reactively considering the orientation angle, the wide-robot needs an algorithm that can decide the orientation angle and rotational velocity command on the spot based on the current obstacle information.

Meanwhile, decision methods of the translational movement have been proposed in many studies (Wang et al., 2000) (Du et al. 2007) (Khatib, 1986) (Borenstein & Koren, 1991) (Dieter, 1997), we employ fuzzy potential method (FPM) (Tsuzaki & Yoshida, 2003). This method realizes some tasks in dynamic environment by fuzzy calculation about desire for each direction of the robot. In this research, it was shown that wheeled robots succeeded getting to the goal with conveying a soccer ball and avoiding obstacles.

Figure 1.

A situation where two robots move on narrow corridor

In this paper, a control method using a capsule-shaped case is described for both translational and rotational movement based on the FPM, and it takes into consideration the width of the robot. With this new approach, real-time control of the orientation angle is easily achieved. Conventional FPM has only been able to deal with translational velocity. The proposed method is able to control the rotational and translational velocity simultaneously within the framework of FPM.

2. Capsule case

2.1. Need to consider the width of mobile robot

In recent years, non-circle robots have been developed, of which vertically long robots, wide-robots, appear. In studies of humanoid robots, the robots have two arms mounted to the stationary torso with wheels because these robots can be used in terms of mobility, manipulation, whole-body activities, and human-robot interaction (Ambrose et al., 2004) (Du et al., 2007). During last two decades, vast number of algorithms of obstacle avoidance for mobile robot, and recently some researches and developments of the mobile robot in practical use have been reported. These robots have problems that conventional methods are inconvenient for applying for the wide-robot because most of conventional methods of obstacle avoidance regard the robot as points or circles. Or due to the postulate of the conventional methods, the robots have needed to be designed in a circle. We also have developed the wide -robot, which has torso with two arms and a head, for making the robot perform not only moving but also communication with human by means of gestures or speeches based on a perspective of human interaction. This robot is also horizontally long. In addition, when the robot opens an arm slightly, as shown in Fig. 1, or both arms, it becomes increasingly harder to apply conventional methods. If these wide-robots are regarded as circles according as conventional approaches, they have possibilities not to go through between two divided objects due to the largest radius of the robot, even if they ought to be able to go through by using their shortest radius. In this study, enable the wide-robots, which move automatically, to move smoothly and safely in the environment with obstacles, a capsule case is introduced.

2.2. Design of capsule case

The capsule shaped case is modeled by two circles and two lines tangent to the circles as shown in Fig.2.

Figure 2.

A capsule case

This closed contour is defined as l(ϕ)with the origin at the point POand as follows equation:

l(ϕ)={Ca/cosϕif0ϕ<ϕ1,ϕ4ϕ<2π,Ca/cosϕifϕ2ϕ<ϕ3,X(ϕ)2+Y(ϕ)2ifϕ1ϕ<ϕ2,ϕ3ϕ<ϕ4,E1

where φis clockwise from the back direction of the robot. ϕiare respectively ϕ1=arctan(CL/Ca),ϕ2=π-arctan(CL/Ca),ϕ3=π+arctan(CR/Ca),ϕ4=2π-arctan(CR/Ca).X(ϕ)and Y(ϕ)are calculated as following equations:

X(ϕ)={-CL-CL2-(CL2-Ca2){1+tan2(π/2-ϕ)}1+tan2(π/2-ϕ)ifϕ1ϕ<ϕ2.CR+CR2-(CR2-Ca2){1+tan2(π/2-ϕ)}1+tan2(π/2-ϕ)ifϕ3ϕ<ϕ4.E2
Y(ϕ)=X(ϕ)tan(π/2-ϕ).E3

In the proposed method, CL, CR, Caare decided in a way that wide-robot shape falls within the capsule case.

Figure 3.

Examples of PMFs

Figure 4.

A concept of fuzzy potential method considering translational and rotational motion with an omni-directional platform

3. Fuzzy potential method (FPM) using capsule case

3.1. Concept

In the fuzzy potential method (FPM), a current command velocity vector that takes into consideration element actions is decided by fuzzy inference. Element actions are represented as potential membership functions (PMFs), and then they are integrated by means of fuzzy inference. The directions on the horizontal axis in Fig. 3 correspond to the directions, which are from -πto πradian measured clockwise from the front direction of the robot. Grades for the directions are represented on the vertical axis. The grades, direction, and configured maximum and minimum speeds, are used to calculate the current command velocity vector. Previously, the FPM dealt with the problem of the translational motion control in the same way as other control methods for autonomous mobile robots. In this paper, it is shown that modifying the FPM enables it to deal with rotational motion control, which is achieved concurrently with translational motions, within the FPM framework. In the modified framework as shown in Fig.4,PMFs for translational motions and rotational motions are designed respectively based not only on the environmental information but also on the robot’s own condition. Environmental information and the robot’s own condition are treated separately and divided into a translation problem and a rotational problem Then the PMFs of each problem are independently integrated using fuzzy inference. Finally, translational and rotational velocities, which are calculated by defuzzification of mixed PMFs, are realized by an omni-directional drive system.

3.2. PMF for translational motions

3.2.1. PMF for obstacles

In order to enable a wide-robot to avoid obstacles safely and efficiently in real time, a concave shaped PMFμojt(j=1,2,,n), which is considering the capsule case, is generated. This PMF is specified by depth and width, which are calculated based on geometrical relation between an obstacle and a robot as shown in Fig.6. By generating based on some variables, which areφL, φR, φL, φR, aand φr,oin Fig. 5, the choice of safe direction becomes possible.

First, φL, φR, φL, φRare calculated as following equations:

Figure 5.

A wide-robot and an obstacle

Figure 6.

PMFs for translational motions: μojtis a PMF for an obstacle (a), μgtis a PMF for a goal (b)

φL=arccos(POQO2+PLQO2-POPL22POQO·PLQO).E4
φR=arccos(POQO2+PRQO2-POPR22POQO·PRQO).E5
φL={arcsin(DPLQO)ifD<PLQO.π-arcsin(PLQO-dsD-ds)ifDPLQO.E6
φR={arcsin(DPRQO)ifD<PRQO.π-arcsin(PRQO-dsD-ds)ifDPRQO.E7

Next, as a measure to decide how far the robot should depart from the obstacle, ais defined as the depth of the concave shaped PMF. ais described as following equation:

a=α-rr,oα-Difrr,o<α.E8

where rr,o=(rx,ry)is current position vector of the obstacle relative to the robot

If the current obstacle position is ins

ide of a circle with radius αfrom the robot position, the PMF for obstacle avoidance is generated. In other words, if a relative distance rr,ois belowα, ais defined and the concave shaped PMF corresponding to the obstacle is generated. Dis decided to ensure the safety distance as following equation:

D=Ca+ro+ds.E9

where Cais the minimum length of capsule case as shown in Fig.2. roand dsdenote respectively the radius of the obstacle and safety distance. φr,ois the angle of direction to the obstacle relative to the robot, which is calculated as following equation:

φr,o=arctan(ryrx).E10

The PMF μojtis generated for all obstacles which the robot has detected. And then, they are all integrated by calculating logical productμot, as following equation:

μot=μo1tμo2tμojt.E11

As mentioned above, by deciding the depth and the base width of concave, PMFμot, which aims to early starting of avoidance behavior and prompt the direction of the velocity vector to be far from obstacle direction in response to the fast-moving obstacle, is generated.

3.2.2. PMF for a goal

To head to the goal, a PMF μgtshaped like triangle as shown in Fig.6 (b). μgtis specified by ga, gb,φr,g. As a measure to decide how much the robot want to head to the goal, ga is defined as the height of the triangular PMF. As a measure to decide how much the robot is allowed to back away from obstacles, gb is defined. μgtgets the maximum value as ga at an angle of the goal direction relative to the front direction of the robot, φr,g, and gets the minimum value as gb at an angle of a direction opposite to the goal direction. ga and gb are described as following equations:

ga={rr,gεifrr,gε1.0ifrr,g>ε,E12
gb=ηga(0η<1),E13

where rr,dis an absolute value of the position vector of the goal relative to the robot. εand ηare constants. If rr,dis belowε, ga is defined. The shorter the distance between the obstacle and the robot is, the smaller ga becomes. The robot decelerates and stops stably.

3.3. Calculation of translational command velocity

The proposed method employs fuzzy inference to calculate the current command velocity vector. Specifically, The PMF μotand the PMF μgtare integrated by fuzzy operation into a mixed PMF μmixtas shown in Fig. 7. μmixtis an algebraic product of μotand μgtas following equation:

μmixt=μgtμot.E14

Figure 7.

A mixed PMF for translational motion

Finally, by defuzzifier, the command velocity vector is calculated as a traveling direction φoutand an absolute value of the reference speed of the robot base on the mixed PMFμmixt. φoutis decided as the direction which makes the PMF μmixt(φ)maximum.

Based onφout, vout is calculated as following equation:

vout=μmixt(φout)(vmax-vmin)+vmin.E15

where μmixt(φout)is the mixed PMF for translational movement corresponding to theφout, vmaxand vminare configured in advance respectively as higher and lower limit of the robot speed.

3.4. PMF for rotational motions

3.4.1. PMF for obstacles

In order to enable a wide-robot to decide the appropriate angle of the direction for obstacle avoidance in real time, PMFμoris generated based on a PMFμer, which considers the environmental information, and another PMFμcras following equation:

μor=μer-μcr.E16

μeris generated based on the information of distances from the center of the robot to obstacles corresponding to all directions, as shown in Fig. 8. The relative information of the distances is obtained by use of range sensors such as the laser range finder, the ultra sonic sensors or the infrared sensors. μcris generated based on the capsule case which is introduced in 2. is calculated with Eq. (1) as following equation:

μcr(φ)=l(φ+π)α.E17

Figure 8.

PMFs for rotational motions: μoris a PMF for an obstacle (a), μgris a PMF for a goal (b)

The aim of the PMF μorin (16) is to search an orientation angle of the robot which enables the distance between a point on capsule case and each obstacle to maximize, by turning front or back side of the robot on the direction that there is a closest point to each obstacle. By considering the capsule case, a design of PMF can deal with the width of the robot for rotational motion.

3.4.2. PMF for a goal

In order to turn front on the goal direction or traveling direction if there is no obstacle to avoid, PMF for a goal for rotational motion is generated asμgr. This shape is decided in same way withμgt, by using (12), (13).

3.5. Calculation of rotational command velocity

As for the rotational movement, like the translational movement, the proposed method employs fuzzy inference to calculate the current rotational command velocity vector. Specifically, The PMFμer, which considers the own condition by using Eq. (1), and the PMFμgr, which is to head to the goal, are integrated by fuzzy operation into a mixed PMF μmixras shown in Fig. 9. μmixris an algebraic product of μorand μgras following equation:

μmixr=μgrμor.E18

Finally, by defuzzifier, the command velocity vector is calculated as a traveling direction φoriand an absolute value of the reference speed of the robot base on the mixed PMFμmixr. φoriis decided as the direction φiwhich makes a following function h(φ)minimum.

h(φ)=φ-ζφ+ζμmixr(ψ)dψE19

Figure 9.

A mixed PMF for rotational motion

where ζis the parameter to avoid choosing undesirable φicaused by such as noises on the sensor data. Based onφori, ωis calculated as following equation:

ω=sgn(φori)E20

where μmixr(φori)is the mixed PMF for translational movement corresponding to theφori, ωmaxand ωminare configured in advance respectively as higher and lower limit of the rotational speed of the robot.

3.6. Calculation of wheel speeds

To realize the movement, in this study, an omni-directional platform is employed for a autonomous mobile robot. The command velocity vector is realized by four DC motors and omni wheels using following equations:

vrx=voutcosφout.E21
vry=voutsinφout.E22
(v1wv2wv3wv4w)=(cosδsinδR-cosδsinδR-cosδ-sinδRcosδ-sinδR)(vrxvryω).E23

where voutand ωare respectively current command translational velocity vector and rotational speed. δis an angle of gradient for each wheel. Ris a half of a distance between two catawampus wheels. viwis a command movement speed of each i-thwheel.

4. Simulation results

The effectiveness of the proposed method was verified by numerical simulations intended for omni-directional autonomous mobile robots. As postulates, the robot supposed to be

Figure 10.

An omni-directional platform with wide body

able to detect obstacles and has information about the relative position vector. The measuring range was assumed to be 4.0m at all directions. Each parameter was as follows:

The wide-robot size was assumed as L=0.4m, W=1.0m, which are in Fig.10. Considering this L and W, Ca, CL and CR in Fig.2 were all set at 0.3m. ro,dsin Fig.5 were both set at 0.3m. Consequently D=0.9m in Eq.(9). αin Eq. (8) was 4.0m. ηin Eq. (13) was 0.2. εin Eq. (12) was 1.0m. vmax, and vmin in Eq. (15) were respectively 0.5m/s and 0.0m/s.

4.1. Performance of capsule case

In this section, the effectiveness of using capsule case and the design method of PMF based on the capsule case are verified, by comparing the results of chosen direction of movement at following two different situations about the orientation angle for a wide-robot. As common assumption, the positions of the robot and two obstacles were immobilized on each point respectively (1.0m, 2.0m), (1.5m, 0.5m) and (3.0m, 2.0m), as shown in Fig.11.

Figure 11.

Simulation results : in Situation I (a), the robot cannot find the direction between the two obstacles, in Situation II (b), the robot can find the direction between the two obstacles.

4.1.1. Situation I

The orientation angle of the robot was fixed to -π/4 radian clockwise from x-axis on the absolute coordinate. Therefore, the robot faced to a goal point, as shown in Fig.11(a), however, the chosen direction of the current movement of the robot was calculated as -1.35 radian, which was clockwise from the front direction of the robot, as shown in Fig.11(a). This value of the chosen direction was calculated based on the mixed PMF μmixtin this situation as shown in Fig.12(a). As a result, the robot chose the direction, which the robot would go the roundabout route.

Figure 12.

Aspects of mixed PMFs for translational motion in two different situations: (a) is in relation to the situation of Fig.11(a), and (b) is in relation to Fig.11(b).

Figure 13.

A simulation result of Method I (conventional) : The robot not using capsule case didn’t succeeded in going through between two divided objects

4.1.2. Situation II

The orientation angle of the robot is fixed to π/4 radian on the absolute coordinate. As contrasted to Situation I, the robot didn’t faced to a goal point, as shown in Fig.11(b), however, the chosen direction of the current movement of the robot was calculated as 1.37 radian, which was clockwise from the front direction of the robot, as shown in Fig.11(b). This value of the chosen direction was calculated based on the mixed PMF μmixtin this situation is shown in Fig.12(b). As a result, the robot chose the direction, which the robot would take a shorter route without collision.

Figure 14.

A simulation result of Method II (proposed): The robot using capsule case succeeded in going through between two objects with translational and rotational motion in real time.

Through these two results, the effectiveness of the capsule case is confirmed. The wide-robot can decide the direction of translational motion with considering the own orientation, goal position and obstacle positions simultaneously in real time.

4.2. Capability of going through between objects

The effectiveness of the proposed method was tested by comparing two design methods, I and II, based on PMF, for obstacle avoidance problem. Start and goal point of the robot are respectively (0.0m,0.0m) and (8.0m,0.0m). The trajectory of the robot and the aspects considering the orientation angle on the position every 1 second are plotted in Fig. 13 and Fig. 14. Obstacles positions are respectively (2.5m,1.8m), (2.5m,1.2m), (2.5m,1.2m), (2.5m,1.8m). In Method I, as a conventional method, the robot was regarded as a circle with radius 0.6m. In Method II, as a proposed method, the capsule case was used and rotational motion of the robot was taken into consideration. When the Method I was used, the robot was based on the maximum radius and did not take into consideration the rotational motion. Therefore, the robot did not succeed in going between two objects as shown in Fig. 13. While the robot did not collide with the obstacles, the robot did not get to the goal.

On the other hand, in the Method II, the capsule case and real-time control based on FPM were used. As shown in Fig. 14, the robot performed translational and rotational motions simultaneously and succeeded in going between two objects. In addition, the robot succeeded in getting to the goal with the orientation angle 0 radian using PMF for rotational motion.

Figure 15.

Experimental results: (a) shows a result of Method I. The robot not using capsule case didn’t succeed in going through between two objects. (b) shows a result of Method II. The robot using capsule case succeeded in going through between two objects with translational and rotational motion in real time.

The effectiveness of the proposed method was verified also by simplified experiments using omni-directional autonomous mobile robots as shown in Fig.15. In each picture of the Fig.15, aspects of the robot every 1 second are plotted. The robot recognized environment by the omni-directional camera. A position of a goal and that of obstacles relative to the robot were calculated by extracting features in images based on objects’ colours. A ball was assumed as the goal and columns were assumed as obstacles, as shown in Fig.15. A dashed circle enveloping a column in Fig.15 corresponds to a dashed circle in Fig.5. As shown in Fig. 15(a), the robot was not able to between two objects without the capsule case. However, as shown in Fig. 15(b), the robot with the capsule case performed translational and rotational motion simultaneously in real time and succeeded in going between two obstacles.

These results showed that motion control without a capsule case made it difficult for the robot to go between two objects due to the largest radius of the robot, even if it would be able to go through by using its shortest radius. Applying the capsule case to a wide robot enhances the possibility of going between two objects.

5. Conclusion

In this paper, the real time control method of simultaneously translational and rotational motions for an autonomous mobile robot, which is horizontally long, has been introduced. This method employs omni-directional platform for the drive system and is based on the fuzzy potential method (FPM). The novel design method of potential membership function (PMF), which is considered the width of the robot by using the capsule case, has been shown. According to this proposed method, the wide-robot can decide the current direction of translational motion to avoid obstacles safely by using capsule case. In addition, by controlling the rotational motion in parallel with the translational motion in real time, the wide-robot can go through narrow distance between two objects. The effectiveness has been verified by numerical simulations and simplified experiments. It has been shown that the proposed method enables simultaneous control of the translational and rotational velocity within the framework of FPM.

How to cite and reference

Link to this chapter Copy to clipboard

Cite this chapter Copy to clipboard

Takafumi Suzuki and Masaki Takahashi (September 9th 2011). Translational and Rotational Motion Control Considering Width for Autonomous Mobile Robots Using Fuzzy Inference, Numerical Analysis - Theory and Application, Jan Awrejcewicz, IntechOpen, DOI: 10.5772/25000. Available from:

chapter statistics

2245total chapter downloads

More statistics for editors and authors

Login to your personal dashboard for more detailed statistics on your publications.

Access personal reporting

Related Content

This Book

Next chapter

Obstacle Avoidance for Autonomous Mobile Robots Based on Position Prediction Using Fuzzy Inference

By Takafumi Suzuki and Masaki Takahashi

Related Book

First chapter

Application of the Lyapunov Exponents and Wavelets to Study and Control of Plates and Shells

By J. Awrejcewicz, V.А. Krysko, I.V. Papkova, Т.V. Yakovleva, N.A. Zagniboroda, М.V. Zhigalov, A.V. Krysko, V. Dobriyan, E.Yu. Krylovа and S.A. Mitskevich

We are IntechOpen, the world's leading publisher of Open Access books. Built by scientists, for scientists. Our readership spans scientists, professors, researchers, librarians, and students, as well as business professionals. We share our knowledge and peer-reveiwed research papers with libraries, scientific and engineering societies, and also work with corporate R&D departments and government entities.

More about us