A Mobile Platform for Nursing Robot 1985

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© 1985 IEEE. Reprinted with permission, from IEEE Transactions on Industrial Electronics, Vol. 32, No. 2, pp. 158-165.

 A Mobil M obile e Pl Platfo atform rm byfor f or N Nurs ursing ing Ro Robots bots JOHANN BORENSTEIN

AND YORAM

KOREN 1

Abstract This paper paper descri describes bes a computer computer-controlled -controlled vehicle v ehicle which is part of a nursing robot system currently under development at the Technion-Israel Technion-Israel Institute Institu te for Technology. Technolo gy. The platfo platform rm of this vehicle can also be used for household household robots. robots. Design considerati considerations, ons, control control algorithms, algorithms, and the the necessar y sensor sen sory y devices devices are discussed. The vehicle applies a motion control strategy which avoids slippage and minimizes minimizes position position errors. errors. Experimental Experimental results, results, perfor p erformed med on a prototy pr ototype pe vehicle, veh icle, are ar e described descr ibed as well.

1. Introduction Even though though experts experts seem seem to disagr disagree ee on the feasibility of the all-around household h ousehold robot [1], [2] some mutants of this species are already about to invade the private home [3]. On one hand, there are personal robots, robo ts, advanced toys for the hobbyist, ho bbyist, which are already commercially commercially available, available, but of litt little le practi practical cal use use in in the household. On the other o ther hand, there are highly specialized, sophisticated robots, which are used for security tasks [4] or, as is the case here, for performing services for the disabled. The "nursing robot" system is designed to serve bedridden patients by performing simple services such as operating electrical appliances or bringing objects to the patient's bedside according to the  patient's  patien t's spoken spoken reques request. t. The nursin nursing g robot robot,, howev however, er, is not suppos supposed ed to to apply any any medical medical treatme treatment nt to the patient. p atient. The workplace workplace of such a robot would would be usually usually confined confined to one one room, either either in in a hospital hos pital or in the patient's home. This definition is important, important, since the the constant presence of th e  patient as a supervisor for the robot's activities activities greatly greatly facili facilitates tates the the design of the the robot in in general and of the robot's mobile base in particular. Most of the design considerations of the nursing robot are also applicable to household househo ld robots. Thus the mobile base will be discussed throughout this paper in general terms as a mobile platform for either nursing or household robots. 1

Manuscript received June 1994; revised January 1985. This paper was supported in part by the Technion under Grant 003-712. The authors are with the Faculty on Mechanical M echanical Engineering, Engineering, Institute for  Technology, Technion, Haifa, Israel.

1

 

II. Design Considerations There are several major differences between industrial robots and nursing or household robots. 1. A nursi nursing ng or household robot must be mobile mobile in order to reach a variety of working sit sites es within the house, house, whereas whereas an industrial industrial robot is usually stationary. statio nary. Even when w hen mobility mo bility is required requ ired of the industri ind ustrial al robot, high accuracy accuracy is usually obtained by means of rails or guided wires imbedde d  beneath the the factory factory floor. floor. Obviously Obviously such means means can not be used used at home. 2. The domestic domestic environment environment is largel largely y disordere disordered. d. This is not not a disrespect disrespectful ful comment o n housewives housew ives,, but rather rather a comparison comparison to the well-defined work wo rk site of a robot in industry. It is for  this reason that object acquisition by a nursing/household robot must be based upon detection by sensors, rather than on absolute memorized locations. 3. While high-speed high -speed and low-cycle time are of major importance for the industrial robot, thes e attributes are less important for the household robot, and even less so for the nursing robot.

A. The Mechanical Design

One design of a mobile robot is used on the HERO 1 personal robot [5]. [5]. It employs a dc motor driven wheel, which is also rotated about the vertical axis with the the help of a stepping motor. Two additio add itional, nal, independent wheels on the rear axle provide stability. Even though the the drive wheel i s supplied supp lied with an optical encoder for position position feedback, it has been found impossi impossible ble to achiev e acceptable path repeatability with this drive configuration [6]. The design frequently used for computer-controlled vehicles consists of two drive wheels, each with its own o wn controlled co ntrolled dc motor or stepping motor [7] - [9]. One or two free-wheeling castor s  providee stabilit  provid stability. y. A similar design des ign was chosen for the the platform platform of the nursing robot, robot, as shown in Fig. 1. Two d c motors with built-in reduction gears and optical encoders drive two rubber wheels, constituting the front axle of the vehicle. In the rear, twotic free-wheel free-w heeling ing Castors castors c astorss provide for  sta static stability. stabi lity. Castor have been said to cause slipping at direction changes [10] , but this is not necessarily so, as shown in the Appendi Appe ndix. x. Another point to consider  is the distance between the two drive wheels which depends on the width of  the platform. It is desirable to place plac e the two drive wheels as far apart as  possible,  possib le, for the the following following reasons. reasons. 1. The stati staticc and dynamic dynamic stabili stability ty of  the vehicle are improved. Figure 1: Bottom view of the mobile platform. 2

 

2. The infl influence uence of the the encoder encoder resolut resolution ion on the the orient orientati ational onal error of the vehicle veh icle is decreased. This may be be seen seen by by assumin assuming g that that the vehicle vehic le is at rest. If one of o f the wheels whe els is then the n turned turne d an amount am ount  just within the encoder's resolution unit, and the other wheel remains at rest, then the vehicl e would change its orientation orientation by rotating rotating about the fixed fixed wheel and cause an error in in th e subsequent motion. The effect of this p henomenon henom enon is reduced reduce d by increasing the distance distan ce between the drive wheels. 3. During straight-line motion, mechanical and electrical dis disturbances turbances will cause the motors to run at different different angular speeds, resulting resulting in a temporarily tempo rarily curved cur ved path. pa th. It can be seen see n by trigonometr trigon ometry y that the radius of the curved path is directly proportional to the distance between drive wheels. In the present design the distance between the two drive wheels is 600 mm.

B. The Controlle Controllerr Hardware Hardware

Fig. 2 shows a block diagram diagram of the platform platform controller. controller. For each eac h motor, mo tor, the th e compute com puterr issues issue s an 8-bit 8 -bit  binary speed command which is converted converted into an analog signal, amplified amplified,, and used to drive th e motor.. An optical encoder produces motor produces two 90 phase-shif phase-shifted ted pulse pulse trains trains which which are fed fed into a directional tional sensi sensing ng circuit (DSC) that issues an appropriate app ropriate pulse train to a 4-b 4-bit, it, up-down counter. co unter. The

Figure 2: The controller hardware.

3

 

counter serves counter serves as a buffe buffer, r, since since the the encoder encoder pulses are transmitted tran smitted faster than can be sampled. Both counters are sampled simultaneously simultaneously and reset before before the next encoder pu lse arrives. An inhibit signal sign al is provide provided d by the DSC DSC in order order to avoid th th e counter cou nter reading readin g at the instant when its state is changed. changed . The optical encoders are mounted on the respective motor shafts and their resolution is such that one pulse represents 2 mm travel of the drive wheel. Several Sev eral mechanically interconnected microswitches microswitches are positioned around the the vehicle so so tha t collisions coll isions may be avoided in time by bringing bringing the vehicle to a stop before hitt hitting ing an object. Th e controller also enables manual steering of the vehicle, with a joystick, which is not shown in Fig. 2. C. The Programming Language

The control con trol algorithm algorithm has been implement implemented ed in FORTH FORTH on a low-cost low-cost personal personal computer. computer. A s opposed to the approach of either writing in assembler language, or using high-level language on a development system and downloading the object code to a task computer, using FORTH offers the following advantages [11]. 1. FORTH is much easier to to write than than assembler assembler language language and and is only slight slightly ly slower slower in mos t applications. 2. Since FORTH FORT H is very very compact, it may be residen residentt in the task task computer computer with with all the the periphera periphera l devices (e.g., (e.g., ADCs, DACs, I/O I/O ports, timers) timers) connected. connected. The programmer may thus thus addres s these devices interactively. This is especially important importan t for eliminating faults (debugging) which occur in conjunction with signal flow between the software and the hardware devices.

III. Motion Control Motion Control means the strategy by which the platform approaches a desired location, and the implementation of this strategy. A. The Control Algorithm

In order to represent represent the the platform platform locatio location n relative r elative to a fixed coordinate system sy stem (Fig. 3), three values mustdrive be given: the and the X X- and Y -coordinate -coordinate of the centerpoint C, which is located midway the two wheels; , which is is the angle between the the vehicle's longitudinal axis andbetween the the  X -axis. -axis. If the vehicle has to travel from a known present location ( x ( x0, y0, 0) to a new location ( x f, yf, f ), then the fol followi lowing ng procedu procedure re is is performed (see Fig. 4). First, the length d  and the slope of the straight line connecting the present and final locations are calculated as  y f   y0 arctan  x f   x0



(1)

(  xx f   x0)2 (  yy f   y0)2

(2)

4

 

Subsequently, the following strategy is performed. 1. The vehicle vehicle first first turns turns about its centerpoint centerpoint through an angle angle - 0. 1 =

1

, which is calculate calculated d by by

2. The vehic vehicle le then then trave travels ls along along a straight line through a distance distanc e d . As a result the centerpoint will  be at ( x f, yf ). 3. Finally, Finally, the vehicle vehicle turns about its centerpoint, centerpoint, through an angle

2

, where

2

=

 f 

-

.

For each of these steps a certain number, called the terminal  pulse  pulse count count (TPC), is calculated. The TPC represents repre sents the number of pulses pulses that each motor has to to produce in order to complete th th e command which can be either rotation or straight-line motion along a distance d . The TPC is always equal for both motors. However, during the platform platform rotation rotation both motors rotate in in opposit e directions, but during straight line motion they rotate in the same direction. Any movement movement between between two given locations loca tions is performed in the sequence described above. The  peculiari  pecul iarity ty of this approach is that it actually actually uses only two distinct distinct kinds of of motion: motion: either motion in a straight line, where both wheels run at the same angular speed in the same direction, or rotation about the the centerpoi centerpoint nt C, where where both both wheels wheels run at the same angular an gular speed spee d but in opposite oppo site directions. This simplification offers numerous advantages. 1. Since Since in either either case case the only only task task of the contr controll oller er is to to maintai maintai n equal angular velocities (measured in pulses pul ses per time time unit), a relatively relatively simple simple control system system may be utilized. utilized. This system will will b e discussed later. 2. Both wheels wheels are either either simultaneously running or standing. Therefore, no case may occur where one wheel is running running while the the other one is is standing, standing, which which would inevi inevi tably cause severe slippage. slippag e. 3. The platform platform path path is always predictable. predictable. 4. The platform platform always travels travels through the shortest shortest possible possible distance distance (straight (straight line or rotation rotation "on the spot").

Figure 4: Procedure for traveling to a new location.

Figure 3: Representation of the platform location relative to a given coordiante system.

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B. The Controll Controller er

A conventional controller controller for a mobile robot would consist consist of two two indepe ndent nde nt velocit ve locity y control co ntrol loops, loop s, one for each motor, similar similar to to the control control loops used to drive the worktable w orktable in CNC milling machines machine s or Cartesian Cartesian robots robots [12]. [12]. Motion coordination coordination in these system system s is achieved by adjusting adjustin g the reference referen ce velocities of the control loops, but the loop of one axis receives no information regarding the other. Any load disturbance disturbance in in one of the axes axes causes an error error which which is correc correc ted only on ly by its own loop, while the other loop carries on as before. This causes an error in the resultant path. An improvement in the path accuracy can can be achieved achieved by providing providing cross-coupl cross-coupling, ing, whereby whereb y an error e rror in either e ither axis a xis affects affe cts the control control loops of both axes. Such a cross-coupling cross-coupling method was applie applie d to a Japanes Jap anesee mobile mob ile robot rob ot [13]. In this design each loop uses the position error of the other loop, but a signal proportional to the resultant path error is not generated. The controller used here, however, applies an approach similar to the cross-coupled cont controller, roller, which whic h has been found found advant advantageou ageouss for for two-axis two-ax is NC and CNC systems [14]. In this design the path error er ror is calculated and fed as a correction signal to the loops. The main difference between th th e  present  prese nt design and the one used in CNC systems systems is that here the controller controller always maintains maintains th e maximum allowable speed of the motors. The block diagram of the proposed proposed controller controller is shown in Fig. Fig. 5. Pulses fr from om the encoders ar e counted in the hardware hardware up-down counters, counters, which which also indicate the the directio directio n of rotat r otation ion of o f the motors mot ors  by coun counti ting ng up for for clockwise rotation rotation and counting down for counterclockwise counterclockwise rotation. rotation. Just prior  to being reset, reset, the contents of both counters are simultaneous simultaneous ly sampled (approxima (a pproximately tely every 50 5 0 ms) and added a dded to the associated software counters. Thus each software software counter counter holds a number tha tha t represents the total number of pulses generated since sin ce the beginning beginn ing of a certain motion. Comparison Compa rison  between the absolute absolute values of both software software counters produces the error error signal  E   E  where i where  E  =  E  | N 1i | -  N   | N 2i | . i = | N 

(3)

This error signal might generate a variable  M   M  defined i defined by  M = K | E |  i

p

(4)

i

where K where  K p is a proportional gain. A nonzero E  nonzero  E  indicates i indicates that one motor has been running faster then the other, and the sign of  E  i identifies identif ies that motor. The speed of the faster motor is then reduced by subtracting  M   M  i  from its reference-velocity  R  R,, and leaving the reference velocity of the slower motor unaltered (this i s

6

 

Figure 5: Cross-coupled control loops.

indicated indicat ed by by the dashed line in Fig. 5). Thus the velocities of both motors are effectively equalized. Unless a disturbance occurs, both motors are fed by their maximum allowable voltage (except for  the acceleration and deceleration phases). However, when a disturbance occurs, an error error signal is generated, which, in turn, produces the correction variable  M   M .i. Since the motors are already fed by their maximum voltage, voltage, the the controller controller always subtracts subtracts t he manipula ma nipulated ted variable var iable from fro m the approp a ppropriate riate ref referen erence-v ce-velo elocit city. y. This is legitimate, since it is the relative velocity of the motors, rather then their  absolute velocities, which is of concern and must, therefore, be controlled. Any temporary disturbance of the steady-state velocities will be successfully corrected by thi s  propor  pro portio tional nal (P)-t (P)-type ype control controller ler.. However However,, a continuous disturbance, as might be caused by different friction forces in the bearings (e.g., due to asymmetric load distribution), requires that different  voltages be supplied continuously to the the motors motors for a straight line motion. The P-type controller will supply different voltages only if a constant difference between difference  between the pulse counts of both motors i s maintained. maint ained. This is the case when the vehicle has traveled through a short arc, and has thus (un desiredly) changed orientation before resuming the straight line motion again (Fig. 6a). In order to overcome this problem, an integration ( I  ( I ) action action must be added added into the controller. contro ller. The  PI -controller -controller provides provides not only only equal velocities, velocities, but also eq ual overall ov erall pulse pu lse count cou nt from the beginning be ginning of each motion. Therefore, this this controller controller guarantees a zero steady-state steady-state orientation error of th e  platform  platfor m for any constant constant continuous continuous disturbance. disturbance. With the PI  the PI -controller, -controller, a continuous disturbance will only cause a temporary change in direction. Afterr correction by the controller, Afte controller, the former direction will be resumed, leaving only a paralle l distorti dist ortion on ( ) of the the actual actual path path (Fig. (Fig. 6b). 6b). 7

 

For the PI  the PI -controll -controller er the equation equation of o f the controller   becomes i

 M i

 K c

n 0

 E n  K  p E i

(5)

where K  where  K  is c is the integration constant. The platform controller is easily implemented and requires req uires only minimal computational effort. Since Sinc e  partt of the required calculations  par calculations (i.e., d , , 1, and 2 ) are performed before the nursing robot actually  begins  begi ns to move, they may be performed performed by the th e  platform  platf orm computer without without affecting affecting the samplin samplin g Figure 6: The effect of a continuous asymmetric load on the path. rate. a. With P -controller.

b. Wi With PI -controller. -controller.

IV. Experimental Results A prototype prototype of the the platform platform has been built and tested. tested. Experiments with the the proto prototype type have hav e shown sho wn that th at the parallel distortion, inherent in the kind of control employed, is very small, on the order o f  magnitude of 10 mm per 10 m straight line travel. Fig. 7 shows shows a typical typical path of the platform, p latform, carrying an asymmetrically distributed load (no (note te the different scales for the Xthe X- and Y -axis -axis in Fig. Fig. 7). The plot was obtained obtaine d by real-time calculation of o f the momentary position position of the the centerpoint, centerpoint, which will will be explained explained later later . As may be seen see n from the graph, gra ph, any disturbance disturbance causes a temporary temporary deviation from the original origin al direction. direction . This disturbance disturb ance however, how ever, is corrected and the vehicle continues in the original direction. During Duri ng the the very last phase of the motion, the vehicle is decelerated. Lowering the input voltage to the motors gradually emphasizes the influence of friction in the bearings, which affects the control loop as a ramp disturbance. disturbance. This effect effect causes causes an orientatio orientatio nal error, erro r, which whic h is corrected cor rected automatica auto matically lly after the deceleration phase by an overshoot correction phase (typically only a few pulses long, thus not recognizabl recognizablee on the graph). graph). The final error of the platform p latform (due to controller-dependent contro ller-dependent effects, e ffects,  but not incl includi uding ng position position errors errors introdu introduced ced by mechanical mechanical inaccura inaccuracies) cies) in this this experiment experiment was less less than 3 mm after traveling a distance of 4 m.  Notee that  Not tha t even though decelerated, decelerated, the vehicle approaches the desired position position with a certai n velocity which causes causes an overshoot by a few few millimeters. millimeters. In order to correct correct this this overshoot, overshoot, eac h motor is independently moved until it reaches the previously calculated TPC which corresponds to the desired location. This action is performed without using the speed speed controller. In another another test, test, the vehicl vehiclee was programm programmed ed to travel along a figure eight path. The path actually followed was calculated in real-time and is shown in Fig. 8. After returning to the original starting location, the vehicle's calculated position error was only 8 to 10 mm transversal and less than 1 rotational. rotation al. Again, this this error does not include mechanically caused inaccuracies. The real position error,, including error including mechanica mechanicall inaccuraci inaccuracies, es, was about 5 to 8 cm transv transversal ersal and 1 rotational. These 8

 

Figure 7: Parallel distortion from the required r equired path. a. With asymmetric load. b. Without load.

res ults compare favorably to the results of a similar experiment, described in [15]. However, th e results vehicle used in [15] was faster, heavier, and did not halt at the corners of the programmed path. The odometr odometric ic techni technique, que, which was employed to calculate the actual momentary location loca tion of the centerpoint centerpo int is described in [8] and [15]. Constantly updating this information information allows us to find the  platform's  platf orm's final final position position,, independent independent of the control control algorithm algorithm.. By compa co mparing ring the actually measured final final position of the vehicle to either either the final positio n achieved by the cross-coupled control or to the odometrically calculated final position, no significant difference of accuracy for determining the the actual final position could be found. This indicates tthat hat any inaccuracies in the final position are caused by control-independent effects, some of which are listed below. 1. The most most ssign ignif ificant icant inaccuracy is caused by directional uncertainty due to the limited resolution of the the encoders. encoders. This problem problem may be partly solved by using encoders with very high resolution,  but this this woul would d require require higher higher sampling sampling rates in in order to maintain maintain smooth smooth control. control. Increasing Increasing the distance dist ance between between drive wheels will also improve accuracy, since a single encoder pulse will w ill have less influence on the platform's direction. 2. It is difficult to obtain rubber wheels with exactly exactly the same diamete diameter. r. In additio addition, n, unequall unequall y distr dis tribut ibuted ed loads loads will will slightly squeeze one wheel wh eel more than the other, thus changing c hanging its diameter. Wheels with w ith different diameters cause the vehicle to travel along an arc, rather then along a straight straight line, line, even if if the motor motorss are running runn ing at exactly equal equa l speeds, The radius of this arc and the orientational error are easily found from trigonometry

9

 

 R

b ( D  D u) u

 L  R

bD u

(6)

 Lu  Db

(7)

where  R = radius of the curved path due to different wheel diameters, = orientation error, in radians,  L = distance traveled, u = difference of diameters of both wheels, b = distance between drive wheels,  D = nominal diameter of drive wheels. In our platform, b = 600 mm and D and D  = 1 14 mm. For an assumed u = 1 mm, the platform performs a curved path with R with R  = 600 × 114/1 114/1 = 68400 68400 mm = 68.4 68.4 m. If the platform travels along a straight line, through a distance L distance L = 10 m, the orientational orientational error becomes = 10/68.4 = 0.14 rad = 8.3 . Obviou Obv iously, sly, this is an intolerably large large error which emphasizes the necessi necessity ty for a rigid whee l design. 3. There is a contact area, area, rrathe atherr than than a contact contact point point,, between between the the wheel wheel and the the floor. floor. Thi s causes an uncertainty about the effective distance distance between the drive wheels, creatin g inaccuracies when turning.

V. Sensors for Absolute Positionin Positioning g

Figure 8: Actual trajectory for the figure-eight shape path.

10

 

As has been pointed out before, before, a wheel-driven wheel-driven vehicle cannot cannot be expected to reach a give n locati loc ation on with absolute reliability, because of wheel slip, slip, errors introduced by crossing smal l obstacles on the floor, etc. Therefore, some some sort of absolute absolute po sition measuremen meas urementt (e.g., by means mean s of navigation navigation beacons) beacons) must must be employed employed in in order to determine determine the the exact position. positi on. Howe However, ver, since si nce these measurements measurements require relativel relatively y long computation times, times, it has been suggested not t o repeatedly perform absolute position measurements in order to control the motion [8]. The relative position measurement, based on the odometric technique as mentioned before, is much faster fas ter and is performed during motion. The disadvantage of this technique is th e accumulati accumu lating ng error due to wheel slip. Therefore, it it is suggested suggested that both techniques techniques be used : relative relati ve position position measurement measurement during during motion, and absolute absolute p osition measurem m easurement ent when wh en the vehicle is at rest (waiting for new commands). Presently, no sensor has been installed on the vehicle, but it is planned to employ an ultrasonic range-finder ran ge-finder for absolute position measurement (such as in [9]). [9]). If non-stationary furnitur e obstructs obstr ucts the walls walls of the room, room, one corner of the room ro om would wou ld have to be cleared c leared and declared as as the "home" "home " positi po sition on of the vehicle. vehicle. After perfor performing ming a few tasks tasks based based solely solely on the relativ relativ e  posit  pos ition ion control, the the robot would return return "home" to update its absol absolute ute position. Obviously, this this is not no t an absolutely reliable solution, solution, applicable in tthe he general case of the the household robot . Howe Ho wever, ver, in the case of the nursing robot, where the patient can fulfill a supervisory functio n (considered a legitimate robotics approach [16]), [16]), no absolute reliability is required. In case the robot "gets lost," the patient would steer the robot manually, with the help of the joystick, to the approximate home position. An ultrasonic device is also advantageous in that it may fulfill fulfill the additional functions o f  collision avoidance and path planning [9].

VI. Conclusions A contr co ntrol ol strategy strategy for mobile robots robots has been presented. presented. Even though some some aspects of thi s strategy strate gy apply only to human-supervised human-supervised robots, the main cont cont rol algorithm algorith m is generally gen erally applicable. ap plicable. Experimental Experiment al results show that the accuracy obtained by the control strategy has exceede d expec tations as well as (in the case of the nursing robot applica expectations application) tion) requirements requirements.. Thes e requirements are defined in relation to positioning errors due to unequal wheel diameters, encoder  resolution, slip, and small obstacles overrun by the wheels. The experiments proved our assumptions regarding the minor influence of the free-wheeling castors to be correct, as given in the Appendix.

11

 

Appendix: Force Analysis of the Free-wheeling Castor For quasi-static equilibrium (Fig. 9)  F x = 0 = F  sin F r sin

- Fs cos

- C   C x 

(8)

 F y = 0 = F   F r  cos + Fs sin sin  Mc = 0 = T - d F   F s 

- C  y

(9) (10)

where T  = moment moment trans transfer ferred red by ffric rictio tion n at the hori horizont zontaa l  bearing of the castor, castor,  F r  = rolling friction of the the castor wheel,  F  s = lateral friction force on the castor wheel, C  x =  x  x component  component of the force force at the vertica verticall castor ca stor bearing, be aring, C  y =  y  y component  component of the force force at the vertica verticall castor ca stor bearing, be aring, d  = length of castor arm. Figure. 9: Forces acting on a

The moment T  is  is determined by the friction in the horizontal free-wheeling castor.  bearing, and the maximal maximal lateral lateral disturbanc disturbancee on the vehicle occur occ urss at = 90 . The Then, n, C x = Fr, Cy = T/d and the equations of equilibrium for the platform can be formulated in terms of T  and F   and F . r . (Fig. 10):  F  =  F  =  Ax + B  + Bx + 2C  2C x  x = 0 = A

(11)

 F y = 0 = 2 C  C   Ay  - B  By  y - A

(12)

 M  B = 0 = -2T  -2 T  +  + 2C 2Cxh  - A  Ayb + Cyb 

(13)

This analysis assumes a distance b between the the castors. castors. Assuming Assuming symmetry symmetry  A x = B  = Bx = C  = C x = F   F r 

(14)

 Ay = 1/b 1/b (  (C C yb + 2C 2Cxh - 2T  2T ) (15) 1 Tb b d 

2 F r h 2T 

1 b T  2 b d 

where

2 F r h

Figure 10:  Forces acting o on n wheels of the platform.

12

 

 A x, Ay  B x, B y b h

= x  x and  and y  y components  components of friction force at wheel A, = x  x and  and y  y components  components of friction force at wheel B, = distance between drive wheels, = distance between drive wheel axis and castors.

Thus in order to prevent prevent slippage, slippage, the following following inequalit inequality y must be satisfied. satisfied.  Nf  >  A x2  A y2

(16)

or   Nf  >

 F r 2

1 b

2



b d 

2

2

(17)

2 F r h

where  N  = normal force on the drive wheels,  f  = coefficient of friction. It may be seen from (17) that in the ideal case of F of  Fr  = 0 and T  = 0, no slippage will will occur. This ideal case does do es not exist, of course. course. However, since since  N, F , F r , and T  depend  depend strongly on the load distribution, distributio n, (17) may be satisfied by positioning positioning the load (normally the robot arm) close to th e dri drive ve wheels wheels (thus far far from the free-wheeling castors). This will Increase  N  and   and decrease F  decrease  F  and r  and T . It is also evident from (17) that b should be as large as possible, while h should be as small as  possible  poss ible.. This desig design n will be limite limited d by static static and and dynamic dynamic stabili stability ty consider consideratio ations. ns.

References [1]

J. F. Engelberger,  Robotics in Practice, Practice, K. Page, Ed. London: Aveburry, Aveburry, 1980, p. 135.

[2]

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