Design of Steering Mechanism and Control of

Proceedings of the 2000 IEEE International Conference on Robotics & Automation San Francisco, CA • April 2000

Design of Steering Mechanism and Control of Nonholonomic Trailer Systems Yoshihiko Nakamura *l Dept. of Mechano-Informatics *l University of Tokyo

7-3-1, Hongo, Bunkyo-ku Tokyo 113-8656, .lapin [nakamura,hide]¢Oynl.t.u-tokyo.ac.jp

Hideaki Ezaki .1

Mechanical and Electrical Engineering .2 Wuhan University of Technology

122 Luoshi Road Wuhan, P.r. China tanzhang~pu blic.wh.h b.cn

Introduction

Nonholonomic mechanical systems have received much attention in tile fields of robolics and control. Laumond[1] proved hy construction that a car-like robot is controllable even wii,h the limited steering angle. Laulnond and Sim6on extended the heuristic approach to a mobile robot with one l.railer[2]. Theoretic approaches, then, followed in control commmfity based on the differential geometric framework. Murray and Sastry[3] named a simple nonlim'a.r system the chained form and proposed to use it. as a canonical form for a class of drifth'ss syst.ems. S0rda.len[4t] showed a mobile robots with n trailers to be convertible t.o the chained form, where a trailer is assumed a passive two-wheeled car without steering and being connected to the previous trailer/tra, ctor with a kingpin hitch at the center of wheel shaft of the previous

0-7803-5886-4/00/$10.00© 2000 IEEE

Woojin Chung *a Robotics Research Center *a

Korea Institute of Science and Technology 39-1, Hawolkok-Dong, Sungbuk-ku

Seoul 136-791, Korea wj('hu ng(@kistmail.kist.re.kr

one. Being converted to the chained form, a tractor and all trailers can be stabilized to desired positions via. esla.I)lished nonholonomie motion control strategies for the chained form[5, 6, 7]. S0rdalen, Naka~ mura and Chung[8] developed "t,he nonholonomic manipulator," whose kinematic model is similar to those of trailer systems. The real-world tra~ject.ory generated by the nonholonomic motion control tends to contain high frequency oscillations or large overshoot, although the mathematical trajectory in the chained form enjoys a nice convergent profile. Chung and Nakamura[9] then proposed to consider mathematical simplicity in the chained-form transformation and designed "the chained form manipulator." On tile other hand, trailers have been developed and widely used in the industry. The main focus of such design was on reducing passive tracking error fi'om t,ractor's trajectory, and little attention was paid on active motion cent.tel. Automated trailer systems will find an important part in automated factories, ports or airports, if the technology is available. Nonholonomic motion planning/control will be applied when loading and unloadtug freighl.. In this paper, we all.erupt to fidfill the gap between t.he nonlinear control theory for nonholonomic trailer systems and the mechanical design for good passive tracking performance of indust.rial trailer systems. Based on the knowledge of clm.ined form transformation, we design a new passive steering mecha.nism and show that it, provides a trailer with the both properties; namely, the chained-form convertihility and a good passive tracking performance. A proretype is fabricated and used t.o verify the proposed design theory.

Abstract

A wheeled mobile robot with trailers has been. studied as a class of nonholonomic systems. It is proved that a system of a tractor and trailers with an app,vpriatc connecting "mechan,ism can be stabilized to desired positions via nonbolonomic motion control. Trailers, on the other hand, have bee,t developed and widely used in lhe industry. The m.ain focus of industrial design is set on reducing tracking error from a reference trajectory, This paper atlempls to bridge over the gap between these lwo approaches. We develop a design theory of trailer systems with passive steering. The designed systems show a good performance in practical path followi.ng, and accept the chained form transformation and nonlinear control st~ttegies for nonholoheroic systems. Keywords: Nonholonomic constraints, Trailer systern, Passive steering, Path following, Practical mechallisln

1

Yuegang Tan .2

247

Kinematic Models with Steering

2

2.1

The Kinematic

1 Oi = "-7(-Gi sin ¢i + sin(0i_l - Oi + ¢i--1) )Vi--1 (6) al

of Trailer System

i

(7)

vi = H Gjvo

Model

j=l

We will compare performances of different mechanical design of trailers with and without, steering. The most general design of trailer system is shown in Fig. 1, from which all the comparative designs we discuss are derived by appropriate choice of parameters.

(i= a,...,,0 where Gi dgr cos(0i_l -- Oi + 0i-1)

Note that the above model is valid ill a limited range of 7r 71--g

r~l

+ s i n ( 1 - r)(Oi_] - Oi) }vi-] 1 = - ~-,.{sin r(Oi - 0i+])

10 ~

t0

is

x

~0

~s

5o

'

0

g

w

i~

1

k0

~s

~

~s

ITS model

Yamamiya model

Figure 10: Motion of trailer system with sinusoidal trajectory.

We then reverse indexing of a tractor and trailers by replacing Oi+a wil,h 0i-1, and vi with vi-1 in Eq.(16). Then, =

rant1

-

,.)(0,:

-

(17)

The reader should notice that the above equation is equivalent to Eq.(6) with 0i = (1 - r)(Oi+a - Oi). In other words a backward motion of the system with sleering gain r is equivalent to a forward motion of the system with steering gain 1 - r. Therefore, if and only if r = 0.5, a trailer system shows non-zigzag motions in both forward and backward directions.

Backward motion of trailer systems has not been discussed in the literature, which seems due to the fact that it apparently tends to show a zigzag or unstable motion. Figure 11 shows the results of simula.lion when the tractor drives trailers backward along the dotted trajectory.

Since the fi-ont trailer in the backward motion has fixed steering On --- 0, the trailer system has to followed the tangential straight line. Therefore, if we provide the n ~h trailer with an active steering, the new trailer system can be completely driven backward direction as well as in the forward direction. In the forward motion, we fix ¢,, to 0. In the backward motion, the angular velocity of the n th trailer is driven by the n th trailer's on-board actuator, while the linear velocity of the n ta trailer is driven by the tractor's pushing force. The tractor is controlled so as t.o keel) the wheel shaft crossing perpendicular to the first, trailer's rod. The pushing velocity v0 is calculated from v,,. via Eq.(7).

Optimal three-poin! mode]

i

5 5.1

Yammniya model

-

+ s i n r(0;_l - 0i))}vi-i

Motion

S¢,rdalen model

1

g{cos,-(0f

i

The behavior of these models, in general, resemble the three-point model with r < 0.5.

Backward

(16)

+ cos r(Oi - Oi+~)tan(1 - r)(0i-] - Oi)}vi

Table 2: Error from reference trajectory Sordalen Y a m a m i y a l ITS elothoid 20.11 10.85 i 2.29 circle 10.75 4.17 ii 1.66 , sinusoid 2.43 1.94 , 2.12

4.3

(15)

Differential Kinematic Analysis and Control of the Three-Point Model Chained

Form

Convertibility

The chained form[3] is a canonical form for a class of driftless nonholonomic systems. Thanks to the simplicity of the chained form, many usefid control schemes have been devek)ped. 'Fhe sufficient conditions[7] For converting of a kinematic model to the chained form are, (A) Kinematic model being convertible into a triangular structm'e. (B) The triangular structure satisfies Eq. (43) in apl)endix A.

ITS model

Figure 11: Backward motion of trailer system with circular trajectory. Except for the optimal three-poinli model, all the trailer models showed the zigzag motion. The optimal three-point mode/ resulted in stable motion, but followed a straight line that is tangentially connected to the circular reference. The reason is explained as

251

Of.+2

Equations (3)-(8) are rewritten as,

1 - r

- -

0aa+l y = v tan0,, 1 Oi = ~ 7 ( - G i sill 0 i + s i n ( 0 i - I

(18) --

i

n

cosOnI-Ij=iG j

0f,,+a (~Xn+ 2

Oi + ¢ i - 1 ))

(i = l , - . . , n )

-

d, cos x,~+~ cos2(1 - r ) ( x , + l - x,~+2) 1

(31)

COS2 J2n+ 2

In the range of Eq.(9),

(19)

COSr(Xi -- X i + l ) :~ 0

def ,

where, v = x. If 0i = (}i(Oi-1), we call converl the kinematic model into an n + 3 dimensional triangular structure by setting (20)

[,,

.., 0,,,

(21)

5.2

(i=l,.

(22)

..-1)

= 3,-..,

n

+ 1)

(32)

Control

for the Chained

Form

We will show, for exanlple, an open-loop control with time polynonlials[13]. The inl)ut of the chained form is written by

(23)

0,, = 0.(0,,-~)

(i

Therefore, condition (B) is satisfied if v # 1. This concludes that tile three point model is chained-form convertible. Scrdalen and ITS nlodels also satisfy tile chained-form convertibility, hut Yamanliya nlodel or most conlmon design of trailer syst(,ms ill industry do not enjoy this property.

Tile tllree point model sat, isfies 0i=0i(0~)

(30)

Next., f i in Eq. (40) are expressed as,

f3=

v2 = co + e i t

d l cos x n + 2 COS r(Xj

n-I-ii

fi ~

vl = b0

tall r(xa - x4) + tan(a~2 - x3)

tau

r(xi

J2i+l) "4"-t a n ( 1

- -

(24)

-- Xj+I)

-

r)(Xi_ 1

- -

COS r(.Xj -- X j + i )

Xi)

(25)

3:z

(i=4,..,,n+l) tan(1 - r ) ( x . + l - x.+2) fn+2 =

+ " " + c,,_~(~)

(34)

.

The coefficients are determined from the boundary conditions of z, which is calculated from the initial and final position. Figure 12 shows that tile optimal three point model with 2 trailers being controlled by the time polynomial inputs. Also shown in Fig. 12 is the result of the S0rdMen's model with the same open loop control strategy. 'File both nlodels exhibited a similar response to the open loop control.

di-2 c o s x n + 2 n+l

(33) r~--2

(~6)

d n COS a~n+ 2

(27)

f n + 3 ---- t, a n x n + 2

which yiehl,

0 .r "e

1 di cos a.,,+,, cos'-'(x=, - x3) ,~+1

-- J ' j + l )

SordMen's model

(28)

T h r e e l)ohlt, model(r = 0.5)

inl0at

Of; Oxi- j

COS r(d'j

-

tnl~

r

di-2 c o s a , + 2 cos'-( 1 -- r ) ( x , _ 1 -- x i ) ,,+1 • II

cos,'(a'j - "'~+A!

(29) 5

S, m d a | e n s model

(i=4,....n.+l)

t

a

2

~

g

T h r e e point, m o d e l ( r = 0..5)

Figure 12: Open loop Control wMl polynondal input.

252

6

o f linear and circular trajectries. Figure 16 s h o w s tile

Prototyping and Experiments

6.1

Prototyping

In this subsection we propose some mechanisms for implementing optimal three-point model. In Example

l

-l.ill ~

~ ~ Rodof~

:

backward motion, where trailer n is steeered as mentioned in subsection 4.3 to follow a circular reference trajectory, Note that. the trailer system keeps n o n z i g z a g motion.

' ~/l g o d

of traileri

"

G ~

tt'fdleri+l !

Figure 13: Mechanism of three point model. 1 of Vig. 13, gears A and B are fixed t,o rods i and i + l, respectively. The body of trailer i is fixed to rod i. If rod i rotates, rod i + 1 also rotates in the opposite direction. If the radius of gear A is k times longer than that of B, we have r = 1/(1 + k). When radii of gears are equal, r is 0.5. If r is limited to 0.5, Exampl< 2 is also frasible. Mechanism D is fixed to E and slides on the axle. Pins F and G are to comlect t o rods i and i + 1 and slide on E. We adopted Example 2 to avoid mechanical complexity and backrnsh due to gear trains. Each intermediate trailer can be switched between r = 0 and 0.5. This allows us to compare path following properties of the optimM three-point model, the Serdalen model and the ITS model. Trailer n has an actuator that rotates rod n about the orogin of frame n. 6.2

Figure 15: Experiment of forward motion.

Figure 16: Experiment of backward m o t i o n .

'7 Conclusion

Experiments

This paper is summarized as follows: (1) As a basis to study trailer systems that show both path-following performance and the chained form convertibility, a kinematic model of trailer systems with steering is formulated. (2) We proposed a passive steering mechanism named the three-point model. Simulation results showed that the optimal three-point model wMl steering gain r = 0.5 is superior to known typical industrial trailer systems in path following perforniance. (3) Path following stal)ility was compared and discussed anlong the industrial trailer designs and lhe optimal three-point model, It is shown by nmnerical analysis and theoretical development that only the optimal t,hree-point model can trace the path of the front car in both forward and backward directions. (4) The steering mechanism of the three point model was proved to make lhe system differentially fiat and convertible to the chained form. (5) We designed and fabricated a prototype of the optimal three-point model with simple mechanisnl. Experiments verified properties that we discussed in this paper.

We fablieated the prototype of the optimal three point, model. Experiments of forward and backward path following were carried out with the prototype. Figure 14 shows photos of the prototype. The prototype consists of a tractor and five trailers. The top two I)hotos in Fig. 14 show the structure of the first. through fourth trailers, The bottom two show the fifth one. The experiments correspond to Example 2 of the previous subsection. Figure 15 shows the opti-

Fib, . . . . . . . . . .

,. . . .

~,

n

real three-point model following a connected reference

253

A

Acknowledgment T h i s research was s u p p o r t e d by t h e " R o b o t Brain P r o j e c t , " C R E S T P r o g r a m of J a p a n Science and Technology C o r p o r a t i o n , J a p a n .

Chained

Form

Conversion[7]

T h e chained form

References

:il = v~

(35)

d2 = v~

(36)

ii = z i - l v l

[1] J. P. Launmnd, "Feasible trajectories for mobile robots with kinematic and environment, constraints", in Proc. lnternatiotml Conference on Intelligent A utinomous Systems, pp, 346-354, 1986.

(i = 3 , . . . , n )

(37)

is a canonical form of a class of driftless nonholonomic s y s t e m s , with two i n p u t s and n d i m e n s i o n a l s t a t e space. All t h e s y s t e m s c o n v e r t i b l e to the chained form are c o n t r o l l a b l e .

[2] J. P. Laumond and T. Sim6on, "Motion planning for a two degrees of freedom mobile robot with towing", i~t Proc. h~tern.tional Conference on Control and Applicatons, 1989.

It is difficult 1.o find a conversion from a. k i n e m a t i c model to the chained form. We first convert it i n t o a t r i a n g u l a r s t r u c t u r e . Namely,

[3] R. Murra.y and S. Saslry, "'Nonholonomie motion planning: Sleering using sinusoids', lEEE Tr, nsactions on Automatic ('ontrol, vol. 38, pp. 700 716. 1993. [4] O. J. Sorda.len. "Teedback control of nonholonomic mobile robots", Ph.D thesis ITK-rapport 1993:5 W, the Norwegian Inslitule of Technology, 1993.

& = ul

(38)

~d'e = u'e

(39)

a;i = f i ( x i - ~ ) u l

(i = 3 , . . - , , )

(40)

where def

x_j. = [ x i , " - , x , ]

[5] M. Sampei, "A control strategy for a class of nonholonomic systems - time-state control form and its application : , in Proc. 3&'d Conference on Decision and Control, pp. 1120-1121, 1994.

z

(41)

f i ( x i - t ) dS [ f i ( x i - 1 ) , " ' ' , f . . ( x , , - 1 ) ] 7~

(42)

where f i t ' ) is a s m o o t h function. A s s u m e t h a t at x = p on the c o n f i g u r a t i o n m a n i f o l d we have

[6] C. Samson, "Time-varying feedback stabilization of nonholonomic car-like mobile robots", Technical Report 1515, INR1A-Sophia Antipolis, September 1991.

Ofi(xi-1)

0~_t

[7] O. J. S0rdalen and K. Y. WicMund, "Exponential stabilization of car with n t,railers", in Proc. 32th Confevence on Decision and Control, pp. 978-983, 1993.

I,:p # 0 (i = a , . . . , , )

(43)

T h e n , at the n e i g h b o r h o o d of x = p, a c o o r d i n a t e t r a n s f o r m a t i o n from t h e t r i a n g u l a r s t r u c t u r e to the chained form is given by

[8] Y. Nakamura O. J. So rdalen and W. J. Chung, "Design of a nonholonomic manipulator", in Proc. 1994 l E E S International Conference on Robotics and Automation, pp. 8--13, 1994,

z. = h,(x,,)

[9] W. J. Chung and Y. Nakamura, "Design of the chained form manipulator", in P~vc. t997 IEEE International (.;onfi'rencc on Robotics and A,~tomation, pp..155-461, 1997.

zi =

(44)

h.~(x_k)de_fOhi+l(xi+l) 0a'i+l

fi+l(a;1 )

def

zl = h i ( x , ) = xt (i= 2,...,~--

[10] K. Yamamiya el. al., ":I'raih.r", .Iapan Patenl Office Unexamined Patent (~azetle B 6 2 D I 3 0 0 1t2 18928 (in Japanese). 1990.

(46) 1)

vl = ~q

(47)

Oh.(a~.,)

[11] O. Gehring and lf. Frilz, "Laleral control concepll,s for lruck platooning in the chauffeur project", it~ Proe. ~¢th l'l'orld ('ogress (m ITS, 1997.

(45)

Oh2('r--Z)

(4S)

where, h,,(x..) is any s m o o t h function which satisfies

[12.] 5'. Umetani and S. llirose, 'q~iomechanical study of active cord mechanism with tactile sensors", in Proc. 6th lnter~mtional Symposium on Industrial Robotics, pp. c l , l - 10, 1976.

0/,,~ (.,,) a.,, I.,.:. ¢ 0

(49)

In this p a p e r , we choose hn = 3% to o b t a i n the simplest e q u a t i o n .

[13] R. Murray D. Tilbury and S. Sastry, "Trajectory generMion for the n-trailer problem using goursat norreal form", in Proc. 32th (:onferencc on Decision and (.'omrol, 1993.

254