Steering Kinematics for a Center-ArticulatedMobile Robot

Steering Kinematics for a Center-Articulated Mobile Robot

Peter I.Corke and Peter Ridley

Abstract—This paper discusses the steering kinematics for a center-ar-ticulated mobile robot.Several models have been recently proposed in the literature and we use experimental data to compare and validate these models.The vehicle heading response to steering command is shown to include a dominant zero due a nonholonomic constraint,and this is verified by experiments conducted using a30-t mobile vehicle at our laboratory.Simulation results are then used to show the significance of this zero and its effect on closed-loop heading angle control.

Index Terms—Autonomous vehicle,load-haul-dump,nonholonomic, steering control.

I.I NTRODUCTION

Load-haul-dump(LHD)vehicles(see Fig.1)transport ore in under-ground metaliferrous(noncoal)mines.Operation is cyclic,typically 15–18trips/h,with round-trip travel distances from100to600m.The full or partial automation of underground truck haulage is considerably attractive to industry[1]since it has the potential to improve safety by removing people from the vehicles,while simultaneously increasing productivity.Automation of this type of vehicle has been reported by many groups including[2]–[4],and[5]provides a recent review. The dynamics of the LHD are different to a normal free-steered ve-hicle such as a car or semi-trailer,which is commonly discussed in the literature.As observed by Altafini[6],the centre articulation gives the vehicle considerable maneuverability within the narrow confines of a tunnel.A number of papers[7]–[9]uses a simple’bicycle-type’model for the LHD.Recently,DeSantis[10]and Altafini[11]have de-veloped more complete models of the LHD,which include a nonholo-nomic constraint due to the articulation joint and the wheels.

The front and back sections of the vehicle are connected by a joint, H,with a vertical axis(see Fig.2).The articulation angle is referred to as .

Two linear hydraulic cylinders acting in a push–pull configuration exert forces S and S0to actuate the joint(see Fig.2).In addition to creating a moment about the pivot point,they exert lateral forces that are opposed by tire adhesion forces.In practice,the actuators are strong enough to break adhesion and scrape the wheels sideways across the ground when the vehicle is parked with brakes applied.

Section II introduces our notation and develops a steady-state model of the turning vehicle.This is extended in Section III to a full kinematic model which describes how the vehicle’s pose evolves with time as a function of steering angle and ground speed v.This full model is compared experimentally with the simpler“bicycle”kinematic model and significant discrepancy is observed.Finally,in Section IV,the two models are compared in the context of a simple heading-angle control system.

We assume throughout that there is only point contact between wheel and ground,that the wheels do not slip,and that all motion is planar. Manuscript received April11,2000;revised December6,2000.This paper was recommended for publication by Associate Editor E.Colgate and Editor A.De Luca upon evaluation of the reviewers’comments.

P.I.Corke is with CSIRO Manufacturing Science and Technology,Kenmore 4069,Australia(e-mail:pic@cat.csiro.au).

P.Ridley is with the School of Mechanical Engineering,Queensland Univer-sity of Technology,Brisbane4001,Australia.

Publisher Item Identifier S

1042-296X(01)04809-1.Fig.1.Picture of a typical LHD.This vehicle is around4m wide,10m long, weighs28.5t,and carries8t of

ore.

Fig.2.Plan view of the LHD steering mechanism.W represents lateral forces due to tire adhesion and S and S

is the distance from O to P

introduced for a nonsymmetric vehicle,i.e.,l1=l2(unlike[9]and [10]).For our particular LHD,l1=1:6m and l2=1:8m.

At the point H we can write

a+b+ =180 (1) and for the right-angled triangles4OHP1and4OHP2we can write

d+b=90 (2)

a+c=90 :(3) Substituting into(1)we determine that

c+d= :(4) Next,consider the right-angle triangle4OP2A,for which we can write

l2+l1

cos

=r2tan (5)

sin (r1+l1tan )=l2+l1

cos (6) which can be simplified to yield the turning radii

r1=l1cos +l2

sin (7)

r2=l2cos +l1

sin

:(8) Interestingly,the two halves of the vehicle follow circles of different radii,and the ratio of radii of curvature

r2 r1=l2cos +l1

l1cos +l2(9)

is identically equal to one only for the case l1=l2or =0.

A consequence of this difference in radii is that the ground speed of the two vehicle halves is also different whilst turning.The vehicle has a separate differential for the front and rear wheels,each driven by the output of the gearbox.The input speed to the differential is the mean speed of its two associated wheels

!1=1

2

+

+

Fig.4.Raw data from the

trial.

(a)

(b)

Fig.5.

Yaw rate prediction errors (

s )0(s

)2(s )0(s

)

Fig.7.Simulated steering response for bicycle and nonholonomic models for plant with no rate or saturation limits and K=10and for two speeds v= 1;5m/s.

to a nonholonomic constraint.The bicycle and exact models were then compared in the context of a heading control loop where the effect of the zero on error and input demand is quite marked,particularly at low speed.As vehicle speed increases,the kinematics tend toward those of a bicycle.However,at low speed,such as required when turning tight corners,the difference is significant and given the confined environ-ment must be taken into account.

A CKNOWLEDGMENT

The first author would like to thank C.Altafini,KTH Stockholm,for discussions on this topic.Experimental data was provided by AMIRA sponsored project P517and the CSIRO LHD project team:G.Win-stanley,J.Roberts,E.Duff,P.Sikka,L.Overs,S.Wolfe,R.McCasker, and J.Cunningham.

R EFERENCES

[1]J.Cunningham,P.Corke,H.Durrant-Whyte,and M.Dalziel,“Auto-

mated LHD’s and underground haulage trucks,”Australian Journal of Mining,1999.

[2]L.Bloomquist and J.P.Huissoon.(1996)Navigation system for

autonomous underground mining vehicles.[Online].Available: http://sail.uwaterloo.ca/labloomq/workshop/info.html

[3]https://www.wendangku.net/doc/d116692635.html,ne,“Automatic steering system for an underground mine haul

truck,”Master’s,Colorado School of Mines,1992.

[4]G.Eriksson and A.Kitok,“Automatic loading and dumping using ve-

hicle guidance in a Swedish mine,”in International Symposium on Mine Mechanization and Automation,Colorado,June1991,pp.15.33–15.40.

[5]J.Roberts and P.Corke,“Automation of underground truck haulage,”

in International Symposium on Mine Mechanization and Automation, Brisbane,July1997,pp.b5.23–32.

[6] C.Altafini,“Why to use an articulated vehicle in underground mining

operations?,”in Proc.IEEE Int.Conf.Robotics and Automation,May 1999,pp.3020–3025.

[7]J.Steele,C.Ganesh,and A.Kleve,“Control and scale model simulation

of sensor-guided LHD mining machines,”IEEE Trans.Industry Appli-cations,vol.29,Nov1993.

[8] A.Hemami and V.Polotski,“Path tracking control problem formulation

of an lhd loader,”Int.J.Robot.Res.,vol.17,pp.193–198,Feb1998.

[9]S.Scheding,G.Dissanyake,and E.M.N.H.Durrant-Whyte,“An ex-

periment in autonomous navigation of an underground mining vehicle,”

IEEE Trans.Robot.Autom.,vol.15,pp.85–95,Feb1999.

[10]R.DeSantis,“Modeling and path-tracking for a load-haul-dump mining

vehicle,”Journal of Dynamic Systems,Measurement and Control,vol.

119,pp.40–47,Mar1997.

[11] C.Altafini,“A path-tracking criterion for an lhd articulated vehicle,”Int.

J.Robotics Research,vol.18,pp.435–441,May1999.

[12]P.Bolzern,R.DeSantis,A.Locatelli,and S.Togno,“Dynamic model of

a two-trailer articulated vehicle subject to nonholonomic constraints,”

Robotica,vol.14,pp.445–450,1996.

[13] A.Hemami and V.Polotski,“Dynamics of a dual-unit articulated ve-

hicle for path tracking control problem formulation,”in Proc.Int.Conf.

Control Applications,Hartford,CT,Oct1997,pp.173–176. Payload Maximization for Open Chained Manipulators:

Finding Weightlifting Motions for a Puma762Robot Chia-Yu E.Wang,Wojciech K.Timoszyk,and James E.Bobrow

Abstract—Although the dynamic equations of motion of open-chained robot systems are well known,they are seldom taken into account during the planning of motions.In this work,we show that the dynamics of a robot can be used to produce motions that extend the payload capability beyond the limit set by traditional methods.In particular,we develop a point-to-point weightlifting motion planner for open-chained robots.The governing optimal control problem is converted into a direct,SQP param-eter optimization in which the gradient is determined analytically.The joint trajectories are defined by B-spline polynomials along with a time-scale factor.The algorithm is applied to a Puma762robot,with its physical limi-tations incorporated into the formulation.The torque limits are formulated as soft constraints added into the objective function while the position and velocity limits are formulated as hard,linear inequality constraints,on the parameters.The solutions obtained with our algorithm extend the robot’s payload capability while reducing the joint torques.Interestingly,nearly all the trajectories found pass through singular configurations,where large in-ternal forces from the robot are applied to the payload and little torque is needed from the motors.A video file of the resulting motions can be found at https://www.wendangku.net/doc/d116692635.html,/~chwang/project/puma762.html.

Index Terms—B-spline,dynamic motion planning,open chained manip-ulator,optimal robot control,optimization.

I.I NTRODUCTION

The majority of current robot motion planning algorithms,such as those found in well-known texts(see,e.g.,[10]),ignore the dynamics of the robot during the motion planning phase.In this paper,we show how the dynamics of a manipulator can be used to extend its payload ca-pability beyond the limits established by the manufacturer.The goal of the research was to develop an algorithm that maximizes the robot pay-load while taking into account realistic constraints such as joint torque limits and velocity bounds.Our solutions also minimize joint torques Manuscript received August7,2000.This paper was recommended for pub-lication by Associate Editor I.Walker and Editor A.De Luca upon evaluation of the reviewers’comments.The work was supported by the National Science Foundation under Grant IIR-9711782.This paper was presented in part at the IEEE International Conference on Robotics and Automation,Detroit,MI,May 1999.

The authors are with the Department of Mechanical and Aerospace Engineering,University of California,Irvine,CA92697USA(e-mail: chwang@https://www.wendangku.net/doc/d116692635.html,;wtimoszy@https://www.wendangku.net/doc/d116692635.html,;bobrow@https://www.wendangku.net/doc/d116692635.html,). Publisher Item Identifier S1042-296X(01)04813-3.

1042–296X/01$10.00?2001IEEE

a rose for Emily 分析

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英语选修六课文翻译Unit5 The power of nature An exciting job的课文原文和翻译

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ARoseforEmily英文分析及简评

“A Rose for Emily” is divided into five sections. The first section opens with a description of the Grierson house in Jefferson. The narrator mentions that over the past 100 years, Miss Emily Grierson’s home has fall into disrepair and become “an eyesore among eyesores.” The first sentence of the story sets the tone of how the citizens of Jefferson felt about Emily: “When Miss Emily Grierson died, our whole town went to the funeral: the men through a sort of respectful affection for a fallen monument, the women mostly out of curiosity to see the inside of her house, which no one save an old manservant – a combined gardener and cook – had seen in at least ten years.” It is known around town that Emily Grierson has not had guests in her home for the past decade, except her black servant who runs errands for her to and from the market. When a new city council takes over, however, they begin to tax her once again. She refuses to pay the taxes and appear before the sheriff, so the city authorities invite themselves into her house. When confronted on her tax evasion, Emily reminds them that she doesn't have to pay taxes in Jefferson and to speak to Colonel Sartoris, although he had died 10 years before. In section two, the narrator explains that the Griersons had always been a very proud Southern family. Mr. Grierson, Emily’s father, believes no man is suitable for his daughter and doesn't allow her to date. Emily is largely dependent upon her father, and is left foundering when he dies. After Mr. Grierson's death, Emily does not allow the authorities to remove his body for three days, claiming he is still alive. She breaks down and allows authorities to take the body away for a quick burial. Section three introduces Emily’s beau, Homer Barron, a foreman from the north. Homer comes to Jefferson with a crew of men to build sidewalks outside the Grierson home. After Emily and Homer are seen driving through town several times, Emily visits a druggist. There, she asks to purchase arsenic. The druggist asks what the arsenic is for since it was required of him to ask by law. Emily does not respond and coldly stares him down until he looks away and gives her the arsenic. When Emily opens the package, underneath the skull and bones sign is written, "For Rats." Citizens of Jefferson believe that Miss Emily is going to commit suicide since Homer has not yet proposed in the beginning of section four. The townspeople contact and invite Emily's two cousins to comfort her. Shortly after their arrival, Homer leaves and then returns after the cousins leave Jefferson. After staying in Jefferson for one night, Homer is never seen again. After Homer’s disappearance, Emily begins to age, gain weight, and is rarely seen outside of her home. Soon, Miss Emily passes away.

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景区讲解员的服务流程-标准化文件发布号：（9456-EUATWK-MWUB-WUNN-INNUL-DDQTY-KII

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