KINEMATICS AND DYNAMICS OF CONFIGURABLE WHEEL-LEG

Transcription

1 8th Intenational DAAAM Baltic onfeence "INDUSTRIAL ENGINEERING 19-1 Apil 01, Tallinn, Estonia KINEMATIS AND DYNAMIS OF ONFIGURABLE WHEEL-LEG Sell, R; Ayassov, G; Petitshenko, A; Kaeeli, M Abstact: In this pape, the kinematics and dynamics of the configuable wheel-leg is consideed. The configuable wheel-leg has been invented by TUT eseaches to incease the mobility of the wheeled vehicles, mostly unmanned gound vehicles, on diffeent teains and to ovecome obstacles that ae unobtainable to the conventional wheels e.g. climbing on stais. The pape pesents an oveview of simila wheels and focuses mostly on the kinematics and dynamics of invented wheel. The esult is a mathematical model which can be used fo futhe analysis and simulations. Key wods: wheel dynamics, wheel-leg, vaiable diamete wheel 1. INTRODUTION Today s wold seveal diffeent types of mobile obots ae in use, both civilian and militay domain. Wheeled obots have advantages of good pefomance when moving on smooth oads. Howeve when obstacle is on the oad o obot needs to tun off-oad the good pefomance is gone and diffeent type of locomotion pinciple have advantages to ovecome of the obstacle o move on ough teain. In escue obots, it is often case that obot have to un elatively long on smooth teain but then needs to go up to stais. It is clea that wheels ae not suitable climbing up to stais. Theefoe these obots have usually two locomotion options mounted on the obot. Tacks ae mounted with combination of wheel. This solution is not efficient in tems of enegy consumption, complexity and stability. Fig. 1. Wheel-leg (Wheg) in closed and open configuation In this pape novel patented [1] solution wheel-leg (Wheg), shown in Fig. 1. is intoduced. The solution combines the advantages of diffeent locomotion pinciples by changing the geomety of the end actuato on the fly. By changing the wheel geomety kinematic and dynamic paametes of the wheel-leg ae also changed consideably. The study gives essential undestanding of kinematic and dynamic popeties of the nonconventional igid wheel-leg and povides necessay infomation to estimate the minimum toque of wheel-leg actuatos. The study is a pat of wide eseach aea, which is connected to the study of mobility, maneuveability and tajectoy contol of wheeled mobile obotics equipped with wheel-legs.. STATE-OF-ART Inventing the wheel has been attacted mankind aleady fo thousands of yeas. Even conventional wheel has been in use without of majo modification a long time seveal inventions have been egisteed 345

2 duing last centuy. The main diving foce to invent new vesions of wheel is to impove the passability on ough teains and climb ove the obstacles o holes. Seveal inventions have been developed in Japan [], in US [3], in Russia [4] and othe counties. Seveal dynamically configuable wheels ae also invented vey ecently, paty diven by space missions whee the ove has to be enegy efficient and uneven teain capability on the same time. Zheng et.al have been intoduced the diamete-vaiable wheel [5] fo out-doo ove which deploys plana polygonal mechanism. Anothe concept is pesented by Xinbo et.al. [6] whee wheel is segmented and by expanding the segments the diamete is also extended. Well known application is so called Galileo wheel [7] which uses elastic long scale expandable tacks as a tie. In addition to vaiable diamete wheel constuction seveal fixed sized but diffeent shape end actuatos ae developed which can be placed between wheel and leg. Most well-known is a Boston Dynamics RHex obot which uses elastic half-cicula wheel-leg as end actuato [8]. All descibed solutions have thei benefits and dawbacks. In next section we ae pesenting the solution integating the wheel and leg locomotion pinciples vey tightly by inceasing the enegy efficiency and constuction dimension by offeing the wide scale of geomety and dynamic pefomance change. 3. WHEEL-LEG Wheel-leg is a mechanism, invented by pape authos, that includes good qualities of both wheels and legs. The esult of that is a good passing ability in diffeent teain including stais and steps. In the smooth teain the wheel egime is used. When teain changes to hadly passable the wheel-leg adjusts its configuation by opening the wheel segments so that the passing ability inceases dastically. In Fig. 1 two diffeent egime of wheel-leg is shown, whee the change fom one egime to othe can be done even duing the nomal opeation. Due to design paametes, the wheel-leg can opeate in diffeent mode of opeation and change its configuation dynamically (Fig. 1.). When blades of the wheel-leg ae closed, it opeates nealy as conventional igid wheel. In case of the blades of the wheel-leg ae open, the wheel-leg opeates as non-cicula wheel and its kinematic and dynamic popeties vay within its position. 4. KINEMATIS OF WHEEL-LEG 4.1 Kinematics of wheel-leg in the atesian efeence fame In Fig. the physical and mathematical model of the fully open vaiable diamete igid wheel-leg constained to move in plane and on the igid suface. The wheelleg is modeled as egula hexagon with cones ounded. The ounded cones ae epesented in Fig. 3 as cicles with continuous lines. The sides of the hexagon coespond to the tangencies to the ounded cones of physical model with constant adius. Fig.. The physical and mathematical model of wheel-leg The position of the igid body in plane is completely defined if it is defined the position of the two points of the igid body at any instant time of its motion. To define the position of the cente of the wheel-leg the two efeence fames in Fig. 3 ae intoduced: gound-fixed efeence fame Oxy and efeence fame Bx 1 y 1 that moves tanslatoy with espect of the gound-fixed efeence fame. 346

3 Bx 1 y 1. The second time deivative fom the both sides of the Eq. () yields to the acceleation of the cente as ϕ ϕ + ϕlcosϕ ϕ lsinϕ = ϕlsinϕ ϕ lcosϕ (3) θ Fig. 3. Rotation of half of the hexagon The position of the wheel-leg is chosen so that at the beginning of the motion the cente of wheel-leg coincides with coodinate axis y of gound efeence fame and angle θ of the tangency to the ounded cones of wheel-leg and coodinate axis x is θ=0. Though the otation of the 1/6 th of one evolution of the hexagon, the angle θ changes fom 0 to π/3 that coesponds to the changes of the angle φ fom -π/6 to π/6 measued fom the y coodinate axes. The position vecto of the cente of the hexagon can then be expessed by it components though the genealized coodinate φ as whee ϕ angula acceleation of the vecto B with espect to the efeence fame Bx 1 y Geneal kinematics of wheel-leg In this section the altenative method to establish the kinematic chaacteistics of the wheel-leg is consideed. It is based on the Eule-Savay fomulation of the moving and fixed centoids of wheel-leg. This method can have advantages in some cases when wheel-leg moves on the cuved o othe sufaces in space. To deive the kinematic chaacteistics of the wheel-leg the Fig. 4 is used. ϕ x0 + ϕ + lsinϕ = (1) + lcosϕ whee x0 - the component of the vecto B with espect of the coodinate axis Ox and coesponds to the initial position of instantaneous cente of otation; - the component of the vecto B with espect of the coodinate axis Oy; l - length of the vecto B ; ϕ - otation angle of vecto B with espect of efeence fame Bx 1 y 1. The fist time deivative fom the both sides of Eq. (1) yields to velocity expession as ϕ + ϕlcosϕ = ϕl sinϕ () whee ϕ the angula velocity vecto of the vecto B espect to the efeence fame Fig. 4. Rotation of the blade of the wheelleg. Let the α-α will be the tajectoy of the cente of the wheel-leg, when blade L olls on igid suface. The motion of the blade epesents the motion of the moving centoid on fixed centoid with angula velocityϕ. At the moment, when the blade is in contact with fixed centoid, the vecto 347

4 B, which connects the cente B of the moving centoid L with cente of the wheel-leg, foms the angle -π/6 φ π/6 with espect of the vetical axis NN. Thus, the olling time of the wheel blade is 0 t π 3 ϕ. To find the adius of cuvatue ρ α of the tajectoy of the cente of the wheel-leg, we daw the line fom point and though instantaneous cente of velocity v (in the following fo instantaneous cente of otation the notation IR will be used). The cente of the cuvatue K of tajectoy of the point has to lie on this line. The angle between the staight lines С v and NN is β. The angles β and φ ae elated to each othe accoding the theoem of sine as follows l sinϕ sin β = (4) l sin ϕ + lcosϕ + ( ) Though the vey small time inteval dt, when blade L olls on the hoizontal line, the cente moves to the position and IR olls to the position v. Nomal K to the tajectoy α-α has to intesect the IR of v. Fom the point v we daw the nomal to the line K and denote the angle K as dα. Fom the angle ΔСK and ΔK v M we eceive Futhe K + = = (5) M K V dγ = dγ = ϕ dt (6) M= β = ϕ dt cos β (7) V V V V cos Substituting the Eq. (6) and (7) to the Eq. (5) we eceive the Eule-Savays equation [9-11] as V + = cosβ (8) Fom the Eq. (8) we eceive cos β = (9) ( cos β ) and adius of cuvatue of the tajectoy of the cente accoding to the Fig. 4 is ρ α cos β = + (10) ( cos β ) whee ρ α = K = +. The length of the vecto we eceive fom the vecto equation as =+ (11) B The length of the vecto though the components of fixed efeence fame can be witten then in the fom sin ( cosϕ) = l ϕ+ + l (1) whee ϕ = ϕt π /6. Paametical epesentation of the tajectoy of the cente accoding to the Eq. (1) can be witten as follows x = x0 + l sin ϕ+ R sin β y = l sin ϕ+ R cos β (13) R= + lcosϕ (14) whee x 0 is initial position of the IR of the v. The velocity of the cente becomes then accoding to the Eq. (1) as V sin (15) v = ϕ = ϕ l ϕ+ R Taking the time deivative fom the Eq. (15) we eceive the tangential acceleation as 348

5 l sin cos Rlsin (16) τ ϕ ϕ ϕ a = ϕ l sin ϕ + R whee ϕ = const. Nomal acceleation of the point taken into account the Eq.(10) and Eq. (15) can be obtained as follows a n ϕ = ( l sin ϕ+ R) cos β + cos β (17) whee angle β accoding to the Eq. (4) is defined as l sinϕ β = acsin (18) l sin ϕ + R Absolute acceleation is defined though Eq. (16) and Eq. (17) as τ ( a ) ( a n ) a = + 5. DYNAMIS OF WHEEL-LEG (19) To evaluate the maximum toque that can be applied to wheel-leg in motion on igid suface the Lagange s equation with undetemined multiplies ae intoduced [10] in the fom d T T f = Q + λ dt q q q (0) whee T kinetic enegy of the wheel-leg, Q genealized foces of wheel-leg, q and q genealized coodinates and thei time deivatives espectively; λ Lagangian undetemined multiplies; f the equation of constaint. As the genealized coodinates the tanslation of the cente of the ounded tip B of the blade in the diection of the x coodinate axis and otation about this cente though the angle φ ae chosen. Accoding to the chosen genealized coodinates the position of cente can be witten as ( sin, cos ) T = x+ l ϕ + l ϕ (1) whee - is constant though the otation of the point with espect to the y- coodinate axis by angle -π/6 to π/6. Taking the time deivative fom the Eq. (1) we eceive expession of the kinetic enegy as follows 1 ( cos I ) z ϕ T = m x + x ϕl ϕ+ ϕ l + () whee I z the moment of inetia of the wheel-leg. The constaint equation f compliant with genealized coodinates is x ϕ x 0 = 0 (3) whee x = x0 + ϕ and x 0 - detemines the initial position of IR. Accoding to Fig. 3 the active foces that do wok as weight P of the wheel-leg and the toque M ae applied to the cente of the wheel-leg. Applying the method of vitual wok we eceive the genealized foces coesponding to the vitual displacement of genealized coodinates as follows Q = 0; Q = M + Plsinϕ (4) x ϕ Taking the patial and time deivatives of Eq. () and Eq. (3) accoding to the Eq. (0) we eceive Lagangian system of equation as mx ( + ϕlcosϕ ϕ lsin ϕ) = λ mxl cos ϕ+ ϕ( ml + I ) = (5) = M + Plsinϕ λ The fist equation in Eq. (5) epesents the maximum fiction foce between the suface and the wheel-leg blade that is constituent with constaint equation Eq. (3), i.e motion of wheel-leg without 349

6 slipping. Fom the second equation of the Eq. (5) it is possible to get the expession fo the maximum toque that can be applied of the wheel-leg. The genealized coodinates in Eq. (1) ae coupled. Dopping out the tem x 0 that epesents the initial position of IR, then the Eq. (5) can be ewitten as m ϕ( + lcos ϕ) m ϕ lsinϕ = λ ϕ( ml cos ϕ+ ml + I ) = (6) = M + Plsinϕ λ The Eq. (6) can be solved numeically fo both equations to detemine the values of the fiction foces and applied toques. 6. ONLUSION In this pape the mathematical fomulation of the plana kinematics and dynamics of the wheel-leg is deived. The kinematic popeties of wheel-leg wee deived by method of igid body mechanics and by altenative method based on the Eule- Savay fomulation of the moving and fixed centoids. The use of Eule-Savay method can have advantageous when wheel-leg moves in cuved o othe sufaces in space. The Lagange s equation with undetemined multiplies has been used to establish elationship between the maximum applied toque and the fiction foce of the wheel-leg. The study of the kinematic and dynamic popeties of the wheel-leg has coveed only the basic theoetical aspects of the wheel-leg motion in hoizontal plane. Thus, to evaluate moe capabilities of the wheel-leg the compehensive analysis of the wheel-leg will be pefomed in futue. 7. AKNOWLEDGEMENT This eseach was suppoted by funding of the Estonian Science Foundation gant No REFERENES 1. Sell, R., Kaeeli, M., Wheel-Leg (Wheg). Patent no EE0583B1, Tsunasawa M., Moving Equipment, Patent no 3. JP , Holmes, M., Vehicle taction assist device, Patent no US , Denisenko, G, Wheel Populsion Unit, Patent no RU8056, Zheng L., et.al, A Novel High Adaptability Out-doo Mobile Robot with Diamete-vaiable Wheels. Poc. of the IEEE Int. onfeence on Infomation and Automation Shenzhen, hina, Xinbo., et. al, Mechanism pinciple and dynamics simulation on vaiable diamete walking wheel. Second Intenational onfeence on Digital Manufactuing & Automation, Galileo Mobility apabilities, 8. ampbell D. and Buehle M., Stai descent in the simple hexapod RHex, Poceedings of the IEEE Int. onfeence on Robotics & Automation, Taipei, Taiwan, Septembe 14-19, Angus, R., W. Theoy of Machines Including the Pinciples of Mechanisms. NABU Pess, leghom, W., L. Mechanics of Machines. Oxfod Univesity Pess, 005. Raivo Sell, Ph.D, senio eseache, TUT, Ehitajate tee 5, Tallinn, aivo.sell@ttu.ee Gennady Ayassov, PhD, Ass. Pof., TUT Ehitajate tee 5, Tallinn, gennadi.ajassov@ttu.ee Andes Petitshenko, PhD, eseache, TUT Ehitajate tee 5, Tallinn, andes.petitsenko@ttu.ee Mati Kaeeli, PhD Student, Tallinn TUT Ehitajate tee 5, Tallinn, mati141@gmail.co 350