Fifth heel Modelling and Testing en Masoy Mechanical Engineeing Depatment Floida Atlantic Univesity Boca aton, FL 4 Lois Malaptias IFMA Institut Fancais De Mechanique Advancee ampus De lemont Feand Les ezeaux B.. 65 675 Aubiee edex illiam ight alm Beach ounty Sheiff ffice est alm Beach, Floida Abstact A two dimensional dynamic model, which povides a full desciption of a vehicle motion based on sensoy input, is pesented. The model has thee sensoy inputs: the velocity of a fifth wheel; the oientation of the fifth wheel with espect to the vehicle; and the yaw ate of the vehicle. Based on these inputs, the model detemines the position, velocity, acceleation and the yaw angle of the vehicle. The esults of expeiment with an instumented vehicle, which wee designed to veify the validity and accuacy of the model, ae epoted. NMENLATUE AND NTATINS oodinate systems (, ˆx, ŷ, ẑ ) - a global fame of efeence (, xˆ, ŷ, ẑ ) - a fame attached to the vehicle s cente of gavity (, xˆ, ŷ, ẑ ) - a fame attached to the 5 th wheel at the pivot point Angles - angle between Θ - angle between β - angle between about ẑ and and V / measued about ẑ measued about ẑ and x measued F - cente of the left font wheel Subscipts - the vehicle - the 5 th heel assembly Vecto notation example V = / u v - Velocity of the vehicle cente of gavity,, which is located on the vehicle,, with espect to fame. oints - vehicle s cente of gavity - cente of the 5 th wheel axis of otation - 5 th wheel pivot point - cente of the left ea wheel
I. INTDUTIN II. SENSS AND SINAL ESSIN Automotive 5 th wheels have been used fo decades in vehicle testing. The 5 th wheel is attached to the ea bumpe is towed by the vehicle. The attachment lows elative otation between the vehicle s and the wheel s axes. An optical encode attached to the wheel s axis eads the angula velocity of the wheel. Knowing the diamete of the wheel, assuming no slip, the vehicle vehicle s speed can be detemined. Usually the 5 th wheel is being used to test the pefomance of the baking systems, to measue coefficient of fiction between the ties and oad suface, vehicle s acceleation etc. bviously, in these applications the vehicle has to tavel along a staight line. the sensos such as acceleometes and gyoscopes ae also being installed on vehicles in ode to extact infomation about thei dynamic behaviou. This poject deals with high yaw ate manoeuves such as shap tuning, high speed passing and side impacts at high speed. Theefoe thee is a need to detemine the yaw ate of the vehicle and its speed. Theefoe, a conventional 5 th wheel, which is instumented only with encode, is not sufficient and additional sensos ae equied. A commecial 5 th wheel was modified to include a senso at its pivot point which measues the angle between the 5 th wheel and the vehicle axes. Also, a gyoscope, measuing the vehicle s yaw ate, was installed. The location of the sensos and the 5 th wheel ae shown in Figue. yoscope The gyoscope A gyoscope, that measues the vehicle s yaw ate, &, was attached to the vehicle close to the pivot point,. A vey low constant speed electical moto was used to calibate the gyoscope. The gyoscope was attached to the moto shaft and the moto was excited with diffeent voltages to poduce diffeent measuable angula velocities. Using the known velocities and the coesponding gyoscope outputs a calibation cuve was established: & o [ ] = 49.9& [V] 5.8 () s The encode The encode, which measues the fifth wheel speed, u, was attached to the wheel s axis. Its digital signal was conveted to an analogue signal popotional to the wheel speed, u. The wheel speed was calibated using the electonic display that was povided with the wheel. The convesion of the analogue signal to wheel velocity found to be: m [ ] = 8.779U [V].48 s U () The otentiomete The potentiomete measues the angle, Θ, between the 5 th wheel and the vehicle axes. It was calibated manually using a potacto: Encode otentiomete Θ[ o ] = 7.667Θ[V] + 77.4 () Filteing the signals Fist, out lia eadings ae detected by compaing the cuent sampled value to the aveage of the last m eadings. If the diffeence exceeds a given theshold the eading is emoved and eplaced in ode to keep the time tack valid: Figue : The fifth wheel and the sensos. if ( m X whee i = n i i X j= i m i k= i n j X X k i ) then (4) X is the measued value and m and n ae integes.
Second a local lineaization was pefomed by fitting the sampled eading to a staight line defined by the pevious samples and the next samples (see Figue ). The above pocessing intoduced delay between the oiginal and the pocessed signal, as shown in Figue 4. Since the signal has to be integated with espect to time, the filteed signal was shifted back in time to match the oiginal one. Figue 4 illustates this opeation. 9 8 7 6 5 4.9.8.7.6.5.4 Figue : Local lineaization. The fist opeation was epeated with theshold value equal to the standad deviation of the next q eadings. An example fo the above pocess is shown in Figue... 4 6 8 iginal and filteed signal.9.8.7.6.5.4.. 4 6 8.9.8.7.6.5.4.. 4 6 8 iginal signal Filteed signal shifted back in time Figue 4: Shifting the filteed signal..9.8.7.6.5.4 Integation and deivatives As will be shown in the poceeding section the integation and deivative of the signal might be needed. Simple algoithms wee used fo both opeations:.. 4 6 8 + (5) f ( n) = f s *( X n X n ) Filteed signal Figue : esults of the filteing opeations. t n x( t) dt X i ( i) (6) f s i= whee f s is the sampling ate.
III. MDELLIN The pupose of the following section is to develop a D model which will descibe the vehicle s motion based on the available eal time sensoy input. In this case the velocity of the vehicle, defined by the components u and v should be detemined by the measued values of the 5 th wheel speed, u, its pivot angle, Θ, and the vehicle angula velocity, &. The dimensions coodinate systems and velocities of the vehicle and the 5 th wheel ae shown in Figues 5 and 6. y y x c x b d a y x Figue 5: Dimensions of the vehicle and the 5 th wheel. The position of Fifth wheel axis,, with espect to given by: = whee these vectos ae given by: (7) is = = = x y x y d cos( Θ + d sin( Θ + (8)
F βf V F / α V / y c y V / β V / β x c x y p Θ x p β y V / x Figue 6: Velocities and angles definitions. Thus, the position of the fifth wheel axis is given by: x y = x = y d cos( Θ + d sin( Θ + The velocity of the fifth wheel with espect to given by: (9) is x& y& = x& = y& + d( )sin( Θ + () d( )cos( Θ + Substituting in Eq. yields: u u cos( Θ + = x& sin( Θ + = y& + d( )sin( Θ + () d( )cos( Θ + V cos( Θ + / = = u sin( Θ + u deiving the velocities fom Eq. : u ) () Similaly, the position of the vehicle s cente,, of gavity, with espect to is given by: = + () o
x y = x = y + ( a + b)cos( + ( a + b)sin( (4) and the velocity of the vehicle s cente of gavity,, is given by: x& y& o: u v = x& = y& & ( a + b)sin( + & ( a + b)cos( (5) = & ( a + b)sin( + u cos( Θ + d( )sin( Θ + (6) = & ( a + b)cos( + u sin( Θ + + d( )cos( Θ + Eqs. 5 and 6 descibe the D motion of the vehicle in tem of the measued vaiables. The yaw angle, β, of the vehicle is impotant since it povides indication whethe o not the vehicle is skidding. In the following an expession fo the yaw angle as function of the measued vaiables, u, Θ and &, will be detemined. The velocity of the vehicle s cente of gavity,, with espect to is given by: V / = V = V / / + Ω / + Ω + Ω / / Substituting the known distances yields: V / = (7) u cos( Θ) ( d )sin( Θ) u sin( Θ) + ( d )cos( Θ) (8) + ( a + b) & And the yaw angle, β, is given by: u sin( Θ) + ( d )cos( Θ) + ( a + b) & β = Ac tan (9) u cos( Θ) d( )sin( Θ) VI. ES ANALYSIS Since u, Θ and & ae being measued, and the tems cosθ, sinθ, and & θ ae used in the above elationships, it is essential to detemine the affect of the measuements eos on the calculated value of the vehicle s cente gavity velocity and position. The theoetical values of u, Θ and & ae given by: u = u + ε Θ = Θ + ε Θ & = & + ε & () whee u, Θ and & ae the measued values and ε, ε Θ and ε ae the coesponding measuement eos. & If contains bias eo, it will accumulate due to time integation as: = & ( + ε ε () whee peiod. ε & ) dt + tε & = + is the eo in at the end of the integation Diffeentiation of a signal that contains a bias eo will not cause an eo. Howeve, diffeentiation of a noisy signal might poduce lage eo and theefoe it has to be filteed to some acceptable level. The execution of tigonomety function may lead to eos: sin( = sin( + ε ) sin( + cos( ε () cos( = cos( + ε ) cos( sin( ε
To educe the eos, due to integation, in the position of the vehicle cente of gavity, only the necessay integations ae pefomed: 6 5 Linea Tajectoy - Sensoy Data Uw [m/s] 4 x v = ( a + b)cos( + + d sin( Θ + = ( a + b)sin( + + d cos( Θ + [ u [ u cos( Θ + ] dt () sin( Θ + ] dt Sensoy Data 5 5 5 - si-dot [ad/sed] Θ [ad] Figue 8: Staight line- sensoy data. V. EXEIMENTS Thee diffeent tajectoies wee used to validate the model and to test the sensoy system. These tajectoies ae shown in Figue 7..5 Yaw angle - Model utput.5 Yaw [ad] 5 5 5 -.5 - -.5 Figue 7: Tajectoies used in the expeiments. Staight line tajectoy A ft long staight line tajectoy was used fo this test. It took about 5 seconds to dive along this line, duing which the sensos wee sampled at Hz (see Figue 8). The model used to detemine the position and oientation of the vehicle. The longitudinal eo was about 6 ft and the maximum lateal eo was about ft. These ae vey good esults consideing the fact the signals wee integated fo a peiod of 5 seconds. The noise at the beginning and at the end of the yaw angle (see Figue 9) is due to the division used in the model. At the beginning and the end of the tajectoy both components of the velocity ae vey small and the pesence of some noise will cause lage eos. Squae tajectoy Figue 9: Yaw angle. The squae tajectoy was diven continuously fou times. The sensoy data obtained fo one squae ae shown in Figue. The tajectoy, detemined by the model, is shown in Figue. The eos in the tajectoy ae due to poo contol of the diving tack, athe than the model itself. It is inteesting to notice the Yaw angle of the vehicle along the tajectoy. Even though diving speed was low, about 9 [mph], the vehicle was skidding at the cones of the squae as indicated in figue when eve the Yaw angle is no zeo.
5 Sensoy Input - Squae Tajectoy.5 icula Tajectoy - Sensoy Data 4.5 Uw[m/s] Θ[degees] Sensoy Input si_dot [ad/s] Uw [m/s] Θ [ad] Sensoy Data.5.5 si_dot[ad/sec] - 8 8 8 -.5 5 5 5 5 4 45 5 - - -.5 Figue : Squae tajectoy - sensoy data. Figue : icula tajectoy sensoy data. 7 4 5 Squae Tajectoy - Model utput icula Tajectoy - Model utput -5 5 5 5 5 9-7 - Figue : Squae tajectoy as constucted by the model. 5 5 7 9.8 Yaw angle [adians] Figue : icula tajectoy constucted by the model. Yaw [ad].6.4. 5 5 5 5 4 45 5 In case of taveling in a constant speed along a cicle the yaw angle should be constant. In this case, as shown in Figue 4, the yaw angle is not constant as expected, due to vaiations in the vehicle speed as shown in Figue 5. -. -.4.9 Yaw Angles [ad] -.6.8 -.8.7 - Figue : Yaw angle along the squae tajectoy. icula tajectoy Yaw angle [ad].6.5.4.. The sensoy inputs to the model, as ecoded along the tajectoy, ae shown in Figue, and the constucted tajectoy in Figue.. 5.5 7.5 9.5.5.5 5.5 Figue 4: Yaw angle along the cicle.
9. Vehicle speed [m/s] 8 7 6 Speed [m/s] 5 4 5.5 7.5 9.5.5.5 5.5 Figue 5: vehicle speed along the cicle VI. NLUSINS A simple, sensoy based, model fo D motion of a vehicle was intoduced. Initial tests show good esults. Bette contolled expeiments with additional sensos ae planned in ode to futhe evaluate the pefomance of the cuent sensos and the model esults.