Capstan Law Motion Planning in Funicular Railways

Transcription

1 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) Capstan Law Motion Planning in Funicular Railways G. MOSCARIELLO (),V. NIOLA (), C. ROSSI (). () Dept. o Matematica ed Applicazioni Univ. di Napoli Federico II () Dept. o Ingegneria Meccanica per l Energetica Univ. di Napoli Federico II Via Claudio, 805 Napoli ITALY Abstract An algorithm to compute suitable laws o motion or the capstan o unicular railways is presented. The aim is to reduce, by these computed capstan laws o motion, the car oscillations that occur during the start stop transients. By the proposed algorithm, once the mathematical model and the equation o motion are deined, the desired car s law is imposed and the equations are solved, computing the law o motion o the capstan and then the law o motion o the other car. A computed example is given by considering the data o an existing unicular railway. These irst computed results show that the railroad car s oscillations, that occur during the start and stop transients, can be signiicantly reduced in some operating conditions. A very simple braking law has been tested: it was supposed a constant deceleration within seconds; it could be possible that, by adopting more suitable deceleration laws, better results can be achieved in all operating conditions. Key-Words: - Motion planning, unicular railways, non constant coeicient systems Introduction As it was observed in previous investigations on the dynamical behaviour o unicular railways [,], when a car is moved by a wire (e.g. unicular railways, cableways, elevators etc.) - because o the elasticity o the wire - the car itsel will not move as the winch moves. What takes place instead, mainly during the start and stop transients, are non negligible cars oscillations. In some cases, due to dynamical eects, these relative motions can reach signiicant amplitudes. From a general point o view, this was well known: oscillations during the start and the stop transient occur in all mechanical systems in which the transmission between an actuator and the mechanical part that receives the motion can not be assumed as rigid. The possibility o computing actuator s laws o motions that can reduce the occurrence o these undesirable oscillations has been investigated by several Authors both or d.o.. systems (see e.g. [3-6]) and or multi d.o.. systems [7]. In unicular railways, these oscillations are obviously uncomortable or passengers and can cause problems to the transmission and other mechanical parts (i.e. backlash, atigue etc.), as it was already observed [,]. For the reasons mentioned above, we thought it was interesting to propose an algorithm to compute suitable capstan laws o motion in order to reduce the occurrence o these undesirable oscillations o both cars that occur during the stop and the start transients. The capstan motion planning The capstan motion planning algorithm proposed starts rom what was observed on the dynamical behaviour o unicular railways [,] and rom an algorithm that was proposed or the motion planning o robots having non rigid transmission between servomotors and links [8,9]. Both investigations start rom a system s dynamical model; hence, in this case, the mathematical model or a unicular railway has to be considered.. System s mathematical model A simple 3 d.o.. damped model, shown in ig., has been considered. As this model is essentially the same that has been adopted in previous investigations, we will conine us to a brie description. The equation o motion are: - m + F x () - σ = 0 c x - σ x- x A - k Mm - (T - T )R - I tot Θ = 0 - m x - σc x - σ x - x ( ) + F = 0 where : A - k (x (x - x A - x ) + A ) + ()

2 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) x i = i-th car position Θ = capstan pulley coordinate R = capstan pulley radius x A = position o a point o the capitan pulley (=RΘ) m c = i-th car mass m p = mass o one passenger n p = number o passengers l 0i = wire initial length ρ = wire mass or unit length ρ(l 0i x i ) = mass o i-th wire m i = m ci + m p n ip + ρ(l 0i + (- i ) x i )/ I c = capstan mass moment o inertia I tot = I c + ρ(l 0 + l 0 )/ R Mm = motor torque on the capstan E = wire Joung modulus A = wire square section k i = EA/(l 0i x i ) wire stiness coeicient σ i = wire damping coeicient = rolling riction coeicient α = gradient o the railway F = - m i g cos α vers(v i ) = rolling riction orce on the i-th car Both railroad cars are moved by steel wires, having internal damping, whose coeicients o stiness change, during the motion, as the wire length changes. The railroad cars are supposed to be rigid (as stiness is considered in the wire) and their mass is the car mass plus the passengers mass plus one hal o the wire mass; as the wire length changes during the motion, the car mass is non constant, too. The riction between rails and wheels is considered as a constant orce, whose sign is opposite to the car speed sign; the riction in the air has been considered by the term σ cx& i. As in eq. (), masses and stiness coeicients are non constant, equations are non constant coeicient derivative equations, hence the system s own requencies change with the car positions. This was already shown in [].. The algorithm As mentioned in the introduction, we thought it could be interesting to extend to cable railways an algorithm that was proposed or the motion planning o robots having non rigid transmission between servomotors and links [8,9]. I we impose the desired law o motion to one o the cars (say car x (t)) in the equations (), rom the irst equation () results the ollowing: m x σc x σ x RΘ m gcosα sign(x ) = 0 EA ( ) ( x R ) l x Hence, the capstan velocity can be deduced: + EAΘ σ ( l0 x) ( σ +σc ) x+ Rσ Θ= m + Rσ EA Rσ x+ x ( l x ) 0 0 Θ + gcosαsign(x ) + () Fig. Scheme o the system The motor moves a capstan C by means o a transmission that is supposed rigid ; consequently, the capstan pulleys will move with the law o motion given by the motor (eventually linked through a gearbox). By (numerically) integrating equation (), it can be obtained that capstan law o motion Θ(t) will move the car, as it was imposed. Once Θ(t) is computed, the term x A (=RΘ) in the third equation in () is known; hence rom this last : x = [-(σ c + σ ) x + σ R Θ - k (x - RΘ)] / [m c + m p n p + ρ(l 0 + x )] - g cos α vers(v )

3 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) by numerical integration, it is possible to compute that car s law o motion comes rom having imposed the desired law o motion to car. Finally, it is possibile to compute the motor torque on the capstan: Mm = (T - T )R + I tot Θ Obviously, at least three points have to be considered: - whether car s law o motion is acceptable or not; - whether the capstan can be moved with the computed law o motion or not; - whether the computed motor torque to the capstan can be achieved or not. 3 Computed results An example was carried on considering an existing cable-railway, whose data are reported in the appendix. As in previous investigations on the same topic, the equations o motion have been solved by means o MatLab code, using the unction ODE45 o the II order Runge Kutta algorithm with variable step o integration. The maximum absolute error (AbsToo) and the relative one (RelToo) did not exceed 0-6 ; or each step o the integration, the error e(i) or each component y(i) o the solution vector y satisies the condition: Case B: Car is approaching the upper end o the railway, thus is linked to the capstan by a short wire length. Case C: Both car and car are in the middle o the run, thereore each car is linked to the capstan by the same wire length. For all cases, the behaviour was computed or 4 seconds in order to observe the oscillations ater the car stop. 3. Case A In this case, the law o motion is imposed to the car when it is linked to the capstan with the highest possible wire length. Fig.3 reports the capstan law o motion, computed by assigning to car the law o motion shown in ig.. Figure 4 reports the law o motion or car. In igures 3 and 4, rom the top to below are reported displacement, velocity and acceleration. e(i) <=max[reltol*abs(y(i)),abstol(i)] On car was imposed a constant deceleration o 0,5 ms -. The law o motion o car is showed in ig.. Fig.3 Capstan law o motion, case A Fig. Law o motion o car Both cars start rom a speed o 3 m/s. When the law o motion described above is imposed on car, it stops ater seconds within 8 metres. Three cases have been simulated that represent three possible operating conditions: Case A: Car is approaching the lower end o the railway, thus is linked by a long wire length. Fig. 4 Car law o motion, case A As expected, the capstan acceleration shows a step at t= s, as car stops. As or car motion, it must be observed that signiicant oscillations take place, in particular an acceleration 3

4 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) amplitude peak, slightly higher than 5 m s -. This is obviously unacceptable or passengers. In ig 5, the displacement o car is compared to the displacement (RΘ) o a point o the capstan pulley. Fig. 6 Wire orces, case A Fig. 5 Car and capstan displacement, case A. It can be observed that the motion o a point o the capstan is rather dierent rom car motion. In particular capstan (RΘ) is in delay with respect to car (x ) motion or 0 t s; at t s, car reaches the capstan motion with a smooth step. To illustrate in detail this aspect, ig 6,a shows an enlarged particular o ig. 5. Fig. 7 Capstan torque, case A Fig 5,a Particular o ig.5 In ig. 6 are reported the wires orces and in ig. 7 the torque on the capstan. From both these last igures, it is evident that a big eort is demanded to the motor and capstan unit. But, most important, in ig.6 it is shown that the wire orce T becomes cyclically negative. Obviously, these operating conditions are physically impossible as the wire can not be stressed by compression orces. 3. Case B From previous observations, it seems reasonable to think that the unacceptable behaviour o case A depends, also, on having imposed the law o motion to a car linked to the capstan by a long wire, having (consequently) big elastic strains. For this reason, operating conditions opposite to case A were simulated. In case B, the law o motion is imposed to the car when it is linked to the capstan with the lowest possible wire length. Fig.8 reports the capstan law o motion, while ig.9 shows the law o motion o car. From igs.8 and 9, it is evident that, in this case, capstan acceleration values are lower than those in case A, both or the capstan and the car. In particular, car s law o motion is more gentle and accelerations are generally lower than 0,4 m s - ; this acceleration amount can be still acceptable or passengers. In ig.0, car s displacement is compared with the displacement (RΘ) o a point o the capstan pulley. As it can be observed, the computed capstan law o motion is practically identical with respect to the imposed law o motion o car. This depends on the act that car (on which the law o motion is imposed) 4

5 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) is linked to the capstan with the lowest possible wire length. Fig. Wire orces, case B Fig.8 Capstan law o motion, case B Fig. Capstan torque, case B Fig. 9 Car law o motion, case B As or car acceleration (ig.9), it must be observed that the y scale limit is ten times lower than the one displayed or case A It is interesting to observe that, in this case, no negative orces on the wire take place; rom this point o view, these operating conditions are physically possible. Moreover, stresses and torques are deinitely lower than those o case A. From both igures above, it can - also - be clearly observed that or t= s, a step in wire orce (hence in the motor torque) occurs. In this instant, in act, the car stops and the braking torque on the capstan is put to zero; so, rom t= s on the capstan itsel, only the torque necessary to hold the car in the assigned position will act. The observed rough step could probably be reduced i dierent car acceleration laws (e.g. linear or sinusoidal) are imposed. Fig. 0 Car and capstan displacement, case B In ig. are reported the wires orces and in ig. the torque on the capstan. 3.3 Case C As mentioned beore, in this case the law o motion is imposed to car when both cars are in the middle o the run. Fig.3 reports the capstan law o motion, while ig. 4 shows the law o motion o car. As or case B, also in this case the y scale limit in the diagram o car acceleration is ten times lower than the one displayed or case A. 5

6 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) Fig.3 Capstan law o motion, case C Fig. 6 Wire orces, case C Fig. 4 Car law o motion, case C As it can be observed in this case, a peak slightly higher than 0,5 m s - occurs in car acceleration; this acceleration amount can not be considered acceptable or passengers. In ig. 5 the displacement o car is compared with the displacement (RΘ) o a point o the capstan pulley. Fig. 5 Car and capstan displacement, case C As it can be observed, or t< s, car motion is slightly in delay with respect to capstan motion. In ig. 6 are reported the wires orces and in ig. 7 the torque on the capstan. Fig. 7 Capstan torque, case C From ig.6, it is possible to observe that no physically impossible working conditions occur, as no negative orces on the wire take place. Moreover, stresses and torques are slightly higher than those in case B, but considerably lower than those in case A.. 4 Conclusions A irst study on an algorithm to compute suitable laws o motion or the capstan o unicular railways has been presented. By the proposed algorithm, once the mathematical model and the equation o motion are deined, the desired car s law o motion is imposed and the equations are solved; thereore, the law o motion o the capstan and, subsequently, the law o motion o the other car are computed. The aim was to reduce, by these computed capstan laws o motion, the car oscillations that occur during the start and stop transients. Computed results are presented by considering the data o an existing unicular railway. These results show that these cars oscillations can be signiicantly reduced i the law o motion is imposed to the car linked to the capstan with a relatively high wire length. 6

7 Proceedings o the 0th WSEAS International Conerence on SYSTEMS, Vouliagmeni, Athens, Greece, July 0-, 006 (pp ) Just a rough deceleration law was imposed (constant value during the stop transient) so it is possible to imagine that, with dierent laws, better results could be obtained. Appendix Tab. reports the meaning and the values o the quantities in eq. (). Table a Car acceleration 0,5 ms -. a max Max car acc. 0, 5 ms - v reg Car velocity 3 ms -. m C Car mass 000 kg M P Passanger s mass 70 kg n P Nr. passeng. car 50 n P Nr. passeng. Car 0 m Car wire mass variable m Car wire mass variable ρ Wire mass or unit length 6 kg/m A Wire section 6,35*0-4 m L 0 Wire lenght at t=0 83 m L 0 Wire lenght at t=0 39 m E Wire elasticity,*0 N/m α Rail inclination 70 R Winch radius,75 m C Friction coeicient 0,0 σ c Car damping coe. 500 Ns/m σ Wire damping coe Ns/m I 0 Winch mass moment 400 kg m o inertia g gravity 9,8 m/s [4] N. C. SINGER, W.P. SEERING Preshaping Command Inputs to Reduce System Vibration - Journ. Dynamic Systems, Measurements and Control. March 990, vol. pp [5] T. SINGH, G.R. HEPPLER Shaped Input Control o a Sistem With Multiple Modes. - Journ. Dynamic Systems, Measurements and Control. Sept 993, vol.5 pp [6] W. E. STINGHOSE et Alii Vibration Reduction Using Multi-Hump Input Shapes. - Journ. Dynamic Systems, Measurements and Control. Sept 993, vol.5 pp [7] M. W. SPONG Modelling and Control o Elastic Joint Robots. - Journ. Dynamic Systems, Measurements and Control. Dec. 987, vol.09 pp [8] C. ROSSI, S. SCOCCA - Inluence o the Transmission Elasticity on the Law o Motion o a Robot Arm. - Invited Paper at CASYS'99, 3 nd International Conerence on Computing Anticipatory Systems, Liege, Belgium, 0-4 Aug.999. Published by International Journal o Computing Anticipatory Systems, vol.6, pp , edited by Daniel M. Dubois. ISBN ISSN [9] R. BRANCATI, C. ROSSI, F. TIMPONE - Motion Planning o a Robot Arm with Non-Rigid Transmission - Proc. o th int. Workshop RAAD 03, Cassino May 7-0, 003. Reerences [] G. MOSCARIELLO, V. NIOLA, C. ROSSI - Funicular Railways Dynamical Behaviour During the Start and Stop Transients - WSEAS TRANSACTIONS ON SYSTEMS, Iusse, Vol. 4, November 005, pagg [] G. MOSCARIELLO, V. NIOLA, C. ROSSI - Law o motion inlence on the start-stop transiet o unicular railways - WSEAS TRANSACTIONS ON SYSTEMS, Iusse, Vol. 4, November 005, pagg [3] D. M. ASPINWALL Acceleration proiles or Minimizing Residual Response Journ. Dynamic Systems, Measurements and Control. March 980, vol.0 pp