Following Performance Evaluation of a Mechanicall Couple Platoon Tetsua SHINOHARA, Masafumi OKADA an Ken-ichi YOSHIMOTO Department ofmechano-informatics, Universit oftoko 7-3-, Hongo, Bunko-ku, Toko, 3-8656 Japan Tel:(+8)3-584-638 Fax:(+8)3-388-835 E-mail:sinohara@nl.t.u-toko.ac.jp Abstract Coupling some trucks with the same estination is calle a truck platoon sstem. The leaing truck is riven manuall an the following trucks are riven automaticall b using sensors on the mechanical link. Mechanical links oes not onl restrict the movabilit of the following trucks, but also are use for sensors. We evaluate the manipulabilit an the following performance on the linke platoon sstem. The relation between the measure of manipulabilit an the following performance is also explaine. Finall, the optimize link formation is propose in this paper. Introuction These as, most of the freight transportation is transacte b trucks, an this situation is expecte not to be change in the near future. However, there are a lot of problems cause b trucks, air pollution, trac congestion, increase of energ consumption an so on. In orer to overcome these problems, truck platoon is now uner investigation[]. Since the trucks are riven b a ver short istance in a platoon, the roa capacit will be increase an the energ consumption will be reuce. In CHAUFFEUR project, an electronicall couple truck platoon has been propose[2][3]. The following truck is riven automaticall b using a CCD camera an the so-calle C (ehicle to ehicle Communication). The electronical coupling has a few problems with respect to the reliabilit in the emergent situations. We propose a mechanicall couple truck platoon that has the avantage of reliable measurement, wire communication an high safet[4][5]. In this paper, we give the esign strateg of mechanical link for mechanicall couple truck platoon base on the measure of manipulabilit an the following performance b using three tpes of mechanical links. An base on the optimization problem, we give the appropriate link formation. 2 Measure of Manipulabilit Evaluation 2. Conguration of the 3 D.O.F link A vehicle has three egree-of-freeom on the horizontal plane, two D.O.F on the position an one D.O.F. on the orientation. If the mechanical link has the same number of freeom, trucks in a platoon coul run with small restriction from the mechanical link. We esign three tpes of 3 D.O.F mechanical links, which have the same formation, as shown in Fig.. All links have tohave two rota- Hf C B Hr A Fig. : Link formation tion joints at H f an H r, which are use as sensors for steering control. These two joints coul measure the relative aw angle between two trucks. The remaining D.O.F is set on the prismatic joint, whose position is ierent for each link as shown in Fig.2. LinkA's prismatic joint is set at the leaing truck's rear bumper, LinkB's is set between the tail of the leaing truck anh r, LinkC's is set between H f an H r.
LinkA LinkB LinkC Fig. 2: Arragement of prismatic joint 2.2 Measure of Manipulabilit Measure of manipulabilit (M.O.M.) is applie to evaluate the movabilit of truck platoon couple with mechanical links. M.O.M. is often use in the evaluation of the robotic arm's manipulabilit performance[6]. In this section, M.O.M. of each link are compare. We set the platoon's coorinate sstem as Fig.3. The kinematics of LinkA is given b the following lf (X,Y) Θ Hf ψ zc Hr Lf zb µ Lr za x In the same wa, the jacobian matrix J B, J C are obtaine. l f sin( + ) J B(2) J B = @ l f cos( + ) J B(22) A (5) ; ; J C = @ J B(2) = L f sin()+l f sin( + ) (6) J B(22) = L f cos()+l f cos( + ) (7) l f sin( + ) J C(2) cos() l f cos( + ) J C(22) ;sin() ; ; J C(2) =(L f ; z C )sin()+l f sin( J C(22) =(L f ; z C )cos()+l f cos( A (8) + )(9) + )() B using the jacobian matrix J i, M.O.M. is ene as: q w i = et(j i J T i ) i= A B C () M.O.M. is epene on the angle of the rotation joint. For the comparison of M.O.M. of each link, M.O.M. is calculate in two situations as follows. Situation Both trucks run along one curve roa. The raius of the curve isset to R = [m]. an are set as =:3[ra] = :9[ra] respectivel. Situation 2 Both trucks run along straight roa. an are set as = [ra] = [ra] respectivel. The M.O.M. of each link in each situation is shown in Table.. LinkC's M.O.M. is the highest in each situation, which means that trucks in a platoon using LinkC have the best movabilit. Fig. 3: Platoon's coorinate sstem equations: X = ;L r ; L f cos() ; l f cos( + ) Y = L f sin()+l f sin( + )+z A =; ; () Table : Measure of Manipulabilit of each Link LinkA LinkB LinkC Situation :38388 4:2428 4:26 Situation2 4:849 ;6 4:26 4:26 Each parameters are set as shown in Fig.3. B partial ierentiation of Eq., we get the jacobian matrix J A as: l f sin( + ) J A(2) J A = @ l f cos( + ) J A(22) A (2) ; ; J A(2) = L f sin()+l f sin( + ) (3) J A(22) = L f cos()+l f cos( + ) (4) 3 Following Performance Evaluation 3. Problem Formulation In this section, we evaluate the following performance of the truck in a platoon b comparing with
the optimize trajector. The optimize trajector is esigne as follows.. Two trucks are not connecte b link. 2. The rst truck runs with velocit. 3. The secon truck is controlle so that it follows the same trajector as the rst truck. 4. The controller is esigne base on the forwar error correction algorithm b linear preiction[7] which is explaine in the following section. B the restriction of the link, the trajector has perturbation b which the following performance inex is evaluate quantativel. Because of the nonlinearit of the vehicle namics or link kinematics, the following performance inex epens on the path of truck. We set one moel, in which the leaing truck is going straight an the secon truck is going to follow the leaing truck asshown in Fig.4. The initial conition of the rotation angle is =:9[ra], =:3[ra]. Initial Conition (m=.9, =.3) as an uncouple truck. The namic equations of the platoon sstem are as follows: a t + a 2 + a 3 = a 4 + a 5 t + a 6 = (2) Here, is the front steering angle, is the slip angle, is the awing rate an coecient a i (i = 2 ::: 6) are constants ene b conguration an velocit of the vehicle. When the vehicle is riven almost straight, an are satise. Approximatel in the moel of Fig.4, Eq.2 can be converte into Eq.3: 2 b + t 2 b 2 + t b 3 + t b 4 = b 5 t + b 2 6 t + 2 b 7 t + b 8 = (3) Here, is the lateral position of the center of gravit, is the awing angle an coecient b i (i = 2 ::: 8) are constants. 3.3 Controller esign Gi -q f m G i- x We esign a controller base on the forwar error correction algorithm b linear preiction[7]. The steering angle of the following truck is etermine b the rotation angle an which are measure b sensors on the mechanical link, an can be formulate as: Final Conition (m=, =) = kz = k + k 2 t + k 3 + k 4 t (4) Gi G i- x Coecients k i (i = 2 3 4) are esign parameters an etermine b the linear quaratic regulator metho. The state space equation of Eq.3 is written as: _x = Ax + B (5) Fig. 4: Platoon moel 3.2 ehicle Dnamics For the evaluation, we esign the vehicle moel. We use the two-wheele moel because the namic equations of the truck in a platoon are same Here, A an B are constant matrices. x is a state vector given b: x = ; T (6) In the moel of Fig.4, the leaing truck is riven straight at constant velocit, the secon truck is going to follow the leaing truck from the initial conition,. Here, an jj
are satise. B using linear quaratic regulator metho, the steering angle is given as follows: = ;Kx = ;K ; K 2 t ; K 3 ; K 4 t (7).7.6 Consiering the steering angle an the following performance, to the leaing truck, we set the cost function as follows: Z J = ( 2 + 2 +2 2 )t (8) Here, we nee to convert the feeback coecient K in Eq.7 into the gain coecient k in Eq.4 to use in the simulation with each mechanical link. Approximatel in the moel of Fig.4, an are written as: Y[m].5.4.3.2. -. -2 2 4 6 8 = L f + l f ( + ) ; = + (9) Fig. 5: Optimize trajector From Eq.4, Eq.7 an Eq.9, k is given as follows: k = B @ ;(L f + l f ) ;(L f + l f ) ;l f ;l f 3.4 Simulation Result C A K (2) First, wesimulate the non-linke platoon moel to obtain the optimize trajector. This simulation is one base on Eq.2, which is substitute Eq.7 into Eq.5. _x =(A ; BK) x (2) The trajector of the following truck is shown in Fig.5. To evaluate the following performance, we simulate three sstems with ierent links. In these simulations, the following truck iscontrolle b the same algorithm as before. The simulation course is the same as the non-linke platoon moel simulation. The result with LinkA is shown in Fig.6, LinkB is in Fig.7, LinkC is in Fig.8 respectivel. The upper part shows the trajector of the following truck an the lower part shows the slie length of the prismatic joint. The trajectories using LinkB an LinkC are similar to that of Fig.5, however the case with LinkA is ierent. Therefore, it seems to be icult to realize the high following performance with LinkA. Y[m] ZA[m].8.6.4.2 -.2 -.4 -.6 -.8-2 2 4 6 8 -. -.2 -.3 -.4 -.5 -.6 -.7.5.5 2 2.5 3 time[s] 3.5 4 4.5 5 Fig. 6: Simulation result with LinkA
Y[m].7.6.5.4.3.2. -. -2 2 4 6 8.4 3.5 Following Performance Evaluation For the evaluation of the following performance, we set the performance inex I, I 2 as follows: Z x(5) I = ()x (22) Z 5 I 2 = (z) 2 t (23) I means the square measure of error of trajector an I 2 means the motion inex of the prismatic joint. The result are shown in Table.2. The result ZB[m].2..8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time[s] Fig. 7: Simulation result with LinkB.7.6 Table 2: Following performance evaluation with LinkA LinkB an LinkC I [m 2 ] I 2 [m 2 ] LinkA 4:796 :627 LinkB :475 ; 7:747 ;4 LinkC 9:248 ;2 7:556 ;4 of the following performance evaluation is that the most ecient link is LinkC. Combining this result with that of Sect.2, we conclue that the following performance becomes better b using a link with a higher M.O.M..5 Y[m] ZC[m].4.3.2. -. -2 2 4 6 8.4.2..8.6.4.2.5.5 2 2.5 3 3.5 4 4.5 5 time[s] Fig. 8: Simulation result with LinkC 4 Improvement of the Following Performance 4. Following Performance with Spring an Damper Since the platoon runs at a constant spee, the link has a bias at the prismatic joint. In this section, a spring an amper are set to make the prismatic quantit smaller an improve the following performance. We simulate about four cases with ierent spring an amper parameter an the result is shown in Table.3. S is a spring constant anc means the coecient of viscosit of the amper. Case S = [N/m] C = [Ns/m] Case 2 S =5 3 [N/m] C = [Ns/m] Case 3 S = 5 [N/m] C = [Ns/m] Case 4 S = 5 [N/m] C = 5 [Ns/m]
Table 3: Comparison of spring constant an amper constant I [m 2 ] I 2 [m 2 ] case 9:248 ;2 7:556 ;4 case 2 9:2564 ;2 8:6 ;4 case 3 9:34 ;2 5:22 ;4 case 4 9:5322 ;2 7:896 ;7 These results show that spring an amper mae the error area bigger because the restriction of the link intensies. 4.2 Optimization of the spring constant an coecient of viscosit We set a cost function to optimize the trae-o relation between the error area an the prismatic quantit. The cost function J is set as follows: Z x(5) Z 5 J = ()x +2 (z) 2 t (24) Here, is the error of the trajector an z is the slie of the prismatic joint. We n the optimize S an C which minimize the cost function J in Eq.24 The result of calculating the cost function J is shown in Fig.9. The minimum of the cost J.96.955.95.945.94.935.93.925 5 4 3 log(s) 2 Fig. 9: Cost function 5 4 3 2 log(c) function occurs in the case of S = 568:99[N/m] an C = 233:[Ns/m]. B using this spring an amper, the performance inex I 2 becomes smaller with less increase of the performance inex I. 5 Conclusion In this paper, we give the esign strateg of mechanical link for mechanicall couple platoon sstem. The result of this paper is as follows.. The movabilit of trucks in mechanicall couple platoon is evaluate b the measure of manipulabilit (M.O.M.). LinkC's M.O.M. is the highest of the three links. 2. The following performance is evaluate b using the performance inex I an I 2. LinkC's I an I 2 are the lowest of the three links. 3. The following performance becomes better when a link with a higher M.O.M is use. 4. The following performance is improve b setting the spring an amper at the prismatic joint. Spring an amper constants are etermine b minimizing the cost function. References [] Christophe Bonnet an Hans Fritz: "Fuel consumption reuction experience b two promotechaueur trucks in electronic tow bar operation" Proc. of the 7th Worl Congress on Intelligent Transport Sstems, 2. [2] Ottmar Gehring an Hans Fritz: "Lateral control concepts for truck platooning in the chaueur project", Proc. of the 4th Worl Congress on Intelligent Transport Sstems, 997. [3] Hans Fritz: "Longituinal an lateral control of two electronicall couple heav-ut trucks in the chaueur project", Proc. of the 6th Worl Congress on Intelligent Transport Sstems, 999. [4] Seree Satukijchai an Ken-ichi Yoshimoto: "A feasibilit stu on the automate platoon freight transportation sstem in Japan", Proc. of the 5th Worl Congress on Intelligent Transport Sstems, 998. [5] Ken-ichi Yoshimoto, Masafumi Okaa an Sigeki Ban: "Lateral control of an automate truck platoon couple with mechanical links", Proc. of the 7th Worl Congress on Intelligent Transport Sstems, 2. [6] Tuneo Yoshikawa: "Manipulabilit of Robot Mechanisms", The secon smposium of Robotics Research, 985, pp.439-446. [7] M. Kono: "On the Basic Relationship Between Automobile Steering an Motion", Trans. JSAE, No.5, 958, p.5(japanese)