The straight-line navigation control of an agricultural tractor subject to input saturation

Transcript

1 DOI: /CTA Control Theory & Applications Vol. 30 No. 10 Oct ,, 1, 1, 3 (1., 1013;., 10096; 3., ) :,,,,,,,, :,, : ; ; ; : TP73 : A The straight-line navigation control of an agricultural tractor subject to input saturation DING Shi-hong 1,, JIANG Yue-xia 1, ZHAO De-an 1, LIN Xiang-ze 3 (1. Key Laboratory of Facility Agriculture Measurement and Control Technology and Equipment of Machinery Industry, Jiangsu University, Zhenjiang Jiangsu 1013, China;. Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education, Southeast University, Nanjing Jiangsu 10096, China; 3. College of Engineering, Nanjing Agricultural University, Nanjing Jiangsu 10031, China) Abstract: For an agricultural tractor, a linear path-tracking navigation algorithm is developed by using the saturation method, providing an effective solution to the ubiquitous input saturation control problem in agricultural tractor navigation. First, based upon the kinematics of the straight-linear navigation system, a linear system model for control design is obtained by linearization. Secondly, a non-saturated controller is designed to stabilize the system globally. Finally, by integrating a saturation function with the non-saturated controller, we obtain a saturated controller under which the closed-loop system is globally asymptotically stable. This controller is simple in structure without the need of sequential decrement of restriction on the saturation levels from outside to inside, thus improving the convergence of the closed-loop system. Simulation results validate the effectiveness of the proposed method. Key words: agricultural tractor; path tracking; controller design; stability 1 (Introduction),,,.,,, [1].,,.,, []., [3 4]. : : ; : dsh@mail.ujs.edu.cn; Tel.: : ( , ); (PAPD, (011)6); ( ); (MCCSE01A01); (BK0183).

2 188 30,,.,, [5].,, [6],. [7] X 804, PID, (difference global positioning system, DGPS),. PID, [8], [9], [10],,., [11],.,,,,,,,,,,, [1].,,., ([13 15]), (3.1).,.,,,,,,,,,,,.,,. (System modelling and problem formulation),, 1 [8, 16]. 1 Fig. 1 The sketch map of agricultural tractor straight-line navigation 1: Ψ(rad), V x (m/s), L(m), δ(rad), x(m), y(m). 1, Ψ = V x tan δ, L ẋ = V x cos Ψ, (1) ẏ = V x sin Ψ, : Ψ, V x, L, δ, x, y1. (1),,,,,,, 1, y 0 Ψ 0, yψ., : ẏ 0 V x 0 Ψ = 0 0 V y 0 x L Ψ + 0 u, () δ δ 1 : u, δ. : u max > 0, u u u max, y 0Ψ 0. 1, PID

3 10 : 189., PID., (1),,, (1), ().,.,,,,.,,. 3 (Controller design) x 1 = y, x = V x Ψ, x 3 = (Vx /L)δ,, () 1 = x, = x 3, 3 = V (3) x L u = v, v. I) v(3). (3), : v = k 3 x 3 k 3 k x k 3 k k 1 x 1, (4) s 3 + k 3 s + k 3 k s + k 3 k k 1 = 0 Hurwitz, (3) (4). II) (4), u.,,,.,,. σ εi (x) = { (sgn x)ε i, x > ε i, x, x ε i, (4), : v = k 3 σ ε3 (x 3 + k σ ε (x + k 1 σ ε1 (x 1 ))), (5) k 3 ε 3 k [ε 3 + (k 1 + k )ε + k1ε 1 ] > 0, k ε > ε 3 + k 1 (ε + k 1 ε 1 ), k 1 ε 1 > ε. S 3 (t) = x 3 + k σ ε (x + k 1 σ ε1 (x 1 )), S (t) = x + k 1 σ ε1 (x 1 ), S 1 (t) = x 1. (6), (5), (3) T 1, t T 1, S 3 (t) = x 3 + k σ ε (S ) ε 3.,, T 1 S 3 (T 1 ) ε 3., t 0, S 3 (t) > ε 3 /, S 3 (t) > ε 3, t > 0, (7) S 3 (t) < ε 3, t > 0. (8) S 3 (t) > ε 3 /, t > 0, (3)(5) 3 = k 3 σ ε3 (S 3 ) < k 3ε 3. (9) (9)x 3 (t) x 3 (0) < k 3 ε 3 t/., t > 0, x 3 (t) < x 3 (0) k 3 ε 3 t/., S 3 S S 3 = x 3 + k σ ε (S ) x 3 (0) k 3ε 3 t + k ε., t, S 3, (7). (7). (8)., T 1 S 3 (T 1 ) ε 3 /. t T 1 S 3 (t) = x 3 + k σ ε (S ) ε 3. Ω 1 = {x : S 3 (t) = x 3 + k σ ε (S ) ε 3 }. Ω 1,,,, P 1 P P 3 P 4 Ω 1, Q 1 Q Q 3 Q 4 Ω 1.. S x 3 Fig. The phase plot of S x 3 x P 1 P, S ε. Ṡ 3 = ẋ 3 k ε = k 3 σ ε3 (S 3 ) = k 3 ε 3 < 0, x P 1 P. (10)

4 Ṡ 3 = ẋ 3 + k ε = k 3 σ ε3 (S 3 ) = k 3 ε 3 < 0, x P 3 P 4. (11) (10) (11), xp 1 P P 3 P 4 Ω 1., x Q 1 Q Q 3 Q 4 Ω 1. xp P 3 Q Q 3 Ω 1. x P P 3, S 3 = ε 3, S ε., Ṡ 3 = ẋ 3 + k Ṡ = k 3 S 3 + k Ṡ = k 3 ε 3 + k Ṡ, x P P 3. (1) x {x : S 3 ε 3, S ε }, Ṡ x 3 + k 1 σ ε1 (x 1 ) x 3 + k 1 σ ε1 (x 1 ). σ ε1 (x 1 ), σ ε1 (x 1 ) = 0, x 1 > ε 1, σ 1 (x 1 ) = x, x 1 ε 1., x {x : S 3 ε 3, S ε }, σ 1 (x 1 ) ε + k 1 ε 1. (13) x {x : S 3 ε 3, S ε }, Ṡ x 3 + k 1 σ ε1 (x 1 ) ε 3 + k ε + k 1 (ε + k 1 ε 1 ) = ε 3 + (k 1 + k )ε + k 1ε 1. (14) x P P 3 x {x : S 3 ε 3, S ε }. (14)(1) S 3 k 3 ε 3 +k [ε 3 +(k 1 +k )ε +k 1ε 1 ], x P P 3. (6)k 3 ε 3 k [ε 3 + (k 1 + k )ε + k 1ε 1 ] > 0., (6) S 3 < 0, x P P 3. (15) S 3 > 0, x Q Q 3. (16) (15) (16), xp P 3 Q Q 3., xω 1. : t T 1, S 3 (t) ε 3. T > T 1 t T, S 3 (t) ε 3, S (t) ε. T > T 1, t T S 3 (t) ε 3.,, T > T 1 t T, S (t) ε., T > T 1 S (T ) < ε., t > T 1, S (t) ε., S (t) ε, t > T 1 S (t) ε, t > T t > T 1, S 3 (t) ε 3. t > T 1, ẋ = x 3 = k σ ε (S ) + x 3 + k σ ε (S ) = k σ ε (S ) + S 3 k σ ε (S ) + ε 3. (17) S (t) ε 3, t > T 1, (17) ẋ k σ ε (S ) + ε 3 k ε + ε 3, x (t) x (0) (k ε ε 3 )t. (18) (6), k ε > ε 3., (18)S (t) S (t) = x (t)+k 1 σ ε1 (x 1 ) x (0) (k ε ε 3 )t + k 1 ε 1. t, S <, S ε, t > T 1., S (t) ε, t > T 1., T > T 1 S (T ) < ε. t > T, S (t) ε. Ω = {x : S (t) = x + k 1 σ ε1 (x 1 ) ε }. Ω,, 3,, H 1 H H 3 H 4 Ω, L 1 L L 3 L 4 Ω. 3 S 1 x Fig. 3 The phase plot of S 1 x x H 1 H, S 1 = x 1 < ε 1, S = x + k 1 σ ε1 (x 1 ) = x k 1 ε 1., x H 1 H, Ṡ = ẋ = x 3. x(t ) Ω. H 1 H Ω, t > T > T 1, x(t ) H 1 H., t = t, S 3 (t ) ε 3. x 3 (t ) = S 3 (t ) k σ ε (S (t )) ε 3 k ε. k ε ε 3 > 0., x H 1 H, Ṡ = ẋ = x 3 < ε 3 k ε < 0, xh 1 H Ω. xh 3 H 4, L 1 L, L 3 L 4 Ω. x H H 3, x 1 ε 1, Ṡ = ẋ + k 1 σ ε (x 1 ) = x 3 + k 1 x = k σ ε (S ) + S 3 + k 1 x.

5 10 : 191 x H H 3, S = ε., x H H 3, x ε + k 1 ε 1., x H H 3, Ṡ k ε + ε 3 + k 1 x (6) k ε + ε 3 + k 1 (ε + k 1 ε 1 ). k ε > ε 3 + k 1 (ε + k 1 ε 1 ), S < 0, x H H 3., H H 3 Ω. L L 3., xh 1 H H 3 H 4 L 1 L L 3 L 4 Ω, t T, 3 S 3 (t) ε 3, S (t) ε. T 3 > T t T 3, S 3 (t) ε 3, S (t) ε, x 1 (t) ε 1. 1, t T, S (t) ε, S 3 (t) ε 3, t T, 1 = x = k 1 σ ε1 (x 1 ) + x + k 1 σ ε1 (x 1 ) = k 1 σ ε1 (x 1 ) + S. (19) (19)(6) ẋ 1 < k 1 ε 1 + ε < 0, x 1 > ε 1, ẋ 1 > k 1 ε 1 ε > 0, x 1 < ε 1, T 3, t T 3, x 1 (t) ε 1, S (t) ε, S 3 (t) ε 3, t > T 3, (5) (4)., t > T 3 (3)(5) 1 = x, = x 3, 3 = k 3 σ ε3 (x 3 + k σ ε (x + k 1 σ ε1 (x 1 ))) = k 3 x 3 k 3 k x k 3 k k 1 x 1. I) s 3 + k 3 s + k 3 k s + k 3 k k 1 = 0 (0) Hurwitz.,, (5), (3). u = k 3L V x σ ε3 ( V x L δ + k σ ε (V x Ψ + k 1 σ ε1 (y))) (1), (), 1 (1), ().,, ()., Teel [13], : u = L Vx σ ε3 (y 3 + σ ε (y + σ ε1 (y 1 ))), () : ε 3 > ε, ε > ε 1, y 1 =y + V xψ + V xδ, y = V xψ + V xδ, y 3 = V xδ. ε i, i = 1,, 3ε 3 > ε > ε 1., y (0), y 3 (0), y 3 (0), (), Lε 1 /V x. ε 1,, 1,. 4 (Simulation), MATLAB,. (1), (1)(),..4 m, V x = 1 m/s., u rad/s. (6), (1). k 1 = 1 5, k = 1, k 3 = 5 3, () ε 1 = 3, ε = 1, ε 3 = ε 3 = 5 6, ε = 1 3, ε = 1 1. (x(0), y(0), Φ(0), δ(0)) = (4, 3, π 4, π 4 ), 5%()., 4 9, : (1), (). 4 6 X = (y, Φ, δ), X = y + Φ + δ 7. 8

6 ,, (), (1)., 4 85%,. 9,., 7 X Fig. 7 The response curves of X 4 Fig. 4 The response curves of lateral deviation 8 y-x Fig. 8 The tractor tracking phase plot of y-x 5 Fig. 5 The response curves of course angle 9 Fig. 9 The response curves of controller 6 Fig. 6 The response curves of steering angle 5 (Conclusions),,,,,.,

7 10 : 193 (References): [1] ZHAO D A, LÜ J D, JI W, et al. Design and control of an apple harvesting robot [J]. Biosystems Engineering, 011, 110(): [] TANG J L, JING X, HE D J, et al. Visual navigation control for agricultural robot using serial BP neural network [J]. Transactions of the Chinese Society of Agricultural Engineering, 011, 7(): [3] MEHTA S, BURKS T, DIXON W. Vision-based localization of a wheeled mobile robot for greenhouse applications: a daisy-chaining approach [J]. Computers and Electronics in Agriculture, 008, 63(1): [4],,,. [J]., 007, 38(5): (CHEN Jun, ZHU Zhongxiang, TORISU Ryo, et al. Automatic ontracking control of farm vehicle based on neural network [J]. Transactions of the Chinese Society for Agricultural Machinery, 007, 38(5): ) [5],,,. [J]., 013, 44(1): (LI Taochang, HU Jingtao, GAO Lei, et al. Agricultural machine path tracking method based on fuzzy adaptive pure pursuit model [J]. Transactions of the Chinese Society for Agricultural Machinery, 013, 44(1): ) [6],,. [J]., 006, 37(1): 0. (DOU Zhiqiang, MAO Zhihuai, WEI Qing. The field target tracking system based on laser passing [J]. Transactions of the Chinese Society for Agricultural Machinery, 006, 37(1): 0.) [7],,,. X-804DGPS [J]., 009, 5(11): (LUO Xiwen, ZHANG Zhigang, ZHAO Zuoxi, et al. Design of DGPS navigation control system for Dongfanghong X-804 tractor [J]. Transactions of the Chinese Society of Agricultural Engineering, 009, 5(11): ) [8],,,. [J]., 01, 8(9): (ZHANG Meina, LIN Xiangze, DING Yongqian, et al. Design of path following controllers based on performance index for agricultural vehicle [J]. Transactions of the Chinese Society of Agricultural Engineering, 01, 8(9): ) [9],,,. [J]., 009, 40(4): (ZHOU Jianjun, ZHANG Man, WANG Maohua, et al. Path tracking for agricultural vehicle based on fuzzy control [J]. Transactions of the Chinese Society for Agricultural Machinery, 009, 40(4): ) [10],,,. [J]., 010, 41(11): (LIU Zhaoxiang, LIU Gang, JI Ying, et al. Autonomous navigation system for agricultural tractor based on self-adapted fuzzy control [J]. Transactions of the Chinese Society for Agricultural Machinery, 010, 41(11): ) [11],,,. [J]., 006, (11): (CHEN Jun, ZHU Zhongxiang, TORISU Ryo, et al. On-tracking control of tractor running along curved paths [J]. Transactions of the Chinese Society of Agricultural Engineering, 006, (11): ) [1] DING S H, LI S H, LI Q. Stability analysis for a second-order continuous finite-time control system subject to a disturbance [J]. Journal of Control Theory and Applications, 009, 7(3): [13] TEEL A R. Global stabilization and restricted tracking for multiple integrators with bounded controls [J]. Systems & Control Letters, 199, 18(3): [14] DING S H, QIAN C, LI S H. Global stabilization of a class of feedforward systems with lower-order nonlinearities [J]. IEEE Transactions on Automatic Control, 010, 55(3): [15],,,. [J]., 01, 9(6): (CHEN Hua, WANG Chaoli, YANG Fang, et al. Finite-time saturated stabilization of nonholonomic mobile robots based on visual servoing [J]. Control Theory & Applications, 01, 9(6): ) [16] O CONNOR M. Carrier-phase differential GPS for automatic control of land vehicles [D]. CA: Stanford University, : (1983 ),,,,, dsh@mail.ujs.edu.cn; (1990 ),,,, jiangyuexiayezi@163.com; (1956 ),,,,, dazhao@mail.ujs.edu.cn; (1978 ),,,,, xzlin@njau.edu.cn.