Simulation of an articulated tractor-implement-trailer model under the influence of lateral disturbances

Transcription

1 Simulation of an articulated tractor-implement-trailer model under the influence of lateral disturbances K. W. Siew, J. Katupitiya and R. Eaton and H.Pota Abstract This paper presents the derivation of the mathematical model for a three-body articulated agricultural vehicle such as a tractor that drags behind two agricultural implements connected in series. It is then used in a simulation to study the effects of slippage. The model is developed with the aim of designing robust controllers that ensure high-precision pathtracking control of such articulated systems. In the simulations, the model was subjected to real conditions experienced in agricultural applications such as disturbances and uncertainties due to ground undulation, gravitational forces due to sloping ground, and lateral wheel slippage. The implement attached to the tractor is assumed to be steerable to enhance the pathtracking capability. This work aims to provide an insight in to the articulated tractor behaviour under the influence of real life farming condition. I. INTRODUCTION The advancement of robotics and control systems is making precision farming a reality. Along with technologies such as Geographic Information System GIS and Global Positioning System GPS there are versatile sensors, monitoring systems and controllers for agricultural equipment. Together they aid in the development of precision farming. Precision farming is greatly facilitated by maintaining a high level of structure in the farming system layout. A structured farming system will minimize the disturbances on the tractorimplement system, thereby enhancing the system s ability to deliver the desired level of precision. The system modeled here is very commonplace in the agricultural industry. In particular, the seeding systems are driven by a prime mover in the form of a large tractor. The tractor is attached to a seeding implement that ploughs the ground and places the seeds and fertilizer. The seeding implement is followed by a seed and fertilizer carrier which appears in the form of a trailer. From a precision point of view, the highest priority is the trajectory following and/or path tracking capability of the seeding point on the implement. From a controlled traffic point of view, the wheels of the tractor must stay within allocated wheel tracks. To study this system, we have a tractor-implement-trailer system modeled in this work. The long term goal is to develop control algorithms that will enable the control of this type of complex system to deliver the desired level of precision. K. W. Siew and J. Katupitiya are with the School of Mechanical and Manufacturing Engineering, The University of New South Wales R. Eaton is with the School of Electrical Engineering and Telecommunication, The University of New South Wales, r.eaton@unsw.edu.au H. Pota is with the School of Information Technology and Electrical Engineering, The University of New South Australian Defence Force Academy, h.pota@unsw.edu.au A lot of research has been done on the path tracking control of a mobile platform [1,[2,[3. Moreover, the path-tracking ability was extended to the involvement of more than one vehicle to form an articulated system [4,[5. This is particularly desirable in agricultural applications as it is the implement that carries out the specific agricultural task. Most of the work has only dealt with non-holonomic systems. This assumption is valid for most mobile platforms under bounded disturbances. However, as one would expect, the system is subjected to a substantial amount of disturbance forces. Among the disturbances are ground undulations, varying soil structure, sloping terrains and significantly large disturbances caused by the uneven ground engagement of the seeding tines. All these forces contribute to drive the implement off course. This issue was noticed and attempts have been made to address the problem [6,[7,[8, and their trajectory tracking ability has shown promising results [9. However, the systems discussed above only guarantee precision guidance of the prime mover. In a farming situation, it is the implement s trajectory or path that needs to be controlled. As an initial step in solving this problem, complete dynamic models have been produced for a tractor-implement system, [1. These models do not include an the effects of an additional trailer. In this paper, as in [1, we also place emphasis on the implement while taking into account the dynamics of the complete tractor-implement-trailer system. The rest of the paper is organized as follows: In section II, two models are presented. Firstly, a slip model that takes into account lateral wheel slippage that may be encountered in practice and a non-slip model that rejects all the elements that give rise to slippage. The simulation results of the model subjected to various conditions are shown in section III. Finally, the concluding remarks are given in section V. II. DYNAMIC MODEL DEVELOPMENT Figure 1 shows the setup of the tractor-implement-trailer articulated system for modeling purposes. The tractor has the steerable front wheels only. The implement is attached to the tractor at an off axle hitch point aft of the rear axle of the tractor. The implement wheels are steerable. The trailer has non-steerable wheels and is attached to the implement at an off axle hitch point aft of the implement axle. A bicycle model representation is adopted for simplicity. The tractor provides propulsion forces T ft and T rt at the front wheels and rear wheels respectively. All wheels are subjected to

2 R H1 F ls F s w s R H2 γ 2 g α s v s ψ f R tine R i v ri β ri δ 2 e F li w i F i α i v i Implement m i, J i F lr R rt T rt w t α t F t R ft F lf δ 1 v rs h Trailer d β ft m s, J s v ft β rs T ft γ 1 φ c v rt b Tractor m t, J t a R s Fig. 1. Tractor-implement-trailer system their corresponding rolling resistances R ft, R rt, R i and R s. Furthermore, the implement experiences a drag force R tine in opposition to its traveling direction. The steering angle of the tractor is δ 1 and for the implement is δ 2. The slip condition of the system is represented by the slip angles β ft, β rt, β ri, β rs with respect to the wheel headings. The tractor s velocities at its centre of mass are in the longitudinal direction and w t in the lateral direction. Likewise, the velocities of the center of mass of the implement are given by v i, w i while v s, andw s denote the velocities of center of mass of the trailer. The tractor mass is m t and that of implement and trailer are m i and m s, respectively. The inertias at the center of mass of the tractor, implement and trailer are J t, J i, J s, respectively. The angular velocities of the tractor, implement and trailer are θ t, θ i, θ s. The reaction force at hitch the point between the tractor and the implement is represented by R H1, while the reaction force at hitch point for the implement and the trailer is denoted by R H2. The misalignment between the tractor and the implement, and the implement and the trailer, is represented by the variables φ and ψ respectively. The parameters a, d and g represent the distances from the front of the tractor, implement and trailer, respectively to their centres of mass. The parameters b, e and h represent the distances from the centres of mass of the tractor, implement and the trailer, respectively to their rear wheels. The parameters c and f are the distances from the rear axles of the tractor and the implement, respectively to their hitch points. By equating velocities at the two hitch points, the following velocity relationships are obtained: = v i cosφ w i + d θ i sin φ 1 w t = v i sin φ + w i + d θ i cos φ + b + c θ t 2 v s = v i cosψ + [w i e + f θ i sin ψ 3 w s = v i sinψ + [w i e + f θ i cosψ g θ s 4 A. Slip Model Three dynamic equations can be written for each body which gives a total of nine equations two translational and one rotational for each body which give a total of nine equations. Equations 1-4 can be used to eliminate the translational components, w t, v s, w s to leave five state variables {v i, w i, θ t, θ i, θ s }. As we are interested in the implement motion, we have chosen to retain v i and w i. The resulting five equations can be combined and expressed in matrix form as, D q + G 1 T + G 2 R + G 3 F l + G 4 F d + G = 5 { where q = v i, w i, θ t, θ i, θ } T s, T = {T ft, T rt } T, R = {R ft, R rt, R i, R tine, R s } T, F l = {F lf, F lr, F li, F ls } T, F d = {F t, F i, F s } T, where F d represents disturbance forces. The force vector F l represent the set of lateral forces on the wheels. The associated D and G matrices are given in Appendix I-A. In addition rate relationships are given by, φ = θ i θ t 6 ψ = θ s θ i 7

3 The steering dynamics are given by, δ 1 = F st 8 δ 2 = F si 9 where F st, F si are the steering inputs of the tractor and implement, respectively. Equations 5, 6-9 form the complete set of dynamic equations for the slip model. The state vector is given by {v i, w i, θ t, θ i, θ s, φ, ψ, δ 1, δ 2 } T and the control input vector is {T ft, T rt, F st, F si } T. The position and orientation of the implement can be obtained by integrating the following expressions, θ i = θ i dt + θ i 1 ẋ i = v i cosθ i w i sinθ i 11 ẏ i = v i sinθ i + w i cosθ i 12 where θ i denotes the initial orientation of the implement. By inspection, slip angles can be calculated using the velocities at each wheel as follow: β ft = tan 1 w t + a θ t + δ 1 13 β rt = tan 1 w t b θ t 14 β ri = tan 1 w i e θ i + δ 2 15 β rs = tan 1 v i w s h θ s v s 16 The lateral forces are assumed to be modeled by the linear representation, F lf = K ft β ft 17 F lr = K rt β rt 18 F li = K ri β ri 19 F ls = K rs β rs 2 where K ft, K rt, K ri, K rs are the cornering stiffness factors. Such convention has been adopted by [4, [11 and [12. The rolling resistance at the tires, on the other hand, are represented by a viscous term that is proportional to the rolling velocity of the tires and another term that is proportional to the normal load on the tires. As such, the rolling resistances can be expressed as, b R ft = C t V 1 + C r a + b 9.81M 1 21 a R rt = C t + C r a + b 9.81M 2 22 d R i = C t V 2 + C r d + e 9.81M 3 23 g R s = C t v s + C r g + h 9.81m s 24 where, V 1 = [ cosδ 1 w t + a θ t sin δ 1 [ ec M 1 = m t m i bd + e [ M 2 = m t + m i ea + b + c g ad + e V 2 = [v i cosδ 2 + w i e θ i sin δ 2 [ hd + e + f M 3 = m i + m s dg + h hf eg + h m s hf eg + h m s where C t and C r are the damping constant and friction coefficient, respectively. The slip model has now been fully described. B. Non-slip model For the non-slip model, disturbance forces have no effect on the model, hence by ignoring the disturbances from the model we get, D q + G 1 T + G 2 R + G 3 F l + G = 25 In the non-slip model, the non-holonomic constraint is such that the β s in equations are equal to zero. From this we obtain four conditions, tan δ 1 = w t + a θ t 26 w t = b θ t 27 tan δ 2 = w i e θ i v i 28 w s = h θ s 29 Along with equations 1-4, the above equations can be solved to obtain a matrix S such that, q = Sv i 3 See Appendix I-B for definition of matrix S. Differentiating gives q = S v i + Ṡv i 31 Substituting 31 into 25 and pre-multiplying by S T gives, S T [ DS v i + Ṡv i + G 1 T + G 2 R + G 3 F l + G = 32 It can be shown that S T G 3 =. Hence the non-slip dynamic model reduces to, v i = S T DS 1 S T DṠv i + S T G 1 T +S T G 2 R + S T G 33 The above equation together with equations 6-9, completes the dynamic model of the non-holonomic system. The state vector is now {v i, φ, ψ, δ 1, δ 2 } T and the control input vector remains unchanged as {T ft, T rt, F st, F si } T.

4 Fig. 2. The compact agricultural tractor being modeled Scenario 1: Without slip, without lateral disturbances. The system is confined to the nonholonomic constraint and have slip angles all equal to zero. Here, the non-slip model is implemented. Fig. 3 shows the trajectories of the system. Scenario 2: With slip, without lateral disturbance. The disturbance forces F t, F i, F s are set to zero. The trajectories are shown in Fig. 4. Scenario 3: With slip, with small lateral disturbance. The magnitude of the disturbance forces reflect that of the gravitational forces acting on the system while it is driven on a slope of grade 2%. In effect, the system starts motion on the slope and drives across the slope, after which turns right down the slope. Scenario 4: With slip, with large lateral disturbance. Similar to scenario 3 except that the disturbance forces correspond to that of having a slope of grade 6%. III. MODEL SIMULATION The models developed in section II are simulated under varying conditions. The parameters and constants have only been partially verified, with currently known parameters based on an existing John Deere compact agricultural tractor used in this research and shown in Fig 2. The remaining unknown parameters are believed to be realistic for the tractor and conditions at hand. Firstly, comparison is made between non-slip and slipping cases. In non-slip cases, conditions stated in subsection II-B are applied so that the articulated system conforms to the non-holonomic constraint. For the slipping case, the same input is given to the model described in subsection II-A. Both cases assume the system is driven on a flat ground without any disturbances. For the third case, the system is subjected to two different degrees of disturbances resulting from the effect of gravity on the system. This is done to imitate the effect of having the articulated system driven on a sloping terrain. In each of the cases, the tractor, implement and trailer are assumed to start motion from rest, and are aligned with each other having orientations of zero degrees. The open loop inputs are defined as T ft = 1N and T rt = 2N held constant throughout the simulation. The steering of the tractor is set to be zero for the first 4s of motion, after which it is actuated by a step input steering rate of 25 o /s to the right for 1s. The steering angle is held at 25 o for a further 1s, which is then actuated in the opposite direction at the same step input steering rate 25 o /s for 1s, resulting the front wheel of the tractor now aligned with the longitudinal axis of the tractor. The gravitational forces are applied to the bodies in the negative y direction with reference to the plots that follow, which corresponds to terrain sloping downwards in the negative y direction. In this case, the system is assumed to start its motion on the slope and drive across the slope. The scenarios can be described briefly as follows: y m Fig. 3. Tractor Implement Trailer x m Articulated system trajectories under non-slip condition. IV. RESULTS The results shown in Figure 3 can be considered as the desired trajectory for the steering commands given. The non-holonomic constraints and hitch point constraints are in force. The square in the figure represents the tractor, the first triangle represents the implement and the second triangle represents the trailer. The results shown in Figure 4 are obtained using exactly the same steering command, however the path followed is significantly different. The nonholonomic constraint is not in force, however, the hitch point constraints are still applicable. Due to slippage, the degree of steering achieved is much less compared to the non-slipping case. Excessive steering will be needed to follow the same path. Figure 5 shows the implement s trajectory of the above two cases compared with different degrees of disturbance forces acting in the negative y direction. In the case of mild

5 5 5 y m 1 15 Non slip Slip, no disturbance Slip, small disturbance Sip, large disturbance x m Fig. 5. Implement s trajectory for all cases. y m Tractor Implement Trailer x m Fig. 4. Trajectories of the system with slip, no disturbances grade 2%, over a distance of 5 meters of straight run, the implement underwent a lateral shift of approximately 1.5 m. In the case of moderate grade 6%, the lateral shift for the same straight run is about 3 m. This demonstrates the need for a steering and propulsion controller for the agricultural tractors to guide their implements to maintain accurate path tracking while subjected to disturbances. V. CONCLUSION This work presents a comprehensive dynamic model of a three-body articulated agricultural vehicle. The model takes in to account various conditions that may be encountered in real farming conditions. Such conditions include disturbances due to lateral ground undulations, sloping terrains, tire slips, rolling resistances and drag forces due to ground engagement of the implement. Both the non-slip and slip model were derived to show the significance of accounting for slips in future path tracking control. As evidenced by the simulation results, the sliding effect gives rise to discrepancies between the trajectories which would cause a problem in high precision guidance of an agricultural mobile platform. In short, the implication of the assumption of nonholonomic constraints in agricultural applications is not feasible, and slip must be taken into consideration when designing a path tracking controller. The model developed lends itself ready for work to be undertaken in designing and testing various robust controllers for three-body articulated agricultural vehicles. REFERENCES [1 R.M. DeSantis. Path-tracking for car-like robots with single and double steering. IEEE Trans. on Vehicular Technology, 442: , May [2 B. d Andrea Novel, G. Bastin, and G. Campion. Modelling and control of non-holonomic wheeled mobile robots. In Proc. IEEE Conf. on Robotics and Automation, volume 2, pages , [3 B. Thuilot, C. Cariou, L. Cordesses, and P. Martinet. Automatic guidance of a farm tractor along curved paths, using a unique cpdgps. In Proc. of IEEE/RSJ of Int. Conf. of Intelligent Robots and Systems, volume 2, pages , 21. [4 R.M. DeSantis. Path-tracking for a tractor-trailer-like robot. The Int. J. of Robotics Research, 136: , December [5 R.M. DeSantis. Path-tracking for articulated vehicles with off-axle hitching. IEEE Trans. on Control System Technology, 64: , July [6 R. Lenain, B. Thuilot, C. Cariou, and P. Martinet. Model predictive control for vehicle guidance in presence of sliding: Application to farm vehicles path tracking. In Proc. of IEEE Int. Conf. on Robotics and Automation, 25. [7 H. Fang, R. Lenain, B. Thuilot, and P. Martinet. Trajectory tracking control of farm vehicles in presence of sliding. In IEEE/RSJ Int.Conf. on Intelligent Robots and Systems, pages 58 63, 25. [8 R. Lenain, B. Thuilot, C. Cariou, and P. Martinet. Adaptive and predictive non linear control for sliding vehicle guidance. In Proc. of IEEE/RSJ Int. Conf. on Intelligent Robots and Systems, pages , September October 24. [9 H. Fang, R. Lenain, B. Thuilot, and P. Martinet. Robust adaptive control of automatic guidance of farm vehicles in the presence of sliding. In Proc.of IEEE Int. Conf. on Robotics and Automation, pages , 25. [1 H. Pota, J Katupitiya, and R. Eaton. Simulation of a tractor-implement model under the influence of lateral disturbances. In Proceedings of the 46th IEEE International Conference on Decision Control, New Orleans, December 27.

6 [11 N. Matsumoto and M. Tomizuka. Vehicle lateral velocity and yaw rate control with two independent control inputs. Journal of Dynamics System Measurement and Control, 114:66 613, [12 Shiang-Lung Koo, Han-Shue Tan, and M. Tomizuka. Nonlinear tire lateral force versus slip angle curve identification. In Proc. of the American Control Conf., volume 3, pages , 24. APPENDIX I A. Detailed Expression of D and G matrices where, D 11 D 13 D 15 D 22 D 23 D 24 D 25 D = D 31 D 32 D 33 D 34 D 42 D 43 D 44 D 45 D 51 D 52 D 54 D 55 D 11 = m t + m i + m s D 13 = b + cm t sin φ D 15 = gm s sin ψ D 22 = m t + m i + m s D 23 = b + cm t cosφ D 24 = dm t e + fm s D 25 = gm s cosψ D 31 = b + cm t sin φ D 32 = b + cm t cosφ D 33 = J t + b + c 2 m t D 34 = b + cdm t cosφ D 42 = dm t e + fm s D 43 = b + cdm t cosφ D 44 = J i + d 2 m t + e + f 2 m s D 45 = ge + fm s cosψ D 51 = gm s sin ψ D 52 = gm s cosψ D 54 = ge + fm s cosψ D 55 = J s + g 2 m s cosφ + δ 1 cosφ sinφ + δ 1 sin φ G 1 = a + b + csin δ 1 d sinφ + δ 1 d sin φ cosφ + δ 1 cosφ cosδ 2 sinφ + δ 1 sinφ sinδ 2 G 2 = a + b + csin δ 1 d sinφ + δ 1 d sin φ e sinδ 2 cosδ 2 β ri cosψ sinδ 2 β ri sin ψ e sinδ 2 β ri e + fsinψ 36 sinφ + δ 1 sin φ cosφ + δ 1 cosφ G 3 = a + b + ccosδ 1 c d cosφ + δ 1 d cosφ sin δ 2 sin ψ cosδ 2 cosψ e cosδ 2 e + fcosψ 37 h + g cosφ α t cosα i sinφ α t sinα i G 4 = b + csin α t d sinφ α t cosψ + α s sinψ + α s e + fsinψ + α s 38 g sin α s G 11 G 21 G = G G 41 G 51 where G 11 = G 21 = m t + m i + m s w i θi [dm t e + fm s θ 2 i b + cm t θ2 t cosφ + gm s θ2 s cosψ m t + m i + m s ν i θi +b + cm t θ2 t sin φ + gm s θ2 s sin ψ G 31 = b + cm t θi v i cosφ w i sin φ d θ i sin φ G 41 = dm t v i θi + db + cm t θ2 t sin φ e + fm s v i θi ge + fm s θ2 s sinψ G 51 = gm s θi [w i sin ψ e + f θ i sin ψ + v i cosψ B. Detailed Expression of S Matrix The matrix S = {s 1, s 2, s 3, s 4, s 5 } T is such that, s 1 = 1 s 2 = e[c cosφtan δ 1 a + bsinφ/s + d tan δ 2 d + e s 3 = tanδ 1 s s 4 = [c cosφtan δ 1 a + bsin φ/s tan δ 2 d + e 1 + f d+e tan δ 2 cosψ sin ψ η/s s 5 = g + h where s = a + bcosφ + c sinφtan δ 1 η = f cosψ d + e [c cosφtan δ 1 a + bsin φ