تجزیه و تحلیل حساسیت محلی و جهانی از یک تراکتور و مدل سیستم پویا سبد دانه تک محور

Tractor and towed implement system models have become increasingly important for model-based guidance controller design, virtual prototyping, and operator-and-hardware-in-loop simulation. Various tractor and towed implement models have been proposed in the literature which contain uncertain or time-varying parameters. Sensitivity analysis was used to identify the effect of system parameter uncertainty/variation on system responses and to identify the most critical parameters of the lateral dynamics model for a tractor and single axle grain cart system. Both local and global sensitivity analyses were performed with respect to three tyre cornering stiffness parameters, three tyre relaxation length parameters, and two implement inertial parameters. Overall, the system was most sensitive to the tyre cornering stiffness parameters and least sensitive to the implement inertial parameters. In general, the uncertainty in the input parameters and the system output responses were related in a non-linear fashion. With the nominal parameter values for a Mechanical Front Wheel Drive (MFWD) tractor, a single axle grain cart, and maize stubble surface conditions, a 10% uncertainty in cornering stiffness parameters caused a 2% average uncertainty in the system responses whereas a 50% uncertainty in cornering stiffness parameters caused a 20% average uncertainty at 4.5 m s−1 forward velocity. If a 5% average uncertainty in system responses is acceptable, the cornering stiffness parameters must be estimated within 25% of actual/nominal values. The output uncertainty increased as the forward velocity was increased.

Off-road vehicle system models are becoming increasingly important as mechatronic engineers increasingly rely on model-based controller design, virtual prototyping, and real-time hardware-and-operator-in-loop simulation in the design process (Antonya and Talaba, 2007, Castillo-Effen et al., 2005, Cremer et al., 1996, Howard and Vance, 2007, Karkee and Steward, 2008, Karkee et al., 2010 and Schulz et al., 1998). As the cost of computational power required for numerical simulation is decreasing, vehicle models have become increasingly complex often possessing dozens of model parameters. Accurate estimation of those parameters is often difficult because of the high variability in field conditions. Because uncertain parameter estimates will have an effect on the model responses, off-road vehicle simulations may often be unrealistic, limiting the applicability of the models and model-based studies (Kioutsioukis, Tarantola, Saltelli, & Gatelli, 2004). It is important to understand and quantify the effect of these parameter uncertainties or variations on the system response (Fales, 2004). Sensitivity analysis is one approach to identify and quantify the relationships between input and output uncertainties (Xu & Gertner, 2007). Sensitivity analysis evaluates the variation in dynamic model outputs with respect to (w.r.t.) variation in model parameters (Crosetto and Tarantola, 2001 and Deif, 1986). Thus, sensitivity analysis can be used to perform uncertainty analysis, estimate model parameters, analyse experimental data, guide future data collection efforts, and suggest the accuracy to which the parameters must be estimated (Rodriguez-Fernandez & Banga, 2009). Sensitivity analysis has been used to optimise vehicle system design (Jang and Han, 1997 and Park et al., 2003). Jang and Han (1997) used a direct differentiation method to study the sensitivity of on-road vehicle lateral dynamics on tyre cornering stiffness, location of vehicle centre of gravity (CG), vehicle mass, and vehicle moment of inertia (MI) using a bicycle model of a front wheel steered vehicle. The study was performed at typical on-road vehicle velocities ranging from 9 m s−1 to 53 m s−1. Park et al. (2003) performed a dynamic sensitivity analysis for a pantograph of a rail vehicle. Dominant design variables were identified using derivative-based state sensitivity measures and were modified for optimal design. Ruta and Wojcicki (2003) also applied sensitivity analysis to a dynamic railroad track vibration model. They focused on developing an analytical solution for derivative-based sensitivity analysis of a system represented by a set of differential equations. Both first and second order derivatives were used to isolate a set of parameters to which the model outputs were the most sensitive. Eberhard, Schiehlen, and Sierts (2007) performed a vehicle model sensitivity analysis with various design variables of a passenger car and found that vehicle dynamics were highly sensitive to MI and the CG location. The study was conducted at 10 m s−1 forward velocity. In off-road vehicle systems, the application of sensitivity analysis to understand the effect of uncertain parameters on a tractor and towed implement lateral dynamics is important to support emerging farm automation technology such as implement guidance and coordinated guidance. Various tractor and towed implement steering models have been proposed in the literature for both on-road (Chen and Tomizuka, 1995, Deng and Kang, 2003 and Kim et al., 2007) and off-road (Feng et al., 2005 and Karkee and Steward, 2008) operations. As suggested by these models, the lateral dynamics of an off-road tractor and towed implement system depend on the lateral forces generated by soil-tyre interactions. A vehicle tyre, when subjected to a steering side force, does not move to the direction it is facing resulting in an angular difference between the two directions called the side slip angle (Wong, 2001). Vehicle tyres go through some level of deformation when they move to a direction different from the direction they are facing. This deformation produces shear stress at the tyre–soil interface. This shear stress causes some level of lateral soil deformation as well, which releases part of the shear stress developed at the interface (Metz, 1993). The resultant shear stress will generate a force at the contact patch called lateral tyre force or cornering force (Crolla & El-Razaz, 1987). In addition to tyre and lateral soil deformation, phenomena such as tyre-soil friction and soil sinkage effects also affect the characteristics of lateral tyre force development (Metz, 1993). Because most of these phenomena are dependent on soil properties such as internal friction angle, cohesion, cone index, and tyre–ground surface friction, the lateral tyre forces and thus the off-road vehicle responses will vary with different soil types (e.g. clay, sand or loam), soil condition (e.g. density and moisture content), and soil surface cover (e.g. bare soil, vegetation or stover; Crolla & El-Razaz, 1987). In addition, the responses may vary with the variation in tyre construction, size, inflation pressure and normal load (Krick, 1973; Raheman and Singh, 2004, Schwanghart, 1968 and Schwanghart and Rott, 1984). It may be difficult to estimate these variables accurately, and it is also difficult to find a widely accepted model to relate these variables to the tyre lateral force. The cornering stiffness coefficient, which is the slope of the lateral tyre force as a function of side slip angle at zero side slip, is the parameter often used to represent the combined effect of these variables (Metz, 1993). Because the soil–tyre parameters are uncertain, varying, and/or inaccurate, the cornering stiffness parameter also tends to be highly uncertain. Vehicle tyre relaxation length, which is defined as the distance a tyre rolls before the steady state side slip angle is reached, is another parameter that could be highly uncertain (Bevly, Gerdes, & Parkinson, 2002). In the case of implements such as grain carts or towed sprayers, implement mass and MI may also be uncertain or may vary substantially over time. Several researchers developed automatic steering controllers for agricultural vehicles and some manufacturers have commercialised these technologies (Bell, 2000, Bevly et al., 2002, Stombaugh et al., 1999 and Whipker and Akridge, 2008). Because implements are often used to perform field operations, it is important to extend the capabilities of these automatic guidance systems to field implements (Bevly, 2001, Karkee et al., 2007, Katupitiya and Eaton, 2008 and Pota et al., 2007). Various emerging automation techniques such as coordinated guidance and autonomous guidance will also have to incorporate the tractor and implement systems. In this regard, sensitivity analysis of the tractor and implement system will be important. In this work, dynamic model sensitivity analysis will be performed to quantify the dependence of tractor and single axle grain cart system responses on the uncertain model parameters. A single axle grain cart will be used as the implement in this work because it represents the general behaviour of a towed implement while offering a well defined system to be modelled. The system provides a good starting point in understanding tractor and towed implement lateral dynamics, which then can be extended to other towed implements or a chain of towed implements. This analysis will increase understanding of the relative significance of the uncertain model parameters, so that more resources can be allocated to more accurately estimate those parameters to which the system is more sensitive. Specific objectives were: • to investigate the changes in sensitivities with the changes in vehicle forward velocity, • to identify the parameters to which the model is most sensitive, and • to evaluate the effect of parameter uncertainties on the system response uncertainty.

Local and global sensitivities of a tractor and single axle grain cart system were calculated using a derivative-based method. Due to the complexity in deriving analytical sensitivity indices, numerical sensitivity analysis was performed using simulation-based dynamic system model responses. The work is important in understanding the effect of parameter variations on the tractor and towed implement system responses. The system parameters were ranked based on the relative importance of the parameters to the system responses. It can be concluded from the work that there is a need for better estimates of the soil-tyre interaction component of off-road vehicle dynamics simulation as it will play an important role in the vehicle responses. It was also determined that changes in implement mass and MI will have little effect on the off-road vehicle responses. Specifically, the following conclusions were drawn from this work. • Overall, cornering stiffness was the most influential parameter of the tractor and a single axle towed implement system dynamics. In terms of global sensitivity, the system was most sensitive to the tractor rear tyre cornering stiffness followed by the tractor rear tyre relaxation length at 1.0 m s−1. At 4.5 m s−1 and 7.5 m s−1, rear and front tyre cornering stiffness were the most important two parameters. In terms of local sensitivity at the nominal point, the system was equally sensitive to both the tractor front and rear tyre cornering stiffness parameters. The system was least sensitive to the implement inertial parameters. • In general, the cornering stiffness parameters influenced the system dynamics more as the forward velocity increased whereas the tyre relaxation length parameters were less influential with the increasing forward velocity. • Uncertainty in system parameters and system outputs were related in a non-linear fashion. About the nominal values and 4.5 m s−1 forward velocity, a 10% uncertainty in the cornering stiffness parameters resulted in a 2% average output uncertainty whereas a 50% cornering stiffness uncertainty resulted in a 20% output uncertainty.