Local and global sensitivity analysis of a tractor and single axle grain cart dynamic system model

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

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Research Paper

Local and global sensitivity analysis of a tractor and single axle grain cart dynamic system model Manoj Karkee, Brian L. Steward* Department of Agricultural and Biosystems Engineering, Iowa State University, Ames, IA 50011, USA

article info

Tractor and towed implement system models have become increasingly important for

Article history:

model-based guidance controller design, virtual prototyping, and operator-and-hardware-

Received 5 July 2009

in-loop simulation. Various tractor and towed implement models have been proposed in

Received in revised form

the literature which contain uncertain or time-varying parameters. Sensitivity analysis

26 March 2010

was used to identify the effect of system parameter uncertainty/variation on system

Accepted 26 April 2010

responses and to identify the most critical parameters of the lateral dynamics model for

Published online 2 June 2010

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. ª 2010 IAgrE. Published by Elsevier Ltd. All rights reserved.

1.

Introduction

Off-road vehicle system models are becoming increasingly important as mechatronic engineers increasingly rely on model-based controller design, virtual prototyping, and realtime hardware-and-operator-in-loop simulation in the design process (Antonya & Talaba, 2007; Castillo-Effen, Castillo, Moreno, & Valavanis, 2005; Cremer, Kearney, & Papelis, 1996; Howard & Vance, 2007; Karkee & Steward, 2008; Karkee,

Steward, Kelkar, & Kemp, 2010; Schulz, Reuding, & Ertl, 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

* Corresponding author. Department of Agricultural and Biosystems Engineering, Iowa State University, 214 D, Davidson Hall, Iowa State University, Ames, IA 50011, USA. Tel.: þ1 515 294 1452; fax: þ1 515 294 2255. E-mail address: [email protected] (B.L. Steward). 1537-5110/$ e see front matter ª 2010 IAgrE. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.biosystemseng.2010.04.006

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

List of variables a

a0 g d l s 4 a b c Ca d e F Iz 0 M

side slip angle or the angle between the direction the tyre is going and the direction it is facing. The velocity vector to the right of the tyre is positive and reverse is negative (rad) steady state side slip angle (rad) yaw-rate (rad s
1) steering angle (rad) angle between tractor heading and implement heading (rad) relaxation length (m) heading angle (rad) distance between front axle and CG of tractor (m) distance between rear axle and CG of tractor (m) distance between hitch point and CG of tractor (m) cornering stiffness (N rad
1) distance between hitch point and CG of implement (m) distance between rear axle and CG of implement (m) force (N) yaw moment of inertia (kg m2) global sensitivity measure (%output/%input)

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 & Tarantola, 2001; 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 (RodriguezFernandez & Banga, 2009). Sensitivity analysis has been used to optimise vehicle system design (Jang & Han, 1997; Park, Han, & Jang, 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,

m p S u v XeY x0 ey0 y

353

mass (kg) parameter vector local sensitivity measure (%output/%input) longitudinal velocity (m s
1) lateral velocity (m s
1) world coordinates vehicle coordinates position of CG in y-axis of the world coordinate system (m)

Superscript t tractor i implement Subscript x y z

1 x axis y axis z axis

Subscript f r c p

2 front tyre axle rear tyre axle centre of gravity hitch point

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 & Tomizuka, 1995; Deng & Kang, 2003; Kim, Yun, Min, Byun, & Mok, 2007) and off-road (Feng, He, Bao, & Fang, 2005; Karkee & 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 soiltyre 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 tyreesoil 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 & ElRazaz, 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

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properties such as internal friction angle, cohesion, cone index, and tyreeground 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 & Singh, 2004; Schwanghart, 1968; Schwanghart & 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 soiletyre 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, Benson, & Hummel, 1999; Whipker & 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, Steward, & Aziz, 2007; Katupitiya & Eaton, 2008; Pota, Katupitiya, & Eaton, 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.

2.

Methods

The sensitivity of a tractor and single axle grain cart steering system to variation in various model parameters was analysed using derivative-based sensitivity analysis. A dynamic bicycle model of the system (Karkee & Steward, in press) was adapted to perform the analysis. Grain cart mass and MI, tyre cornering stiffness, and tyre relaxation length parameters were assumed to have some level of uncertainty and/or variation over the field operation. Expanded ranges of these uncertain parameter values were used to define a parameter space for the study. A numerical simulation-based approach was used to calculate local and global sensitivities to those parameters.

2.1.

Vehicle model

A tractor and single axle towed implement model was required to study the sensitivity of the system states and outputs on various system parameters. Karkee and Steward (2008) studied three different models of a tractor and single axle grain cart system namely a kinematic model, a dynamic model and a higher fidelity model (a dynamic bicycle model with tyre relaxation length dynamics). Karkee and Steward (in press) extended the work by performing field experiments to validate the model responses. Among those three models, the higher fidelity model best fitted the experimental data (see also Karkee, 2009). This model was used in the present work to study the sensitivity of the system w.r.t. eight system parameters including three tyre cornering stiffness parameters, three tyre relaxation length parameters and two implement inertial parameters. Common vehicle dynamics symbols were used to describe the dynamics of the system (for notation, see the list of the variables). This model is described by Eqs. (1)e(8) (Fig. 1):



  mt þ mi v_ tc
mi cg_ t
mi dg_ i ¼
mi þ mt utc gt
Cta;f atf
Cta;r atr
Cia;r air




(1)

 Itz þ mi c2 g_ t
mi cv_ tc þ mi cdg_ i ¼ mi cutc gt
aCta;f atf þbCta;r atr þ cCia;r air



 Iiz þ mi d2 g_ i
mi dv_ tc þ mi cdg_ t ¼ mi dutc gt þ ðd þ eÞCia;r air

(2) (3)

a_ tf ¼

vtc agt utc ut þ t
t d
tc atf stf sf sf sf

(4)

a_ tr ¼

vtc bgt utc t
t
t ar str sr sr

(5)

a_ ir ¼

vtc cgt ðd þ eÞgi utc t utc i utc i
i
þ i 4
i 4
i ar si s si s s s

(6)

t j_ ¼ gt

(7)

i j_ ¼ gi

(8)

Eqs. (1)e(8) can be represented in matrix form as,

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y_ tc ¼ utc sinð4t Þ þ vtc cosð4t Þ

a

y

t c

Lt

 Small steering angle d: This assumption is reasonable for the intended application of tracking a line as required in automatic guidance of agricultural vehicles.  “Bicycle” approach meaning that the lateral forces in the left and right wheels were assumed to be equal and summed together. This assumption will be reasonable for small steering angle and small velocity operations expected in the automatic steering applications.  Roll, pitch and vertical dynamics were neglected. This assumption will be reasonable for relatively level fields with small steering angle and small velocity operations expected in automatic steering applications.  Small angle l, between the tractor and implement heading angles at the hitch point: This assumption will be valid for small steering angle operations.  The model was linearised about a zero steady state lateral velocity and zero yaw-rate with a constant longitudinal velocity. This linearisation is reasonable for the application of navigating the vehicle to track a straight line path.  Small side slip angle a leading to linear tyre model. Because the side slip angles remain small for small steering angles, this assumption will be valid for small steering angle operations.

Fxt, f

a

t

x’

Fyt, f

b

ct

Fxt,r

c

y’

Fxt, p P

d

F

t y ,r

Fxi, p

Ft

i

P

Fyi, p

Li ci

e

i

Fxi,r Fyi,r

Fxi,r

i c

Fyi,r

Y

b u tf

Lt

a

u C

u

c

u tp

u

t f

t

t

t r

vct v rt

t r

v tp

i p

v ip

P d

u ci Ci

e

V ft

Vct

b

P

v tf

t c

X

Li

i

v ci

u ri

i r

v

i r

Y

Fig. 1 e Dynamic bicycle model of a tractor and implement system; a) forces on the system, and b) velocities at different locations of the system. MX_ ¼ NX þ PU

(9)

where the state vector is X¼[vtcgtgiatfatrair4t4i]T and the input is U¼[d]. The tractor CG trajectory was calculated as follows. x_ tc ¼ utc cosð4t Þ
vtc sinð4t Þ

(11)

The higher fidelity model described by Eqs. (9)e(11) was developed based on the following assumptions.

X

y

355

(10)

Because the vehicle models and analysis performed in this work were intended for automatic guidance applications, the vehicle system will be controlled about a line and thus the small angle assumptions will be valid. If this model is used outside the range of operation for which the model was intended, i.e. the small steering angle (<10 ) and small forward velocity (<7.5 m s
1), the assumptions made in developing the model would be violated and the sensitivity results obtained based on this model may not be realistic. The parameter values suggested by Karkee and Steward (2008) were used as the nominal values in this study (Table 1). The parameters were based on a mechanical front wheel drive (MFWD) tractor (model 7930, Deere and Co., Moline, IL) and a single axle 18 m3 grain cart (model 500, Alliance Product Group, Kalida, OH). As mentioned before, the front tyre cornering stiffness Ctaf, rear tyre cornering stiffness Ctar, implement tyre cornering stiffness Ciar, front tyre relaxation length stf, rear tyre relaxation length str, implement tyre relaxation length sif, implement mass mi and implement moment of inertia Iiz were considered to be uncertain or varying parameters. Cornering stiffness and relaxation length parameters were dependent on the soiletyre properties, which can be highly uncertain. In the case of the tractor and grain cart system model, the implement mass and MI could also vary substantially depending on the load on the cart. Implement CG was considered to be fixed assuming that the shape of the grain cart and the load on it are symmetrical about the implement lateral axis. Geometric parameters of the tractor and implement system, tractor mass, tractor MI and the tractor CG were also considered fixed as they can be known or estimated with reasonable accuracy. A range of

356

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Table 1 e Dynamic bicycle model parameters for the JD 7930 tractor and Parker 500 grain cart system. Tractor

Implement

Parameters Nominal values Parameters Nominal values a b c mt Itz Ctaf Ctar stf str

1.7 m 1.2 m 2.1 m 9391 kg 35,709 kg m2 220 kN rad
1a 486 kN rad
1a 1.1 mb 1.5 mb

d e

3.62 m 0.1 m

mi Iiz

2127 kg 6402 kg m2

Ciar sir

167 kN rad
1a 1.1 mb

a Values based on the work of Metz (1993) and Schwanghart and Rott (1984). b Values based on the work of Bevly et al. (2002).

parameter values was defined for each of the uncertain/ varying model parameters (Table 2). These parameter ranges were used to define a parameter hyperspace used in the global sensitivity analysis. An uncertainty region of roughly 100% about the nominal values (Table 1) was defined for cornering stiffness and tyre relaxation length parameters. Because zero values for these parameters are not realistic in real-life conditions, small positive numbers (w10% of the nominal values) were used as the minimum values. In the case of the implement inertial parameters, the lowest limit roughly represented an empty grain cart (w10 m3) and upper limit roughly represented a 35 m3 grain cart with a full load of maize.

2.2.

Sensitivity analysis

Sensitivity is the measure of the variation in system responses w.r.t. small perturbations in a system parameter. When using a dynamic model, the more a state or output variable changes with a small change in a model parameter, the more sensitive it is to this parameter (Rodriguez-Fernandez & Banga, 2009). The sensitivity measure calculated at a particular location in the parameter space is called the local sensitivity. This measure will be valuable in studying the parameter uncertainty about a nominal point or a point of interest. The average sensitivity calculated over the entire parameter space is called the global sensitivity. This type of sensitivity is useful to identify the overall ranking of parameters in terms of their influence on the model responses within the parameter space as the method accounts for the interaction between model parameters (Kucherenko, Rodriguez-Fernandez, Pantelides, & Shah, 2009).

Table 2 e Ranges of the uncertain parameters of the JD 7930 tractor and Parker 500 grain cart system dynamic model. Tractor parameters

Implement parameters

20  Ctaf  450 kN rad
1 50  Ctar  980 kN rad
1 0.1  stf  2.2 m 0.15  str  3.0 m

1100  mi  7000 kg 3200  Iiz  21,000 kg m2 10  Ciar  340 KN rad
1 0.1  sir  2.2 m

Various sensitivity analysis techniques have been proposed in the literature to calculate both local and global sensitivity measures or indices. Primarily, these techniques can be classified into three groups, namely variance-based methods, screening methods, and derivative-based methods. In the variance-based methods, the contribution of a model parameter to the output variance is estimated (Kioutsioukis et al., 2004). Some of the widely used variance-based methods are Fourier Amplitude Sensitivity Test (FAST) (Cukier, Fortuin, Shuler, Petschek, & Schaibly, 1973; McRae, Tilden, & Seinfield, 1982), extended FAST (Saltelli, Tarantola, & Chan, 1999) and the method of Sobol (1993). The drawback of the variance-based methods is their computational inefficiency requiring a large number of function evaluations to calculate the sensitivity with a reasonable accuracy (Kucherenko et al., 2009). Screening methods are used to qualitatively rank the model parameters from the most influential to the least influential order. These methods are computationally attractive, but are less accurate (Kucherenko et al., 2009). The method of Morris (1991) is one such method. Derivative-based methods apply first order differentiation of the model outputs w.r.t. the model parameters. Local and global sensitivity measures are defined based on those derivatives. Jang and Han, 1997 and Kucherenko et al. (2009), among others, have proposed and used derivative-based methods. Kucherenko et al. (2009) used the method to analyse the sensitivity of seven algebraic equations to the equation parameter variations and compare the results with those from the method of Sobol and Morris. This method was more accurate than the method of Morris and was computationally more efficient than Sobol’s method. In addition, derivativebased methods provided both local and global sensitivity measures within the same framework. Because both local and global sensitivities were of interest, the derivative-based sensitivity analysis method was used in this work. Moreover, the derivative-based sensitivity analysis method was preferred over other methods in analysing continuoussystem dynamic models (Eberhard et al., 2007; Jang & Han, 1997; Park et al., 2003; Ruta & Wojcicki, 2003). In this work, some modifications were applied to the derivative-based formulation developed by Kucherenko et al. (2009) so that the sensitivity analysis of a complex dynamic model would be possible.

2.2.1.

Local sensitivity measures

A dynamic system model such as one presented in Section 2.1 can be represented as, X_ ¼ f ðX; p; t; uÞ

(12)

where X is the state variable vector, p is the parameter vector t is time, and u is the input. The local sensitivity of the system to a parameter pi is defined as the partial derivative of the function f w.r.t. the parameter. The local sensitivity function about a nominal point p* in the parameter space on parameter pi is then represented as, Si ðpÞ ¼

vf j vpi p¼p

(13)

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where i ¼ 1, ., n and n is the dimension of the parameter vector. As the point p* moves in the parameter space, the local sensitivity measures will also change. The sensitivities given by Eq. (13) were calculated numerically using the simulated responses of the dynamic model (Ruta & Wojcicki, 2003). Six system variables, namely tractor lateral velocity and yaw-rate, implement yaw-rate, tractor CG position, tractor heading, and implement heading, were used in the analysis, which accounted for sensitivities of important state variables as well as output variables. The tractor and grain cart dynamic system (Eq. (9)) was then represented in the state space form as, X_ ¼ AX þ BU

(14)

Y ¼ CX þ DU

(15)

where A ¼ M
1N, B ¼ M
1P, C ¼ identity matrix of size 8, and D ¼ [0]. M and N were defined in Eq. (9). First, the model was simulated with a 10 step steering input at the nominal point p* in the parameter space. The simulation was repeated at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities to perform sensitivity analysis at those three velocities. The simulation times were twice the settling times of the slowest non-zero eigenvalue of the system model, which were 30 s, 6 s and 3 s at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities respectively. With this simulation time, the sensitivity analysis accounted for both the transient and steady state sensitivities. The output responses from this simulation were used as the references. Parameters of interest were then varied one at a time by 1% of the nominal values to get the perturbed output responses. Finally, the local sensitivity of an individual output/state variable with the parameters at the nominal point was calculated using the finite difference equation:
 Yj p1 ; ...; pi
1 ; pi þ Dpi ; piþ1 ; ...; pn
Yj ðp Þ Sij ðp Þ ¼  Dpi

of the CG position, the distance between the end points of the reference and perturbed trajectories was normalised by the steady state turn radius of the reference trajectory. In the case of the heading angles, the heading difference at the end points of the reference and perturbed trajectories was normalised by the absolute heading difference between the start and end point of the reference trajectory. The normalised local sensitivity measures had units of percentage-of-output-response/ percentage-of-input-parameter (or %output/%input). Finally, the overall local sensitivity to a parameter pi was calculated using a mean over the output/state variables, Si ¼

m 1X Sij m j¼1

2.2.2.

where j ¼ 1,.,m; m ¼ number of output or state variables; Dpi* ¼ small change in the parameter of interest. The variable Yj represented time series data of an individual output response of the dynamic system. A root mean squared (RMS) change in the output variable over the simulation time was used to calculate the local sensitivity of an individual output/ state variable. This local sensitivity measure was calculated with all parameters of interest at various points in the parameter space. The changes in input parameters were divided by the nominal values of the parameters to normalise the results as it was not appropriate to directly compare or average the output sensitivities with various input parameters whose numerical ranges differed by several orders of magnitude. Output variables were also of different orders of magnitudes. An approach to define the sensitivity in this case might also be to normalise the changes in the output variables by the nominal output values. However, because the dynamic system responses included zero crossings over the simulation time, the three state variables were normalised only after the RMS changes in output variables were calculated. In the case

Global sensitivity measures

A global sensitivity measure can be calculated by averaging the local sensitivities (Eq. (13)) over the whole parameter space using (Kucherenko et al., 2009), 0

Z

Mi ¼

Si dp

(18)

Hn 0

where Mi is the global sensitivity of the system with the parameter pi, Hn is the parameter space, and p is the parameter vector. In this work, an 8-dimensional parameter space was defined by using ranges of uncertain model parameters about the nominal values (Table 2). A finite summation technique was applied to the model simulation-based local sensitivities (Eq. (16)) to calculate the global sensitivities of the output/state variables with each model parameter pi, 0

(16)

(17)

The overall local sensitivity measures (Eq. (17)) also had a unit of %output/%input.

Mij ¼



357

n 1X Sijk n k¼1

(19)

where k ¼ 1,.,n and n ¼ number of points in the parameter space. An overall global sensitivity measure was then calculated as M0i ¼

m 1X M0 m j¼1 ij

(20)

Hyperspace gridding was required to calculate the global sensitivity measures (Eqs. (19) and (20)). For an eight-dimensional parameter space used in this work, the computational load was tremendously high when a small grid resolution was used. Therefore, the sensitivity analysis was started with a fairly large grid size, and it was gradually reduced to achieve a good trade-off between the computational time and the accuracy achieved. The global sensitivities were also computed at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities. The average global sensitivities of the system over a simulation period of twice the settling times were used to rank the parameters from high to low importance in terms of the estimation accuracy required (Sobol, 2001). In summary, the local sensitivities of all six individual output variables w.r.t. each of the eight system parameters were calculated using the difference between the corresponding

358

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Fig. 2 e Propagation of uncertainty from input variables/ parameters to the output variables (adapted from Wittwer, 2009). Monte Carlo simulation can be used to iteratively evaluate the model and estimate the output uncertainties.

nominal and perturbed outputs (Eq. (16)). Corresponding local sensitivities of individual output variable were then averaged together to get the overall local sensitivities (Eq. (17)). A total of 390,000 of such overall local sensitivities were evaluated over the regularly gridded 8-dimensional parameter space. These local sensitivities were then averaged together to determine the global sensitivities w.r.t. the corresponding system parameters (Eq. (20)).

2.2.3.

Parameter uncertainty analysis

Parameter uncertainty analysis was performed to study the effect of the model parameter uncertainties on the system responses. Monte Carlo simulation is widely used to analyse uncertainty propagation from the input parameters pi to the

Fig. 3 e Arbitrary probability distributions of an output/ state variable. For the normally distributed input parameter, standard deviation was used to define the input parameter uncertainty u0 whereas half of the 68% interval was used to define the uncertainties associated with the output variables having distributions different than the normal distribution.

output variables yi (Istva´n, 1999). In Monte Carlo simulation, sets of randomly generated numbers are used as inputs to iteratively evaluate a system model and generate the distribution of output variations (Wittwer, 2009). Depending on the existing knowledge and/or estimates of the variables, different types of distributions can be used to define the variation in the input parameters (Fig. 2). In this study, input parameters were assumed to be normally distributed. Because the system was most sensitive to cornering stiffness parameters, the analysis was performed with just three cornering stiffness parameters. The three

Fig. 4 e Individual state/output local sensitivities at the nominal point in the parameter space w.r.t. a) implement mass, and b) implement MI. The sensitivities were calculated at 1.0 m sL1, 4.5 m sL1 and 7.5 m sL1 forward velocities.

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

359

Fig. 5 e Individual state/output local sensitivities at the nominal point in the parameter space w.r.t. cornering stiffness parameters; a) front tyre cornering stiffness, b) rear tyre cornering stiffness, and c) implement tyre cornering stiffness.

cornering stiffness parameters varied together because the parameters were positively correlated. Monte Carlo simulation was performed at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities with a set of 2500 cornering stiffness parameter values randomly selected from normal distributions. Then, the variations in the six individual output/state responses were averaged to generate the distribution of the output variation. The same numerical simulation technique used in the sensitivity analysis method was used to evaluate the tractor and towed single axle grain cart model for the Monte Carlo simulation. The uncertainty u0 associated with a parameter or a variable can be quantitatively defined by the absolute or relative error about the estimated value (Ellison, Rosslein, & Williams, 2000; Wittwer, 2009). In this work, uncertainties were represented by the relative error written as a percentage. For the normally distributed input parameters, the standard deviation, s, was used to define the uncertainty as this was the most widely used uncertainty measure (Ellison et al., 2000). With this uncertainty measure, 68% of the total estimates of an uncertain variable were used to define the level of uncertainty in the variable. Assuming that the estimated values of the tractor and single axle grain cart system parameters were

normally distributed with means representing the expected parameter values, standard deviations of the distributions were used as the uncertainties in the input parameters. However, the distribution of the average output variation w.r.t. the normally distributed input parameters did not follow a normal distribution at any forward velocity. In this case, a 68% interval in the central lobe of the output distribution was calculated leaving 16% samples of the distribution in each tail. Half of this interval was used as the uncertainty measure for the output responses (Fig. 3). To model the relationship between input and output uncertainties (Grau, 2002), the uncertainty propagation study was performed with varying level of uncertainties in cornering stiffness parameters.

3.

Results and discussions

3.1.

Local sensitivity analysis

The local sensitivities at the nominal point (Table 1) in the parameter space showed the effect of variations in the system parameters on the system responses (Figs. 4e6). In general, lateral velocity and yaw-rate states were more sensitive to the

360

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Fig. 6 e Individual output/state local sensitivities at the nominal parameter point to a) front tyre relaxation length, b) rear tyre relaxation length, and c) implement tyre relaxation length.

input parameters at 1.0 m s
1 forward velocity than at 4.5 m s
1 and 7.5 m s
1 forward velocities. As expected, the implement mass, implement MI, and implement tyre cornering stiffness parameters primarily influenced the implement yaw-rate and heading responses. The tractor CG position, tractor heading and implement heading outputs

were most sensitive to the front and rear tyre cornering stiffness parameters. Local sensitivities with implement inertial parameters were relatively small at all forward velocities with a maximum of 0.14 for the tractor lateral velocity (Fig. 4). Implement yaw-rate and tractor lateral velocity were the most

b Tractor Yaw-rate (rad s -1 )

(rad s -1 )

a 0.12 Reference Perturbed

Tractor Yaw-rate

0.08

0.04

0

0

2

4

Time (s)

6

0.4

Reference Perturbed

0.3 0.2 0.1 0

0

2

Time (s)

4

6

Fig. 7 e Tractor yaw-rate time histories for the nominal parameters (reference) and 10% change in the rear tyre relaxation length parameter (perturbed); a) at 1.0 m sL1 forward velocity, b) at 4.5 m sL1 forward velocity.

361

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

b

0.4

Tractor Yaw-rate (rads-1)

Tractor Yaw-rate (rads-1)

a

0.3 0.2

Reference Perturbed

0.1 0

0

1

2

3 Time (s)

4

5

6

0.5 0.4 0.3

Reference Perturbed

0.2 0.1 0

0

0.5

1

1.5 Time (s)

2

2.5

3

Fig. 8 e Tractor yaw-rates for the nominal parameters (reference) and 10% change in the rear tyre cornering stiffness (perturbed); a) at 1.0 m sL1 and b) at 4.5 m sL1 forward velocities.

sensitive states to both the implement mass and implement MI. This result was expected as the implement inertial parameters will have a higher influence on the implement dynamics, and the tractor lateral velocity was coupled with the implement yaw-rate state. Because the implement MI influenced only the transient implement yaw-rate, the influence on the implement heading was diminished when the yaw-rate was integrated to get the heading angle. In general, velocity response sensitivities (tractor lateral velocity, tractor yaw-rate and implement yaw-rate) with tyre cornering stiffness parameters decreased as the forward velocity increased to 4.5 m s
1 from 1.0 m s
1 and increased as the forward velocity increased to 7.5 m s
1 from 4.5 m s
1 (Fig. 5). However, the positional response sensitivities (tractor heading, implement heading and tractor position) increased consistently with increasing forward velocity. All responses were equally sensitive to tractor front tyre cornering stiffness variation. In general, the system was more sensitive to the cornering stiffness parameters than to the other uncertain parameters (Figs. 4 and 6). Tyre relaxation lengths only influenced the velocity responses of the system (Fig. 6). The sensitivities were much higher at 1.0 m s
1 than at higher velocities. The lateral velocity and yaw-rate sensitivities to tyre relaxation length variations were caused by the changes in the damped natural frequency of the under-damped transient responses as a relaxation length does not affect the steady state system responses. The changes in the natural frequency of the underdamped transient responses were cancelled out when the

lateral velocity and yaw-rate responses were integrated to get the position and heading responses. The tractor CG position, tractor heading and implement heading sensitivities with relaxation length parameters were thus negligible even when the lateral velocity and yaw-rate states were highly sensitive. As discussed above, the system velocity states were highly sensitive to all input parameters at 1.0 m s
1 forward velocity. The system responses were less damped at 1.0 m s
1 than at 4.5 m s
1 and 7.5 m s
1 forward velocities (Fig. 7), and the interaction between tyre relaxation length and other parameters caused a larger change in un-damped natural frequencies between the reference and perturbed responses at 1.0 m s
1 (Fig. 7a). The increased natural frequency resulted in the higher system sensitivities at the low forward velocity. Physically, as the forward velocity is increased, the time taken by the tyre to reach the steady state side slip angle (which is the time taken to travel the equivalent relaxation length) would decrease and thus the effect of relaxation length diminishes. Generally, the local sensitivities with the tyre relaxation length parameters at 7.5 m s
1 forward velocity were higher than those at 4.5 m s
1. This result was caused by the larger change in the steady state output responses at higher forward velocity as compared to the changes in the steady state responses at lower forward velocities (Fig. 8). This result also indicated that the cornering stiffness parameters will be more important to the system responses as these parameters will influence the steady state responses in addition to the transient responses of the system.

Fig. 9 e Overall local sensitivities at nominal point in the parameter space w.r.t. eight system parameters.

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Fig. 10 e Overall local sensitivities at a sample point in the parameter space w.r.t. various system parameters.

At the nominal parameter point, the overall system was most sensitive to tractor front and rear tyre cornering stiffness variations (Fig. 9). The front and rear tyre relaxation lengths were the next two parameters to which the system response was more sensitive. The system was least sensitive to the implement mass and MI. At 1.0 m s
1, the system was almost equally sensitive to cornering stiffness and corresponding relaxation length parameters. At 4.5 m s
1 and 7.5 m s
1, however, the tractor and implement system was twice as sensitive to cornering stiffness parameters as it was to relaxation length parameters. The system was less sensitive to the cornering stiffness parameters at 4.5 m s
1 than it was at 1.0 m s
1 forward velocity. When the forward velocity was increased to 7.5 m s
1, the system became more sensitive to all three cornering stiffness parameters. This result can be explained because the cornering stiffness parameters define the velocity state of the eigenvalues, which dominate the system response at higher forward velocities (Karkee & Steward, 2008). However, because the interaction between cornering stiffness and relaxation length parameters was substantial at lower velocities (Fig. 5), the system was highly sensitive to cornering stiffness parameters at 1.0 m s
1 forward velocity. Similar trends were observed with the implement inertial parameters. The system was most sensitive to variations in implement mass and MI at 1.0 m s
1.

The interactions between input parameters were observed by computing local sensitivities at various points in the parameter space. At a sample point with small relaxation length parameters (10% of the nominal values), large implement inertial parameters (200% of the nominal values), and nominal cornering stiffness parameters, the interaction between relaxation length and other input parameters was not substantial even at 1.0 m s
1 forward velocity (Fig. 10). As expected, the sensitivities with cornering stiffness and implement inertial parameters were smaller at 1.0 m s
1. The system sensitivities with cornering stiffness parameters increased consistently as the forward velocity increased. At this point, the system was most sensitive to cornering stiffness parameters at all velocities. The front and rear tyre relaxation length parameters were the second most influential parameters at 1.0 m s
1 and 4.5 m s
1 forward velocities whereas implement inertial parameters were the second most influential parameters at 7.5 m s
1 forward velocity.

3.2.

Global sensitivity analysis

The local sensitivities at different points in the parameter space varied due to the interaction between the input parameters (Figs. 9 and 10). The overall system sensitivity within the parameter space was studied with global sensitivity measure. The grid resolution selected for the parameter

Fig. 11 e Average global sensitivities over 390,000 regularly gridded points in the parameter space.

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

Fig. 12 e Average global sensitivities of a tractor without the towed implement to cornering stiffness and relaxation length parameters.

hyperspace gridding had an effect on the computational load and accuracy of the global sensitivity calculation. As the resolution was decreased, the accuracy of the global sensitivity improved, but with an O(N8) increase in the computation time. The global sensitivities did not change much after the resolution was decreased to 1/5th of the parameter range, i.e. when there were five grid points in each parameter axis. With this grid resolution, 390,000 points in the parameter space were used to calculate global sensitivities w.r.t. various model parameters (Fig. 11). In terms of overall global sensitivities, generally the system was more sensitive to the cornering stiffness parameters than

363

to the implement inertial parameters and the tyre relaxation length parameters. This result indicated that the soil-tyre interaction parameters will be more influential to the lateral dynamic behaviour of the system than the inertial parameters. At 1.0 m s
1 and 4.5 m s
1, the system was most sensitive to the tractor rear tyre cornering stiffness (Fig. 11). At 7.5 m s
1, the tractor front tyre relaxation length was the most influential parameter. At lower velocity, similar to the local sensitivities at the nominal point, the tyre relaxation length parameters remained highly influential. The second most influencing parameter at 4.5 m s
1 was the tractor front tyre relaxation length whereas at 7.5 m s
1 the tractor rear tyre cornering stiffness was the second most influencing parameter. The system sensitivity to the variation in cornering stiffness parameters increased as the forward velocity increased. This result was expected as the higher order dynamics influenced by the cornering stiffness parameter dominate the system response as the forward velocity is increased. Similar trends were observed with the implement inertial parameters (implement mass and MI). However, the variation in the system sensitivities with the inertial parameters was much smaller than the variation with cornering stiffness parameters. With respect to the relaxation length parameters, the system was most sensitive at 1.0 m s
1. These sensitivities decreased substantially when the velocity increased to 4.5 m s
1 but did not change much when the velocity increased to 7.5 m s
1 from 4.5 m s
1. The system sensitivity of a tractor model (without an implement) to cornering stiffness and relaxation length parameters also showed the similar pattern as in the tractor and towed implement case (Fig. 12). In general, the result indicated that the cornering stiffness parameter will have a substantial influence on the responses

Fig. 13 e Probability density function of the average output variation w.r.t. the variation in the cornering stiffness parameters about their nominal values at different forward velocities; a) variation in cornering stiffness parameters (u0 [ 10%), b) average output variation at 1.0 m sL1(u0 [ 1.5%), c) average output variation at 4.5 m sL1(u0 [ 2.0%), and d) average output variation at 7.5 m sL1(u0 [ 4.0%). Scaled relative frequency was used to represent the histograms (bars) so that the histograms also represented the probability distribution function. Lines represent the normal distribution fit (a) and skewed normal distribution fits (bed) of the probability distribution functions.

364

b i o s y s t e m s e n g i n e e r i n g 1 0 6 ( 2 0 1 0 ) 3 5 2 e3 6 6

with more confidence for applications where the system is operating with higher forward velocities. The result also showed that if a 5% average uncertainty in system responses is acceptable, the cornering stiffness parameters must be estimated within 50%, 25% and 15% of actual/nominal values at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities, respectively.

40

Output Uncertainty (%)

35 30

-1

Vel = 1.0

ms

Vel = 4.5

ms

Vel = 7.5

ms

25

-1

-1

20 15

4.

10 5 0

0

10

20

30

40

50

60

Uncertainty in Cornering Stiffness (%) Fig. 14 e The relationship between average uncertainty over the six output variables of the system and the uncertainty in cornering stiffness parameters. Lines in the figure represent the polynomial fit of the relationship between input and output uncertainties at 1.0 m sL1, 4.5 m sL1 and 7.5 m sL1 forward velocities.

of the lateral dynamics model of an agricultural vehicle and need to be estimated with reasonable accuracy.

3.3.

Parameter uncertainty analysis

The distributions of the average output variation were skewed to the right when the variation in the cornering stiffness parameters was normally distributed (Fig. 13). A goodness of fit test at 1% significance level did not support the hypothesis that the output variations were normally distributed. The distribution of the average output variation at 1.0 m s
1 was not very skewed about the reference output (skewness of 0.04), but followed an approximately uniform distribution (Fig. 13b). The skewness of the distributions at 4.5 m s
1 and 7.5 m s
1 forward velocities were 0.46 and 0.56 (Fig. 13c and d). In general, the uncertainty u0 in output variables, as defined in Section 2.3, increased as the forward velocity was increased. For a 10% uncertainty in the cornering stiffness parameters, the output variables had an average of 1.5%, 2.0% and 4.0% uncertainties respectively at 1.0 m s
1, 4.5 m s
1 and 7.5 m s
1 forward velocities. The six state/output variables (Fig. 4) were used to calculate the effect of varying system parameter uncertainties on the system responses. In general, the parameter uncertainties and uncertainties in the system responses were related in a nonlinear fashion (Fig. 14). The trend lines relating these uncertainties had r-squared values of 0.98 or more at each velocity. Generally, the output uncertainties increased most rapidly at 7.5 m s
1 and least rapidly at 1.0 m s
1 forward velocities with the increasing input uncertainties. The average output uncertainty followed a second order polynomial relationship with the uncertainties in cornering stiffness parameters at 4.5 m s
1 and 7.5 m s
1 forward velocities. The same relationship was linear at 1.0 m s
1 forward velocity. The results indicated that these system parameters should be estimated

Conclusions

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.

Acknowledgements This research was supported by Hatch Act and State of Iowa funds. Journal paper of the Iowa Agriculture and Home Economics Experiment Station, Ames, Iowa, Project No. 3612.

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