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Optimized Chassis Stability Relative to Dynamic Terrain Profiles in a Self-Propelled Sprayer Multibody Dynamics Model
Bailey Adams1,*, Matthew Darr1, Aditya Shah2
Published in Journal of the ASABE 66(1): 127-139 (doi: 10.13031/ja.15230). Copyright 2023 American Society of Agricultural and Biological Engineers.
1Agricultural and Biosystems Engineering, Iowa State University, Ames, Iowa, USA.
2Virtual Design & Verification, John Deere Des Moines Works, Ankeny, Iowa, USA.
* Correspondence: adamsba@iastate.edu
The authors have paid for open access for this article. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License https://creative commons.org/licenses/by-nc-nd/4.0/
Submitted for review on 9 June 2022 as manuscript number MS 15230; approved for publication as a Research Article by Associate Editor Dr. Yanbo Huang and Community Editor Dr. Heping Zhu of the Machinery Systems Community of ASABE on 7 December 2022.
Highlights
Abstract. Multibody dynamics (MBD) models are continuing to be valuable for engineering design and product development, especially regarding subsystem optimization. Most MBD optimization processes begin with a sensitivity analysis of treatment factors and levels to understand how uncertainty in model inputs can be attributed to different sources of uncertainty within model outputs; however, this study developed a new MBD methodology to automatically determine the optimized dynamic chassis suspension responses on each corner of the vehicle from a single simulation for a self-propelled sprayer model as the chosen application use-case. This technique leveraged a prismatic joint (with a high spring stiffness and damping coefficient) connected between the chassis mainframe and the simplified optimization tire to create a distance constraint that held the chassis body at a near-consistent height above the ground. Then the solver optimized the response of the chassis suspension system to maintain a stable chassis platform relative to the terrain beneath it as the vehicle traversed across dynamic terrain conditions. This optimization response was also accomplished by replacing the baseline chassis suspension components with a free-floating cylinder, which permitted the unrestricted, optimized motion needed to keep the chassis body at a near-level position with respect to the roll and pitch profiles of the terrain. For a simulation with an aggressive terrain configuration, the analysis showed that an optimized suspension system resulted in a 46% decrease in operator comfort and a 19.5% increase in overall boom height stability as the boom height control system better maintained a dynamic position closer to the specified target height.
In recent years, there has been an increasing trend toward using model-based design strategies and simulations to provide data-driven decisions early in the engineering design process (Sargent, 2013). Especially for off-road vehicle applications, model-based design has been successfully utilized to simulate a wide variety of agricultural equipment and to generate virtual data that supports analyzing aspects like safety, mobility, stability, and operating loads (Kading, 2006). The underlying objective of most of these strategies is to provide an accurate basis for the virtual dynamic behavior of a given agricultural machine so that design modifications and experimental concepts that improve performance can be studied.
The multibody dynamics (MBD) framework is a common model-based design structure for dynamic performance analysis of mechanical systems (Zhu, 2014). An MBD system is generally defined as a rigid and/or flexible mechanical system composed of mass bodies and kinematic joints (Pappalardo, 2015). When a joint is introduced into an MBD model, it imposes a kinematic constraint that determines how two bodies move translationally and/or rotationally to each other (MathWorks, 2022a). Translational degrees of freedom (DoF) and/or rotational degrees of freedom can be modeled through a joint, but any given joint cannot exceed a total of six degrees of freedom (i.e., three translational plus three rotational DoF). The MBD environment can be very helpful for a technical understanding of the interactions between dynamic subsystems, joints, bodies, etc., and the inherent characteristics that govern the respective dynamic behavior for specific agricultural machines that are represented by complex DoF.
With state-of-the-art MBD off-road vehicle models, one interesting application is the design optimization of certain vehicle subsystems (Zhu et al., 2013). In general, optimization is defined as a process that determines a design that minimizes (or maximizes) an objective function while fulfilling predefined design constraints. For agricultural machine applications, the design is often defined by variables like mass properties, stiffness/damping characteristics, control system PID gains, operational parameters, etc.
With respect to MBD systems, design optimization presents challenges induced by sensitivity analysis, high computational effort, and transient behaviors (Wehrle et al., 2021). “Sensitivity analysis, which quantifies the effect of specific design parameters of interest on the dynamic response of a multibody dynamic system, is often performed before the optimization or in parallel with it” (Zhu, 2014). The fundamental purpose of sensitivity analysis is to study how uncertainty in model outputs can be attributed to different sources of uncertainty within model inputs. However, a robust sensitivity analysis in MBD models typically necessitates a large number of simulations, which can be costly in terms of computation and time, depending on the number of design factors and levels included in an overall experimental design matrix (Zhu et al., 2013).
There have been a number of previous MBD-focused studies that have performed various aspects of optimization procedures:
- Yang et al. utilized an MBD environment to perform sensitivity analysis on the design parameters of a tractor tandem suspension system to optimize ride comfort (Yang et al., 2009).
- Cepon created an algorithm that identifies an optimal equilibrium point for minimizing the interplay between different joint parameters on the natural frequencies of an MBD system.
- Tazaki and Suzuki (2014) demonstrated a trajectory-planning technique where a constraint was leveraged to automatically determine the optimized dynamics of rigid bodies and kinematic conditions of joints that allow the MBD system to follow the planned trajectory.
- Hong et al. (2010) presented an “optimization methodology that iteratively links the results of MBD simulations and structural analysis software” to an optimization method so that flexible MBD systems under dynamic loading conditions can be designed (Hong et al., 2010).
Most of these studies relied on sensitivity analysis as part of the optimization procedure. This research, however, steered away from typical sensitivity analysis and instead implemented a new methodology to automatically compute an optimized dynamic response from a single simulation for a self-propelled sprayer MBD model. More specifically, this study developed an optimization subsystem that applied values for internal mechanics conditions within a prismatic joint, which controls only one translation DoF in the vertical Z direction (MathWorks, 2022b). This prismatic joint connected the chassis body and a simplified, optimized tire, which was in turn linked to the virtual terrain mesh. The solver then automatically calculated and administered an optimized chassis suspension response through a free-floating cylinder on each corner of the vehicle to keep the chassis mainframe at a near-constant vertical distance above the terrain. Therefore, the optimization objective function for this study was to maintain a constant distance from each corner of the chassis mainframe to the dynamic terrain underneath.
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| Figure 1. Diagram of categorized self-propelled sprayer subsystems on single-ramp (height = 355.6 mm, length = 10,089.2 mm, inclination angle = 6.6 degrees) virtual terrain. |
In other words, a prismatic joint, with high spring stiffness and damping internal mechanics values, was used to hold the chassis body stable at a relatively consistent height above the ground as the vehicle traversed across dynamic terrain. Furthermore, by replacing the baseline chassis suspension components with a free-floating cylinder on each corner of the vehicle, the chassis suspension was “free-to-float” over the terrain, which resulted in an optimized chassis suspension response required to keep the chassis at a level position with respect to the instantaneous roll and pitch profile of the terrain. This optimized chassis suspension innovation allowed for the performance evaluation of a few aspects, specifically the evaluation of operator comfort and boom height stability.
Materials and Methods
Validated Self-propelled Sprayer MBD Model
As previously stated, this study used a high-fidelity MBD representation of a self-propelled sprayer, which in this use case was a John Deere R4038 machine that was modeled in Simscape Multibody (MathWorks, 2022c). As shown in figure 1, the vehicle system could be categorized into four main subsystems: (1) chassis suspension, (2) chassis, engine, cab, and solution tank, (3) fixed and suspended centerframes, and (4) left and right booms. The chassis suspension attempts to suspend the sprung mass, isolate the rest of the vehicle from terrain disturbances, and maintain a consistent contact patch with the terrain. The chassis mainframe acts as a platform for the engine, cab, solution tank, and other large mass bodies to reside upon. The fixed centerframe is rigidly attached to the chassis mainframe, and the suspended centerframe acts as a secondary vehicle suspension to mitigate dynamic disturbances from reaching and impacting the booms. In the physical realm, the booms serve as structures that are used to help deliver chemicals across wide swaths. Boom movement is generally determined through passive mechanical dynamics or through boom height control, which is a control system that attempts to maintain the booms at a specified reference target height above the terrain. This control system is dependent upon sensing the height of the boom in relation to the ground through ultrasonic sensors, which are depicted by the strategically placed yellow lines across the width of the booms in figure 1. Out of these main subsystems, the chassis suspension and chassis mainframe were of particular interest for the following optimization modeling methodology and implementation.
Before this model could be employed for confident optimization analysis, it was subjected to a thorough validation procedure where a variety of tests were conducted to compare the virtual response against a physical ground-truth benchmark. These tests included an extensive breadth of magnitude and frequency domains and employed both steady-state and transient maneuvers. The main validation methodology and results are outlined in a study by Adams and Darr (2022).
Virtual Chassis Suspension Subsystem Architecture
A virtual representation of the suspension system for this self-propelled sprayer is depicted in figure 2. The model's left side corresponds to the top of the physical chassis suspension assembly, where the knee body connects with and slides into the tube of the chassis mainframe. The right model's right side represents the bottom of the chassis suspension assembly, which encompasses the rim and tire virtual subsystems. As a summary, this suspension system is classified as a modified, independent MacPherson strut where two parallel spindles, separated by a fixed, arbitrary distance from each other, guide the near vertical displacement in an overall range of ±101.6 mm. This ±101.6 mm travel range is restricted by two separate hard stops, which are also incorporated into the model. One hard stop is activated when the bottom of the spindle contacts the wheel motor casting body outlined to the left of the rim body. The other hard stop acts on the system when the internal bump stop of the air spring is reached. Furthermore, the air spring subsystem model exists between the king pin and the wheel motor casting body, and it contains a mathematical function that produces a nonlinear, dynamic amount of force corresponding to the suspension stroke that acts between these two bodies.
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| Figure 2. Virtual MBD overview of per-corner suspension system model with labeled components and subsystems. |
Optimization Subsystem Model with Prismatic Joint
To achieve a stable chassis that is held at a constant height with respect to the terrain at each corner of the mainframe, an optimization subsystem was developed (fig. 3). This optimization subsystem is mainly dependent upon a prismatic joint, which allows one translational degree of freedom relative motion between two connection frames and physically connects each corner of the mainframe to the terrain at a specified distance of 1614 mm. This 1614 mm resulted from measuring the vertical height at which each chassis suspension system was in a nominal stroke position if the vehicle was sitting over completely flat terrain. Additionally, this 1614 mm value also corresponded to the equilibrium position input field within the prismatic joint block parameter settings, which separates the base and follower frames (denoted by B and F of the prismatic joint block in fig. 3) apart. To ensure that the follower frame of the prismatic joint maintained an approximately constant contact with the terrain, a simplified optimization tire model was also linked to the follower connection line. This simplified tire model is essentially a copy of the regular tires that are connected to each rim, with the exception that the geometric size properties are substantially reduced, as depicted at the bottom of the green sensor direction line in figure 4. The reduction of geometric size properties has no major implications or ramifications for the methodology established here; instead, the small, simplified optimization tire is used to ensure that the prismatic joint's follower frame maintained contact with a dynamic terrain mesh of variable elevation.
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| Figure 3. Optimization subsystem top-level viewpoint and internal components. |
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| Figure 4. Mechanics Explorer visuals of optimization subsystem reference frames, sensor direction, simple tire model, prismatic joint, and free-floating cylinder. |
For the prismatic joint to hold up each corner of the chassis mainframe to a constant separation distance of 1614 mm, a high amount of spring stiffness and damping within the internal mechanics section of the block properties had to be configured (table 1). From thorough experience and understanding of the overall mass and inertia composition of the virtual self-propelled sprayer model, 1e8 N/m for the spring stiffness and 1e6 N/(m/s) for the damping coefficient were selected and deemed appropriate for this application. By setting the equilibrium position value to 1614 mm, we introduced a fixed constraint around displacement for the prismatic joint, which in turn allowed the solver to compute an optimized chassis suspension dynamic response required to maintain that equilibrium position between the chassis mainframe and the simplified optimization tire. The solver achieved this in part by applying a corresponding amount of force and damping to the joint, each of which was dynamically computed using the set spring stiffness and damping coefficient parameters. Furthermore, the equilibrium position of 1614 mm is the position where the spring force is zero. In summary, the prismatic joint imparts dynamic amounts of force and damping to maintain the 1614 mm translational constraint set within the joint, and the solver, therefore, calculates an optimized chassis suspension position response for the free-floating cylinder to meet this equilibrium position constraint. Therefore, the free-floating cylinder is free to move, and it extends and compresses on an order of magnitude higher than that of the prismatic joint, which is essentially fixed at a 1614 mm translational constraint.
Some additional modifications to the baseline chassis suspension model were made to produce an optimized chassis suspension position, velocity, and acceleration response needed to hold the chassis at a consistent height at each corner. These alterations included the following:
- The air spring model’s (fig. 2) force output was disabled to prevent the air spring from acting on the system. This air spring force output was replaced with a free-floating cylinder (fig. 2) to allow the suspension system to “free-float” and extend/compress over the dynamic terrain and not conflict with the prismatic joint fixing the chassis mainframe at 1614 mm. Additionally, through a comparison of the optimization subsystem and the baseline suspension dynamic properties, the prismatic joint block property values maintain substantially higher spring stiffness and damping coefficient values as compared to those of the air spring within the baseline system. This substantial increase in magnitude is needed to produce enough dynamic force to hold the vehicle up and maintain the 1614-mm translational constraint between the mainframe and the terrain below the vehicle. Overall, the 1e8 N/m spring stiffness of the prismatic joint remains unchanged in the optimization configuration, whereas the baseline air spring stiffness changes slightly depending on the stroke of the air spring but is still substantially lower than the 1e8 N/m.
- The hard stops (fig. 2) were also disabled to allow the chassis suspension/free-floating cylinder to move freely to any optimal position, which was dictated by the prismatic joint, as needed. If the hard stops were not disabled, then there would be another conflict with the prismatic joint, and the regular tires would either lose contact with the terrain or be forced through the terrain mesh in certain situations so that the 1614 mm equilibrium condition would be maintained. In other words, if the hard stops were included, there would be certain terrain conditions with highly variable elevation changes where there would be a suspension stroke limitation issue, which would prohibit the chassis from maintaining a near-level position above the terrain.
- Since the prismatic joint now handled the sprung-mass static load acting on each corner, this mass was re-introduced back onto the un-sprung mass near the rim to result in the same amount of static load acting on the regular tire.
| Table 1. Prismatic joint block property values. | |||
| Z Prismatic Position Target (mm)[a] | Equilibrium Position (mm) | Spring Stiffness (N/m) | Damping Coefficient (N/(m/s)) |
| 1614 | 1614 | 1e8 | 1e6 |
[a]Z Prismatic Position Target corresponds to the initial position value of the prismatic joint when the simulation enters the initialization phase at t = 0 seconds. | |||
There is one main disadvantage of this modeling methodology, which results in a non-perfect 1614 mm distance between the chassis mainframe and the contact point with the terrain. In this case, the 1614 mm equilibrium distance is maintained between the chassis mainframe and the simplified optimization tire, and not between the chassis mainframe and the terrain. Therefore, the solver will continuously calculate force and damping values for the prismatic joint to maintain that 1614 mm equilibrium distance. Although the optimization system can follow the dynamic terrain undulations, which is achieved through the employment of the simplified optimization tire model, this simplified optimization tire is still capable of moving freely over the terrain. This means that this simplified tire still experiences deflection, as any regular tire does, and it can lose contact with the terrain underneath it. Therefore, in certain low-frequency, high-dynamic situations, like when there is excessive positive sprung-mass vertical acceleration (fig. 5), the chassis mainframe picks up the simplified optimization tire off the terrain and causes it to lose contact with the terrain mesh.
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| Figure 5. Example of simplified optimization tire model that lost contact with terrain mesh due to excessive positive sprung-mass vertical acceleration. |
Conceptually, if there were no external influences from the rigid chassis mainframe acting on each suspension corner like previously described (fig. 5) and no other suspension systems, besides the main chassis suspension, then the resulting optimized chassis suspension response from the free-floating cylinder would essentially mirror the terrain undulations underneath the tire. However, there is a secondary suspension system present within the suspended centerframe (fig. 1) that also affects the dynamic behavior of the chassis mainframe and therefore further affects the chassis suspension’s optimal response as well. For example, figure 6 depicts a state in which there is significant boom motion that imparts a small level of dynamic inputs back onto the rear right (RR) corner of the chassis mainframe. In this condition, the free-floating cylinder is extending and compressing as expected to follow the terrain, but it also must extend to an additional level to compensate for the boom inertia lifting the chassis mainframe upwards. Basically, the boom slightly lifts the chassis mainframe and causes the simplified optimization tire to displace while the prismatic joint holds the same 1614 mm equilibrium position, which in turn causes the chassis suspension free-floating cylinder to extend even further because of these supplementary dynamic conditions. Therefore, the calculated optimized chassis suspension response is not only a function of the terrain, but it is also of sprung-mass dynamics.
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| Figure 6. Influence of terrain input and boom dynamics on rear right (RR) chassis suspension optimization subsystem. |
Virtual Testing Experimental Design
As part of this study, a total of four unique simulation tests were conducted, as outlined in table 2. The first two simulations ran the baseline suspension configuration without the optimization subsystem and free-floating cylinder active. These two simulations had the air spring model and hard stops enabled. The last two simulations were carried out with the optimization configuration and free-floating cylinder active, and the air spring model and hard stops were disabled. The first and third simulations utilized the single ramp (fig. 1) terrain with a vehicle speed of 4.47 m/s where the front right and rear right suspension corners would contact the single ramp, and the second and fourth simulations operated over an aggressive terrain configuration (fig. 7) at a vehicle speed of 5.36 m/s. Furthermore, all four simulation tests were configured with the boom height control system ON and a target height of 1270 mm relative to the terrain. Although lower target heights are common for most spraying operations, an increased target height of 1270 mm is required for this type of aggressive terrain to prevent boom strikes with the ground. With boom height control activated for all four tests, the performance impact and potential of an optimized chassis suspension system on improving boom height stability could be examined.
| Table 2. Virtual test configuration matrix for experimental design. | |||||||
| Test Index | Suspension Configuration | Terrain Configuration | Vehicle Speed (m/s) | Free- Floating Cylinder | Air Spring Model | Hard Stops | Boom Height Control Target Height (mm) |
| 1 | Baseline | Single Ramp | 4.47 | Disabled | Enabled | Enabled | 1270 |
| 2 | Baseline | Aggressive | 5.36 | Disabled | Enabled | Enabled | 1270 |
| 3 | Optimized | Single Ramp | 4.47 | Enabled | Disabled | Disabled | 1270 |
| 4 | Optimized | Aggressive | 5.36 | Enabled | Disabled | Disabled | 1270 |
Results and Discussion
Single Ramp Terrain Baseline versus Optimization Configuration Data
Figure 8 depicts the front right corner suspension displacement time series response over the single ramp terrain (fig. 1) for both the baseline and optimized chassis suspension configurations. For this test, the machine sits stationary and initializes for five seconds before rapidly increasing speed to reach a set point of 4.47 m/s. During this initialization period, various subsystems of the virtual machine are allowed a window to reach a steady-state with a variety of initial values that are incorporated throughout. For example, the suspension system mathematical model uses this timeframe, specifically in the first second of the simulation as shown in figure 8, to settle at the suspension’s nominal height. Another example would be the boom-height ultrasonic sensors, where these sensors utilize this timeframe to initialize and detect the terrain underneath the sensor mounting location.
As shown in the plot, the sprayer’s front right chassis suspension corner first contacts the inclined portion of the ramp at about 11 seconds. For the baseline test, the suspension system compresses by about 50 mm after it first encounters the single ramp and then enters a new phase of extension caused by the chassis mainframe rolling hard to the left (i.e., negative chassis roll in degrees) and separating the suspension system apart. This new phase of extension continues until it eventually reaches the extension hard stop present at the bottom of the spindles, which is also denoted by the top dashed line. For the optimization test, the suspension position data is practically a mirrored version of the single ramp geometry. In this optimization configuration, the free-floating cylinder compresses significantly beyond the nominal operating range of ±101.6 mm as the tire traverses up the incline, then holds steady over the flat portion of the ramp, and then finally extends again as the tire travels down the ramp. This optimization response can travel outside of the nominal operating range because, as previously stated, the hard stops are disabled in this optimization configuration to prevent stroke limitation issues. The absence of hard stops is practically infeasible in the physical realm, but it is plausible in the virtual realm in terms of understanding the extent of unrestricted suspension travel needed to traverse specific types of terrain.
Although there are considerable differences between the two baseline and optimized responses, especially when the optimized response goes opposite of the baseline in the 11 through 13 seconds timeframe, this is a positive result. This optimized suspension response demonstrates that the suspension can free-float and follow the geometry of the single ramp terrain underneath it while still maintaining the 1614 mm constraint distance with respect to the flat terrain underneath the front right corner of the mainframe, which is where the prismatic joint indicated by the green line (fig. 3) exists. Previously, with the baseline response, the suspension compressed until about 200 mm at 11 seconds, and the amount of air spring force increased almost exponentially, which in turn caused chassis instability where the chassis rolled at a fast rate to the left.
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| Figure 7. Side view of terrain object for aggressive terrain configuration. |
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| Figure 8. Time series responses for baseline and optimized front right (FR) suspension displacements over single ramp terrain. |
Furthermore, the fact that the optimized suspension response goes beyond -101.6 mm (fig. 8) means that there is enough compression stroke needed for this single ramp disturbance to cause the top face of the wheel motor casting to go through the bottom face of the king pin. In other words, the suspension position is positively measured as the distance between the wheel motor casting top face and king pin bottom face reference frames. So, when the wheel motor housing goes through the king pin, as shown on the right side of figure 9, then the corresponding suspension position values will be negative.
Again, this is the optimized response of the suspension system required for the prismatic joint to maintain a near-constant chassis mainframe height (1614 mm) above the terrain, as shown in figure 9. Although this example is quite simple, it visually provides a clear understanding of the combined functionality of the optimization system with the free-floating cylinder integrated within the chassis suspension system before showing results of more complex dynamic conditions, such as with the aggressive terrain configuration.
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| Figure 9. Mechanics Explorer comparison of baseline and optimized responses on peak of single ramp. |
This optimized suspension system can follow the single-ramp terrain and provide a near-stable chassis mainframe. Although the prismatic joint is holding that 1614 mm equilibrium position constraint, there are some dynamic influences that cause the chassis to experience slight deviations away from 0 deg of chassis roll/pitch and 0 deg/s of chassis roll rate/pitch rate, as shown in figures 10 and 11. These deviations are caused by the previously described points about the disadvantages of the modeling methodology. Although this optimized suspension system can follow the ramp geometry and the prismatic joint can hold the 1614 mm constraint between the mainframe and the simplified optimization tire outlined in red in figure 9, the chassis mainframe still experiences some slight roll and pitch. This results because the suspension system experiences high vertical acceleration while traveling up the inclined portion of the ramp at the indicated travel speed (table 2), which causes the vertical inertial effect of the suspension system to generate some slight upward displacement of the chassis mainframe/sprung mass. As previously described, the chassis mainframe slightly picked up the simplified tire off the terrain and caused it to barely lose contact with the terrain mesh.
Aggressive Terrain Baseline versus Optimization Configuration Data
The front right (FR) suspension displacement data for the baseline suspension versus the optimized suspension system across the aggressive terrain simulation are depicted in figure 12. It is shown that the two time series responses are quite different from each other, especially when the virtual machine crosses over the first and second terraces in the 17.5-second and 27.5-second timeframes, respectively. The optimization suspension is extending and compressing to a great extent, as compared to the baseline, to compensate for the harsh terrain conditions and high variability in the terrain elevation, especially for the terraces. For the baseline simulation, the suspension system is constrained by a hard stop that prevents further extension, which is why the baseline time series reaches a plateau twice at approximately 17 and 28 seconds of simulation. However, because the hard stops are disabled in the optimization suspension system configuration, the optimization model allows this suspension to travel beyond the hard stop threshold while still maintaining a constant height of 1614 mm from the chassis mainframe to the simplified optimization tire and an overall near level mainframe with respect to the terrain underneath it.
From an engineering design and product development perspective, the resulting optimization data could be leveraged to influence the component selection and overall geometry specifications on future suspension system designs. For example, if some future suspension design was able to achieve the optimized, predicted response from the simulation shown in figure 12, the suspension operating range would have to increase to even achieve a greater level of stroke required for the aggressive terrain. If the operating range was expanded to ±152.4 mm instead of the nominal of ±101.6 mm, then approximately (95.4% instead of 81.0%) of the time the suspension would be able to achieve its optimized position. This increase in operating range is needed mainly because of the terrace portions of the aggressive terrain (fig. 7). Additionally, figure 13 shows another example of this, where the aggregate distribution of optimized suspension velocities across all four machine corners could be utilized to ensure that a prospective suspension component in a future design is able to travel and function properly at the indicated output velocities.
More importantly, the performance impact of an optimized chassis suspension system on both chassis dynamics and boom height stability across an aggressive terrain setting was analyzed for this investigation. The time series responses for baseline versus optimized suspension chassis roll/roll rate and chassis pitch/pitch rate across the aggressive terrain simulation are shown in figures 14 and 15. The common finding between both plots is that the optimized chassis suspension configuration resulted in less roll and pitch for the chassis mainframe, but there was an increase in the rate of both responses, as indicated by the sharp high-frequency spikes in the rate data. This makes sense when relating the responses back to the optimization objective function. Overall, with the optimization subsystem constraining the chassis to a fixed distance of 1614 mm above the
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| Figure 10. Time series responses for baseline and optimized chassis roll and roll rate over single ramp terrain. |
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| Figure 11. Time series responses for baseline and optimized chassis pitch and pitch rate over single ramp terrain. |
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| Figure 12. Time series responses for baseline and optimized front right (FR) suspension displacements over aggressive terrain. |
simplified optimization tire, which is in contact with the terrain, the chassis mainframe dynamics should mimic more of the shape of the terrain profile beneath it. With the optimization subsystem, the model is essentially locking the mainframe with respect to the terrain profile, so it must angularly rotate faster in certain portions of the terrain to ensure that the distance constraint is continuously achieved. The baseline configuration, on the other hand, is less restricted as compared to the optimization configuration, and the chassis mainframe rotates in the roll and pitch directions at a smoother rate across the terrain. In some conditions, the baseline chassis mainframe will naturally rotate beyond the instantaneous roll and pitch profile of the terrain due to dynamics imparted from the chassis suspension.
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| Figure 13. Distribution of optimized suspension velocities across all four corners over aggressive terrain. |
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| Figure 14. Time series responses for baseline and optimized chassis roll and roll rate over aggressive terrain. |
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| Figure 15. Time series responses for baseline and optimized chassis pitch and pitch rate over aggressive terrain. |
Figure 16 illustrates the resulting boom height data recorded from one of the outermost virtual ultrasonic sensors placed on the outer boom section. When comparing the baseline response against the optimized chassis suspension response, it was concluded that the optimized chassis suspension was able to assist the boom height control system and create more stability out of the boom platform, which was deduced from less boom height variation, especially for the extreme conditions induced from the terraces, around the 1270 mm target height for the optimized chassis suspension configuration. The two terraces for this aggressive terrain configuration pose challenges for boom height control, which can be visualized by the extreme upward movement of the booms as these components traverse over the peaks of the terraces and attempt to return back to the target height following rapid elevation changes with the machine traveling at relatively high vehicle speeds.
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| Figure 16. Time series responses for baseline and optimized boom height positions over aggressive terrain. |
Fundamentally, boom height control is a control algorithm that attempts to control the booms to a set target relative position above the terrain. Considering the relationship between boom height control and the chassis suspension system in the baseline system, boom height control performs better, and with increased boom height precision, when the chassis suspension generates dynamics that help stabilize the vehicle platform and reduce overall chassis roll with respect to the terrain profile. With regards to the optimization suspension configuration, the chassis is essentially fixed at a certain separation distance to the terrain, which naturally gives an added advantage to the boom platform as the main vehicle system is already following the terrain better than the baseline vehicle configuration.
| Table 3. Maximum vertical tire dynamic load, RMS acceleration comfort, and boom height standard deviation metric values for baseline versus optimized suspension system across aggressive terrain configuration. | ||||
| Suspension Configuration | Terrain Configuration | Maximum Vertical Tire Dynamic Load (N) | RMS Acceleration Comfort (m/s2) | Boom Height Standard Deviation (mm) |
| Baseline | Aggressive | 81,029 | 1.50 | 970.47 |
| Optimized | Aggressive | 95,476 | 2.19 | 781.46 |
Furthermore, it is observed that in certain timeframes, like the 15-second timeframe, the optimized chassis suspension configuration does not provide any benefit for the boom height response. This similarity is mainly a function of a high vehicle speed over aggressive terrain disturbances, where the boom height control response speed has difficulties reaching its target over rapidly changing terrain. In other timeframes, like at 18 seconds, where the vehicle has just crossed over the first terrace, the optimized chassis suspension configuration does reduce variability in the boom height response. This is mainly due to the previously stated point about less chassis roll and pitch experienced with the optimized chassis suspension. In this specific example, the optimized chassis suspension model caused the chassis mainframe to not roll as much to the right (i.e., positive roll) as it crossed over the terrace, which helped better stabilize the boom platform and limit the boom height variability as compared to the baseline.
To quantitatively evaluate the system-wide performance effects on tire load reactions, chassis responses through an operator comfort perspective, and boom height stability and chassis responses through an operator comfort perspective, summary metrics, which are outlined in table 3, were calculated. To provide some context around tire reaction forces and their impact on soil compaction, maximum vertical tire dynamic loads were calculated. The maximum corresponds to the maximum recorded vertical tire force output between all four chassis suspension corner tires. The baseline configuration resulted in an 81,029 N maximum force, and the optimized configuration resulted in a 95,476 N maximum force, which is approximately a 17.8% increase. To summarize the influence on the level of operator comfort, ISO standard 2631-1 was referenced to compute the corresponding RMS Acceleration Comfort (eq. 1) key performance indicator (ISO Standards, 1997). This comfort metric calculates the square root of the sum of squares between each acceleration time series array centered around zero m/s2, and it analyzes the level of operator comfortability. Between the baseline and optimized suspension configurations, there was a 46% increase in the RMS Acceleration Comfort metric, meaning that the operator would be 46% less comfortable under the optimization system. For both the baseline and optimized configurations, the resulting absolute magnitudes of the RMS Acceleration Comfort values are already classified as “very uncomfortable” according to the ISO standard. This is due to the extreme terrain conditions and rapid elevation changes of the aggressive configuration that the sprayer is subjected to. Furthermore, this increase in accelerations for the chassis suspension would also result in higher loads for the chassis mainframe, and the overall durability of this subsystem would be negatively impacted if the design structure stayed the same.
(1)
where
RMS Accln X (m/s2) = root mean square (X Longitudinal Acceleration)
RMS Accln Y (m/s2) = root mean square (Y Lateral Acceleration)
RMS Accln Z (m/s2) = root mean square (Z Vertical Acceleration).
To numerically describe the variability of the boom height dynamic response, a standard deviation of the boom height position data channel was employed. Under an optimized suspension configuration that holds the chassis mainframe at a near-consistent height, the boom height standard deviation was decreased from 970.5 mm to 781.5 mm, which equates to a 19.5% increase in the overall stability of the boom system.
Even though there is a tradeoff present here between operator comfort and boom height stability, this tradeoff was expected. In a baseline configuration, the chassis suspension better isolates the cab from terrain disturbances as the air spring and the compressed air contained within naturally reduce vibrations and work to provide a relatively smooth chassis mainframe platform. This is more comfortable for the operator, but it increases boom height variability since the chassis mainframe is not following the roll and pitch profiles of the terrain as well. With an optimized chassis mainframe that holds a constant height as the vehicle travels across the terrain, the prismatic joint causes more sudden movements and rapid accelerations out of the chassis to ensure that height constraint is maintained, which is less comfortable for the operator. However, for the boom system, since the chassis is more or less "fixed" to the terrain underneath it in the optimized configuration, there are fewer unwanted chassis roll and pitch disturbances that reach the boom, which in turn allows the boom height control system to better sustain a dynamic position closer to the specified target height.
The overall results from this study are constructive for a few reasons as they relate to guiding engineering design. If an active suspension control system were to be developed that focused on maintaining a stable chassis height relative to terrain, these optimization results would inform the potential performance impact on operator comfort and boom height variation. This theoretical active suspension control system would fundamentally operate under the methodology of “pushing” and “pulling” on the suspension components, likely through low-latency hydraulic capabilities, to maintain a fixed height of the chassis relative to the dynamic terrain profile below the chassis. No physical implementation or exploration of such a system was conducted; this virtual research study was completed to gain some fundamental understanding and direction of system-wide sprayer performance under a system of this nature. Additionally, the associated data with the optimization configuration would help inform future component selection. A brief example of that was described previously when understanding the distribution of suspension velocity needed to achieve the optimized dynamic suspension response. Furthermore, although this was not discussed previously, another example of useful data from the optimized suspension model includes the output forces from the free-floating cylinder needed to compress and extend the suspension system to the optimized stroke position. This would assist in specifying the characteristics of a physical component required to achieve that level of force distribution across an aggressive terrain configuration.
Conclusions
This study presented insight into a new multibody dynamics methodology to optimize the chassis platform with respect to the terrain through free-floating chassis suspension motion and a 1614 mm translational constraint between the mainframe and the terrain. This technique was accomplished through the disabling of some of the baseline suspension components and the utilization of a free-floating cylinder and a prismatic joint with high stiffness and damping internal mechanics values that were connected between the chassis mainframe and a simplified optimization tire that was in contact with the terrain. The prismatic joint essentially tried to hold the chassis mainframe at a constant height as the self-propelled sprayer traveled across dynamic terrain conditions. In a simulation with an aggressive terrain configuration, the data showed that an optimized chassis mainframe resulted in a 46% decrease in operator comfort and a 19.5% increase in boom height stability.
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