1. General Vehicle Model on Incline
Recall from the Basics Dynamics of Ground Vehicle, the free-body diagram and all the forces acting on the body is shown in Figure 1.
Fig 1. Two-Axle Longitudinal Model
Table 1. Vehicle Parameters
2. Equations of Motion
Let us assume that for a passenger car, ha ≈ hd ≈ h and the vehicle does not leave the ground with only motion along the longitudinal axis.
The governing equations for a general planar vehicle model on an inline are
with the following assumptions
We can deduce our states equations after doing some algebra
3. Frictional Force
Up until now, we have considered a constant friction coefficient between the tire and the road. In reality, this friction coefficient is a function of slip. Using a lookup table, we can formulate a mu-slip curve for this particular simulation. It's important to note that slip has a range from -1 (skidding) to 1 (slipping). Trying to keep tires operating within a desired range of slip is the basis of anti-lock braking systems (ABS). We will discuss the concept of ABS in a later tutorial.
Fig 2. Mu-Slip Curve
Since slip and skid decouple the dynamics of the rotational components from the translational dynamics of the vehicle, it is common to formulate slip and skid in a single function
4. Motor Characteristics
For this particular vehicle model, we assume that the driveline has two permanent-magnet DC motors that teach drive a worm-gear reducer connected to the wheel shafts. For a given DC Motor, there's a specific torque/speed curve shown in Figure 3.
Fig 3. DC Motor Torque Speed Curve
In this case, the Mabuchi motor specifications can be implemented in LabVIEW as shown in Figure 4.
Fig 4. Motor Specification in LabVIEW
5. Performance Comparison
Let us compare the performance difference between the zero traction (slipping) and the traction model. In the zero traction case, we will set slip = 1 as shown in Figure 5. We can see from Figure 6 that the traction model takes a bit longer to reach steady-state speed. You can also change the motor specifications to compare steady-state speed difference between different motors.
Fig 5. Block Diagram of Vehicle Simulation Zero Traction
Fig 6. Steady-State Vehicle Speed Comparison
6. Maximum Incline
In order to determine the maximum grade that the vehicle will climb, we will look at the sum of the forces in the x-direction equation, namely
At maximum grade, the vehicle cannot accelerate anymore hence
For the simulation, let's assumed a nominal coefficient of friction, µ=0.5 and motor torque T = 0.000185 by recording the steady state value of the torque constant. As shown in Figure 7, both the zero traction model and the traction model have a maximum incline of about 28.4 degrees.
Fig 7. Max. Incline Analysis: Zero Traction Model (Left) Traction Model (Right)
Fig 8. Block Diagram of Finding Maximum Incline Angle
7. Weight Distribution
From the model, we can see that the weight distribution contributes to the traction force of the model.
and recall that the mouse has a front caster wheel to help distributing the weight. We can see that as l1 (distance from front pivot to CG) increases, the relative traction force will also increase. This is illustrated in Figure 9. Notice that this only affects the model with traction as the zero traction model does not take distance from pivot to CG into consideration.
Fig 9. Weight Distribution Comparison at Zero Incline
In this tutorial, we added motor dynamics to the longitudinal vehicle model and briefly discussed the concept mu-slip function. By adding additional dynamics to the model, we are able to examine different motor specifications and their effects on the performance. Utilizing both LabVIEW Control Design and Simulation Module and LabVIEW MathScript RT Module, students can implement their kinematic equations and simulate with various vehicle and environmental parameters.
You can download an evaluation copy of the modules here
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ME 390: Vehicle Dynamics and Controls (Spring 2011)
Prof. Raul. G. Longoria, ME, University of Texas, Austin