1. Braking Dynamics
Let us begin by drawing the free-body diagram of a tire with all frictional and contact forces acting on the body shown in Figure 1.
Fig 1. Free-body Diagram of a Simple Wheel
Recall the rotational dynamics of a wheel in traction or braking can be described by
The two state variables, namely vehicle speed Vx and wheel speed ωw, are very important in determining the correct friction coefficient between the tire and the ground. Recall in the tutorial of Vehicle Performance Analysis, we have described that 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
Using LabVIEW Control Design and Simulation Module and LabVIEW MathScript RT Module, we can model the braking dynamics, slip function, and characterize the mu-slip curve as shown in Figure 2 and Figure 3. Noticed we have implemented two different mu-slip models to emulate different tire-road dynamics. Figure 4 demonstrates the overall simulation diagram for simulating simple braking dynamics.
Fig 2. Braking Dynamics (Left) Slip Function (Right)
Fig 3. Mu-Slip Model Formulations
Fig 4. Block Diagram of Simple Braking Simulation
2. Simulation of Braking Dynamics
Let's setup the wheel parameters and initial conditions for vehicle speed. In this simple braking simulation, a constant braking torque is applied throughout.
Fig 5. Simple Braking Simulation Results
The simulation results are shown in Figure 5 showing that the stopping distance is about 175 m. In this particular tutorial, we will focus on the Speed and Slip curves. The wheel speed goes to zero which indicates that the wheel is locked-up. It is extremely important to know whether lock-up occurs or not, especially for assessing vehicle yaw stability. This motivates us to build an anti-lock braking system to adjust braking torque preventing wheel lock-up.
3. Anti-lock Braking Systems (ABS)
The basis of ABS is to monitor carefully the operating conditions of the wheels and adjust the applied braking torque. As shown in Figure 6, the ABS is typically designed to keep the tries operating within a desired range of slip. This will prevent the wheel from locking thus maintain steering and vehicle stability.
Fig 6. Generic Mu-Slip Curve
In our ABS, we will adapt a bang-bang (on-off) controller, a very common and inexpensive way to control a system. A simple bang-bang control could follow the basic rule,
Fig 7. ABS Formulation
4. LabVIEW ABS Simulation
The bang-bang controller schematic is shown in Figure 8. Notice in this particular bang-bang controller, a differential gap is also implemented. The differential gap introduces hysteresis in the system which is intended to slow down and switching between two states. The overall ABS simulation model in LabVIEW is shown in Figure 9.
Fig 8. Block Diagram of Bang-Bang Controller
Fig 9. Block Diagram of ABS Simulation
As shown in Figure 10, the bang-bang controller successfully regulate braking torque and ensure that the wheel will never lock-up. It is interesting to note that the braking distance with ABS is slightly longer than without ABS. Nevertheless, having ABS ensures that the vehicle can maintain steering and yaw stability during braking.
Fig 10. ABS Braking Simulation Results
In this tutorial, we developed a single braking-wheel model in LabVIEW Control Design and Simulation Module and LabVIEW MathScript RT Module and analyzed the lock-up condition. By applying bang-bang controller, the ABS system ensures that the wheel will never be locked during braking. Further analysis could be done to optimize braking distance utilizing ABS system.
You can download an evaluation copy of the modules here
Please contact firstname.lastname@example.org to request more information about this article.
6. Related Links
ME 390: Vehicle Dynamics and Controls (Spring 2011)
Prof. Raul. G. Longoria, ME, University of Texas, Austin