1. Differential Steer Vehicle Dynamics
Recall that a typical differentially steered vehicle will maneuver in various directions with the center of rotation anywhere in the line joining the two wheels. We can represent a simple 2D vehicle turning with kinematic states q = [X Y Ψ] shown in Figure 1
Fig 1. Single Axis Differential Drive
Let's assume that there is no slip and each wheel has controllable speeds, ω1 and ω2. By applying a transformation matrix to transform the body-fixed reference frame velocities into a global frame, the velocities in the global reference frame are,
2. Open Loop Lane Change Maneuver
Suppose the vehicle is given a lane change trajectory and a constraint that the vehicle will have a constant velocity. The open loop schematic is shown in Figure 2
Fig 2. Open Loop Schematic
Notice the transformation from reference X,Y to angle psi is a way to modify the MIMO system into a SISO system. This will simplify the controls problem when designing our closed-loop controller. The open loop LabVIEW simulation is shown in Figure 4. As shown in Figure 5, the vehicle is not able to track the trajectory correctly. Let's now design a closed-loop controller to minimize the error.
Fig 3. Transformation from XY to Psi
Fig 4. Block Diagram of Open Loop Simulation
Fig 5. Open Loop Response
3. Closed-Loop Lane Change Maneuver on Rotation
The undesired open loop performance motivates us to design a closed-loop controller to minimize the error. Let's consider a simple proportional-integral (PI) controller on the turning angle psi. From Figure 8, the vehicle now tracks the desired path much better than the open loop response, nevertheless, additional improvement could be made.
Fig 6. Closed-Loop Psi Control Schematic
Fig 7. Block Diagram of Closed-Loop Psi Control
Fig 8. Closed-Loop Psi Control Response
4. Modified Closed-Loop Lane Change Maneuver
Various control architectures were considered in order to make improvement to the vehicle's tracking ability. Figure 9 considered that the vehicle should not have an instantaneous longitudinal velocity, but rather it should have an initial velocity at rest, then speeds up to the desired cruising speed. The schematic in Figure 10 tracks both the vehicle rotation angle as well as the Y-position. We can approximate the desired Y-position of the vehicle given its current X position through interpolation.
Fig 9. Closed-Loop Rotation and Velocity Control Schematic
Fig 10. Closed-Loop Parallel Tracking Schematic
Each of the proposed control strategy has made incremental improvement; however, a simple tracking on the Y-position proves to give the best closed-loop response. The schematic of the final implementation and the LabVIEW block diagram are shown in Figure 12. It is interesting to note that the open loop response using this 1D mapping approach is worse than the open loop response using XY to rotation transformation.
Fig 11. Closed-Loop Y Position Control Schematic
Fig 12. Block Diagram of Closed-Loop Y Control
Fig 13. Closed-Loop (Left) Open Loop (Right) Y Control Response
This article utilizes the concepts of vehicle turning through differential steer and apply to a lane change maneuver problem. Through various closed-loop formulation, it was shown that a simple proportional-derivative (PD) control on vehicle's Y-position is the most effective approach among the ones that were considered in this tutorial. The parameters of the vehicle is based on LabVIEW Robotics Starter Kit (DaNI) and the kinematic equations and control algorithms are implemented using LabVIEW Control Design and Simulation Module and LabVIEW MathScript RT Module. You can download this example program below.
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