Control of Vehicle Motion: Lane Change Maneuver

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,

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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

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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

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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

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5. Conclusion

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

LabVIEW Control Design and Simulation Module 

LabVIEW MathScript RT Module

NI LabVIEW Robotics Starter Kit

Please contact andy.chang@ni.com to request more information about this article.

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6. Related Links

NI Automotive Applications

Hybrid Vehicle Test and Simulation using NI's Hardware-In-The-Loop (HIL) Platform

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7. Reference

ME 390: Vehicle Dynamics and Controls (Spring 2011)

Prof. Raul. G. Longoria, ME, University of Texas, Austin

http://www.me.utexas.edu/~longoria/VSDC/