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Imaginary Axis Imaginary Axis Imaginary Axis Imaginary Axis Imaginary Axis Sustainable Construction and Design <strong>2011</strong> a 8 6 System Poles b 8 6 System Poles 4 4 2 2 0 0 -2 -2 -4 -4 -6 -6 -8 -10 -8 -6 -4 -2 0 2 Real Axis -8 -10 -8 -6 -4 -2 0 2 Real Axis c 6 4 2 System Poles d 8 6 4 2 a b System Poles a b 0 0 -2 -2 -4 -4 -6 -6 -12 -10 -8 -6 -4 -2 0 Real Axis -8 -10 -8 -6 -4 -2 0 2 Real Axis System Poles 10 e 8 a. increasing ground speed: loosing 6 4 2 0 -2 -4 -6 -8 -10 -10 -8 -6 -4 -2 0 2 4 Real Axis Figure 2: System pole locations with changing model parameters. stability at 25mps (positive real part of system matrix eigenvalues) b. System poles at decreasing rear wheel cornering stiffness: loosing stability c. decreasing pneumatic trail, remaining viscous friction d. decreasing viscous friction, at 10 m/s (a) and 20 m/s (b) e. unstable Rear Wheel Steering poles at increasing speed Combining the mechanical and dynamic differential equations we get for the haptic feedback: 70 Copyright © <strong>2011</strong> by Laboratory Soete
Sustainable Construction and Design <strong>2011</strong> The state equations for the haptic feedback device: Because of the limited dynamics of the whole system, in the simulation we ignore the inductive impedance. And therefore we can reduce the number of state variables to angular speed and angle position. The reduced state equations are: The input of the haptic feedback is the armature voltage which is controlled by the state feedback controller. 3.2 Explicit calculation of feedback gain for emulated front wheel steering The control system is shown on Figure 3. There are several methods for optimal design for the feedback gain, e.g. the concept of the linear quadratic regulator (LQR). It will give optimal solution for full state feedback at given transient and steady-state constrains. t D Haptic Feedback Device d R Rear Wheel Steering Vehicle r f f v State Feedback Controller Figure 3: State feedback controller for rear wheel steered vehicle (without steering actuator dynamics) based on emulated front wheel aligning moment and added viscous friction We follow a different method by explicit calculating the feedback gain, because we would like to emulate a steering system which has a front wheel steering. So the system may work suboptimal compared to the LQR design but gives the operator the feeling of an front wheel steered vehicle. The controller has to calculate the necessary voltage to create realistic steering torque: The emulated hand wheel torque can be calculated on the virtual aligning moment on the front wheels in case of an front wheel steering plus the synthetic viscous friction in the system, which is used to modify system dynamics. 71 Copyright © <strong>2011</strong> by Laboratory Soete
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