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Chapter 2. Vehicle Dynamics and Preliminary Analysis
Dynamic Model of the Biplanar Bicycle
The dynamic model derived for the purposes of this work follows the approach in previous works [2,3,4]. The final state space form of the dynamic model is convenient for simulation and control design. A planar model is used to simplify the dynamics. The analysis is performed using extended Lagrange techniques [5,6,7]. The variables are defined below in Figure 2.1, and listed for convenience in Table 2.1.
The behavior of the vehicle can be separated into two general cases for the purposes of the work at hand. The first such case is a static equilibrium (all velocities and accelerations set to zero) on a constant slope. The second condition of interest is the behavior of the vehicle near an operating point defined by a constant translational velocity on a constant slope. We now proceed by analyzing the first case.
Preliminary Analysis
We now take a largely qualitative look at the vehicle in order to establish an intuitive fell for how the biplanar bicycle works. The static equilibrium case supports the ability of the vehicle to climb modest slopes (and traverse discontinuous terrain). The discussion of dynamic properties establishes the idea of an equilibrium reaction mass angle at speed in order to overcome pseudo-constant disturbances such as friction and viscous damping.
Static Equilibrium on a Constant Slope
Assume that the vehicle is standing at rest on a constant slope of angle a from the horizontal. By definition, we know that q& = q&& = f& = f&& = 0 . Combining these facts and the nonlinear state model above we are able to solve for the following equilibrium conditions.
We know that the sine of the equilibrium reaction mass angle is bounded by ± 1 , and we therefore apply this constraint to our system. This allows us to solve for the bounds on a. These bounds specify the maximum slope at which the vehicle is able to remain at rest. The bounds on a are:
If we examine the equilibrium reaction mass angle as a function of slope (see Figure 2.2), we see that the equilibrium angle approaches 90 degrees as the slope approaches the critical value. This becomes intuitive if we observe how the vehicle actually develops drive torque to move forward. The biplanar bicycle can be thought of as continuously ‘falling’ forward. This is due to the fact that the center of mass of the vehicle must be moved in front of the vehicle, causing a moment about the axle. It is obvious that when the reaction mass is at 90 degrees, the maximum moment is created because the center of mass of the vehicle is as far forward as possible.
This simple explanation of how the vehicle actually moves fails to touch on the cases beyond 90 degrees. It would seem that these positions would also be stable, but in practice they are generally undesirable. The work performed by Reinholtz et al. [2,3] demonstrates a phenomenon that is labeled as ‘whirling’. This occurs when a vehicle attempts to climb a slope that is beyond the maximum angle allowed by the vehicle configuration. The consequence of whirling is a descent back down the incline in a somewhat uncontrolled manner. The best way to avoid this in practice is to attempt to confine the reaction mass angle to less than 90 degrees in either direction. We will elaborate on this further in a velocity control discussion later.
Constant Velocity on a Constant Slope
Assume that the vehicle is traveling at a constant velocity on a slope of incline a with respect to the horizontal. It has been shown in [3] that the equilibrium reaction mass angle will obey the relation above for a constant velocity. This would lead us to assume that the equilibrium reaction mass angle would be zero for a constant velocity on a slope of zero degrees, but this result ignores the fact that torque (although a small amount) is still necessary to keep the vehicle moving forward. This torque overcomes the resistance imposed on the vehicle by the terrain that it is traversing. Tall grass, for example, would require a greater equilibrium reaction mass angle at a constant speed than would a smooth surface such as a gymnasium floor.
Open Loop Velocity Response
In order to have a standard to compare our control designs to, we will now look at the open loop behavior of the vehicle. The model that we have derived accepts a voltage input to the drive motors, supplying the necessary torque to move the reaction mass, and thus move the vehicle either forward or backward. A constant voltage causes the vehicle to travel at a constant speed. Figure 3.1 shows the camera and reaction mass angle (f) as a function of 5V step input, along with the vehicle velocity (q’). Note the transient behavior and then the steady-state displacement. The vehicle velocity is directly related to the angular velocity of the wheels, and is given in the lower plot of Figure 3.1. Note that the reaction mass angle can intuitively be related to the acceleration of the vehicle.
We desire to design a control system that will reduce the transient disturbance of the camera. This requires that we minimize the displacement of the reaction mass from the vertical. We propose that some improvement may be gained by controlling the plant as it has been modeled, but will later introduce a second solution that can achieve greater results by adding a secondary actuator to control a camera mast. We will proceed now with an attempt to improve performance without a second actuator.
Linearization by Taylor Series Expansion
Attempting to design a controller based on a linearized plant model is a first step in many control designs and is presented here for thoroughness. Linearization is discussed in any senior level controls textbook, and will not be discussed here in depth. For a further discussion see [9] or [10].
(ABSTRACT)
ACKNOWLEDGMENTS
CHAPTER 1. INTRODUCTION
GENERAL MOTIVATION
PROBLEM STATEMENT
DESIGN CRITERIA
OUTLINE OF THE FOLLOWING CHAPTERS
CHAPTER 2. VEHICLE DYNAMICS AND PRELIMINARY ANALYSIS
DYNAMIC MODEL OF THE BIPLANAR BICYCLE
PRELIMINARY ANALYSIS
Static Equilibrium on a Constant Slope
Constant Velocity on a Constant Slope
CHAPTER 3. VELOCITY CONTROL
OPEN LOOP VELOCITY RESPONSE
LINEARIZATION BY TAYLOR SERIES EXPANSION
LQR CONTROL OF VELOCITY
DISCUSSION OF VELOCITY CONTROL
CHAPTER 4. PROPOSED ACTUATOR CONFIGURATIONS
COUNTER-TORQUE ACTUATOR
DOUBLE PENDULUM ACTUATOR
GYROSCOPIC ACTUATOR
CHAPTER 5. CONTROL DESIGN AND SIMULATION
PROCEDURE AND SIMULATION CRITERIA
COUNTER-TORQUE ACTUATOR
Linearized System Model and LQR Design
Simulation Results
Discussion
DOUBLE PENDULUM ACTUATOR.
Linearized System Model and LQR Design
ALTERNATIVE METHODS.
CHAPTER 6. PRACTICAL CONSIDERATIONS
DOUBLE PENDULUM ACTUATOR
COUNTER-TORQUE ACTUATOR
SENSORS AND SENSOR CONFIGURATION
Tilt Sensors
Accelerometers
Rotary Encoders
Sensor Configuration on the Biplanar Bicycle
CHAPTER 7. CONCLUSIONS
BIBLIOGRAPHY
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Reference Frame Regulation for a Biplanar Bicycle
