Get Complete Project Material File(s) Now! »

**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

GET THE COMPLETE PROJECT

Reference Frame Regulation for a Biplanar Bicycle