Dale L. Peterson

Yaw Rate and Velocity Tracking Control of a Hands-Free Bicycle

The control of a bicycle has been well studied when a steer torque is used as the control input. Less has been done to investigate the control of a hands free bicycle through the rider’s lean relative to the bicycle frame. In this work, we extend a verified benchmark bicycle model to include a rider with the ability to lean in and out of the plane of the bicycle frame. A multi-input multi-output LQR state feedback controller is designed with the control objective being the tracking of a reference yaw rate and rear wheel angular velocity through the use of rider lean torque and rear wheel (pedaling) torque. The LQR controller is tested on the nonlinear model and numerical simulation results are presented. Conclusions regarding the required lean angle of the rider relative to the bicycle frame necessary to execute a steady turn are made, as well as observations of the effects of right half plane zeros in the transfer function from rider lean torque to yaw rate.© 2008 ASME

Constrained Multibody Dynamics With Python: From Symbolic Equation Generation to Publication

Symbolic equations of motion (EOMs) for multibody systems are desirable for simulation, stability analyses, control system design, and parameter studies. Despite this, the majority of engineering software designed to analyze multibody systems are numeric in nature (or present a purely numeric user interface). To our knowledge, none of the existing software packages are 1) fully symbolic, 2) open source, and 3) implemented in a popular, general, purpose high level programming language. In response, we extended SymPy (an existing computer algebra system implemented in Python) with functionality for derivation of symbolic EOMs for constrained multibody systems with many degrees of freedom. We present the design and implementation of the software and cover the basic usage and workflow for solving and analyzing problems. The intended audience is the academic research community, graduate and advanced undergraduate students, and those in industry analyzing multibody systems. We demonstrate the software by deriving the EOMs of a N-link pendulum, show its capabilities for LATEX output, and how it integrates with other Python scientific libraries allowing for numerical simulation, publication quality plotting, animation, and online notebooks designed for sharing results. This software fills a unique role in dynamics and is attractive to academics and industry because of its BSD open source license which permits open source or commercial use of the code. ∗Address all correspondence to this author INTRODUCTION There are many dynamic systems which can be better or more effectively studied when their EOMs are accessible in a symbolic form. For equations that may be visually inspected (i.e., of reasonable length), symbolics are generally preferable because the interrelations of the variables and constants can give clear understanding to the nature of the problem without the need for numerical simulation. Many classic problems fit this category, such as the mass-spring-damper, double pendulum, rolling disc, rattleback, and tippy-top. The benefits of symbolic equations of motion are not limited to these basic problems though. Larger, more complicated multibody systems can also be studied more effectively when the equations of motion are available symbolically. Advanced simplification routines can sometimes reduce the length of the equations such that they are human readable and the intermediate derivation steps are often short enough that symbolic checks can be used to validate the correctness. Furthermore, the symbolic form of the EOMs often evaluate much faster than their numerical counterparts, which is a significant advantage for real time computations. Problems in biomechanics, spacecraft dynamics, and single-track vehicles have all been successfully studied using symbolic EOMs. Having the symbolic equations of motion available permits numerical simulation, but also allows for a more mathematical study of the system in question. System behavior can be studied parametrically by examining coefficients in the differential equations. This includes symbolic expressions for equiProceedings of the ASME 2013 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference IDETC/CIE 2013 August 4-7, 2013, Portland, Oregon, USA

General Steady Turning of a Benchmark Bicycle Model

We analyze general steady turns of a benchmark bicycle model in the case of nonzero applied steer torque. In a general steady turn, the lean and steer angles are constant, the velocity of the bicycle must ensure moment balance about the contact line, and some torque must be applied to maintain the constant steer angle. We identify two boundaries in lean–steer plane: first, the region of kinematic feasibility, and second, the region where steady turns are feasible. In the region of feasible steady turns, we present level curves of these steady turning velocities and steer torques. Additionally, we present level curves of mechanical trail in the lean–steer plane.© 2009 ASME

Analysis of the Holonomic Constraint in the Whipple Bicycle Model (P267)

In this work, we examine the holonomic constraint of the Whipple bicycle model on a level surface. In the Whipple model, one must enforce a constraint which ensures that the front and rear wheels touch the ground. We first derive the constraint in an intuitive geometric fashion and then verify that the constraint can be expressed as quartic polynomial in the sine of the frame pitch. We present three methods for enforcing the constraint in a dynamic model and comment on the differences and practical issues involved in each method.