Keith W. Buffinton

Formulation of Equations of Motion for Systems Subject to Constraints

A method for constructing equations of motion governing constrained systems is presented. The method, which is particularly useful when equations of motion have a/ready been formulated, and new equations of motion, reflecting the presence of additional constraints are needed, allow the new equations to be written as a recombination of terms comprising the original equations. An explicit form in which the new dynamical equations may be cast for the purpose of numerical integration is developed, along with special cases that demonstrate how the procedure may be simplified in two commonly occurring situations. An illustrative example from the field of robotics is presented, and several areas of application are identified.

Stability Analysis of Passive Dynamic Walking of Quadrupeds

We introduce a detailed numerical simulation and analysis framework to extend the principles of passive dynamic walking to quadrupedal locomotion. Non-linear limit cycle methods are used to identify possible gaits and to analyze the stability and efficiency of quadrupedal passive dynamic walking. In doing so, special attention is paid to issues that are inherent to quadrupedal locomotion, such as the occurrence of simultaneous contact collisions and the implications of the phase difference between front and back leg pairs. Limit cycles identified within this framework correspond to periodic gaits and can be placed into two categories: in-phase gaits in which front and back legs hit the ground at roughly the same time, and out-of-phase gaits with a ±90° phase shift between the back and front leg pairs. The latter are, in comparison, energetically more efficient but exhibit one unstable eigenvalue that leads to a phase divergence and results in a gait-transition to a less efficient in-phase gait. A detailed analysis examines the influence of various parameters on stability and locomotion speed, with the ultimate goal of determining a stable solution for the energy-efficient, out-of-phase gait. This was achieved through the use of a wobbling mass, i.e. an additional mass that is elastically attached to the main body of the quadruped. The methods, results, and gaits presented in this paper additionally provide a point of departure for the exploration of the considerably richer range of quadrupedal locomotion found in nature.

Dynamics of a Beam Moving Over Supports

Abstract The behavior of a uniform beam moving longitudinally at a prescribed rate over two bilateral supports is studied. Equations of motion are formulated by regarding the supports as kinematical constraints imposed on an unrestrained beam and by discretizing the beam via the assumed modes technique. The results of numerical simulations are displayed for cases in which the beam undergoes no longitudinal motion, sinusoidal longitudinal motion, or longitudinal motion for the purpose of repositioning. The instability of harmonic longitudinal motion is investigated, and methods are presented for predicting whether or not a given motion will give rise to an unstable response.

PROJECT TEAM DYNAMICS AND COGNITIVE STYLE

Abstract Problem-solving styles and interpersonal dynamics of project teams are often critical factors for a team to function effectively. To study problem-solving styles and track intra-team interactions, the Kirton Adaption-Innovation Inventory (KAI) was used to determine the cognitive styles of engineering and management students in Bucknells Institute for Leadership in Technology and Management (ILTM). KAI scores allowed interpretation and characterization of data from student journaling assignments that recorded observations about project team members abilities to work and communicate with each other. KAI results show correlations with both positive and negative aspects of project team experiences. The results indicate potential sources of conflicts in project teams comprised of mature individuals working in a corporate environment.

Extended Kalman Filtering Applied to a Two-Axis Robotic Arm with Flexible Links

An industrial robot today uses measurements of its joint positions and models of its kinematics and dynamics to estimate and control its end-effector position. Substantially better end-effector position estimation and control performance would be obtainable if direct measurements of its end-effector position were also used. The subject of this paper is extended Kalman filtering for precise estimation of the position of the end-effector of a robot using, in addition to the usual measurements of the joint positions, direct measurements of the end-effector position. The estimation performances of extended Kalman filters are compared in applications to a planar two-axis robotic arm with very flexible links. The comparisons shed new light on the dependence of extended Kalman filter estimation performance on the quality of the model of the arm dynamics that the extended Kalman filter operates with.

Pulse Width Control for Precise Positioning of Structurally Flexible Systems Subject to Stiction and Coulomb Friction

Pulse width control refers to the use of a control law to determine the duration of fixed-height force pulses for point-to-point position control of a plant that is subject to mechanical friction, including stiction. The use of constant-gain pulse width control laws for precise positioning of structurally flexible plants subject to stiction and Coulomb friction is analyzed. It is shown that when the plant is a simple two-mass system subject to stiction and Coulomb friction, a position error limit cycle can result. Sufficient conditions for stability and self-sustained oscillation of this closed-loop system are derived. The sufficient conditions for stability are used to determine conditions on the plant parameters and the control gain that guarantee closed-loop stability and thus limit-cycle-free operation and zero steady-state position error. The analysis methods that are introduced are demonstrated in applications to the control of the position of the end-effector of an industrial robot.

A MATLAB framework for efficient gait creation

This work introduces a framework for the creation and analysis of efficient gaits for legged systems based on the exploitation of natural dynamics. It summarizes the theory behind hybrid dynamic modeling, the identification of optimal periodic motions with single shooting and direct collocation, and the analysis of first order stability. Three examples introduce various aspects of gait creation and analysis: a stability study of a passive dynamic walker determines the ideal position of the legs center of mass, the cost of transportation is minimized for a prismatic monopod hopper based on series elastic actuators, and a basic controller is created for the model of a bounding robot. The presented tools and examples are freely available at

Piecewise-Linear-Gain Pulse Width Control for Precise Positioning of Structurally Flexible Systems Subject to Stiction and Coulomb Friction

Pulse width control refers to the use of a control law to determine the duration of fixed-height force pulses for point-to-point position control of a plant that is subject to mechanical friction, including stiction. A quantitative measure of the performance of a pulse width control system is introduced. Applications of this measure suggest that piecewise-linear-gain pulse width control laws will often provide better performance than constant-gain pulse width control laws. A method for designing piecewise-linear-gain pulse width control laws is introduced. The performance measure and piecewise-linear-gain control law design method are demonstrated in applications to the control of the position of the end-effector of an industrial robot.

Laboratory, computational and field studies of snowboard dynamics

While many studies have documented the dynamic behaviour of skis, similar studies for snowboards have been rare. Characteristics such as board stiffness and damping are acknowledged to be linked to performance, but a quantitative determination of corresponding natural frequencies and damping ratios has to date not been published. The present work uses laboratory, computational and field studies to develop and document an in-depth understanding and quantification of snowboard dynamics. In particular, laboratory tests are used to determine the first three bending and first two torsional natural frequencies and modal damping ratios for eight snowboards from two manufacturers. Computer models are developed using the software packages Pro/ENGINEER and Pro/MECHANICA that accurately reproduce the experimentally measured natural frequencies and that facilitate visualization of mode shapes. Field tests are discussed that provide insights into the strains and accelerations experienced by snowboards while subject to turns, stops and jumps. Quantitative results are shown to correlate well with qualitative descriptions offered by manufacturers and riders. Medium-quality boards designed for beginner riders and characterized as ‘soft’ have lower natural frequencies and larger damping ratios than similar boards designed for advanced riders and characterized as ‘stiff.’ Moreover, boards designed for advanced riders and characterized as ‘high-quality’ have natural frequencies higher than ‘medium-quality’ boards while still exhibiting high damping ratios.

Energetics of passivity-based running with high-compliance series elastic actuation

The efficiency of running gaits in nature results in large part from passive elastic oscillations on springy legs. In this paper, this principle is applied to robotic systems by endowing them with high compliance series elastic actuators in which the electric motors are decoupled from the joints by elastic elements. Periodic motor inputs excite the natural dynamic motion of the robot and create a passivity-based running motion. An optimisation algorithm minimised energy expenditure and estimated the necessary initial model states and the coefficients of a parameterised excitation function for the simulations of a two-dimensional hopping monopod and a planar bounding quadruped. Gait synthesis within this framework was analysed with respect to energy consumption, particularly as a function of running speed. Different solution groups were found, each of them corresponding to a characteristically different movement which proved to be most efficient for the corresponding speed range. This shines a different light on the meaning of ‘gait’ in the context of robotics, and directly contributes to a better understanding of the creation and exploitation of different modes of locomotion in legged robotics.