EFFECT OF VERTICAL ACCELERATION ON THE FREQUENCY OF A PENDULUM: IMPACT ON INPUT SHAPING John Huey, William Singhose zyxwv Woodniff zyxwvutsr School zyxwvuts of Mechanical Engineering Georgia Institute of Technology Atlanta, GA 30332-0405 bill.sinphose~me.entech.edu Abstract zyxwvutsrqpon ~ The vibration of a pendulum is often linearized to obtain zyxwvutsrqponm a natural frequency that can he used by vibration control techniques to reduce or eliminate pendulum sway. Vertical acceleration will alter this effective natural frequency. Therefore, this paper studies.the impact that vertical acceleration has on a pendulum's linearized natural frequency and studies how input shaping can he used to counteract this effect. Simulation results show that robust input shapers perform well in reducing vibration in the 'midst of frequency variation, and that new input shapers can be specifically designed based on studying how vertical acceleration influences pendulum oscillations. Keywords - Input Shaping, Pendulum, Natural Frequency I. INTRODUCTION The pendulum is a widely studied flexible system. Although a three-dimensional pendulum is non-linear, under small displacements, its natural frequency of vibration can be characterized by the well known equation: zyxwvutsrq 0" =a (1) where, g is the acceleration due to gravity and L is the suspension length. Pendulums can be found in a variety of engineering applications. For example, large gantry, boom, and tower cranes support loads with vertically suspended cables. There has been much study dedicated to vibration- reducing control of such pendulum systems. Some control strategies use state feedback or pre-computed trajectories. However, sensor systems are costly, and accurate state information is ofien difficult to obtain on cranes. In addition, cranes are typically controlled by humans. This makes feedback control difficult; as the human operator is a separate feedback system. If two feedback systems are used (operator and sensor), they can often come into conflict. Human operators also make pre-computed trajectories implausible. Trajectories and boundary conditions are seldom known before a move is executed, because human operators often create the trajectory as they go along. Therefore, pre-computed and time optimal control have limited applications [l]. Fortunately, input shaping has been shown to effectively eliminate residual vibration in pendulum motion for real time maneuvers [2-61. Input shaping is implemented by convolving a sequence of impulses, known as the input shaper, with a desired system command to produce a shaped input that is then used to drive the system. This process is demonstrated in Figure 1. The amplitudes and time locations of the impulses are determined by solving a set of constraint equations which attempt to limit the unwanted dynamic response of the system. Although vibration control techniques increase the potential speed of cargo transfer, one important speed limitation is generally unaddressed in traditional crane design: inertia. Typically, cranes require a large portion of the crane structure itself to move around the workspace. In many cases, the mass of the payload is insignificant compared to the mass of the moving crane structure. This means that much of the actuator effort goes into moving the crane structure. Consequently, cranes are not able to quickly manipulate payloads, even in the presence of good vibration control techniques. One recent concept which attempts to avert this problem is the cable-driven mechanism, like the one shown in Figure 2. The conceptual basis of cable-driven mechanisms is to control the payload position by lengthening and shortening multiple suspension cables that come together at the payload rigging. Although this new technology is primarily being applied to small scale, high speed movements [7], some research is beginning to study the extension of cable-driven mechanisms over large workspaces [S-lo]. Although cable-driven mechanisms promise to reduce Figure zyxwv 1: The Input Shaping Process. 0-7803-7729-X/03/$17.00 02003 IEEE zyxwvuts 532