# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 1 Introduction  
Input shaped commands with standard on-off actuators have been effectively used without large transient and residual deflections of linear flexible systems such as those applied in large cranes, robot manipulators, microstages and space structures [1; 2; 3; 4]. However, the conventional input shaping approach [5; 6; 7] should be modified to account for asymmetric first-order dynamics resulting from the nonlinear effects of a separate actuation mechanism, non-co-located actuators, electric motors, or unilaterally dispersed dynamics. Even if an input-shaped command is properly designed for elastic modes, a shaped command deviated by the asymmetric first- order actuator could deteriorate control performance and may even lead to system instability in precision process control. As a result, large transient and residual deflections may occur in flexible systems. The objective was to develop two command-shaping approaches with the limited-state operation of asymmetric first-order actuators while reducing the transient and residual deflections of elastic modes. Many techniques have been proposed to generate input shaped commands that are time optimal [8; 9; 10; 11], limit transient deflection [12], and limit fuel consumption [13; 14] using reliable optimization algorithms. Some practical developments based on input shaping have been made by considering robustness [15], limitation of transient deflection [16], and fuel efficiency [17]. As one of the open-loop control approaches, the input shaping process is implemented by convolving a referencecommand with a sequence of impulses of an input shaper. While many input shaping techniques have been derived primarily from linear system theory [5; 6], there is much interest in the application of input shaping to non-ideal systems with control performance constraints. A deflection-reduction shaping approach for on-off commands was initially formulated via parameter optimization with a deflection constraint [18]. Several alternative approaches to this formulation were presented by applying a deflection limit to either all or a sampling of local extrema points throughout the entire transient deflection. Thereafter, a closed-form solution for limited-state commands was proposed by adjusting the final impulse magnitude of the unit magnitude zero vibration (UMZV) shaper with the summation of deflection responses to multiple step inputs [19]. As an extension of their previous approach, a robust approach was proposed with parameter optimization to determine additional impulse times [20]. An analytical deflection-limiting shaping approach was also presented under the velocity constraint of electric motors [21]. To constrain the transient deflection of the flexible mode, a time-optimal control profile was parameterized in terms of a permissible deflection value. These deflection reduction shaper approaches have been developed only under ideal on-off actuator operations to limit the transient and residual deflections within a specified bound. Nonlinear systems can be problematic in the application of input shapers because they are based on linear system theory. As a nonlinearity source, actuator nonlinearities can be described by discontinuous nonlinearities within a system, such as backlash, saturation, rate limiting, and dead zone. When an actuator nonlinearity deviates an input-shaped command, the oscillation-reducing performance of the shaped command can be significantly degraded. To reduce the detrimental effects of actuator nonlinearities, input shaper redesign techniques have been established from the perspective of rate limits [22; 23], backlash [24], and Coulomb friction [25; 26]. In addition, on-off shaping techniques were developed for non-ideal input commands distorted by asymmetrical delayed first-order actuators [27] and second-order actuators [28].