# Deflection Reduction Shaping Commands with Asymmetric First-Order Actuators
## 2 Materials and Methods
### Analytical Deflection Reduction Shaping Command  
A closed-form deflection reduction input shaper, the so-called UMZV\({}_{\text{DRF}}\) shaper, is developed to generate on-off command profiles under asymmetric first-order actuators based on a deflection-limiting input shaper, the UMZV\({}_{\text{DL}}\) shaper, as follows [18]:  
\[\left\{ \begin{array}{l}A_{i}\\ t_{i}\end{array} \right\}=\left[ \begin{array}{cc}1&-1&K_{3}\\ 0&\left\{cos^{-1}\left(1-K_{3}^{2}/2\right)\right\}/\omega_{n}&\left\{cos^{-1} \left(-K_{3}/2\right)\right\}/\omega_{n}\end{array} \right] \tag{1}\]  
where \(\omega_{n}\), \(A_{i}\), and \(t_{i}\) are the natural frequency of a mass-spring-mass system and the magnitude and time of the \(i^{th}\) impulse, respectively; the final limited impulse magnitude is defined as \(K_{3}=2\cdot D_{lim}\); and the deflection limit, \(0<D_{lim}\leq 0.5\), is defined as the ratio between the allowable and maximum deflections. The UMZV\({}_{\text{DL}}\) shaper is convolved with a pulse input such that the shaped commands are shown as on-off command profiles and lead to zero transient and residual oscillations for ideal flexible systems. However, if the shaped command is applied to a flexible system with asymmetric first-order actuators, the control performance is degraded because of the distorted command, as shown in Figure 1. Therefore, a deflection reduction shaped command is newly required to account for asymmetric first-order actuators. First, for the deflection limit of a linear pendulum system, an expression for the deflection as a function of the input shaper must be obtained to limit the transient deflection of the flexible mode. The desired function can be derived using superposition of deflections from individual first-order response functions to a step input. The derivation begins with the equation for modelling the payload for the pendulum system in Figure 2 as follows:  
\[L\ \ddot{\theta}(t)+g\ sin\ \theta(t)=u(t)\ cos\ \theta(t) \tag{2}\]  
where \(u(t)=\dot{v}_{f}(t)\) is the time derivative of velocity input command, \(L\) is the length of a cable, and \(g\) is the gravity constant. With the assumption of a small angle \(\theta(t)\) and zero initial conditions for the  
Figure 1: Effect of asymmetric first-order actuators. (a) UMZV\({}_{\text{DL}}\) shaped command; (b) Distorted command. development of input shaped commands, Equation (2) can be expressed with the Laplace transform as follows:  
\[\Theta(s)=-\frac{s\ V_{f}(s)}{L}\frac{1}{s^{2}+\omega_{n}^{2}} \tag{3}\]  
where \(\omega_{n}=\sqrt{L/g}\) is the natural frequency of the pendulum system and \(V_{f}(s)\) is the Laplace transform of \(v_{f}(t)\). The start-up portion of the velocity profile in Figure 1b consists of a first-order response to a step command. The velocity command can be expressed as \(V_{f}(s)=A\cdot V_{d}\left\{1/s-1/(s+1/\tau)\right\}\) where \(A\), \(\tau\), and \(V_{d}\) are the magnitude of the shaped input command, time constant, and desired velocity magnitude, respectively. Substituting \(V_{f}(s)\) into Equation (3) and taking the inverse Laplace transformation with zero initial conditions yields the deflection as follows:  
\[\theta(t)=-\frac{A\cdot V_{d}}{L\ \omega_{n}\Big{(}1+\omega_{n}^{2}\tau^{2} \Big{)}}\Big{\{}sin(\omega_{n}t)-\omega_{n}\cdot\tau\cdot cos(\omega_{n}t)+ \omega_{n}\cdot\tau\cdot e^{-t/\tau}\Big{\}} \tag{4}\]  
The deflection is approximately presented with the property of a sinusoidal function and the simplified expression assuming \(e^{-t/\tau}\to 0\) rapidly in time is as follows:  
\[\theta(t)=-\frac{A\cdot V_{d}\sqrt{1+\omega_{n}^{2}\ \tau^{2}}}{L\ \omega_{n} \Big{(}1+\ \omega_{n}^{2}\ \tau^{2}\Big{)}}\Big{[}\sin\Big{(}\omega_{n}t-\tan^{-1}( \omega_{n}\cdot\tau)\Big{)}\Big{]} \tag{5}\]  
With this approximated expression of a single first-order response to a step input, the final limited magnitude can be easily obtained to limit the transient deflections. Later, the negligence of the exponential term will be justified by comparing the control performance to an optimized solution.