Introduction:

Precise and fast position control is needed in many areas of industries but generally, that demands high rigidity from the system. This rigidity comes at a cost of payload capacity. In this project, Input shaping is used to suppress the vibration in a flexible beam that is being rotated by a stepper motor. Using the method will allow designers to reduce the robot weight while maintaining accuracy and performance (for ex. topologically optimized parts for robotic arm). This method is also very useful for applications with very low friction or damping coefficients, for eg in cranes, space applications.

Quick Demo of input shaping in action.

Objective:

The objection of this project is to implement input shaping and reduce vibrations in the flexible beam due to point-point motion.

Theory:

An uncoupled, linear, vibratory system of any order can be specified as a cascaded set of second-order poles with the decaying sinusoidal response [4] -

$y(t) = \left [A \frac{w_{0}}{\sqrt{1-\zeta^{2} }} e^{-\zeta w_{0} (t-t_{0})} \right ] sin(w_{0} \sqrt{1- \zeta^{2}}(t-t_{0}) )$

where A is the amplitude of the impulse, $w_{0}$ is the undamped natural frequency of the plant, $\zeta$ is the damping ratio, t is time, and $t_{0}$ is the time of the impulse input.

To reduce vibrations, responses from the multiple impulses can be superimposed so that system can move without any vibrations. hence, if we can estimate w0 and $\zeta$ then we can calculate the Amplitudes which will result in zero vibration.

Ref: [1]

The superimposed response from multiple impulses is given by -

$V(w,\zeta) = e^{-\zeta wt_n} \sqrt{C(w,\zeta)^{2} + S(w,\zeta)^{2}}$..............(1)

where,

$C(w,\zeta) = \sum_{i=1}^{n} A_i e^{\zeta w t_i} cos(w_d t_i)$

$S(w,\zeta) = \sum_{i=1}^{n} A_i e^{\zeta w t_i} sin(w_d t_i)$

$A_i$ and $t_i$ are the amplitudes and time locations of the impulses, n is the number of impulses in the impulse sequence, and $w_d = w \sqrt{1 - \zeta^2}$.

$\therefore V(w, \zeta) = 0$.......................................................(2)

Moreover, to avoid the trivial solution and to obtain normalized result,

$\sum A_i = 1$ and i > 1 ................................................(3)

Using the (2) and (3) constraints, and solving for two impulse response (n=2), results in -

$A_0 = \frac{1}{1+K}$, $A_1 = \frac{K}{1+K}$, $t_0 = 0$, $t_1 = 0.5 T_d$

where, $K = exp (\frac{-\zeta \pi}{\sqrt{1-\zeta^2}})$

Using these values, we can generate the input to the system which will move without any vibrations. But this assumes that we have perfectly modeled the system and would result in residual vibrations in case of modelling errors. This method is called Zero Vibration (ZV) Shaper.

For this project, we use the 4 impulse input that is derived in [1] and the Amplitudes are mentioned below -