Effect of Uncertainty on the payload’s dynamics on the isolation platform dynamics

If we write the equation of motion of the system in Figure 1, we obtain:

After eliminating \(x^\prime\), we obtain:

Let the payload have:

• a nominal mass of \(m^\prime = 50\ [kg]\)
• a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)
• a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)

mpi = 50;
kpi = 5e6;
cpi = 3e3;

kpi = (2*pi*50)^2*mpi;
cpi = 0.2*sqrt(kpi*mpi);

Let’s also consider some uncertainty in those parameters:

mp = ureal('m', mpi, 'Range', [1, 100]);
cp = ureal('c', cpi, 'Percentage', 30);
kp = ureal('k', kpi, 'Percentage', 30);

The compliance of the payload without the isolation platform is \(\frac{1}{m^\prime s^2 + c^\prime s + k^\prime}\) and its bode plot is shown in Figure 2.

One can see that the payload has a resonance frequency of \(\omega_0^\prime = 250\ Hz\).

Let’s first fix the mass of the isolation platform:

m = 10;

And we generate three isolation platforms:

• A soft one with \(\omega_0 = 0.1 \omega_0^\prime = 5\ Hz\)
• A medium stiff one with \(\omega_0 = \omega_0^\prime = 50\ Hz\)
• A stiff one with \(\omega_0 = 10 \omega_0^\prime = 500\ Hz\)

The obtained dynamics from \(F\) to \(x\) for the three isolation platform are shown in Figure 3.

Let’s now consider a general payload described by its impedance \(G^\prime(s) = \frac{F^\prime}{x}\) as shown in Figure 4.

Now let’s consider the system consisting of a mass-spring-system (the isolation platform) supporting the general payload as shown in Figure 5.

We have to following equations of motion:

And by eliminating \(F^\prime\), we find the plant dynamics \(G(s) = \frac{x}{F}\).

We can learn few things about the obtained transfer function:

• the zeros of \(x/F\) will be the poles of \(G^\prime(s)\).
• if the impedance of the payload is small \(|G^\prime(s)| \ll |ms^2 + cs + k|\), then the payload will have small influence on the obtained dynamics

In order to verify that the formula is correct, let’s take the same mass-spring-damper system used in the system shown in Figure 1:

By eliminating \(x^\prime\) of the equations, we obtain:

And we obtain

Which is the same transfer function that was obtained in section 1 (Eq. \eqref{eq:plant_simple_system}).

The impedance of the mass-spring-damper system is shown in Figure 6.

• Before the resonance frequency \(\omega_0^\prime\), the impedance follows the mass line
• After the resonance, the impedance will follow the stiffness line (depending on the relative values of the stiffness and damping)
• At high frequency, it will follow the damping line

To summarize, we consider:

• an Isolation platform represented by a mass \(m\), a damper \(c\) and a stiffness \(k\). This system resonate at \(\omega_0 = \sqrt{\frac{k}{m}}\)
• A payload represented by a mass \(m^\prime\), a damper \(c^\prime\) and a stiffness \(k^\prime\). The payload resonate at \(\omega_0^\prime = \sqrt{\frac{k^\prime}{m^\prime}}\)

The “impedance” of the payload is represented by: \[ G^\prime(s) = \frac{m^\prime s^2 (c^\prime s + k^\prime)}{m^\prime s^2 + c^\prime s + k^\prime} \]

And the plant is: \[ G(s) = \frac{x}{F} = \frac{1}{ms^2 + cs + k + G^\prime(s)} \]

Let’s write the asymptotic behavior of \(|G^\prime(j\omega)|\):

• \(\lim_{\omega \to 0} |G^\prime(j\omega)| = m^\prime s^2\)
• \(|G^\prime(j\omega_0)| = \frac{k^\prime \sqrt{1 + (2\xi^\prime)^2}}{2 \xi^\prime}\)
• \(\lim_{\omega \to \infty} |G^\prime(j\omega)| = c^\prime s + k\)

Let’s find some conditions in order to have that the dynamics of the payload does not influence to much the dynamics of the plant: \[ |G^\prime(s)| \ll |ms^2 + cs + k| \]

Let’s take the case of a stiff payload (\(\omega_0^\prime \gg \omega_0\)).

Below \(\omega_0\), the condition becomes: \[ |G^\prime(s)| \ll k \Leftrightarrow m^\prime \omega_0^2 \ll k \Leftrightarrow m^\prime \ll m \] The payload mass should be small with respect to the isolation platform mass.

Above \(\omega_0\): \[ |G^\prime(j\omega)| \ll m \omega^2 \]

Until \(\omega_0^\prime\), we have \(m^\prime \ll m\) which is the same condition as before. Above \(\omega_0^\prime\), we obtain \(|jc^\prime \omega + k| \ll m \omega^2\).

We model the payload by a mass-spring-damper model with some uncertainty.

Let the payload have:

• a nominal mass of \(m^\prime = 50\ [kg]\)
• a nominal stiffness of \(k^\prime = 5 \cdot 10^6\ [N/m]\)
• a nominal damping of \(c^\prime = 3 \cdot 10^3\ [N/(m/s)]\)

The main resonance of the payload is then \(\omega^\prime = \sqrt{\frac{m^\prime}{k^\prime}} \approx 50\ Hz\).

m0 = 10;
c0 = 3e2;
k0 = 5e5;

Gp0 = (m0*s^2 * (c0*s + k0))/(m0*s^2 + c0*s + k0);

Let’s represent the uncertainty on the impedance of the payload by a multiplicative uncertainty (Figure 7): \[ G^\prime(s) = G_0^\prime(s)(1 + w_I^\prime(s)\Delta_I(s)) \quad |\Delta_I(j\omega)| < 1\ \forall \omega \]

This could represent unmodelled dynamics or unknown parameters of the payload.

We choose a simple uncertainty weight: \[ w_I(s) = \frac{\tau s + r_0}{(\tau/r_\infty) s + 1} \] where \(r_0\) is the relative uncertainty at steady-state, \(1/\tau\) is the frequency at which the relative uncertainty reaches \(100\ \%\), and \(r_\infty\) is the magnitude of the weight at high frequency.

The parameters are defined below.

r0 = 0.5;
tau = 1/(50*2*pi);
rinf = 10;

wI = (tau*s + r0)/((tau/rinf)*s + 1);

We then generate a complex \(\Delta\).

DeltaI = ucomplex('A',0);

We generate the uncertain plant \(G^\prime(s)\).

Gp = Gp0*(1+wI*DeltaI);

A set of uncertainty payload’s impedance transfer functions is shown in Figure 8.

Let’s express the uncertainty of the plant \(x/F\) as a function of the parameters as well as of the uncertainty on the platform’s compliance:

Let’s first fix the mass of the isolation platform:

m = 20;

And we generate three isolation platforms:

• A soft one with \(\omega_0 = 5\ Hz\)
• A medium stiff one with \(\omega_0 = 50\ Hz\)
• A stiff one with \(\omega_0 = 500\ Hz\)

Soft Isolation Platform:

k_soft = m*(2*pi*5)^2;
c_soft = 0.1*sqrt(m*k_soft);

G_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp);
G0_soft = 1/(m*s^2 + c_soft*s + k_soft + Gp0);
wiI_soft = Gp0*G0_soft*wI;

Mid Isolation Platform

k_mid = m*(2*pi*50)^2;
c_mid = 0.1*sqrt(m*k_mid);

G_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp);
G0_mid = 1/(m*s^2 + c_mid*s + k_mid + Gp0);
wiI_mid = Gp0*G0_mid*wI;

Stiff Isolation Platform

k_stiff = m*(2*pi*500)^2;
c_stiff = 0.1*sqrt(m*k_stiff);

G_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp);
G0_stiff = 1/(m*s^2 + c_stiff*s + k_stiff + Gp0);
wiI_stiff = Gp0*G0_stiff*wI;

The obtained transfer functions \(x/F\) for each of the three platforms are shown in Figure 10.

Now that we know the expression of the uncertainty on the plant, we can wonder what parameters of the isolation platform would lower the plant uncertainty, or at least bring the uncertainty to reasonable level.

The uncertainty of the plant is described by an inverse multiplicative uncertainty with the following weight: \[ w_{iI}(s) = \frac{G_0^\prime(s) w_I(s)}{ms^2 + cs + k + G_0^\prime(s)} \]

Let’s study separately the effect of the platform’s mass, damping and stiffness.