Reaction Cancellation

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1 Introduction to Reaction Cancellation

Ground and space based astronomy require the position of their optics to be stable to an unprecendented level. This level of stability often relies on feedback sensors that are pushing the edge of what is possible. This level of stability also requires a stable platform. This requires reaction cancellation.

If we control the system's line-of-sight (LOS) by moving a mirror and that mirror has enough inertia then reaction cancellation is used to cancel out the mirror's motion. The mirror's motion induces a torque on the platform it is mounted to. A dummy mass is moved so that it produces an equal and opposite torque. This reaction mass cancels out the mirror's motion leading to the name reaction cancellation.

2 Rigid Body Model for Reaction Cancellation

A mass attached to a spring and a damper. The F in the diagram denotes an external force, which this example does not include.

For a system of linear translation

 $LaTeX: \sum F = ma = m \ddot{x} = m \frac{d^2x}{dt^2}$

where

$LaTeX: a$ is the acceleration (in m/s2) of the mass,
$LaTeX: m$ is mass and
$LaTeX: x$ is the displacement (in m) of the mass relative to a fixed point of reference.

For a rotational system

 $LaTeX: \sum \Tau = I \ddot{\theta} = I \frac{d^2\theta}{dt^2}$

where

$LaTeX: \theta$ is the angular displacement (in rad) of the inertia relative to a fixed point of reference and
$LaTeX: I$ is the inertia.

The above equations combine to form the equation of motion, a second-order differential equation for displacement θ as a function of time t (in seconds).

 $LaTeX: I \ddot{\theta} + B \dot{\theta} + k \theta = 0.\,$

where

$LaTeX: B$ is the damping coefficient and
$LaTeX: k$ is the spring force constant.

Rearranging, we have

 $LaTeX: \ddot{\theta} + \frac{B}{I} \dot{\theta} + \frac{k}{I} \theta,$

then

 $LaTeX: \ddot{\theta} = -\frac{B}{I} \dot{\theta} + -\frac{k}{I} \theta$

2.1 State Space Model of Rigid Body Reaction Cancellation

Figure 2: Reaction Cancellation double Mass

Figure 2 shows a 2 DOF Rigid Body model. This example is for rotation but Figure 2 is good enough. Figure 2 breaks up the rigid body diagram into Left and Right halves and the Mounting plate. The state space form of this equation for a single axis is

 $LaTeX: \begin{bmatrix}\ddot{\theta} \\ \dot{\theta}\end{bmatrix}=\begin{bmatrix}-\frac{B}{I} & -\frac{k}{I} \\ 1 & 0\end{bmatrix} \begin{bmatrix}\dot{\theta} \\ \theta\end{bmatrix}+\begin{bmatrix}\frac{1}{I} \\ 0\end{bmatrix}\begin{bmatrix}\tau\end{bmatrix}$

In order to calculate the reaction cancellation we need to know the torque imposed on the mounting plate from both the mirror and the reaction mass. For the mounting plate the torque is

 $LaTeX: \Tau_{Mount}=\ddot{\theta}_{Mount}I_{Mount}=\Tau_{Mirror}+\Tau_{ReactionMass}$

Therefore

 $LaTeX: \ddot{\theta}_{Mount}=\frac{1}{I_{Mount}}\left(I_{Mirror}\ddot{\theta}_{Mirror}+I_{ReactionMass}\ddot{\theta}_{ReactionMass}\right)$

So for the mirror's contribution to the imposed torque on the mounting plate

 $LaTeX: \begin{bmatrix}\ddot{\theta}_{Mount, Mirror} \\ \dot{\theta}_{Mount, Mirror}\end{bmatrix}=\frac{I_{Mirror}}{I_{Mount}}\begin{bmatrix}-\frac{B_{Mirror}}{I_{Mirror}} & -\frac{k_{Mirror}}{I_{Mirror}} \\ 1 & 0\end{bmatrix} \begin{bmatrix}\dot{\theta}_{Mirror} \\ \theta_{Mirror}\end{bmatrix}+\begin{bmatrix}\frac{1}{I_{Mirror}} \\ 0\end{bmatrix}\begin{bmatrix}\tau_{Mirror}\end{bmatrix}$

Using the following abbreviations Mount = Mnt, Mirror = Mr, and ReactionMass = RM the full state space reaction cancellation model for a single axis is

 $LaTeX: \begin{bmatrix}\ddot{\theta}_{Mr} \\ \dot{\theta}_{Mr} \\ \ddot{\theta}_{Mnt} \\ \dot{\theta}_{Mnt} \\ \ddot{\theta}_{RM} \\ \dot{\theta}_{RM}\end{bmatrix}$$LaTeX: =\begin{bmatrix}-\frac{B_{Mr}}{I_{Mr}} & -\frac{k_{Mr}}{I_{Mr}} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \frac{B_{Mr}}{I_{Mnt}} & \frac{k_{Mr}}{I_{Mnt}} & 0 & 0 & \frac{B_{RM}}{I_{Mnt}} & \frac{k_{RM}}{I_{Mnt}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{B_{RM}}{I_{RM}} & -\frac{k_{RM}}{I_{RM}} \\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}$$LaTeX: \begin{bmatrix}\dot{\theta}_{Mr} \\ \theta_{Mr} \\ \dot{\theta}_{Mnt} \\ \theta_{Mnt} \\ \dot{\theta}_{RM} \\ \theta_{RM}\end{bmatrix}+$$LaTeX: \begin{bmatrix}\frac{1}{I_{Mr}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{1}{I_{RM}} \\ 0 & 0\end{bmatrix}$$LaTeX: \begin{bmatrix}\tau_{Mr} \\ \tau_{RM}\end{bmatrix}$

Simplifying for the reaction cancellation in the x-axis

 $LaTeX: \dot{x}_{x}=\begin{bmatrix}\ddot{\theta}_{Mr} \\ \dot{\theta}_{Mr} \\ \ddot{\theta}_{Mnt} \\ \dot{\theta}_{Mnt} \\ \ddot{\theta}_{RM} \\ \dot{\theta}_{RM}\end{bmatrix}_{x}$

 $LaTeX: A_{x}=\begin{bmatrix}-\frac{B_{Mr}}{I_{Mr}} & -\frac{k_{Mr}}{I_{Mr}} & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 \\ \frac{B_{Mr}}{I_{Mnt}} & \frac{k_{Mr}}{I_{Mnt}} & 0 & 0 & \frac{B_{RM}}{I_{Mnt}} & \frac{k_{RM}}{I_{Mnt}} \\ 0 & 0 & 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & -\frac{B_{RM}}{I_{RM}} & -\frac{k_{RM}}{I_{RM}} \\ 0 & 0 & 0 & 0 & 1 & 0\end{bmatrix}_{x}$

 $LaTeX: B_{x}=\begin{bmatrix}\frac{1}{I_{Mr}} & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & 0 \\ 0 & \frac{1}{I_{RM}} \\ 0 & 0\end{bmatrix}_{x}$

 $LaTeX: u_{x}=\begin{bmatrix}\tau_{Mr} \\ \tau_{RM}\end{bmatrix}_{x}$

 $LaTeX: \dot{x}_{x}=A_{x}x_{x}+B_{x}u_{x}$

For 2 axes the reaction cancellation state space equations become

 $LaTeX: \begin{bmatrix}\dot{x}_{x} \\ \dot{x}_{y}\end{bmatrix}=\begin{bmatrix}A_{x} & 0 \\ 0 & A_{y}\end{bmatrix} \begin{bmatrix}x_{x} \\ x_{y}\end{bmatrix}+\begin{bmatrix}B_{x} \\ B_{y}\end{bmatrix} \begin{bmatrix}u_{x} \\ u_{y}\end{bmatrix}$

For a rigid body model the mounting plate imposed torque can be determined with the mounting plate inertia and angular acceleration therefore

 $LaTeX: C_{x}=\begin{bmatrix}0 & 0 & 0 & 1 & 0 & 0\end{bmatrix}_{x}$

This Cx provides the angular position of the mounting plate

 $LaTeX: y=\begin{bmatrix}C_{x} & 0 \\ 0 & C_{y}\end{bmatrix}\begin{bmatrix}x_{x} \\ x_{y}\end{bmatrix}+0\begin{bmatrix}\tau_{x} \\ \tau_{y}\end{bmatrix}$

This y is only the x and y angular position of the mounting plate. Using MATLAB or Simulink we can add 2 derivatives to the state space output providing mounting plate angular accelerations. This angular acceleration can be multiplied by the mounting plate inertia to determine the residual torque in the mounting plate. Thus the error in the reaction cancellation.