Update Content - 2023-06-28

content/zettels/position_jitter_due_to_asynchronous_acquisition.md

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+title = "Position Jitter due to Asynchronous Acquisition"
+author = ["Dehaeze Thomas"]
+draft = false
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+
+Tags
+:
+
+
+## Observed issue {#observed-issue}
+
+Sometime the controller is not compatible with the encoder protocol.
+In that case a PEPU can be used in between the encoder and the controller to convert the encoder value to something readable by the controller.
+This is illustrated in Figure <fig:position_jitter_issue>.
+
+<a id="figure--fig:position-jitter-issue"></a>
+
+{{< figure src="/ox-hugo/position_jitter_issue.png" caption="<span class=\"figure-number\">Figure 1: </span>Measurement Setup: an encoder working with BISS protocol is read by a PEPU (every \\(T\_{s,\text{pepu}}\\) seconds) and the controller reads the stored encoder value in the PEPU every \\(T\_{s,\text{ctrl}}\\) seconds" >}}
+
+When scanning the device (i.e. changing rapidly the read value on the encoder), some "jumps" on the encoder value read by the controller can be seen.
+This effect is due to some "jitter" between the PEPU acquisition rate and the controller rate as will be explained bellow.
+
+
+## Visual display of jitter issue {#visual-display-of-jitter-issue}
+
+Let's make a simulation to understand what is going on.
+Let's choose the following parameters:
+
+-   \\(T\_{s,\text{pepu}} = 60\\,\mu s\\): the acquisition rate of the encoder on the PEPU (very typical for a 32bit absolute encoder)
+-   \\(v = 1\\,mm/s\\): the scan velocity
+-   \\(T\_{s,\text{ctrl}} = 100\\,\mu s\\) the "sampling rate" of the controller
+
+The encoder position as well as the stored value on the PEPU and the position used in the controller are shown in Figure <fig:jitter_error_example>, left.
+The errors associated with the "jitter" is shown in Figure <fig:jitter_error_example>, right.
+
+```matlab
+%% Simulation parameters
+v = 1e-3; % Scanning velocity [m/s or rad/s]
+Ts_pepu = 60e-6; % Sampling time of PEPU [s] i.e. time to get a new encoder value
+Ts_ctrl = 1e-4; % Sampling time of controller [s]
+
+t_sim = 10*Ts_ctrl; % Total simulation time [s]
+t = 0:1e-6:t_sim; % Time vector used for simulation [s]
+x = v*t; % Suppose linear position [m, rad]
+
+%% Compute position stored in PEPU as every time step
+x_pepu = Ts_pepu*floor(t/Ts_pepu)*v;
+
+%% Compute position get on the controller at each control period
+t_ctrl = 0:Ts_ctrl:t_sim; % Time vector [s]
+x_ctrl = zeros(size(t_ctrl)); % Position stored on the controller [m]
+x_error = zeros(size(t_ctrl)); % Position error due to "jitter" [m]
+for i = 1:length(t_ctrl)
+    [~, i_t] = min(abs(t - t_ctrl(i))); % Find the stored encoder value in the PEPU at the time of the controller period
+    x_ctrl(i) = x_pepu(i_t);
+    x_error(i) = x_pepu(i_t) - x(i_t);
+end
+```
+
+<a id="figure--fig:jitter-error-example"></a>
+
+{{< figure src="/ox-hugo/jitter_error_example.png" caption="<span class=\"figure-number\">Figure 2: </span>Measurement error due to Jitter. 1mm/s velocity scan, Ts_pepu is 60us and Ts_ctrl is 100us" >}}
+
+
+## Expected error induced by "jitter" {#expected-error-induced-by-jitter}
+
+Th "position jitter" depends on:
+
+-   \\(T\_{s,\text{pepu}}\\): the "sampling time" of the encoder on the PEPU
+-   \\(v\\): the scan velocity in [unit/s]
+
+The obtain "jitter" can be as large as (expressed in the same units as \\(v\\)):
+
+\begin{equation}
+dx = v \cdot T\_{s,\text{pepu}}
+\end{equation}
+
+Let's make a numerical example:
+
+```matlab
+%% Simulation parameters
+v = 1e-3; % Scanning velocity [m/s or rad/s]
+Ts_pepu = 60e-6; % Sampling time of PEPU [s] i.e. time to get a new encoder value
+```
+
+```text
+dx = 60 [nm] with v = 1.0 [mm/s] and Ts_pepu = 60 [us]
+```
+
+
+## Is there an optimal PEPU acquisition rate? {#is-there-an-optimal-pepu-acquisition-rate}
+
+Changing the readout time of the PEPU (its clock for instance) changes the jitter amplitude as well as its "pattern":
+
+```matlab
+%% Longer simulation than before to better see the pattern
+t_sim = 50*Ts_ctrl; % Total simulation time [s]
+t = 0:1e-6:t_sim; % Time vector used for simulation [s]
+x = v*t; % Suppose linear position [m, rad]
+t_ctrl = 0:Ts_ctrl:t_sim; % Time vector [s]
+
+%% Large Ts_pepu to try to match with controller sampling time
+Ts_pepu = 95e-6; % Sampling time of PEPU [s] i.e. time to get a new encoder value
+x_pepu = Ts_pepu*floor(t/Ts_pepu)*v;
+x_error_1 = zeros(size(t_ctrl)); % Position error due to "jitter" [m]
+for i = 1:length(t_ctrl)
+    [~, i_t] = min(abs(t - t_ctrl(i))); % Find the stored encoder value in the PEPU at the time of the controller period
+    x_error_1(i) = x_pepu(i_t) - x(i_t);
+end
+
+%% Small Ts_pepu as possible to reduce jitter amplitude
+Ts_pepu = 30e-6; % Sampling time of PEPU [s] i.e. time to get a new encoder value
+x_pepu = Ts_pepu*floor(t/Ts_pepu)*v;
+x_error_2 = zeros(size(t_ctrl)); % Position error due to "jitter" [m]
+for i = 1:length(t_ctrl)
+    [~, i_t] = min(abs(t - t_ctrl(i))); % Find the stored encoder value in the PEPU at the time of the controller period
+    x_error_2(i) = x_pepu(i_t) - x(i_t);
+end
+
+%% Small Ts_pepu as possible to reduce jitter amplitude
+Ts_pepu = 10e-6; % Sampling time of PEPU [s] i.e. time to get a new encoder value
+x_pepu = Ts_pepu*floor(t/Ts_pepu)*v;
+x_error_3 = zeros(size(t_ctrl)); % Position error due to "jitter" [m]
+for i = 1:length(t_ctrl)
+    [~, i_t] = min(abs(t - t_ctrl(i))); % Find the stored encoder value in the PEPU at the time of the controller period
+    x_error_3(i) = x_pepu(i_t) - x(i_t);
+end
+```
+
+<a id="figure--fig:jitter-errors-effect-Ts-pepu"></a>
+
+{{< figure src="/ox-hugo/jitter_errors_effect_Ts_pepu.png" caption="<span class=\"figure-number\">Figure 3: </span>Measurement errors due to Jitter. Effect of the refresh rate on the PEPU, different \"patterns\" can appear. Velocity scan is 1mm/s" >}}
+
+<div class="important">
+
+For high velocity / high precision scans, it is important to reduce the timing jitter of the measured position (i.e. "position jitter").
+
+Ideally, this is the control system (i.e. where the feedback controller is implemented) that triggers the readout of all the sensors.
+
+Having a intermediate electronic device (here the PEPU) that triggers the readout of the encoders not in sync with the controller can affect quite negatively the quality of the motion.
+
+</div>
+
+
+## Bibliography {#bibliography}
+
+<style>.csl-entry{text-indent: -1.5em; margin-left: 1.5em;}</style><div class="csl-bib-body">
+</div>