Starting CAN cycles after sleep or after action #452

stephane-caron · 2024-10-31T16:05:37Z

stephane-caron
Oct 31, 2024
Maintainer

In 8000 this note, for two function calls f and g, we write f . g when f and g run sequentially and f // g when f and g run in parallel.

Case 1: plan . (can // sleep)

Currently, Spine::cycle_actuation is implemented as plan . (can // sleep), like so:

where we denote by:

$T$ the spine period,
$T_p$ the duration of the "planning" (spine observation and spine action) steps, for which we assume $T_p < T$,
$T_c$ the duration of a CAN cycle, for which we assume $T_c < T$.

Case 2: (plan // can) . sleep

There is an alternative (plan // can) . sleep:

where again we denote by:

$T$ the spine period,
$T_p$ the duration of the "planning" (spine observation and spine action) steps, for which we assume $T_p < T$,
$T_c$ the duration of a CAN cycle, for which we assume $T_c < T$.

Comparison of the two

To understand the tradeoff between the two, we consider the action forward to actuators during the k-th spine cycle $[kT, (k+1)T]$. We can define:

The time $t_{send}[k]$ at which the action is sent (i.e. the CAN cycle starts).
The time $t_{act}[k]$ at which that action was computed.
The time $t_{obs}[k]$ at which the observation used to compute that action was made.

Let's also denote by $T_o \in [0, T_c]$ the time, during a CAN cycle, where the last observation is made.

In Case 1:

$$\begin{align} t_{send}^{(1)}[k] & = kT + T_p \\\ t_{act}^{(1)}[k] & = kT + T_p \\\ t_{obs}^{(1)}[k] & = (k-2)T + T_p + T_o \end{align}$$

In Case 2:

$$\begin{align} t_{send}^{(2)}[k] & = kT \\\ t_{act}^{(2)}[k] & = (k-1) T + T_p \\\ t_{obs}^{(2)}[k] & = (k-2)T + T_o \end{align}$$

Regardless of when the action was computed ($t_{act}$), one thing to observe is that, in both cases:

$$ t_{send} - t_{obs} = 2 T + T_o $$

Thus, both solutions are equivalent in terms of the duration $t_{send} - t_{obs}$.

On a side note, $t_{send} - t_{obs} > 2T$, which is why we execute 3 cycles when resetting a simulation spine. The action in Case 2 is sent $T_p$ earlier, but it is also based on an observation made $T_p$ earlier compared to Case 1.

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Upkie

Starting CAN cycles after sleep or after action #452

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Upkie

Starting CAN cycles after sleep or after action #452

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stephane-caron Oct 31, 2024 Maintainer

Case 1: plan . (can // sleep)

Case 2: (plan // can) . sleep

Comparison of the two

See also

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stephane-caron
Oct 31, 2024
Maintainer