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Drake
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Drake
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This page describes discrete systems modeled by difference equations (contrast to continuous systems modeled by ordinary differential equations) as well as considerations for implementing these systems in Drake.
The state space dynamics of a discrete system is:
(We're using LaTeX underscore notation for subscripts, where x_0 means x₀.)
Here n ∈ ℕ is the step number (typically starting at zero), x is the discrete-time state variable ("discrete time" refers to the countability of the elements of the sequence, x_0, x_1, ..., x_n and not the values that x can take), y is the desired output, and u is an external input. f(.) and g(.) are the update and output functions, respectively. Any of these quantities can be vector-valued. The subscript notation (e.g., x_0) is used to show that the state, input, and output result from a discrete process. We use square bracket notation, e.g. x[1] to designate particular elements of a vector-valued quantity (indexing from 0). Combined, x_1[3] would be the value of the fourth element of the x vector, evaluated at step n=1.
The following class implements in Drake the simple discrete system
which should generate the sequence S = 0 10 20 30 ... (that is, S_n = 10*n).
The following code fragment can be used to step this system forward:
The above yields the following output:
Purely-discrete systems produce values only intermittently. For example, the system above generates values only at integer values of n:
y_n
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30 | ●
| ┆ Figure 1. The discrete-valued system
20 | ● ┆ from the example above.
| ┆ ┆
10 | ● ┆ ┆
| ┆ ┆ ┆
0 ●----+----+----+--
0 1 2 3 n
+----+----+----+--
0 .02 .04 .06 t
Drake's simulator is for hybrid systems, that is, systems that advance through time, and evolve in time both continuously (flow) and discretely (jump). It is easy enough to use time to represent the discrete steps n, by the conversion t=n*h where h is a periodic sampling time. In Figure 1 we've shown the conversion to time used by the example above as a second horizontal axis, assuming h=0.02 seconds.
However, since Drake simulations advance through continuous time, it must be possible to obtain the values of all state variables and outputs at any time t, not just at discrete times. So the question arises: what is the value of y(t) for values of t in between the sample times shown above? The answer doesn't matter for the example above, but becomes significant when we mix continuous and discrete systems, since they are typically interdependent.
Sample-and-hold is the most common way to go from a discrete value to a continuous one. There are two equally-plausible ways to use sample-and-hold to fill in the gaps between the discrete sample times above, shown in Figure 2:
y(t) ●━ y(t)
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30 | ●━━━━○ 30 | ●━
| ┆ | ┆
20 | ●━━━━○ 20 | ●━━━━○
| ┆ | ┆
10 ●━━━━○ 10 | ●━━━━○
┆ | ┆
0 ○----+----+----+-- t 0 ●━━━━○----+----+-- t
0 .02 .04 .06 0 .02 .04 .06 (a) Sample at start of step. (b) Sample at end of step.
(Drake uses this method.) (Not used in Drake.) Figure 2: two ways to make a continuous function from
a discrete one, using sample-and-hold.
In the figure, the ○ markers show the function value at time t before the update function is invoked, while the ● markers show the value after the update. In (a), the ○ markers coincide with the original discrete values, while in (b), the ● markers do.
You might expect that 2(b) would be the most natural mapping from the discrete system to a continuous one. In practice, however, it is problematic for mixed discrete/continuous (hybrid) systems so Drake uses the mapping in 2(a). The advantage of 2(a) is that the hybrid update function x_{n+1} = f(t,n,x_n,u(t)) is invoked at time t=n*h, while in 2(b) it would be invoked at time t=(n+1)*h. That would make it difficult to coordinate discrete and continuous signals.
Drake's choice of 2(a) dictates what value a discrete quantity will have when evaluated at times between update times. In particular, consider a discrete variable x evaluated during a simulation from a publish, update, or derivative function at times t ∈ (n*h, (n+1)*h]. x will be seen to have value x(t) = x_{n+1} (not x_n). You can see that clearly by inspection of Figure 2(a).
A discrete system viewed in continuous time does not have a unique value at its sample times. In Figure 2, each pair of ○ and ● symbols shows two possible values at the same time. For a given sample time t, we use the notation x⁻(t) to denote the "pre-update" value of the state x, and x⁺(t) to denote the "post-update" value of x. So x⁻(t) is the value of x at time t before discrete variables are updated, and x⁺(t) the value of x at time t after they are updated. Thus if we have t = n*h as in the discussion above, then x⁻(t) = x_n and x⁺(t) = x_{n+1}. State-dependent computations are affected by the scheduling of these updates. For example, evaluating an input u(t) yields u⁻(t) before discrete updates, and u⁺(t) afterwards, meaning that the input evaluation is carried out using x⁻(t) or x⁺(t), respectively.
With those distinctions drawn, we can define Drake's state update behavior during a time step:
Publish events at time t see x⁻(t), so if a publish event handler evaluates an input it sees u⁻(t). This occurs at the end of a step, shown as ○ markers in Figure 2.Update events (of all kinds) at time t also see x⁻(t) and u⁻(t), and produce x⁺(t). This occurs at the start of the next step, shown as ● markers in Figure 2.Continuous update (numerical integration and time advancement) starts with x⁺(t). Input and derivative evaluations will occur repeatedly as the time and continuous state advance. Each evaluation will be performed using updated values for continuous states xc and time. However, the discrete variables (state partitions xd and xa) are held constant at their x⁺(t) values, that is, at xd⁺(t) and xa⁺(t).If you define periodic events starting at t=0 as we did in the example above, the first publish event occurs at the end of initialization, while the first discrete update event occurs at the beginning of the first step, followed by continuous time and state advancement.