# CtcEval: \(y_i=x(t_i)\)

\(y_i=x(t_i)\) is a constraint that links a value \(y_i\) to the evaluation of the trajectory \(x(\cdot)\) at time \(t_i\). A dedicated contractor \(\mathcal{C}_{\textrm{eval}}\) implements this constraint. One of its applications is the correction of the positions of a robot based on observations.

## Definition

Important

```
ctc.eval.contract(ti, yi, x, v)
```

```
ctc::eval.contract(ti, yi, x, v);
```

Prerequisite

The tubes \([x](\cdot)\) and \([v](\cdot)\) must share:

the same slicing (same sampling of time)

the same

*t*-domain \([t_0,t_f]\)the same dimension of \([y_i]\) in the vector case

## 1d theoretical presentation

The \(\mathcal{C}_{\textrm{eval}}\) contractor aims at intersecting a tube \([x](\cdot)\) by the envelope of all trajectories compliant with the bounded observation \([t_i]\times[y_i]\). In other words, \([x](\cdot)\) will be contracted by the tube of all \(x(\cdot)\in[x](\cdot)\) going through the box \([t_i]\times[y_i]\), as shown in Fig. 22. Some trajectories may partially cross the box at some point over \([t_i]\): the contractor must take into account the feasible propagations during the intersection process. To this end, the knowledge of the derivative \(\dot{x}(\cdot)\) is required to depict the evolution of \(x(\cdot)\) over \([t_i]\).

The contractor is proposed in the most generic way; the derivative \(\dot{x}(\cdot)\) is also bounded within a tube denoted \([v](\cdot)\), thus allowing the \([x](\cdot)\) contraction even if the derivative signal is uncertain.

The code leading to the contraction presented in this figure is:

```
dt = 0.01
tdomain = Interval(-math.pi, math.pi/2)
v = Tube(tdomain, dt, TFunction("cos(t)+[-0.1,0.1]")) # uncertain derivative (not displayed)
x = v.primitive() + Interval(-0.1, 0.1) # x is the primitive of v
# Bounded observation
ti = Interval(-0.5,0.3)
yi = Interval(0.3,1.1)
# Contraction
ctc_eval = CtcEval()
ctc_eval.contract(ti, yi, x, v)
# Note that we could use directly ctc.eval.contract(ti, yi, x, v)
```

```
double dt = 0.01;
Interval tdomain(-M_PI, M_PI/2.);
Tube v(tdomain, dt, TFunction("cos(t)+[-0.1,0.1]")); // uncertain derivative (not displayed)
Tube x = v.primitive() + Interval(-0.1, 0.1); // x is the primitive of v
// Bounded observation
Interval ti(-0.5,0.3);
Interval yi(0.3,1.1);
// Contraction
CtcEval ctc_eval;
ctc_eval.contract(ti, yi, x, v);
// Note that we could use directly ctc::eval.contract(ti, yi, x, v)
```

Restrict the temporal propagation (save computation time)

\(\mathcal{C}_{\textrm{eval}}\) may contract the tube \([x](\cdot)\) and in this case, the contraction will be temporally propagated forward/backward in time from the \([t_i]\) *t*-domain. In Fig. 22, we can see that the contraction occurs over \([-1.9,t_f]\). The `.enable_time_propag(false)`

method can be used to limit the contraction to the \([t_i]\) *t*-domain only. This is useful when dealing with several observations \([t_i]\times[z_i]\) on the same tube: it becomes faster to first perform all the *local* contractions over each \([t_i]\) and then smooth the tube only once with, for instance, the \(\mathcal{C}_{\frac{d}{dt}}\) contractor presented before.

For instance, we now consider three constraints on the tube:

```
ti = [Interval(-0.5,0.3), Interval(-0.6,0.8), Interval(-2.3,-2.2)]
yi = [Interval(0.3,1.1), Interval(-0.5,-0.4), Interval(-0.8,-0.7)]
```

```
Interval ti[3], yi[3];
ti[0] = Interval(-0.5,0.3); yi[0] = Interval(0.3,1.1);
ti[1] = Interval(-0.6,0.8); yi[1] = Interval(-0.5,-0.4);
ti[2] = Interval(-2.3,-2.2); yi[2] = Interval(-0.8,-0.7);
```

Then we use the contractor configured for the limited contraction:

```
ctc_eval.enable_time_propag(False)
for i in range (0,3):
ctc_eval.contract(ti[i], yi[i], x, v)
ctc.deriv.contract(x, v) # for smoothing the tube
for i in range (0,3): # for contracting the [ti]×[yi] boxes
ctc_eval.contract(ti[i], yi[i], x, v)
```

```
ctc_eval.enable_time_propag(false);
for(int i = 0 ; i < 3 ; i++)
ctc_eval.contract(ti[i], yi[i], x, v);
ctc::deriv.contract(x, v); // for smoothing the tube
for(int i = 0 ; i < 3 ; i++) // for contracting the [ti]×[yi] boxes
ctc_eval.contract(ti[i], yi[i], x, v);
```

The following animation presents the results before and after the \(\mathcal{C}_{\frac{d}{dt}}\) contraction:

Fixed point propagation

When dealing with several constraints on the same tube, a single application of \(\mathcal{C}_{\textrm{eval}}\) for each \([t_i]\times[y_i]\) may not provide optimal results. Indeed, \(\mathcal{C}_{\textrm{eval}}\) propagates an evaluation along the whole domain of \([x](\cdot)\) which may lead to new possible contractions. It is preferable to use an iterative method that applies all contractors indefinitely until they become ineffective on \([x](\cdot)\) and the \([t_i]\times[y_i]\)’s:

See also

The CN chapter for constraint propagation.

## 2d localization example

Contracting the tube

Let us come back to the Lissajous example of the previous page.

Assume now that we have no knowledge on \([\mathbf{x}](\cdot)\), except that the feasible trajectories go through the box \([\mathbf{b}]=[-0.73,-0.69]\times[0.64,0.68]\) at some time \(t\in[4.3,4.4]\).

The tube is contracted over \([t_0,t_f]\) with its uncertain derivative \([\mathbf{v}](\cdot)\) given by:

```
dt = 0.01
tdomain = Interval(0,5)
# No initial knowledge on [x](·)
x = TubeVector(tdomain, dt, 2) # initialization with [-∞,∞]×[-∞,∞]
# New values for the temporal evaluation of [x](·)
t = Interval(4.3,4.4)
b = IntervalVector([[-0.73,-0.69],[0.64,0.68]])
# Uncertain derivative of [x](·)
v = TubeVector(tdomain, dt, TFunction("(-2*sin(t) ; 2*cos(2*t))"))
v.inflate(0.01)
# Contraction
ctc_eval = CtcEval()
ctc_eval.contract(t, b, x, v)
# Note that in this case, no contraction is performed on [t] and [b]
# Note also that we could use directly ctc.eval.contract(t, b, x, v)
```

```
double dt = 0.01;
Interval tdomain(0.,5.);
// No initial knowledge on [x](·)
TubeVector x(tdomain, dt, 2); // initialization with [-∞,∞]×[-∞,∞]
// New values for the temporal evaluation of [x](·)
Interval t(4.3,4.4);
IntervalVector b({{-0.73,-0.69},{0.64,0.68}});
// Uncertain derivative of [x](·)
TubeVector v(tdomain, dt, TFunction("(-2*sin(t) ; 2*cos(2*t))"));
v.inflate(0.01);
// Contraction
CtcEval ctc_eval;
ctc_eval.contract(t, b, x, v);
// Note that in this case, no contraction is performed on [t] and [b]
// Note also that we could use directly ctc::eval.contract(t, b, x, v)
```

The obtained tube is blue painted on the figure, the contraction to keep the trajectories going through \([\mathbf{b}]\) (red box) over \([t]=[4.3,4.4]\) is propagated over the whole *t*-domain:

Contracting the evaluation box

Assume now that we know the actual trajectory to be bounded within the tube:

```
x = TubeVector(tdomain, dt, TFunction("(2*cos(t) ; sin(2*t))"))
x.inflate(0.05)
```

```
TubeVector x(tdomain, dt, TFunction("(2*cos(t) ; sin(2*t))"));
x.inflate(0.05);
```

The tube is blue painted on the figure:

The yellow robot depicts an unknown position \(\mathbf{x}\) in the box \([-1,0]\times[0.4,1.2]\) at an unknown \(t\in[t_0,t_f]\). The \(\mathcal{C}_{\textrm{eval}}\) can be used to evaluate the position time and reduce the uncertainty on the possible positions.

```
t = Interval(0,oo) # new initialization
b = IntervalVector([[-1,0],[0.4,1.2]]) # (blue box on the figure)
ctc_eval.contract(t, b, x)
# [t] estimated to [4.15, 4.54]
# [b] contracted to ([-1, -0.29] ; [0.4, 0.95]) (red on the figure)
```

```
t = Interval(); // new initialization
b = {{-1.,0.},{0.4,1.2}}; // (blue box on the figure)
ctc_eval.contract(t, b, x);
// [t] estimated to [4.15, 4.54]
// [b] contracted to ([-1, -0.29] ; [0.4, 0.95]) (red on the figure)
```