CtcDist: \(\sqrt{x^2+y^2}=d\)

The \(\mathcal{C}_{\textrm{dist}}\) contractor is mainly used in robotics to express the distance constraint \(d\) between two vectors \(\mathbf{a}\) and \(\mathbf{b}\). Its implementation currently stands on the \(\mathcal{C}_{\mathbf{f}}\) contractor previously presented.

Definition

Important

\[\sqrt{(a_1-b_1)^2+(a_2-b_2)^2}=d \longrightarrow \mathcal{C}_{\textrm{dist}}\big([\mathbf{a}],[\mathbf{b}],[d]\big)\]

PythonC++

ctc.dist.contract(a, b, d)

Optimality

This contractor is optimal.

Example

Suppose that we want to estimate the distance \(d\) between a vector \(\mathbf{x}=(0,0)^\intercal\) and several vectors \(\mathbf{b}^1\), \(\mathbf{b}^2\), \(\mathbf{b}^3\).

We define domains (intervals and boxes):

• \([d]=[7,8]\)

• \([\mathbf{x}]=[0,0]^2\)

• \([\mathbf{b}^1]=[1.5,2.5]\times[4,11]\)

• \([\mathbf{b}^2]=[3,4]\times[4,6.5]\)

• \([\mathbf{b}^3]=[5,7]\times[5.5,8]\)

PythonC++

d = Interval(7,8)
x = IntervalVector(2,Interval(0))
b1 = IntervalVector([[1.5,2.5],[4,11]])
b2 = IntervalVector([[3,4],[4,6.5]])
b3 = IntervalVector([[5,7],[5.5,8]])

Several calls to \(\mathcal{C}_{\textrm{dist}}\) will allow the contraction of both the \([\mathbf{b}^i]\) and \([d]\). Because domains are involved in several contractions, an iterative contraction loop is necessary in order to reach a consistency state (in our case, two iterations are sufficient):

PythonC++

ctc_dist = CtcDist()

for i in range(0,2): # iterative contractions
  ctc_dist.contract(x, b1, d)
  ctc_dist.contract(x, b2, d)
  ctc_dist.contract(x, b3, d)

# note that we could also use directly the ctc.dist object already available

Fig. 16 Illustration of several contracted boxes with the above ctc_dist contractor. The blue boxes have been contracted as well as the ring.

Technical documentation

See the C++ API documentation of this class.