Chapter 3 Polar method and obstacle avoidance
3.4 Polar algorithm
The key challenge of the supervisory control is to find an avoidable set that contains the infeasible set. To solve this problem, a polar algorithm is proposed. The infeasible set is represented by a bounded polytope containing the collision set as described in (3.9), then the polar algorithm solves for another polytope that contains the infeasible set and satisfies the boundary condition introduced in (3.12). The polar algorithm is applicable to a dynamic model in the following form
, ,
,
E G
x u d x u d , (3.15)
where , and are polytope; E and G are constant matrices. Note that there is no state dependent term in the state derivative. The dynamic model in Section 3.2 is simplified to this form by viewing all state dependent terms as disturbance, which is explained in detail in Section 3.5.2.
3.4.1 Polar of a polytope
In this section, the polar algorithm is introduced, which is built based on the dual property of polytope. A polytope P has two important elements: vertices and facets. For simplicity, only bounded polytopes with a finite number of vertices and bounding hyperplanes are considered. A bounded polytope n
P with the origin in its interior can be represented as a set of the convex combination of its vertices:
1, | , 0, i i i i i i v i Px x
, (3.16)where
vi P V. is the set of vertices of P . It can also be represented as an intersection of finitely many closed half spaces:
| 1
i T i H P x H x , (3.17)where
x H x | iT 1
are the bounding half spaces.
Hi X# denotes the set of linear functionalsthat corresponds to the half spaces:
T i i H x H x, (3.18) where #
| : fX f X linear is the algebraic dual of the vector space X . Since the algebraic dual of a Euclidean space is also a Euclidean space, the linear functionals in #
X are treated as vectors in the dual space and the set
Hi is denoted asP H . In addition, half-spaces and .polytopes can also be defined in #
X . The bounding hyperplane corresponding to a bounding half- space
x H x | iT 1
is defined as
x H x | iT 1
, (3.19) and the corresponding facet is defined as
| T 1
i i
F H P x H x . (3.20)
For polytopes, the normal vector of the facet F H
i is simply Hi, which simplifies the conditionin (3.14). The polar of Pis a polytope in the dual space #
X defined as
| x , T 1
P H P H x . (3.21)
Because of the convexity and linearity of polytopes, a simpler definition is
| vi P. , T i 1
P H V H v . (3.22)
The vertices of Pare mapped to the facets of P, and the facets of the Pare mapped to the vertices of P, as shown in the following example.
Figure 3.3 Example of polar of polytopes
For a bounded polytope Pwith the origin in its interior, the following hold:
(1) The polar of P , denoted asP, is a bounded polytope with the same dimension as P and containing the origin in the interior.
(2) The polar of P, denoted as, is the original polytope, i.e., P P.
(3) For anyHP, P is completely contained in the half-space
x H x | T 1
, i.e.,, , T 1
H P x P H x
. (3.23)
, , T 1
H P x P H x
. (3.24)
Because of the properties above, the polar concept provides a clear condition for polytope inclusion:
1 2 2. 1
P P P H P. (3.25)
This property is one of the building blocks of the polar algorithm. For more detailed properties of polar, see the textbook [73].
When finding an avoidable set that contains the infeasible set
In
X , the set inclusion condition is enforced by the following constraint:
. n
B I
P HX. (3.26)
3.4.2 Hyperplane orientation and boundary condition
The boundary condition of the avoidable set is interpreted as an orientation condition for the bounding hyperplanes. For each facet of a polytope P , suppose that the corresponding bounding hyperplane is H x T 1, the normal vector that points outwards from P is simply H .
For the dynamic system in (3.15), the avoidable set boundary condition in (3.14) becomes
. . , . . , , 0 H P H d u s t Eu Gd H (3.27)
In order to find all bounding hyperplane orientations that are valid for an avoidable set, input u is fixed first. For a bounding hyperplane H x T 1to satisfy the boundary condition of the avoidable set with a fixed input u, the following inequality must hold:
, T 0
d H Eu Gd
. (3.28)
Since is a polytope and the system dynamics is linear, (3.28) is simplified to checking only its vertices:
. , T 0
d V H Eu Gd
. (3.29)
Condition (3.29) defines a polytope in X# and it is easy to check that
| . , T 0
u Hs
P H d V Eu Gd H contains all the functionals corresponding to the bounding hyperplanes valid under input u. Define
. u H V s Hs u P P , (3.30)
this union may not be convex, but using the linearity of the dynamics, the following theorem is true:
Theorem 3.1: PHsis the maximal set of linear functionals that satisfies the boundary condition of an avoidable set for the dynamic system shown in (3.15).
See Appendix A for proof.
Recall that there are two requirements for the avoidable set: set inclusion and boundary conditions. The set inclusion requirement is simplified to selecting bounding hyperplanes from the polar of the infeasible set
In
X ; the boundary conditions requirement is simplified to selecting bounding hyperplanes from PHs. Therefore, it is natural to intersect the two sets. Define
H Hs In
P P X. (3.31)
This set may not be convex since it is the intersection of a convex polytope and a nonconvex union of polytopes. Further, define
B H
P Conv P , (3.32)
where Conv denotes the convex hull of a polytope. Then
PBpossess the following properties:Theorem 3.2: PB is an avoidable set that contains XIn.
Theorem 3.3: PB is the minimal avoidable set.
Theorem 3.4: If the origin is in the interior of
In
X , and XIn XIn , where c is a constant c
shifting vector, then PB PB wherec PB is the constructed avoidable set based onXIn .
The above claims mean that
B
P is the minimal avoidable set that satisfies both set inclusion condition and boundary condition, and it is invariant with respect to the change of origin position.
See Appendix B, Appendix C, and Appendix D for proofs.