This article proposes an improved A* algorithm aimed at improving the logistics path quality of automated guided vehicles (AGVs) in digital production workshops, solving the problems of excessive path turns and long transportation time. The traditional A* algorithm is improved internally and externally. In the internal improvement process, we propose an improved node search method within the A* algorithm to avoid generating invalid paths; offer a heuristic function which uses diagonal distance instead of traditional heuristic functions to reduce the number of turns in the path; and add turning weights in the A* algorithm formula, further reducing the number of turns in the path and reducing the number of node searches. In the process of external improvement, the output path of the internally improved A* algorithm is further optimized externally by the improved forward search optimization algorithm and the Bessel curve method, which reduces path length and turns and creates a path with fewer turns and a shorter distance. The experimental results demonstrate that the internally modified A* algorithm suggested in this research performs better when compared to six conventional path planning methods. Based on the internally improved A* algorithm path, the full improved A* algorithm reduces the turning angle by approximately 69% and shortens the path by approximately 10%; based on the simulation results, the improved A* algorithm in this paper can reduce the running time of AGV and improve the logistics efficiency in the workshop. Specifically, the walking time of AGV on the improved A* algorithm path is reduced by 12s compared to the traditional A* algorithm.
Citation: Na Liu, Chiyue Ma, Zihang Hu, Pengfei Guo, Yun Ge, Min Tian. Workshop AGV path planning based on improved A* algorithm[J]. Mathematical Biosciences and Engineering, 2024, 21(2): 2137-2162. doi: 10.3934/mbe.2024094
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This article proposes an improved A* algorithm aimed at improving the logistics path quality of automated guided vehicles (AGVs) in digital production workshops, solving the problems of excessive path turns and long transportation time. The traditional A* algorithm is improved internally and externally. In the internal improvement process, we propose an improved node search method within the A* algorithm to avoid generating invalid paths; offer a heuristic function which uses diagonal distance instead of traditional heuristic functions to reduce the number of turns in the path; and add turning weights in the A* algorithm formula, further reducing the number of turns in the path and reducing the number of node searches. In the process of external improvement, the output path of the internally improved A* algorithm is further optimized externally by the improved forward search optimization algorithm and the Bessel curve method, which reduces path length and turns and creates a path with fewer turns and a shorter distance. The experimental results demonstrate that the internally modified A* algorithm suggested in this research performs better when compared to six conventional path planning methods. Based on the internally improved A* algorithm path, the full improved A* algorithm reduces the turning angle by approximately 69% and shortens the path by approximately 10%; based on the simulation results, the improved A* algorithm in this paper can reduce the running time of AGV and improve the logistics efficiency in the workshop. Specifically, the walking time of AGV on the improved A* algorithm path is reduced by 12s compared to the traditional A* algorithm.
Throughout the paper, we work over an algebraically closed field
Σk=Σk(C,L)⊆Pr |
of
Assume that
σk+1:Ck×C⟶Ck+1 |
be the morphism sending
Ek+1,L:=σk+1,∗p∗L, |
which is a locally free sheaf of rank
Bk(L):=P(Ek+1,L) |
equipped with the natural projection
H0(Bk(L),OBk(L)(1))=H0(Ck+1,Ek+1,)=H0(C,L), |
and therefore, the complete linear system
βk:Bk(L)⟶Pr=P(H0(C,L)). |
The
It is clear that there are natural inclusions
C=Σ0⊆Σ1⊆⋯⊆Σk−1⊆Σk⊆Pr. |
The preimage of
Theorem 1.1. Let
To prove the theorem, we utilize several line bundles defined on symmetric products of the curve. Let us recall the definitions here and refer the reader to [2] for further details. Let
Ck+1=C×⋯×C⏟k+1times |
be the
Ak+1,L:=Tk+1(L)(−2δk+1) |
be a line bundle on
The main ingredient in the proof of Theorem 1.1 is to study the positivity of the line bundle
Proposition 1.2. Let
In particular, if
In this section, we prove Theorem 1.1. We begin with showing Proposition 1.2.
Proof of Proposition 1.2. We proceed by induction on
Assume that
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
is surjective. We can choose a point
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
where all rows and columns are short exact sequences. By tensoring with
rz,k+1,L:H0(Ck+1,Ak+1,L)⟶H0(z,Ak+1,L|z) |
in which we use the fact that
Since
Lemma 2.1. Let
Proof. Note that
B′/A′⊗A′A′/m′q=B′/(m′qB′+A′)=B′/(m′p+A′)=0. |
By Nakayama lemma, we obtain
We keep using the notations used in the introduction. Recall that
αk,1:Bk−1(L)×C⟶Bk(L). |
To see it in details, we refer to [1,p.432,line –5]. We define the relative secant variety
Proposition 2.2. ([2,Proposition 3.15,Theorem 5.2,and Proposition 5.13]) Recall the situation described in the diagram
αk,1:Bk−1(L)×C⟶Bk(L). |
Let
1.
2.
3.
As a direct consequence of the above proposition, we have an identification
H0(Ck+1,Ak+1,L)=H0(Σk,IΣk−1|Σk(k+1)). |
We are now ready to give the proof of Theorem 1.1.
Proof of Theorem 1.1. Let
b:˜Σk:=BlΣk−1Σk⟶Σk |
be the blowup of
b:˜Σk:=BlΣk−1Σk⟶Σk |
We shall show that
Write
γ:˜Σk⟶P(V). |
On the other hand, one has an identification
ψ:Ck+1⟶P(V). |
Also note that
ψ:Ck+1⟶P(V). |
Take an arbitrary closed point
α−1(x)⊆π−1k(x″)∩β−1k(x′). |
However, the restriction of the morphism
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