尝试DS理论应用到自动驾驶地图众包更新。
地图特征变化判断
a mass function is applied to quantify the evidence of the existence.
existence state: existenct、non-existent、tenative、conflict
∃ ∄ Ω ϕ \exist \\ \not\exist \\ \Omega \\ \phi ∃∃Ωϕ
mass function: quantify the evidence of the existence.
- mass functions of the measurement
m a s s z t ( ∃ ) = λ m a s s z t ( ∄ ) = 0 m a s s z t ( ϕ ) = 0 m a s s z t ( Ω ) = 1 − λ mass_{z_t}( \exist ) = \lambda \\ mass_{z_t}( \not \exist ) = 0 \\ mass_{z_t}( \phi ) = 0 \\ mass_{z_t}( \Omega ) = 1-\lambda masszt(∃)=λmasszt(∃)=0masszt(ϕ)=0masszt(Ω)=1−λ
- mass functions of the non-measurement
m a s s z t ( ∃ ) = 0 m a s s z t ( ∄ ) = λ m a s s z t ( ϕ ) = 0 m a s s z t ( Ω ) = 1 − λ mass_{z_t}( \exist ) = 0 \\ mass_{z_t}( \not \exist ) = \lambda \\ mass_{z_t}( \phi ) = 0 \\ mass_{z_t}( \Omega ) = 1-\lambda masszt(∃)=0masszt(∃)=λmasszt(ϕ)=0masszt(Ω)=1−λ
Inference of the map feature existence based Dempster Combination Rule
- mass functions of map features and new map features
初始化使用第i个地图特征的先验置信度 λ H D \lambda_{HD} λHD
m a s s H D 0 { i } ( ∃ ) = λ H D m a s s H D 0 { i } ( ∄ ) = 0 m a s s H D 0 { i } ( ϕ ) = 0 m a s s H D 0 { i } ( Ω ) = 1 − λ H D mass_{HD_{0\{i\}}}( \exist ) = \lambda_{HD} \\ mass_{HD_{0\{i\}}}( \not \exist ) = 0 \\ mass_{HD_{0\{i\}}}( \phi ) = 0 \\ mass_{HD_{0\{i\}}}( \Omega ) = 1 - \lambda_{HD} massHD0{i}(∃)=λHDmassHD0{i}(∃)=0massHD0{i}(ϕ)=0massHD0{i}(Ω)=1−λHD
新增加的第j个地图特征,按下列式初始化
m a s s n e w 0 { j } ( ∃ ) = 0 m a s s n e w 0 { j } ( ∄ ) = 0 m a s s n e w 0 { j } ( ϕ ) = 0 m a s s n e w 0 { j } ( Ω ) = 1 mass_{new_{0\{j\}}}( \exist ) = 0 \\ mass_{new_{0\{j\}}}( \not \exist ) = 0 \\ mass_{new_{0\{j\}}}( \phi ) = 0 \\ mass_{new_{0\{j\}}}( \Omega ) = 1 \\ massnew0{j}(∃)=0massnew0{j}(∃)=0massnew0{j}(ϕ)=0massnew0{j}(Ω)=1
- Usd Dempster combination rule ⊕ \oplus ⊕ to accumulate the measurement existence m a s s z t mass_{z_t} massztto the each map feature existence at time t − 1 t-1 t−1,
m a s s H D t { i } = m a s s H D t − 1 { i } ⊕ m a s s z t m a s s n e w t { j } = m a s s n e w t − 1 { j } ⊕ m a s s z t mass_{HD_{t\{i\}}} = mass_{HD_{t-1\{i\}}}\oplus mass_{z_t} \\ mass_{new_{t\{j\}}} = mass_{new_{t-1\{j\}}}\oplus mass_{z_t} massHDt{i}=massHDt−1{i}⊕massztmassnewt{j}=massnewt−1{j}⊕masszt
其中,
m a s s 1 ⊕ 2 ( A ) = m a s s 1 ∩ 2 ( A ) 1 − m a s s 1 ∩ 2 ( ϕ ) , ∀ A ⊆ Ω , A ≠ ϕ m a s s 1 ⊕ 2 ( ϕ ) = 0 ∀ A ⊆ Ω , m a s s 1 ∩ 2 ( A ) = ∑ B ∩ C = A ∣ B , C ⊆ Ω m a s s 1 ( B ) m a s s 2 ( C ) mass_{1\oplus2}(A) = \frac{mass_{1\cap2}(A)}{1-mass_{1\cap2}(\phi)}, \forall A\subseteq\Omega,A\neq\phi \\ mass_{1\oplus2}(\phi) = 0 \\ \forall A\subseteq\Omega, mass_{1\cap2}(A) = \sum_{B\cap C=A|B,C\subseteq\Omega}mass_1(B)mass_2(C) mass1⊕2(A)=1−mass1∩2(ϕ)mass1∩2(A),∀A⊆Ω,A=ϕmass1⊕2(ϕ)=0∀A⊆Ω,mass1∩2(A)=B∩C=A∣B,C⊆Ω∑mass1(B)mass2(C)
注:公式 m a s s 1 ⊕ 2 ( A ) mass_{1\oplus2}(A) mass1⊕2(A)即 m a s s 1 ( A ) ⊕ m a s s 2 ( A ) mass_{1}(A)\oplus mass_2(A) mass1(A)⊕mass2(A)
求和条件中的 ∣ | ∣为并列含义, Ω \Omega Ω为超集 2 X 2^X 2X,
∃ ∩ ∃ = ∃ ∃ ∩ Ω = ∃ ∃ ∩ ∄ = ∅ ∅ ∩ ∃ = ∅ ∅ ∩ ∄ = ∅ ∅ ∩ Ω = ∅ ∅ ∩ ∅ = ∅ \exist \cap \exist = \exist\\ \exist \cap \Omega = \exist\\ \exist \cap \not\exist = \emptyset \\ \emptyset \cap \exist = \emptyset \\ \emptyset \cap \not \exist = \emptyset \\ \emptyset \cap \Omega = \emptyset \\ \emptyset \cap \emptyset = \emptyset \\ ∃∩∃=∃∃∩Ω=∃∃∩∃=∅∅∩∃=∅∅∩∃=∅∅∩Ω=∅∅∩∅=∅
集合运算满足交换律。