Abstract

A set of algorithms is proposed for determining the first-order derivatives of a skew ray with respect to all the independent variables of an aspherical boundary surface in a general 3D optical system. Ellipsoidal, paraboloidal, hyperboloidal, and cylindrical boundary surfaces are given as examples. The proposed method has important applications in the analysis of aspherical surfaces and enables a detailed understanding of a wide variety of optical effects, including reflection and refraction, the modulation transfer function, and aberrations.

© 2012 Optical Society of America

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References

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  1. B. D. Stone, “Determination of initial ray configurations for asymmetric systems,” J. Opt. Soc. Am. A 14, 3415–3429 (1997).
    [CrossRef]
  2. R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
    [CrossRef]
  3. T. B. Andersen, “Optical aberration functions: derivatives with respect to axial distances for symmetrical systems,” Appl. Opt. 21, 1817–1823 (1982).
    [CrossRef]
  4. T. B. Andersen, “Optical aberration functions: chromatic aberrations and derivatives with respect to refractive indices for symmetrical systems,” Appl. Opt. 21, 4040–4044 (1982).
    [CrossRef]
  5. T. B. Andersen, “Optical aberration functions: derivatives with respect to surface parameters for symmetrical systems,” Appl. Opt. 24, 1122–1129 (1985).
    [CrossRef]
  6. D. P. Feder, “Calculation of an optical merit function and its derivatives with respect to the system parameters,” J. Opt. Soc. Am. 47, 913–925 (1957).
    [CrossRef]
  7. D. P. Feder, “Differentiation of ray-tracing equations with respect to constructional parameters of rotationally symmetric systems,” J. Opt. Soc. Am. 58, 1494–1505 (1968).
    [CrossRef]
  8. O. Stavroudis, “A simpler derivation of the formulas for generalized ray tracing,” J. Opt. Soc. Am. 66, 1330–1333 (1976).
    [CrossRef]
  9. P. D. Lin and C. Y. Tsai, “ First-order gradients of skew rays of axis-symmetrical optical systems,“ J. Opt. Soc. Am. A 24, 776–784 (2007).
    [CrossRef]
  10. P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
    [CrossRef]
  11. W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).
  12. W. A. Allen and J. R. Snyder, “ Ray tracing through uncentered and aspherical surfaces,” J. Opt. Soc. Am. A 42, 243–249(1952).
    [CrossRef]
  13. W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).
  14. C. S. Liu and P. D. Lin, “Computational method for deriving the geometrical point spread function of an optical system,” Appl. Opt. 49, 126–136, (2010).
    [CrossRef]
  15. W. Wu and P. D. Lin, “Numerical approach for computing the Jacobian matrix between boundary variable vector and system variable vector for optical systems containing prisms,” J. Opt. Soc. Am. A 28, 747–758 (2011).
    [CrossRef]

2011 (1)

2010 (1)

2008 (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

2007 (1)

1999 (1)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

1997 (1)

1988 (1)

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

1985 (1)

1982 (2)

1976 (1)

1968 (1)

1957 (1)

1952 (1)

W. A. Allen and J. R. Snyder, “ Ray tracing through uncentered and aspherical surfaces,” J. Opt. Soc. Am. A 42, 243–249(1952).
[CrossRef]

Allen, W. A.

W. A. Allen and J. R. Snyder, “ Ray tracing through uncentered and aspherical surfaces,” J. Opt. Soc. Am. A 42, 243–249(1952).
[CrossRef]

Andersen, T. B.

Feder, D. P.

Kross, J.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Lin, P. D.

Liu, C. S.

Oertmann, W.

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

Shi, R.

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).

Snyder, J. R.

W. A. Allen and J. R. Snyder, “ Ray tracing through uncentered and aspherical surfaces,” J. Opt. Soc. Am. A 42, 243–249(1952).
[CrossRef]

Stavroudis, O.

Stone, B. D.

Tsai, C. Y.

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

P. D. Lin and C. Y. Tsai, “ First-order gradients of skew rays of axis-symmetrical optical systems,“ J. Opt. Soc. Am. A 24, 776–784 (2007).
[CrossRef]

Wu, W.

Appl. Opt. (4)

Appl. Phys. B (1)

P. D. Lin and C. Y. Tsai, “General method for determining the first order gradients of skew-rays of optical systems with non-coplanar optical axes,” Appl. Phys. B 91, 621–628 (2008).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (4)

Proc. SPIE (2)

R. Shi and J. Kross, “Differential ray tracing for optical design,” Proc. SPIE 3737, 149–160 (1999).
[CrossRef]

W. Oertmann, “Differential ray tracing formulae; applications especially to aspheric optical systems,” Proc. SPIE 1013, 20–26 (1988).

Other (1)

W. J. Smith, Modern Optical Engineering, 3rd ed. (Edmund Industrial Optics, 2001).

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Figures (10)

Fig. 1.
Fig. 1.

The generating curve of an aspherical boundary surface.

Fig. 2.
Fig. 2.

Ray tracing at an aspherical boundary surface.

Fig. 3.
Fig. 3.

Schematic representation of the unit directional vector ¯ 0 of a source ray.

Fig. 4.
Fig. 4.

Change in refracted ray [ Δ P ¯ i Δ ¯ i ] T as result of change in incidence point Δ P ¯ i 1 on previous boundary surface.

Fig. 5.
Fig. 5.

Change in refracted ray [ Δ P ¯ i Δ ¯ i ] T as result of change in unit directional vector Δ ¯ i 1 at previous boundary surface.

Fig. 6.
Fig. 6.

Change in refracted/reflected ray Δ R i ¯ = [ Δ P ¯ i Δ ¯ i ] T as result of change in boundary variables, Δ X ¯ i .

Fig. 7.
Fig. 7.

Computation of change in refracted/reflected ray Δ R ¯ i = [ Δ P ¯ i Δ ¯ i ] T via summation of matrices M ¯ i Δ R ¯ i 1 and S ¯ i Δ X ¯ i .

Fig. 8.
Fig. 8.

An optical element is a block of optical material having constant refractive index ξ e j of medium. Each element j contains L j number of boundary surfaces labeled from i = m j L j + 1 to i = m j .

Fig. 9.
Fig. 9.

Flowchart showing solution procedure for computing change in exit ray at j th element, i.e., [ Δ P ¯ e j Δ ¯ e j ] T .

Fig. 10.
Fig. 10.

Computation of change in source ray [ Δ P ¯ 0 Δ ¯ 0 ] T via matrix multiplication of M ¯ 0 and Δ X ¯ e 0 .

Equations (119)

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q ¯ i i = [ 0 y i ( β i ) z i ( β i ) 1 ] ( 0 z i ( β i ) ) .
e i ¯ i = [ e i x e i y e i z 0 ] = s i y i 2 + z i 2 [ 0 z i y i 0 ] ,
r ¯ i i = y rot ( α i ) q ¯ i i = [ C α i 0 S α i 0 0 1 0 0 S α i 0 C α i 0 0 0 0 1 ] [ 0 y i ( β i ) z i ( β i ) 1 ] = [ z i ( β i ) S α i y i ( β i ) z i ( β i ) C α i 1 ] ,
n ¯ i i = y rot ( α i ) e ¯ i i = [ C α i 0 S α i 0 0 1 0 0 S α i 0 C α i 0 0 0 0 1 ] [ e i x e i y e i z 0 ] = s i y i 2 + z i 2 [ y i S α i z i y i C α i 0 ] ,
A i 0 = tran ( t i x , 0 , 0 ) tran ( 0 , t i y , 0 ) tran ( 0 , 0 , t i z ) z rot ( ω i z ) y rot ( ω i y ) x rot ( ω i x ) = [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] = [ C ω i z C ω i y C ω i z S ω i y S ω i x S ω i z C ω i x C ω i z S ω i y C ω i x + S ω i z S ω i x t i x S ω i z C ω i y S ω i z S ω i y S ω i x + C ω i z C ω i x S ω i z S ω i y C ω i x C ω i z S ω i x t i y S ω i y C ω i y S ω i x C ω i y C ω i x t i z 0 0 0 1 ] ,
n ¯ i = [ n i x n i y n i z 0 ] = A i 0 n ¯ i i = s i y i 2 + z i 2 [ I i x J i x K i x t i x I i y J i y K i y t i y I i z J i z K i z t i z 0 0 0 1 ] [ y i S α i z i y i C α i 0 ] = s i y i 2 + z i 2 [ I i x y i S α i J i x z i + K i x y i C α i I i y y i S α i J i y z i + K i y y i C α i I i z y i S α i J i z z i + K i z y i C α i 0 ] .
P ¯ i = [ P i x P i y P i z 1 ] = [ P i 1 x + i 1 x λ i P i 1 y + i 1 y λ i P i 1 z + i 1 z λ i 1 ] .
P ¯ i i = A 0 i P ¯ i = ( A i 0 ) 1 P ¯ i = [ σ i ρ i τ i 1 ] = r i ¯ i = [ z i ( β i ) S α i y i ( β i ) z i ( β i ) C α i 1 ] ,
σ i = I i x ( P i 1 x + i 1 x λ i ) + I i y ( P i 1 y + i 1 y λ i ) + I i z ( P i 1 z + i 1 z λ i ) ( I i x t i x + I i y t i y + I i z t i z ) = I i x P i x + I i y P i y + I i z P i z ( I i x t i x + I i y t i y + I i z t i z ) ,
ρ i = J i x ( P i 1 x + i 1 x λ i ) + J i y ( P i 1 y + i 1 y λ i ) + J i z ( P i 1 z + i 1 z λ i ) ( J i x t i x + J i y t i y + J i z t i z ) = J i x P i x + J i y P i y + J i z P i z ( J i x t i x + J i y t i y + J i z t i z ) ,
τ i = K i x ( P i 1 x + i 1 x λ i ) + K i y ( P i 1 y + i 1 y λ i ) + K i z ( P i 1 z + i 1 z λ i ) ( K i x t i x + K i y t i y + K i z t i z ) = K i x P i x + K i y P i y + K i z P i z ( K i x t i x + K i y t i y + K i z t i z ) .
{ α i = atan 2 σ i , τ i ) when z i ( β i ) > 0 α i = 0 when z i ( β i ) = 0 ,
z i ( β i ) 2 = σ i 2 + τ i 2 ,
y i ( β i ) = ρ i .
C θ i = ¯ i 1 n ¯ i = ( i 1 x n i x + i 1 y n i y + i 1 z n i z ) = s i y i 2 + z i 2 [ i 1 x ( I i x y i S α i J i x z i + K i x y i C α i ) + i 1 y ( I i y y i S α i J i y z i + K i y y i C α i ) + i 1 z ( I i z y i S α i J i z z i + K i z y i C α i ) ] .
S θ ̲ i = ξ i 1 ξ i S θ i = N i S θ i ,
¯ i = [ i x i y i z 0 ] = [ n i x 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 x + n i x C θ i ) n i y 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 y + n i y C θ i ) n i z 1 N i 2 + ( N i C θ i ) 2 + N i ( i 1 z + n i z C θ i ) 0 ] ,
¯ i = [ i x i y i z 0 ] = [ i 1 x + 2 n i x C θ i i 1 y + 2 n i y C θ i i 1 z + 2 n i z C θ i 0 ] ,
X ¯ i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i coef ¯ i ] T ,
X ¯ e 0 = [ P 0 x P 0 y P 0 z α 0 β 0 ] T .
P ¯ i R ¯ i 1 = [ ¯ i P ¯ i 1 ¯ i ¯ i 1 ] = [ P i x / R ¯ i 1 P i y / R ¯ i 1 P i z / R ¯ i 1 ] = [ 1 0 0 λ i 0 0 0 1 0 0 λ i 0 0 0 1 0 0 λ i ] + [ i 1 x i 1 y i 1 z ] λ i R ¯ i 1 .
λ i R ¯ i = [ λ i , 1 λ i , 2 λ i , 3 λ i , 4 λ i , 5 λ i , 6 ] .
¯ i R ¯ i 1 = [ ¯ i P ¯ i 1 ¯ i ¯ i 1 ] = [ i x / R ¯ i 1 i y / R ¯ i 1 i z / R ¯ i 1 ] = ( N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 + N i ) [ n i x n i y n i z ] C θ i R ¯ i 1 + N i [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] + ( 1 N i 2 + ( N i C θ i ) 2 + N i C θ i ) [ n i x / R ¯ i 1 n i y / R ¯ i 1 n i z / R ¯ i 1 ] ,
C θ i R ¯ i 1 = [ C θ i , 1 C θ i , 2 C θ i , 3 C θ i , 4 C θ i , 5 C θ i , 6 ]
C θ i , 1 = ( i 1 x n i x , 1 + i 1 y n i y , 1 + i 1 z n i z , 1 ) ,
C θ i , 2 = ( i 1 x n i x , 2 + i 1 y n i y , 2 + i 1 z n i z , 2 ) ,
C θ i , 3 = ( i 1 x n i x , 3 + i 1 y n i y , 3 + i 1 z n i z , 3 ) ,
C θ i , 4 = ( i 1 x n i x , 4 + i 1 y n i y , 4 + i 1 z n i z , 4 ) n i x ,
C θ i , 5 = ( i 1 x n i x , 5 + i 1 y n i y , 5 + i 1 z n i z , 5 ) n i y ,
C θ i , 6 = ( i 1 x n i x , 6 + i 1 y n i y , 6 + i 1 z n i z , 6 ) n i z .
[ n i x / R ¯ i 1 n i y / R ¯ i 1 n i z / R ¯ i 1 ] = [ n i x , 1 n i x , 2 n i x , 3 n i x , 4 n i x , 5 n i x , 6 n i y , 1 n i y , 2 n i y , 3 n i y , 4 n i y , 5 n i y , 6 n i z , 1 n i z , 2 n i z , 3 n i z , 4 n i z , 5 n i z , 6 ] = s i y i 2 + z i 2 [ I i x J i x K i x I i y J i y K i y I i z J i z K i z ] [ y i C α i 0 y i S α i ] α i R ¯ i 1 + s i y i 2 + z i 2 [ I i x J i x K i x I i y J i y K i y I i z J i z K i z ] [ y i S α i z i y i C α i ] β i R ¯ i 1 s i ( y i y i + z i z i ) ( y i 2 + z i 2 ) y i 2 + z i 2 [ I i x J i x K i x I i y J i y K i y I i z J i z K i z ] [ y i S α i z i y i C α i ] β i R ¯ i 1 .
α i R ¯ i 1 = 1 τ i 2 + σ i 2 ( τ i σ i R ¯ i 1 σ i τ i R ¯ i 1 ) ,
σ i R ¯ i 1 = [ σ i , 1 σ i , 2 σ i , 3 σ i , 4 σ i , 5 σ i , 6 ] ,
σ i , 1 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 1 + I i x ,
σ i , 2 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 2 + I i y ,
σ i , 3 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 3 + I i z ,
σ i , 4 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 4 + I i x λ i ,
σ i , 5 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 5 + I i y λ i ,
σ i , 6 = ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i , 6 + I i z λ i ,
τ i R ¯ i 1 = [ τ i , 1 τ i , 2 τ i , 3 τ i , 4 τ i , 5 τ i , 6 ] ,
τ i , 1 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 1 + K i x ,
τ i , 2 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 2 + K i y ,
τ i , 3 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 3 + K i z ,
τ i , 4 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 4 + K i x λ i ,
τ i , 5 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 5 + K i y λ i ,
τ i , 6 = ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i , 6 + K i z λ i .
β i R ¯ i 1 = [ β i , 1 β i , 2 β i , 3 β i , 4 β i , 5 β i , 6 ] = 1 y i ρ i R ¯ i 1 = 1 y i [ ρ i , 1 ρ i , 2 ρ i , 3 ρ i , 4 ρ i , 5 ρ i , 6 ] ,
ρ i , 1 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 1 + J i x ,
ρ i , 2 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 2 + J i y ,
ρ i , 3 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 3 + J i z ,
ρ i , 4 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 4 + J i x λ i ,
ρ i , 5 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 5 + J i y λ i ,
ρ i , 6 = ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i , 6 + J i z λ i .
¯ i R ¯ i 1 = [ ¯ i P ¯ i 1 ¯ i ¯ i 1 ] = [ i x / R ¯ i 1 i y / R ¯ i 1 i z / R ¯ i 1 ] = 2 C θ i [ n i x / R ¯ i 1 n i y / R ¯ i 1 n i z / R ¯ i 1 ] + [ 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 ] + [ 2 n i x 2 n i y 2 n i z ] C θ i R ¯ i 1 ,
R ¯ i R ¯ i 1 = M ¯ i = [ P ¯ i P ¯ i 1 P ¯ i ¯ i 1 ¯ i P ¯ i 1 ¯ i ¯ i 1 ] .
Q ¯ i = [ I i x I i y I i z J i x J i y J i z K i x K i y K i z t i x t i y t i z ξ i 1 ξ i coef ¯ i ] T
S ¯ i = R ¯ i X ¯ i = [ P ¯ i X ¯ i ¯ i X ¯ i ] = [ P ¯ i Q ¯ i ¯ i Q ¯ i ] 6 × ( 14 + q ) [ Q ¯ i X ¯ i ] ( 14 + q ) × ( 8 + q ) ,
P ¯ i Q ¯ i = [ P i x / Q ¯ i P i y / Q ¯ i P i z / Q ¯ i ] = [ i 1 x i 1 y i 1 z ] λ i Q ¯ i .
¯ i Q ¯ i = [ i x / Q ¯ i i y / Q ¯ i i z / Q ¯ i ] = N i 2 C θ i 1 N i 2 + ( N i C θ i ) 2 [ n i x n i y n i z ] C θ i Q ¯ i + ( N i N i C θ i 2 ) 1 N i 2 + ( N i C θ i ) 2 [ n i x n i y n i z ] N i Q ¯ i 1 N i 2 + ( N i C θ i ) 2 [ n i x / Q ¯ i n i y / Q ¯ i n i z / Q ¯ i ] + [ i 1 x i 1 y i 1 z ] N i Q ¯ i + C θ i [ n i x n i y n i z ] N i Q ¯ i + N i C θ i [ n i x / Q ¯ i n i y / Q ¯ i n i z / Q ¯ i ] + N i [ n i x n i y n i z ] C θ i Q ¯ i .
C θ i Q ¯ i = ( i 1 x n i x Q ¯ i + i 1 y n i y Q ¯ i + i 1 z n i z Q ¯ i ) ,
N i Q ¯ i = [ 0 ¯ 1 × 12 1 ξ i N i ξ i 0 ¯ 1 × q ] .
n i x Q ¯ i = s i y i 2 + z i 2 [ y i S α i 0 0 z i 0 0 y i C α i 0 0 0 ¯ 1 × ( 5 + q ) ] + s i y i 2 + z i 2 ( I i x y i C α i K i x y i S α i ) α i Q ¯ i + s i y i 2 + z i 2 ( I i x y i S α i J i x z i + K i x y i C α i ) β i Q ¯ i s i ( y i y i + z i z i ) ( y i 2 + z i 2 ) y i 2 + z i 2 ( I i x y i S α i J i x z i + K i x y i C α i ) β i Q ¯ i ,
n i y Q ¯ i = s i y i 2 + z i 2 [ 0 y i S α i 0 0 z i 0 0 y i C α i 0 0 ¯ 1 × ( 5 + q ) ] + s i y i 2 + z i 2 ( I i y y i C α i K i y y i S α i ) α i Q ¯ i + s i y i 2 + z i 2 ( I i y y i S α i J i y z i + K i y y i C α i ) β i Q ¯ i s i ( y i y i + z i z i ) ( y i 2 + z i 2 ) y i 2 + z i 2 ( I i y y i S α i J i y z i + K i y y i C α i ) β i Q ¯ i ,
n i z Q ¯ i = s i y i 2 + z i 2 [ 0 0 y i S α i 0 0 z i 0 0 y i C α i 0 ¯ 1 × ( 5 + q ) ] + s i y i 2 + z i 2 ( I i z y i C α i K i z y i S α i ) α i Q ¯ i + s i y i 2 + z i 2 ( I i z y i S α i J i z z i + K i z y i C α i ) β i Q ¯ i s i ( y i y i + z i z i ) ( y i 2 + z i 2 ) y i 2 + z i 2 ( I i z y i S α i J i z z i + K i z y i C α i ) β i Q ¯ i .
α i Q ¯ i = 1 τ i 2 + σ i 2 ( τ i σ i Q ¯ i σ i τ i Q ¯ i ) .
σ i Q ¯ i = [ P i x t i x P i y t i y P i z t i z 0 ¯ 1 × 6 I i x I i y I i z 0 0 0 ¯ 1 × q ] + ( I i x i 1 x + I i y i 1 y + I i z i 1 z ) λ i Q ¯ i ,
τ i Q ¯ i = [ 0 ¯ 1 x 6 P i x t i x P i y t i y P i z t i z K i x K i y K i z 0 0 0 ¯ 1 x q ] + ( K i x i 1 x + K i y i 1 y + K i z i 1 z ) λ i Q ¯ i .
β i Q ¯ i = 1 y i ρ i Q ¯ i = 1 y i [ 0 ¯ 1 x 3 P i x t i x P i y t i y P i z t i z 0 ¯ 1 x 3 J i x J i y J i z 0 0 0 ¯ 1 x q ] + 1 y i ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) λ i Q ¯ i .
¯ i Q ¯ i = [ i x / Q ¯ i i y / Q ¯ i i z / Q ¯ i ] = [ 2 n i x 2 n i y 2 n i z ] C θ i Q ¯ i + 2 C θ i [ n i x / Q ¯ i n i y / Q ¯ i n i z / Q ¯ i ] .
Q ¯ i X ¯ i = [ Q ¯ i t i x Q ¯ i t i y Q ¯ i t i z Q ¯ i ω i x Q ¯ i ω i y Q ¯ i ω i z Q ¯ i ξ i 1 Q ¯ i ξ i Q ¯ i coef ¯ i ] ( 14 + q ) × ( 8 + q )
Q ¯ i t i x = [ 0 ¯ 1 × 9 1 0 0 0 ¯ 1 × ( 2 + q ) ] T ,
Q ¯ i t i y = [ 0 ¯ 1 × 9 0 1 0 0 ¯ 1 × ( 2 + q ) ] T ,
Q ¯ i t i z = [ 0 ¯ 1 × 9 0 0 1 0 ¯ 1 × ( 2 + q ) ] T ,
Q ¯ i ω i x = [ 0 ¯ 1 x 3 C ω i z S ω i y C ω i x + S ω i z S ω i x S ω i z S ω i y C ω i x C ω i z S ω i x C ω i y C ω i x C ω i z S ω i y S ω i x + S ω i z C ω i x S ω i z S ω i y S ω i x C ω i z C ω i x C ω i y S ω i x 0 ¯ 1 x ( 5 + q ) ] T ,
Q ¯ i ω i y = [ C ω i z S ω i y S ω i z S ω i y C ω i y C ω i z C ω i y S ω i x S ω i z C ω i y S ω i x S ω i y S ω i x C ω i z C ω i y C ω i x S ω i z C ω i y C ω i x S ω i y C ω i x 0 ¯ 1 × ( 5 + q ) ] T ,
Q ¯ i ω i z = [ S ω i z C ω i y C ω i z C ω i y 0 S ω i z S ω i y S ω i x C ω i z C ω i x C ω i z S ω i y S ω i x S ω i z C ω i x 0 S ω i z S ω i y C ω i x + C ω i z S ω i x C ω i z S ω i y C ω i x + S ω i z S ω i x 0 0 ¯ 1 × ( 5 + q ) ] T ,
Q ¯ i ξ i 1 = [ 0 ¯ 1 × 12 1 0 0 ¯ 1 × q ] T ,
Q ¯ i ξ i = [ 0 ¯ 1 × 12 0 1 0 ¯ 1 × q ] T ,
Q ¯ i coef ¯ i = [ 0 ¯ q × 12 0 ¯ q × 1 0 ¯ q × 1 I ¯ q × q ] T .
[ Δ P ¯ i Δ ¯ i ] = S ¯ i Δ X ¯ i .
[ Δ P ¯ i Δ ¯ i ] = [ P ¯ i P ¯ i 1 P ¯ i ¯ i 1 ¯ i P ¯ i 1 ¯ i ¯ i 1 ] [ Δ P ¯ i 1 Δ ¯ i 1 ] + [ P ¯ i X ¯ i ̲ i X ¯ i ] Δ X ¯ i = M ¯ i [ Δ P ¯ i 1 Δ ¯ i 1 ] + S ¯ i Δ X ¯ i .
[ Δ P ¯ i Δ ¯ i ] = M ¯ i [ Δ P ¯ i 1 Δ ¯ i 1 ] + S ¯ i X ¯ i X ¯ sys Δ X ¯ sys .
[ Δ P ¯ e j Δ ¯ e j ] = [ Δ P ¯ m j Δ ¯ m j ] = M ¯ m j [ Δ P ¯ m j 1 Δ ¯ m j 1 ] + S ¯ m j X ¯ m j X ¯ sys Δ X ¯ sys = M ¯ m j M ¯ m j 1 [ Δ P ¯ m j 2 Δ ¯ m j 2 ] + M ¯ m j S ¯ m j 1 X ¯ m j 1 X ¯ sys Δ X ¯ sys + S ¯ m j X ¯ m j X ¯ sys Δ X ¯ sys = = M ¯ m j M ¯ m j 1 M ¯ m j L j + 2 M ¯ m j L j + 1 [ Δ P ¯ e j 1 Δ ¯ e j 1 ] + M ¯ m j M ¯ m j 1 M ¯ m j L j + 2 S ¯ m j L j + 1 X ¯ m j L j + 1 X ¯ sys Δ X ¯ sys + + M ¯ m j S ¯ m j 1 X ¯ m j 1 X ¯ sys Δ X ¯ sys + S ¯ m j X ¯ m j X ¯ sys Δ X ¯ sys = M ¯ m j M ¯ m j 1 M ¯ m j L j + 2 M ¯ m j L j + 1 [ Δ P ¯ e j 1 Δ ¯ e j 1 ] + i = m j L j + 1 i = m j M ¯ m j M ¯ m j 1 M ¯ i + 2 M ¯ i + 1 S ¯ i X ¯ i X ¯ sys Δ X ¯ sys .
[ Δ P ¯ e j Δ ¯ e j ] = [ Δ P ¯ m j Δ ¯ m j ] = M ¯ m j M ¯ m j 1 M ¯ 2 M ¯ 1 [ Δ P ¯ 0 Δ ¯ 0 ] = [ P ¯ m j P ¯ 0 P ¯ m j ¯ 0 ¯ m j P ¯ 0 ¯ m j ¯ 0 ] [ Δ P ¯ 0 Δ ¯ 0 ] ,
[ Δ P ¯ 0 Δ ¯ 0 ] = [ 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 C β 0 S ( 90 ° + α 0 ) S β 0 C ( 90 ° + α 0 ) 0 0 0 C β 0 C ( 90 ° + α 0 ) S β 0 S ( 90 ° + α 0 ) 0 0 0 0 C β 0 ] [ Δ P 0 x Δ P 0 y Δ P 0 z Δ α 0 Δ β 0 ] = M ¯ 0 Δ X ¯ e 0 .
q ¯ i i = [ 0 a i S β i b i C β i 1 ] T ( 0 < a i , 0 < b i , π 2 β i π 2 ) ,
λ i = D i ± D i 2 H i E i H i ,
H i = 2 ( 1 a i 2 1 b i 2 ) ( J i x J i y i 1 x i 1 y + J i x J i z i 1 x i 1 z + J i y J i z i 1 y i 1 z ) + 1 b i 2 + ( 1 a i 2 1 b i 2 ) ( J i x 2 i 1 x 2 + J i y 2 i 1 y 2 + J i z 2 i 1 z 2 ) ,
D i = ( 1 b i 2 J i x 2 b i 2 + J i x 2 a i 2 ) P i 1 x i 1 x + ( 1 b i 2 J i y 2 b i 2 + J i y 2 a i 2 ) P i 1 y i 1 y + ( 1 b i 2 J i z 2 b i 2 + J i z 2 a i 2 ) P i 1 z i 1 z + ( 1 a i 2 1 b i 2 ) [ J i x J i y ( P i 1 x i 1 y + i 1 x P i 1 y ) + J i x J i z ( P i 1 x i 1 z + i 1 x P i 1 z ) + J i y J i z ( P i 1 y i 1 z + i 1 y P i 1 z ) ] ( t i x i 1 x + t i y i 1 y + t i z i 1 z ) b i 2 + ( 1 b i 2 1 a i 2 ) ( J i x t i x + J i y t i y + J i z t i z ) ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) ,
E i = ( 1 b i 2 J i x 2 b i 2 + J i x 2 a i 2 ) P i 1 x 2 + ( 1 b i 2 J i y 2 b i 2 + J i y 2 a i 2 ) P i 1 y 2 + ( 1 b i 2 J i z 2 b i 2 + J i z 2 a i 2 ) P i 1 z 2 + t i x 2 b i 2 + t i y 2 a i 2 + t i z 2 b i 2 2 b i 2 ( t i x P i 1 x + t i y P i 1 y + t i z P i 1 z ) 1 + 2 ( 1 a i 2 1 b i 2 ) ( J i x J i y P i 1 x P i 1 y + J i x J i z P i 1 x P i 1 z + J i y J i z P i 1 y P i 1 z ) 2 ( 1 a i 2 1 b i 2 ) ( t i x J i x + t i y J i y + t i z J i z ) ( J i x P i 1 x + J i y P i 1 y + J i z P i 1 z ) ,
β i = atan 2 ( ρ i a i , σ i 2 + τ i 2 b i ) ,
X ¯ i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i a i b i ] T .
q ¯ i i = [ 0 a i β i 2 β i 1 ] ( 0 β i , 0 < a i ) .
λ i = D i ± D i 2 H i E i H i ,
H i = 1 ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) 2 ,
D i = ( 1 J i x 2 ) P i 1 x i 1 x + ( 1 J i y 2 ) P i 1 y i 1 y + ( 1 J i z 2 ) P i 1 z i 1 z ( t i x i 1 x + t i y i 1 y + t i z i 1 z ) J i x J i y ( P i 1 x i 1 y + P i 1 y i 1 x ) J i x J i z ( P i 1 x i 1 z + P i 1 z i 1 x ) J i y J i z ( P i 1 y i 1 z + P i 1 z i 1 y ) + ( J i x t i x + J i y t i y + J i z t i z ) ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) 1 2 a i ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) ,
E i = ( P i 1 x 2 + P i 1 y 2 + P i 1 z 2 ) + t i x 2 + t i y 2 + t i z 2 2 ( t i x P i 1 x + t i y P i 1 y + t i z P i 1 z ) [ J i x ( t i x + P i 1 x ) + J i y ( t i y + P i 1 y ) + J i z ( t i z + P i 1 z ) ] 2 + 1 a i [ J i x ( t i x P i 1 x ) + J i y ( t i y P i 1 y ) + J i z ( t i z P i 1 z ) ] .
λ i = D i H i ,
D i = J i x ( t i x P i 1 x ) + J i y ( t i y P i 1 y ) + J i z ( t i z P i 1 z ) + a i [ ( t i x P i 1 x ) 2 + ( t i y P i 1 y ) 2 + ( t i z P i 1 z ) 2 ] a i [ J i x ( t i x P i 1 x ) + J i y ( t i y P i 1 y ) + J i z ( t i z P i 1 z ) ] 2 ,
H i = J i x i 1 x + J i y i 1 y + J i z i 1 z ,
X ¯ i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i a i ] T .
q ¯ i i = [ 0 a i C β i b i S β i C β i 1 ] T ( 0 β i < π / 2 , 0 < a i , 0 < b i ) ,
λ i = D i ± D i 2 H i E i H i ,
H i = ( 1 b i 2 J i x 2 b i 2 J i x 2 a i 2 ) i 1 x 2 + ( 1 b i 2 J i y 2 b i 2 J i y 2 a i 2 ) i 1 y 2 + ( 1 b i 2 J i z 2 b i 2 J i z 2 a i 2 ) i 1 z 2 2 ( 1 a i 2 + 1 b i 2 ) ( J i x J i y i 1 x i 1 y + J i x J i z i 1 x i 1 z + J i y J i z i 1 y i 1 z ) ,
D i = ( 1 b i 2 J i x 2 b i 2 J i x 2 a i 2 ) P i 1 x i 1 x + ( 1 b i 2 J i y 2 b i 2 J i y 2 a i 2 ) P i 1 y i 1 y + ( 1 b i 2 J i z 2 b i 2 J i z 2 a i 2 ) P i 1 z i 1 z ( 1 a i 2 + 1 b i 2 ) [ J i x J i y ( P i 1 x i 1 y + P i 1 y i 1 x ) + J i x J i z ( P i 1 x i 1 z + P i 1 z i 1 x ) + J i y J i z ( P i 1 y i 1 z + P i 1 z i 1 y ) ] 1 b i 2 ( i 1 x t i x + i 1 y t i y + i 1 z t i z ) + ( 1 a i 2 + 1 b i 2 ) ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) ( J i x t i x + J i y t i y + J i z t i z ) ,
E i = ( 1 b i 2 J i x 2 b i 2 J i x 2 a i 2 ) P i 1 x 2 + ( 1 b i 2 J i y 2 b i 2 J i y 2 a i 2 ) P i 1 y 2 + ( 1 b i 2 J i z 2 b i 2 J i z 2 a i 2 ) P i 1 z 2 2 ( 1 a i 2 + 1 b i 2 ) ( J i x J i y P i 1 x P i 1 y + J i x J i z P i 1 x P i 1 z + J i y J i z P i 1 y P i 1 z ) + 2 ( 1 a i 2 + 1 b i 2 ) ( J i x P i 1 x + J i y P i 1 y + J i z P i 1 z ) ( J i x t i x + J i y t i y + J i z t i z ) 2 b i 2 ( P i 1 x t i x + P i 1 y t i y + P i 1 z t i z ) + t i x 2 + t i y 2 + t i z 2 a i 2 + 1 ,
X ¯ i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i a i b i ] T .
q ¯ i i = [ 0 β i R i 1 ] T ( 0 β i ) ,
λ i = D i ± D i 2 H i E i H i ,
H i = 1 ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) 2 ,
D i = ( P i 1 x t i x ) i 1 x + ( P i 1 y t i y ) i 1 y + ( P i 1 z t i z ) i 1 z + [ J i x ( t i x P i 1 x ) + J i y ( t i y P i 1 y ) + J i z ( t i z P i 1 z ) ] ( J i x i 1 x + J i y i 1 y + J i z i 1 z ) ,
E i = P i 1 x 2 + P i 1 y 2 + P i 1 z 2 + t i x 2 + t i y 2 + t i z 2 R i 2 2 ( t i x P i 1 x + t i y P i 1 y + t i z P i 1 z ) [ J i x ( t i x P i 1 x ) + J i y ( t i y P i 1 y ) + J i z ( t i z P i 1 z ) ] 2 ,
X ¯ i = [ t i x t i y t i z ω i x ω i y ω i z ξ i 1 ξ i R i ] T .
tran ( ε 1 , 0 , 0 ) = [ 1 0 0 ε 1 0 1 0 0 0 0 1 0 0 0 0 1 ] ,
tran ( 0 , ε 2 , 0 ) = [ 1 0 0 0 0 1 0 ε 2 0 0 1 0 0 0 0 1 ] ,
tran ( 0 , 0 , ε 3 ) = [ 1 0 0 0 0 1 0 0 0 0 1 ε 3 0 0 0 1 ] ,
x rot ( ε 4 ) = [ 1 0 0 0 0 C ε 4 S ε 4 0 0 S ε 4 C ε 4 0 0 0 0 1 ] ,
y rot ( ε 5 ) = [ C ε 5 0 S ε 5 0 0 1 0 0 S ε 5 0 C ε 5 0 0 0 0 1 ] ,
z rot ( ε 6 ) = [ C ε 6 S ε 6 0 0 S ε 6 C ε 6 0 0 0 0 1 0 0 0 0 1 ] .

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