实时视觉惯性里程计的On-Manifold预集成技术
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"On-Manifold Preintegration for Real-Time vio 是一篇关于实时视觉惯性里程计技术的学术论文，由Christian Forster, Luca Carlone, Frank Dellaert 和 Davide Scaramuzza等人撰写，并发表在IEEE Transactions on Robotics上。该论文提出了一种新的方法来解决视觉惯性里程计（VIO）在实时优化中的效率问题。
在当前的视觉惯性里程计系统中，通过非线性优化可以获得高度精确的状态估计。然而，随着轨迹的增长，实时优化的计算负担会迅速增加，特别是由于惯性传感器（如陀螺仪和加速度计）以高频率提供测量数据，导致优化问题中的变量数量快速增加，这使得实时处理变得困难。
针对这一挑战，论文提出了"on-manifold preintegration"的概念，即在选定的关键帧之间预先整合惯性测量数据，将其转换为单一的相对运动约束。这种方法的主要贡献在于它能够在保持精度的同时，减少在线优化中的变量数量，提高了实时性。
On-manifold preintegration的核心思想是保持运算在惯性测量的自然流形（如旋转群SO(3)和加速度空间）上进行，以避免不必要的坐标变换和数值误差。通过这种方式，可以预先计算出两个关键帧之间的旋转和平移增量，形成紧凑的约束，用于后续的优化过程。
论文详细介绍了该方法的数学框架，包括如何在流形上执行积分以及如何处理漂移误差。此外，还讨论了如何将这些预积分约束集成到现有的视觉惯性滤波或滑窗优化框架中。实验结果证明，这种方法能够有效地减少计算负担，提高系统的实时性能，同时保持高精度的定位结果。
"On-Manifold Preintegration for Real-Time vio"为解决视觉惯性导航系统中的实时性和计算效率问题提供了一个创新解决方案，对移动机器人、无人机导航以及其他需要精确实时定位的应用具有重要价值。"
5
Later, we will use the following first-order approximation:
Exp(φ + δφ) ≈ Exp(φ) Exp(J
r
(φ)δφ). (7)
The term J
r
(φ) is the right Jacobian of SO(3) [43, p.40] and
relates additive increments in the tangent space to multiplica-
tive increments applied on the right-hand-side (Fig. 1):
J
r
(φ) = I −
1−cos(kφk)
kφk
2
φ
∧
+
kφk−sin(kφk)
kφ
3
k
(φ
∧
)
2
. (8)
A similar first-order approximation holds for the logarithm:
Log
Exp(φ) Exp(δφ)
≈ φ + J
−1
r
(φ)δφ. (9)
Where the inverse of the right Jacobian is
J
−1
r
(φ) = I +
1
2
φ
∧
+
1
kφk
2
+
1 + cos(kφk)
2kφk sin(kφk)
(φ
∧
)
2
.
The right Jacobian J
r
(φ) and its inverse J
−1
r
(φ) reduce to the
identity matrix for kφk=0.
Another useful property of the exponential map is:
R Exp(φ) R
T
= exp(Rφ
∧
R
T
) = Exp(Rφ) (10)
⇔ Exp(φ) R = R Exp(R
T
φ). (11)
b) Special Euclidean Group: SE(3) describes the group
of rigid motion in 3D, which is the semi-direct product of
SO(3) and R
3
, and it is defined as SE(3)
.
= {(R, p) : R ∈
SO(3), p ∈ R
3
}. Given T
1
, T
2
∈ SE(3), the group operation
is T
1
· T
2
= (R
1
R
2
, p
1
+ R
1
p
2
), and the inverse is T
−1
1
=
(R
T
1
, −R
T
1
p
1
). The exponential map and the logarithm map
for SE(3) are defined in [44]. However, these are not needed
in this paper for reasons that will be clear in Section III-C.
B. Uncertainty Description in SO(3)
A natural definition of uncertainty in SO(3) is to define a
distribution in the tangent space, and then map it to SO(3) via
the exponential map (6) [44, 46, 47]:
˜
R = R Exp(), ∼ N (0, Σ), (12)
where R is a given noise-free rotation (the mean) and is a
small normally distributed perturbation with zero mean and
covariance Σ.
To obtain an explicit expression for the distribution of
˜
R,
we start from the integral of the Gaussian distribution in R
3
:
Z
R
3
p()d =
Z
R
3
αe
−
1
2
kk
2
Σ
d = 1, (13)
where α = 1/
p
(2π)
3
det(Σ) and kk
2
Σ
.
=
T
Σ
−1
is the
squared Mahalanobis distance with covariance Σ. Then, ap-
plying the change of coordinates = Log(R
−1
˜
R) (this is the
inverse of (12) when kk < π), the integral (13) becomes:
Z
SO(3)
β(
˜
R) e
−
1
2
k
Log(R
−1
˜
R)
k
2
Σ
d
˜
R = 1, (14)
where β(
˜
R) is a normalization factor. The normalization factor
assumes the form β(
˜
R) = α/| det(J (
˜
R)|, where J (
˜
R)
.
=
J
r
(Log(R
−1
˜
R)) and J
r
(·) is the right Jacobian (8); J (
˜
R) is a
by-product of the change of variables, see [46] for a derivation.
From the argument of (14) we can directly read our “Gaus-
sian” distribution in SO(3):
p(
˜
R) = β(
˜
R) e
−
1
2
k
Log(R
−1
˜
R)
k
2
Σ
. (15)
For small covariances we can approximate β ' α, as
J
r
(Log(R
−1
˜
R)) is well approximated by the identity matrix
when
˜
R is close to R. Note that (14) already assumes relatively
a small covariance Σ, since it “clips” the probability tails
outside the open ball of radius π (this is due to the re-
parametrization = Log(R
−1
˜
R), which restricts to kk < π).
Approximating β as a constant, the negative log-likelihood of
a rotation R, given a measurement
˜
R distributed as in (15), is:
L(R) =
1
2
Log(R
−1
˜
R)
2
Σ
+const =
1
2
Log(
˜
R
−1
R)
2
Σ
+const,
(16)
which geometrically can be interpreted as the squared angle
(geodesic distance in SO(3)) between
˜
R and R weighted by
the inverse uncertainty Σ
−1
.
C. Gauss-Newton Method on Manifold
A standard Gauss-Newton method in Euclidean space
works by repeatedly optimizing a quadratic approximation
of the (generally non-convex) objective function. Solving the
quadratic approximation reduces to solving a set of linear
equations (normal equations), and the solution of this local
approximation is used to update the current estimate. Here we
recall how to extend this approach to (unconstrained) optimiza-
tion problems whose variables belong to some manifold M.
Let us consider the following optimization problem:
min
x∈M
f(x), (17)
where the variable x belongs to a manifold M; for the sake
of simplicity we consider a single variable in (17), while the
description easily generalizes to multiple variables.
Contrarily to the Euclidean case, one cannot directly ap-
proximate (17) as a quadratic function of x. This is due
to two main reasons. First, working directly on x leads to
an over-parametrization of the problem (e.g., we parametrize
a rotation matrix with 9 elements, while a 3D rotation is
completely defined by a vector in R
3
) and this can make the
normal equations under-determined. Second, the solution of
the resulting approximation does not belong to M in general.
A standard approach for optimization on manifold [45, 48],
consists of defining a retraction R
x
, which is a bijective map
between an element δx of the tangent space (at x) and a
neighborhood of x ∈ M. Using the retraction, we can re-
parametrize our problem as follows:
min
x∈M
f(x) ⇒ min
δx∈R
n
f(R
x
(δx)). (18)
The re-parametrization is usually called lifting [45]. Roughly
speaking, we work in the tangent space defined at the current
estimate, which locally behaves as an Euclidean space. The use
of the retraction allows framing the optimization problem over
an Euclidean space of suitable dimension (e.g., δx ∈ R
3
when
we work in SO(3)). We can now apply standard optimization
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