开发工具:
文件大小: 2mb
下载次数: 0
上传时间: 2019-07-02
详细说明：非常好的graph-slam资料Contents
1 Intro(duction)
2 Robot motion
2.1 Rigid body transformations
2.1.1 Frame specification
2.1.2 Frame transformation
577789
2.1.3 Frame composition
静
2.2 Modeling robot motion
10
2.2.1 Integrating robot motion uncertainty
10
2.3 Popular motion models and their variants
10
2.3.1 Constant-velocity model and variants
10
2.3.2 Odometry models
12
2.3.3 Differential drive model(2D only)
2.3.4 Twist control model(2D only)
14
2.3.5 IMU-driven model(3D only)
15
3 Environment perception
17
3.1 Geometry of robot-centered measurements
17
3.2 Modeling environment perception
3.3 Popular sensors and their models
19
3.3.1 2D laser range scanner
19
3.3.2 3D laser range scanner
20
3.3.3 Monocular camera
22
3.3.4 Stereo camera
25
3.3.5 Vision+depth cameras(RGBD)
3.3.6 GPS fixes, compasses, altimeters, and the like
4 Graph-based SLAM
31
4.1 Problem formulation
4.1.1 SLAM as a Dynamic Bayes Network
4.1.2 SLAM as a factor graph
34
I because it is so short!
4
CONTENTS
A note on motion errors
36
4.2 Iterative non-linear optimization
37
4.2.1 The general case and the Newton method
4.2.2 The least squares case and the Gauss-Newton methog
37
4.2.3 Improving convergence with the Levenberg-Marquardt algorithm
40
4.2.4 The sparse structure of the SLAM problem
41
4.2.5 Optimization on a manifold
42
Case of unit quaternion
45
Case of pose states with translation and quaternion
47
5 Solving by matrix factorization
49
5.1 The sparse QR factorization method
49
5. 1.1 Triangulating using reordering and Givens rotations
51
5.1.2 Incremental operation
53
5.1.3 Variants to the QR factorization method
55
5.2 The sparse Cholesky factorization method
58
5.2.1 The basic Cholesky decomposition
59
5.2.2 Sparse structure of the problem
59
5.2.3 Extra sparsity of landmark-based SLAM
60
Sparse structure of the Schur complement
62
5.3 Links between methods
63
a Brief on quaternion algebra
67
A1 Definition of quaternion
67
A2 Quaternion properties
68
A 3 Quaternion identities
70
Chapter 1
Intro(duction)
Fig. 1. 1 is a 3d visualization of a truck building a map of its environment while simul
taneously getting localized in it. It represents the trajectory as an ordered set of past
poses(yellow boxes) and a set of measurements(yellow lines)to landmarks(poles ). Poles
correspond to corners in the environnent. They are extracted by the vehicle by anal y zing
laser scans(colored profiles)
This document formalizes this situation into mathematical problems that can be solved
I because it is so short
CHAPTER 1. INTRO(DUCTION)
Figure 1.1: 3D visualization of a truck building a map of its environment while simultane-
ously getting localized in it
Chapter 2
Robot motion
2. 1 Rigid body transformations
Essential to every work in mobile robotics is the notion to rigid body, and its motion. We
briefly detail here the specification and manipulations of the position and orientation of
rigid bodies in space, through the notion of reference frame. For rigid bodies, all their fea
tures(pieces, parts, points, planes, or whatsoever) are rigidly specified in a local reference
frame(they constitute constant parameters in this frame). The motion of the body, then
can be completely specified by the motion of its reference frame
2.1.1 Frame specification
A reference frame is specified by the position of its origin of coordinates, and its orientation
always relative to another frame, which is called its parent frame. We call the parent of
all frames the global frame, which is typically considered static and attached arbitrarily
somewhere in the world. We name pose the position and orientation of a rigid body. We
use a light notation, as follows
e Points and vectors in the body frame b have this indicator as a sub-or super-index
depending on the situation), pg, VH, p,vB
Points and vectors in the global frame have no indicator, P, V
The pose of the body B relative to a frame F is denoted as Bp and specified by
its position and orientation, Br=(tFB, pr B), where trB is a translation vector
indicating the position of the origin of B in frame F, and FB is an orientation
specification of our choice, of frame B with respect to frame F
In the typical case, only the global and body frames are present. Then, the pose o
the body b can be specified with no indicators, (t, dp)
CHAPTER 2. ROBOT MOTION
2D In 2D, positions are 2D points, and orientations dp are specified by an angle 6
B
∈R
2.
3D In 3D, positions are 3D points, and orientations p admit a number of variants. In
this document, we use the unit quaternion representation, q=[u, Qa, y, q2], so that
B
R
q‖|=
(22)
NOTE: A brief but sufficient material on quaternion algebra is necessary. It is provided in
App. A
2.1.2 Frame transformation
Points and vectors in a reference frame can be expressed in another frame through the
operation of frame transformation. In particular, points and vectors in the body frame
B=(t, have expressions in the global frame given respectively by
p=R{亚}pB+t
R{更}v
(24)
whereas the opposite relations are
pB=R(
B
R{Φ
Here, R is the rotation matrix associated to the orientation p. Its expression depends
on the dimension of the space(2D or 3D) and on the orientation specification, as follows
2D Re is the rotation matrix associated to the orientation angle g
s 0- sin e
R{}=
(27)
3D Rq is the rotation matrix associated to the quaternion q, given by(A 27)as
y
2(q92+quay
R{q}=2(m9+g9)9-2+92-22(99:-99)
(28)
2(
y)2(92+a1qn
2.1. RIGID BODY TRANSFORMATIONS
SR=SOR
S=RESR
Figure 2.1: Representation of frame compositions as if they were vectors
2.1.3 Frame composition
Let us assume a robot with a reference frame r. and a sensor with a reference frame s
The sensor frame, when expressed in the robot frame, is denoted Se. We are interested in
two kinds of compositiOns. The additive composition, denoted by is a concatenation of
R and Se, providing the sensor in the global frame, S
S=ROSe
(29)
The subtractive composition, denoted by e, is the inverse, and expresses the local sensor
frame Se given the globals R and s
(2.10)
These frame relations can be sketched as if they were vectors, as in Fig 2.1
Let us denote the three involved frame definitions by
R
t
rs
更
(2.11)
Rs
The expressions of their compositions are detailed as follows
2D
R+rOotS
R+8
(212)
rs
RORI (ts-ti
8S-OR
(2.13)
rs
3D
t尸+R{qR}tPs
qr qRS
(2.14)
as
Rs
RaRI(ts-tR
R
2
qrs
qR 0 qs
where q is the quaternion conjugate and o is the quaternion product, as defined in App. A
10
CHAPTER 2. ROBOT MOTION
2.2 Modeling robot motion
Motion models express the state of the robot at a given time as a function of the state in
the last time step, the control input, and possible perturbations. We write motion models
generically with the state's time-prediction form
fn1(xn-1,1u2,i),i~N{0,Q}
(2.16)
where xm is the robot state at time tn= nAt, fn is generally a non-linear function, un is the
control signal producing the motion, and i is a. vector of randorm impulses that perturb the
desired trajectory, generally considered Gaussian with covariance Q. In some occasions, in
order to alleviate our notation, we introduce the arrow assignment operator "3
meaning
that the left-hand term has been updated with the result of the right-hand operation
x<-f(x,u,i),i~N{0,Q}
(2.17)
2.2.1 Integrating robot motion uncertainty
Motion data comes in the form of small increments or velocities, measured at regular time
intervals ot. In the framework of motion estimation, we are interested in integrating the
motion data, u, but also the uncertainties generated by the random perturbation i. We
do this by linearizing the motion model, and integrating a Gaussian estimate of the state
MiX, P, as follows,
又←f(,u,0)
2.18
P←F2PF+F2QF
(2.19)
where x is the mean of x, and p its covariances matrix. The matrices f and f are the
Jacobians of fo with respect to the state x and the perturbation impulse i
0
di
x ui=0
2.3 Popular motion models and their variants
We provide a collection of popular motion models. They come in the general form(2. 17)
and we specify, for each one of them, the contents of the state vector x, the control vector
u,the perturbations vector i, and the nonlinear algebra implementing the model f().For
a small selection of models, we also provide the Jacobians of f (
2.3.1 Constant-velocity model and variants
Useful when no control signals are available. For example, for a hand-held camera
(系统自动生成,下载前可以参看下载内容)
下载文件列表
相关说明
- 本站资源为会员上传分享交流与学习,如有侵犯您的权益,请联系我们删除.
- 本站是交换下载平台，提供交流渠道，下载内容来自于网络，除下载问题外，其它问题请自行百度。
- 本站已设置防盗链，请勿用迅雷、QQ旋风等多线程下载软件下载资源，下载后用WinRAR最新版进行解压.
- 如果您发现内容无法下载，请稍后再次尝试；或者到消费记录里找到下载记录反馈给我们.
- 下载后发现下载的内容跟说明不相乎，请到消费记录里找到下载记录反馈给我们，经确认后退回积分.
- 如下载前有疑问，可以通过点击"提供者"的名字，查看对方的联系方式，联系对方咨询.
