Foundations of Robotics (WIP)
[Edit]If you find any error you can edit article using edit link $\uparrow$.
 References
 History
 Kinematics
 Motion model
 Localization
 Satellite localization
 Environment representation
 Planning
 Multirobot systems
 Satellite localization
References
 Karel Čapek: R.U.R.
 Isaac Asimov: Runaround
 B. Siciliano, O. Khatib: Handbook of Robotics
 S.M. LaValle: Planning Algorithms
 R.J. Schilling: Fundamentals of robotics: Analysis & Control
History
Robots are old and in movies. There is also robotic fish.
Prehistory
There is book Leonardos robots. Japan had mechanic dolls.
Robotic structures
 Manipulators
 40’s of 20$^{\text{th}}$ century  remote manipulation with radioactive material
 1961  H.A. Ernst (MIT)  founder of company making manipulator robots
 Mobile robots  remotely controlled
 since ca. 70’s of 20$^{\text{th}}$ century
 Mobile robots  autonomous
 since ca. 90’s of 20$^{\text{th}}$ century
Three laws of robotics
1. Law
A robot may not injure a human being or, through inaction, allow a human being to come to harm
2. Law
A robot must obey orders given to it by human beings, except where such orders would conflict with the first law.
3. Law
A root must protect its own existence as long as such protection does not conflict with the first or second laws.
by Isaac Asimov, 1942, Runaround
Kinematics
 Motion and transformation
 translation, rotation, spherical movement
 Movement of individual robot parts without relation to forces which move it.
 Representation of the location and orientation in space

Forward vs. Inverse kinematics
 Calculate position of end effector vs. Calculate motion to reach desired position
Degrees of freedom (DOF)
 Basic directions of translation and rotation
 2D  3 DOF [$x,y,\alpha$]

3D  6 DOF [$x,y,z,\alpha,\beta,\gamma$]
 Alternatively “$x,y,z$ position + rotation i.r.t $xy$ and $zy$ planes and i.r.t. the tool rotation axis”
 Human body has 244 DOF
Manipulators
 Manipulation of an object in space
 at least 6 DOF for 3D
 Arms, elbows, joints, hinges
 Joint variable $q_i$
 denotes joint setup, orientation
 Joints State $q = [q_1, q_2,...,q_n]$
 DOF = $n$
 Working space
 Local vs. Global coordinate system (LCS, GCS)
 coordinates transformation
 lokální je užitečnější
Lower pair joints
 Many variants possible, but usually:
 revolute (DOF = 1)
 prismatic (DOF = 1)
 helical (DOF = 1)
 cylindrical (DOF = 2)
 spherical (DOF = 3)
 planar (DOF = 3)
 Primitive joint types:
 prismatic
 revolute
 Most robots are build using only these two
Rotation
$P' = R . P$Rotation $x$ axis, angle $\phi$
$R_{x, \phi} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos\phi & \sin\phi \\ 0 & \sin\phi & \sin\phi \end{bmatrix}$Rotation $y$ axis, angle $\psi$
$R_{y, \psi} = \begin{bmatrix} \cos\psi & 0 & \sin\psi \\ 0 & 1 & 0 \\ \sin\psi & 0 & \sin\psi \end{bmatrix}$Rotation $z$ axis, angle $\xi$
$R_{z, \xi} = \begin{bmatrix} \cos\xi & \sin\xi & 0 \\ \sin\xi & \sin\xi & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}$Rotation along $x$,$y$,$z$ by angles $\phi$, $\psi$, $\xi$
$R_{\phi, \psi, \xi}=\left[\begin{array}{ccc} \cos (\psi) \cos (\xi) & \cos (\psi) \sin (\xi) & \sin (\psi) \\ {} & {} & {} \\ \sin (\phi) \sin (\psi) \cos (\xi) & \sin (\phi) \sin (\psi) \sin (\xi) & \\ +\cos (\phi) \sin (\xi) & +\cos (\phi) \cos (\xi) & \sin (\phi) \cos (\psi) \\ & & \\ \cos (\phi) \sin (\psi) \cos (\xi) & \cos (\phi) \sin (\psi) \sin (\xi) & \cos (\phi) \cos (\psi) \\ +\sin (\phi) \sin (\xi) & +\sin (\phi) \cos (\xi) & \end{array}\right]$Homogenní souřadnice
Bod $p = (x,y,z,w)^{T}, x,y,z \in \R, w = 1$
2D Rotation
$\left(x^{\prime}, y^{\prime}, w^{\prime}\right)^{T}= \left[\begin{array}{ccc} \cos (\alpha) & \sin (\alpha) & 0 \\ \sin (\alpha) & \cos (\alpha) & 0 \\ 0 & 0 & 1 \end{array}\right] \begin{bmatrix}x \\ y \\ w\end{bmatrix}$2D Translation
$\left(x^{\prime}, y^{\prime}, w^{\prime}\right)^{T}= \left[\begin{array}{ccc} 1 & 0 & d_{x} \\ 0 & 1 & d_{y} \\ 0 & 0 & 1 \end{array}\right] \begin{bmatrix}x \\ y \\ w\end{bmatrix}$Rotation + translation
$P^{\prime}=R \cdot P + T \rightarrow P^{\prime} = \Tau . P$
$\begin{bmatrix} P' \\ 1 \end{bmatrix} = \begin{bmatrix} R & T \\ 0\cdots0 & 1 \end{bmatrix} \cdot P$ $\begin{bmatrix} x' \\ y' \\ z' \\ 1 \end{bmatrix} = \begin{bmatrix} \begin{bmatrix} \square & \square & \square \\ \square & \square & \square \\ \square & \square & \square \\ \end{bmatrix} &\kern1em \begin{array}{c} t_{x} \\ t_{y} \\ t_{z} \end{array} \\ 0 \kern1.3em 0 \kern1.3em 0 &\kern1em 1 \end{bmatrix} \cdot \begin{bmatrix} x \\ y \\ z \\ 1 \end{bmatrix}$Forward kinematics (3D)
 $P = f(q)$$$
 $q = [q_{1},q_{2},\ldots,q_{6}]$$$
 $P = [x,y,z,\alpha,\beta,\gamma]$$$
 General
 each link represented by its geometric transformation
 transition between $LCS$s (local coordinates system)
 may be hard to construct the full combined transformation matrix
System composition
 General a. each link represented by its geometric transformation b. translation between $LCS$s
 May be hard to construct the full combined transformation matrix
DenavitHartengerg
Fictive movements which connect two systems:
rotate, move, move, rotate.
That can be generalized to any sequence.

Typical system composition
 Chains composed of rotational and translatioal joints only:
 joint $k$ connects links $k_{i1}$ and $k_i$
 link $k$ connects joints $k_i$ and $k_{i+1}$
 Chains composed of rotational and translatioal joints only:
DenavitHartenberg overview
Definice
Relation between $L C S_{i1}$ and $L C S_{i}$ is a composed transformation:
1. Rotate $x_{i1}$ axis along $z_{i1}$ by angle $\vartheta_{i}$  $A_{z_{i1},\vartheta_{i}}$ 
2. Translate $x_{i1}$ axis towards $z_{i1}$ by distance $d_{i}$  $A_{z_{i1}, d_{i}}$ 
3. Translate $L C S_{i1}$ along $x_{i}$ axis by distance $a_{i}$  $A_{x, a_{i}}$ 
4. Rotate $z_{i1}$ axis along $x_{i}$ by angle $\alpha_{i}$  $A_{x, \alpha_{i}}$ 
DH parameters: $\vartheta_{i}, d_{i}, a_{i}, \alpha_{i}$
DH transformation

Rotate $x_{i1}$ axis along $z_{i1}$ by angle $\vartheta_{i}$
$A_{z_{i1}, \vartheta_{i}}=\left[\begin{array}{cccc} \cos \left(\vartheta_{i}\right) & \sin \left(\vartheta_{i}\right) & 0 & 0 \\ \sin \left(\vartheta_{i}\right) & \cos \left(\vartheta_{i}\right) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$ 
Translate $x_{i1}$ axis towards $z_{i1}$ by distance $d_{i}$
$A_{z_{i1}, d_{i}}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & d_{i} \\ 0 & 0 & 0 & 1 \end{array}\right]$ 
Translate $L C S_{i1}$ along $x_{i}$ axis by distance $a_{i}$
$A_{x_{i}, a_{i}}=\left[\begin{array}{cccc} 1 & 0 & 0 & a_{i} \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$ 
Rotate $z_{i1}$ axis along $x_{i}$ by angle $\alpha_{i}$
$A_{z_{i1}, d_{i}}=\left[\begin{array}{cccc} 1 & 0 & 0 & 0 \\ 0 & \cos \left(\alpha_{i}\right) & \sin \left(\alpha_{i}\right) & 0 \\ 0 & \sin \left(\alpha_{i}\right) & \cos \left(\alpha_{i}\right) & 0 \\ 0 & 0 & 0 & 1 \end{array}\right]$
Finálně vše zkombinujeme
$A_{i1}^{i}=A_{z_{i1}, \vartheta_{i}} \cdot A_{z_{i1}, d_{i}} \cdot A_{x, a_{i}} \cdot A_{x, \alpha_{i}}$ $A_{i1}^{i}= \left[\begin{array}{cccc} \cos \left(\vartheta_{i}\right) & \sin \left(\vartheta_{i}\right) \cos \left(\alpha_{i}\right) & \sin \left(\vartheta_{i}\right) \sin \left(\alpha_{i}\right) & a_{i} \cos \left(\vartheta_{i}\right) \\ \sin \left(\vartheta_{i}\right) & \cos \left(\vartheta_{i}\right) \cos \left(\alpha_{i}\right) & \cos \left(\vartheta_{i}\right) \sin \left(\alpha_{i}\right) & a_{i} \sin \left(\vartheta_{i}\right) \\ 0 & \sin \left(\alpha_{i}\right) & \cos \left(\alpha_{i}\right) & d_{i} \\ 0 & 0 & 0 & 1 \end{array}\right]$ DH parameters: $\vartheta_{i}, d_{i}, a_{i}, \alpha_{i}$
 $\vartheta_{i}$ angle between $x$ axes about $z_{i1}$ (joint angle)
 $d_{i}$ distance between $x$ axes (link offset)
 $a_{i}$ distance between $z$ axes (link length)
 $\alpha_{i}$ angle between $z$ axes about $x_{i}$ (link twist)
$A^i_{i1}$ usage
 universal transformation between two adjacent $LCS$
 Same format independent to joint type:
 Rotation  $\vartheta_{i}$ variable, others are constant
 Traslation  $d_{i}$ variable, others are constant
 Forward kinematics is easy:
 while iterating, we use always one variable and 3 constants
DH system construction
 Joints numbered $0\ldots n$ ($0$ is the first, fixed, $1\ldots n$ are the rest, moving)
 Links numbered $1\ldots n$ (link $i$ connects joints $i1$ and $i$)
 Righthanded orthonormal coordinate system
 Let axis $z_{i1}$ be the axis of joint $i$ movement, positive direction towards positive quadrant of the basic system
 Let axis $x_{i}$ be perpendicular to $z_{i1}$ and $z_{i}$ :
 $z_{i1}$ and $z_{i}$ identical  endpoint of joint $i$, parallel to $x_{i1}$
 skew $x_{i}$ share the normal $z_{i1}$ a $z_{i}$, positive direction from $z_{i1}$ towards $z_{i}$.
 intersecting $x_{i}$ perpendicular to $z_{i1}$ and $z_{i}$, in the intersection, positive direction so that it moves along $x_{i}$ from $z_{i1}$ to $z_{i}$ in positive sense
 Set $y_{i}$ axis to complete the righthanded orthonormal $LCS_{i}$
 Set $LCS_{i}$ origin at intersection of $z_{i1}$ and $z_{i}$ or (if they do not intersect) at intersection of their common normal and $z_{i}$
 Determine the four parameters:
 $\vartheta_{i} \ldots$ angle of rotation from $x_{i1}$ to $x_{i}$ about $z_{i1}$
 $d_{i} \ldots$ distance from $L C S_{i}$ origin to $b_{i}$ along $z_{i1}, b_{i}$ is the intersection of $x_{i}$ and $z_{i1}$ (or $x_{i}$ and their common normal)
 $a_{i} \ldots$ distance from $b_{i}$ to $L C S_{i}$ origin along $x_{i}$
 $\alpha_{i} \ldots$ angle of rotation from $z_{i1}$ to $z_{i}$ about $x_{i}$
 $z_{n}$ from the endpoint of last link either parallel to $z_{n1}$ or to some significant direction (e.g. supply cable)
 $x_{n}$ from the endpoint of the last link so that it intersects $z_{n1}$, positive direction towards the workspace.
Motion model
Holonomic / nonholonomic drive
Definice
A vehicle is holonomic if the number of local degrees of freedom of movement equals the number of global degrees of freedom.
Ackermann steering
Basic principle  Simplified model 
 Nonholonomic
 Constant $\omega \Rightarrow$ movement along a circle or straight line
 Circle: $r_{s}=\frac{d}{\sin \omega}, r_{0}=\frac{d}{\tg\omega}$
 Orientation change $\Delta \theta=\frac{\Delta S}{r_{s}}$
 Manoeuvring abilities depend on $d$ and $\omega_{\max }$ (wheelbase and wheel turn)
Differential steering
 nonholonomic
 Approximation:
 more precisely, integrating:
 Differential steering for real life  General trajectory is replaced by a series of
 line segments $(\Delta R=\Delta L, r=\infty)$
 arcs $\left(\Delta R \neq \Delta L, r=\frac{b}{2} \frac{\Delta R+\Delta L}{\Delta R\Delta L}\right)$
 Wisely set $\Delta t$
 Odometry
Omnidirectional steering
 Wheels are able to run in any direction
 Swerve/Crab drive
 Omniwheel
 Mecanum wheel
 Syncro drive
 Holds yaw (body rotation) independently to movement direction
 Arbitrary movement as for omnidrive
 High manoeuvring ability
 sideways run
 start in any direction
 body orientation independent on movement direction
 onplace turning
 Simpler mechanical construction for omniwheels and mecanum wheels (fixed mount)
1. Holonomic
 Movement easily calculated by vector combination
2. Killough / Ilon wheels
3. Omniwheel drive
Mecanum \& swerve drive
Robot movement
 Got $\overrightarrow{v_{t}}$ (translation speed) and $\vec{\omega}$ (rotation speeds)
 Need $\vec{v}$  specific point speed
 vector approach
Swerve drive
 Resolve $\overrightarrow{v_{t}}$ ($x$, $y$ components = axes velocities ) into wheel speed $v_{w}$ and steering angle $\theta$
 Resolve velocity into parallel and perpendicular components; magnitude $v$ of parallel component is wheel speed $v_{w}$
 $\hat{u}$ is a unit vector in the direction of the wheel (whichever direction is assumed to be “forwards”)
Mecanum drive
 Similar to omniwheel drive
 Conceptually: Resolve velocity into components parallel to wheel and parallel to roller
 Not easy to calculate directly (directions are not perpendicular), so do it in two steps:
 Resolve to roller
 Resolve to wheel
 Resolve velocity into components parallel and perpendicular to roller axis
 $\hat{u}$ is not the same for each wheel; pick direction parallel to roller axis, in forwards direction
 Perpendicular component can be discarded
 Use component parallel to roller axis and resolve it into components parallel to wheel and parallel to roller
 $v_{w}$ is the component parallel to the wheel
 When the angle is known, we can calculate $v_{w}$ directly.
 E.g. for $45^{\circ}$ inclination:
Localization
 The goal is to find the position and orientation of the robot (pose)
 relative to the map
 relative to the environment
 Sometimes we also want to determine the position and orientation of individual parts
Coordinate transformation
 The map is independent of the robot’s movement
 The sensors move with the robot  it has local coordinates
 Localization tries to map the local and global position onto each other
 The pose cannot be perceived directly
 We divide localization into
 absolute (global) / relative (local)
 passive / active
 static / dynamic (lighting, obstacles,… do (not) change )
Absolute localization
 Measurement / Approximation regardless of previous state, typically more demanding on performance / technology
 Localization after reset / relocation is problematic (wake up / kidnap problem)
 Uses landmarks, or GPS and other services
Relative localization
 Track position and update (odometry)
 we know the initial position
 we measure / estimate the change compared to the previous state (change in rotation and orientation)
 The problem is the accumulation of the error  after a long time the error can be too large
Dead reckoning
 the easiest localization
 direction + speed + time of movement
 accumulates an error
Odometry
 Measurement of wheel rotation (in different places we measure slightly different values) (TODO to explain)
 Systematic errors:
 losses
 different wheel radii
 other specified and actual parameters
 asymmetry
 measurement resolution
 sampling frequency
 etc.
 Unsystematic errors:
 terrain irregularities
 obstacles
 poor contact with the surface
Robot centric sensors
Most sensors move with the robot, so we have to do further data processing (e.g. recalculate coordinates relative to the global system,$\ldots$).
Types of localization

Passive
 does not affect the control of the robot

Active localization
 If the measurement is needed, it will affect the behavior of the robot
 active measurement and navigation

Realistic localization
 Inputs are not trustworthy, actuators are not reliable, external influences must be taken into account
A possible solutions
 More accurate measurement  better sensors and actuators, data filtering, the most accurate model of the robot and the environment
 Do not rely on the fact that localization is accurate  probabilistic methods, fuzzy logic, interval algebra
 We can consider the data as random variables (inputs, pose)
 The position estimate is then a distribution
 Localization is then the search for a distribution that best matches the real position of the robot
 In 1D Belief $Bel(x)$ where $\int_{\infty}^{\infty}Bel(x)dx=1$ Ideally, $Bel(x) = 1$ for exact but we can’t reach the position
 We want to model $Bel(x) = P(x\mid l_{1},\dots,l_{k})$
 Random modeling can be used with particle filters, Bayesian methods,…
 We estimate the posterior by measuring the prior
Representation
 Continuous representation is more accurate but harder to obtain and maintain (Kalman filter)
 That’s why we use a rather discrete representation
 This way we get probability grids, topological graphs,…
Probability Grid
 $Bel(l)$ indicates the probability of the appearance of a robot in the $L_{ij}$ field
 Difficulty depends on grid fineness and map size
 It can be reduced using
 selective updates (only interesting part of the grid)
 Hierarchical models (quadtree)
 nonorthogonal graphs
Topological graphs
 Nodes  positions, edges  possible transitions
 $Bel(l)$ indicates the probability of occurrence in a given node
 More nodes can be made in the place where the robot is more likely to be found
 The chart does not need to contain geometric data
Monte Carlo localization
 A set of weighted points in space
 $Bel(l)$ is the sum of the weights of points at a given distance from l
 More points can be created for a more likely location
 Algorithm:
 Sample movement, weights do not change
 Correction, the positions of the points do not change, recalculate the weights
 Oversampling, cleaning  discard unnecessary points, add new ones,…
 Doesn’t work well for too accurate sensors, but not too imprecise either
 There is a lot of noise when navigating outside  MC can combine multiple measurements well
Continuous representation (Kalman filter)
 based on a normal distribution
 cannot record multiple hypotheses
 strict assumption: measurements must have a normal distribution
Other methods
 fuzzy logic
 interval algebra
Solution 3 no localization
 hardcoded automata
 reactive systems
 evolutionary algorithms
SLAM
 simultaneous location and mapping
 browsing a static environment
 we want to create a map and find the position from it
 a poorly recognized landmark can cause a huge error
 when using more landmarks, the difficulty increases ($n^2$, with $n \log n$ improvements)
 map errors and movement errors can be correlated
 knowledge of the angle with respect to two landmarks determines the part of the circle
 in addition, the distance can be calculated while moving
Satellite localization
If we know the distance of the aiming point from the satellite, it will determine the sphere. Two specifies a circle and three a point. In practice, at least four satellites are typically used.
Simple idea: measuring the signal timeofflight $\rightarrow$ distance
History
 Werner von Braun (V1, V2,…)
 Sputnik(1957) The Doppler effect on the signal was another idea for a method of measurement.

Transit
 US Navy
 proof of concept, but didn’t cover the whole country, slow, 2D only
 TODO technology

Timeation
 uses precise (later atomic) clocks
 2D only

621B
 US air force
 Can also target planes (3D)
 Pseudorandom noise increases resilience
 Needs constant satelliteground connection

NAVSEG
 a group formed by the cooperation of several sectors
 Uses ideas from Transit (orbit + prediction) and Timetion (accurate clock) and 621B (resilient signal)
 In 1973, Navstar GPS was created, the basis of today’s GPS launched in 1978
 Accuracy about 10 m
 Initially intended only for the military, later a less accurate version of the signal was broadcast for civilian purposes.
 Since 2000, even the more accurate army version is available to everyone
 Navigation failure caused several air accidents (typically wrong trajectory > shot down USSR)
GPS satellites
 6 Orbit $60{\degree}$ gap declination $55\degree$ after four slots
 There are at least four satellites in each orbit
 At least 4 satellites are visible from every location on earth
 Due to atmospheric phenomena and calculation errors, the GPS error is about 15m with post processing, better accuracy around 10m can be achieved
Competitive global systems
Compass(China), Galileo(EU), GLONASS(Russia)
Regional systems
Beidou(China), DORIS(France), IRNSS(India)
Local systems
 EGNOS, GAGAN, MSAS, WAAS
 Typically a stationary satellite
Galileo
European project, more accurate (about 1 m)
Ublox
 centimeter accuracy
 uses RTK technology, which combines several techniques and thus achieves higher accuracy (verify)
Indoor GPS
 Local location determination can be done using Wifi or bluetooth
 More accurate technology can achieve about 1m accuracy
Environment representation
Maps
we divide according to several aspects

according to production
 manual (expensive)
 (semi)automatic

According to production time
 online
 offline

According to time of use
 Firm
 adaptive

basic division
 Metric
 Topological

By level of abstraction
 Sensory
 Geometric
 Topological
 Symbolic
Types of maps

Metric maps
 Based on the given coordinate system
 Objects are represented by coordinates
 tourist map, (city plan)…

Topological maps
 They don’t have positional information only about relationships between objects.
 For example connections, neighboring objects…
 Additional information: names, attributes and other data important to the robot.
 Typically chart + labels + metadata.
 Maps in robotics tend to be a combination of both approaches.

Sensory maps
 Uses sensor input or preprocessed data (filtered,…).
 For example occupancy grid  regular (square, hex,…) or irregular (various sizes and shapes).
 Mostly they are only local

Geometric maps
 Description based on geom objects: curves, cubes, cylinders,…
 Automatic production is demanding

Topological maps
 typically a geom map generalized to a graph
 mostly used by:
 nodes = objects and edges = adjacent obj
 nodes = objects and connections, edges = affiliations (TODO meaning)

Symbolic maps
 for direct robot communication with people, e.g. prolog(?)

Occupancy grid
 directly it is difficult to maintain
 probabilistic methods
 fuzzy approach
 Neurons, genetic alg.  a problem with learning and anticipating unexpected events
 directly it is difficult to maintain
Planning
Prerequisites
 Mapping
 Knowledge of location and destination
 Movement model
 Metric
Description of the problem
 go from start to finish
 find a path or notice that it does not exist
 reacts to the surroundings  avoiding obstacles,…
 Sometimes we want navigation: planning and avoiding obstacles
 Planning is the opposite of avoiding obstacles, however both complement each other
Scheduling algorithms
 Graph methods
 Use of grid, potential field
 independent of map type
 Exact, Approximate, Probabilistic, $\ldots$
Bug algorithm
 has no global model  no map, does not know the location of obstacles or if there is a solution
 knows the destination location relative to the starting location
 only local detection of the surroundings (contact), possibly at a short distance
 Reactive
Prerequisites bug alg
 2D static environment
 Final parameters: the number of obstacles and their perimeters,
 nonzero thickness of obstacles
 Straight line crosses with obstacles
 Obstacles do not touch, they can be united when overlapping
Bug 0
 Go straight to the target
 When you hit an obstacle, go around the obstacle until you can’t go straight to the goal
 It may not work  for example, for a circle with a small hole (where the target is on the opposite side to the center), the robot will go inside and before it comes out, it will get to the state where it can go to the target again
Bug 1
 Go straight to the target
 When you hit an obstacle, go around the obstacle
 After you go around the obstacle, go to the place that was closest to the goal and go straight again
 The robot will never return to the obstacle from which it left (it will exit from the place that is closest to the target towards it)
 Because the robot’s position is always in a sufficiently large circle around the target, it will encounter only finitely many obstacles, so the algorithm is finite.
 The robot does not ignore obstacles, so the paths found are valid
 If the path existed but the robot did not find it, then the path from the minimum to the goal must lead directly to the obstacle

But each straight line passes through the obstacle $2k$ times. Therefore there is another intersection, this intersection must be closer to the goal because it lies on the way to the goal. If there is a path to this place, the robot would choose this  the dispute cannot get closer to the goal. (TODO check)

Distance Estimate
 Best case $\lvert \text{target}  \text{start} \rvert$
 Worst case
 For each obstacle in the circle $\lvert \text{target}\text{start}\rvert$ from the target, we count 1.5 times the perimeter (he goes around it and chooses a shorter path to the minimum) + $\lvert \text{target}\text{start}\rvert$
Bug 2
Go to the goal, if you come across an obstacle, go around it until you can’t go straight to the goal again
Lemma
Bug 2 finally encounters many obstacles and passes them all through the startfinish junction
Bug 2 passes each point on the obstacle at most $\frac{n_i}{2}$ times, where $n_i$ is the number of crossings of the obstacle $i$ with the startfinish line.

Distance Estimate
 Best case $\lvert \text{target}  \text{start} \rvert$
 Worst case  For each obstacle on the connecting line between the start and the finish line, we count half the circumference
Comparison of Bug 1 and Bug 2 (TODO)
There are cases for which 1 is better than two and vice versa
Estimates for length can be improved if the robot can see some of its surroundings (area inside the circle around itself)
Tangent bug
selects by angle of direction to direction to target (TODO) It is correct
Dijkstra
alternative path search A* is a Dijkstra + heuristic, if the heuristic satisfies something it can be proven that it does not process vertices twice. E.g. price + distance from destination D* also handles replanning when an obstacle appears, starting from the destination
Rapidly exploring random trees
Randomized algorithm It quickly finds some solution and iteratively adjusts
Potential field
The robot is in sweat. fields, obstacles are peaks  repel the goal is the minimum
Multirobot systems
Why use multirobot systems
 Too challenging for one robot
 The task is naturally distributed
 One universal robot would be too complex, more specialized ones are easier to make
 Parallelism accelerates the solution of the task
 Greater robustness

Application
 Warehouses
 Modeling the real world
 A swarm of robots
 robots can cooperate or act independently
Types of systems

Centralized Architecture
 One control center
 Possible but problematic: the failure of the center may mean the failure of all robots, too much communication load on the center.

Hierarchical Architecture
 Divide and rule
 Robots have small groups of robots that he gives orders to
 Scales better, problematic when higher level control fails

Decentralized Architecture
 robots act according to (semi) local perceptions
 Every robot must know high level target, problematic target changes

Hybrid driving
 combination of local control with control of multiple robots (TODO check)
Satellite localization
If we know the distance of the aiming point from the satellite, it will determine the sphere. Two specifies a circle and three a point. In practice, at least four satellites are typically used.
Simple idea: measuring the signal timeofflight $\rightarrow$ distance
History
 Werner von Braun (V1, V2,…)
 Sputnik(1957) The Doppler effect on the signal was another idea for a method of measurement.

Transit
 US Navy
 proof of concept, but didn’t cover the whole country, slow, 2D only
 TODO technology

Timeation
 uses precise (later atomic) clocks
 2D only

621B
 US air force
 Can also target planes (3D)
 Pseudorandom noise increases resilience
 Needs constant satelliteground connection

NAVSEG
 a group formed by the cooperation of several sectors
 Uses ideas from Transit (orbit + prediction) and Timetion (accurate clock) and 621B (resilient signal)
 In 1973, Navstar GPS was created, the basis of today’s GPS launched in 1978
 Accuracy about 10 m
 Initially intended only for the military, later a less accurate version of the signal was broadcast for civilian purposes.
 Since 2000, even the more accurate army version is available to everyone
 Navigation failure caused several air accidents (typically wrong trajectory $\rightarrow$ shot down USSR)
GPS satellites
 6 Orbit $60{\degree}$ gap declination $55\degree$ after four slots
 There are at least four satellites in each orbit
 At least 4 satellites are visible from every location on earth
 Due to atmospheric phenomena and calculation errors, the GPS error is about 15m with post processing, better accuracy around 10m can be achieved
Competitive global systems
Compass(China), Galileo(EU), GLONASS(Russia)
Regional systems
Beidou(China), DORIS(France), IRNSS(India)
Local systems
 EGNOS, GAGAN, MSAS, WAAS
 Typically a stationary satellite
Galileo
European project, more accurate (about 1 m)
Ublox
 centimeter accuracy
 uses RTK technology, which combines several techniques and thus achieves higher accuracy (verify)
Indoor GPS
 Local location determination can be done using Wifi or bluetooth
 More accurate technology can achieve about 1m accuracy