Fundamental Problems In Robotics
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- Shannon Freeman
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1 Fundamenal Problems In Roboics Wha does he world looks like? (mapping sense from various posiions inegrae measuremens o produce map assumes perfec knowledge of posiion Where am I in he world? (localizaion Sense relae sensor readings o a world model compue locaion relaive o model assumes a perfec world model Togeher, hese are SLAM (Simulaneous Localizaion and Mapping
2 Sensors Propriocepive Sensors (monior sae of robo IMU (accels & gyros Wheel encoders Doppler radar Eerocepive Sensors (monior environmen Cameras (single, sereo, omni, FLIR Laser scanner MW radar Sonar Tacile CS-47 Inroducion o Roboics and Inelligen Sysems 2
3 Specific eamples acile close-range proimiy angular posiion infrared Sonar laser (various ypes radar compasses, gyroscopes Force GPS vision Types of sensor CS-47 Inroducion o Roboics and Inelligen Sysems 3
4 Orienaion Represenaions Describes he roaion of one coordinae sysem wih respec o anoher B Z A A X A Y A CS-47 Inroducion o Roboics and Inelligen Sysems 4
5 Roaion Mari Roaion Mari: (9 variables Euler Angles [roll, pich, yaw] (Gimbal Lock, disconinuiy, muliple represenaions Angle-Ais [V,q] Quaernions A B R q q4 qi q2 j q3k r r r 2 3 r r r r r r B Z A A X A Y A CS-47 Inroducion o Roboics and Inelligen Sysems 5
6 Pah Planning Visibiliy Graph Bug Algorihms Poenial Fields Skeleons/Voronoi Graphs C-Space PRM s RRT s
7 Generalized Voronoi Graph (GVG Free Space wih Topological Map (GVG
8 Generalized Voronoi Graph (GVG Access GVG Free Space wih Topological Map (GVG
9 Generalized Voronoi Graph (GVG Access GVG Follow Edge Free Space wih Topological Map (GVG
10 Generalized Voronoi Graph (GVG Access GVG Follow Edge Home o he MeePoin Free Space wih Topological Map (GVG
11 Generalized Voronoi Graph (GVG Access GVG Follow Edge Home o he MeePoin Selec Edge Free Space wih Topological Map (GVG
12 Local echniques Poenial Field mehods compue a repulsive force away from obsacles compue an aracive force oward he goal le he sum of he forces conrol he robo To a large een, his is compuable from sensor readings
13 SONAR modeling using Occupancy Grids The key o making accurae maps is combining los of daa. Bu combining hese numbers means we have o know wha hey are! Wha should our map conain? small cells each represens a bi of he robo s environmen larger values => obsacle smaller values => free wha is in each cell of his sonar model / map?
14 Configuraion Space
15 Tool: Configuraion Space (C-Space C q 2 q q 2 q
16 Tool: Configuraion Space (C-Space C q 2 q q 2 q
17 Tool: Configuraion Space (C-Space C
18 RRT-Connec: eample Connecion made!
19 Coverage Firs Disincion Deerminisic Random Second Disincion Complee No Guaranee Third Disincion Known Environmen Unknown Environmen Demining Vacuum Cleaning
20 Cellular Decomposiion Criical Poin (CP Cell 0 Direcion of Coverage
21 Single Cell Coverage Direcion of Coverage
22 Cellular Decomposiion Cell 2 CP 0 CP Cell 0 Cell CP 2 CP 3 Cell N CP m CP m- Direcion of Coverage Reeb Graph
23 Human Robo Ineracion Acive roles Human or Robo Supervisor Operaor Mechanic / Assisan Peer Slave
24 Human Robo Ineracion Levels of Auonomy (LOA [Sheridan 978]. Compuer offers no assisance; human does i all 2. Compuer offers a complee se of acion alernaives 3. Compuer narrows he selecion down o a few choices 4. Compuer suggess a single acion 5. Compuer eecues ha acion if human approves 6. Compuer allows he human limied ime o veo before auomaic eecuion 7. Compuer eecues auomaically hen always informs he human 8. Compuer informs human afer auo-eecuion only if human asks 9. Compuer informs human afer eecuion only if i decides o 0. Compuer decides everyhing and acs auonomously, ignoring he human direc conrol dynamic auonomy
25 Human Robo Ineracion Sliding Auonomy Teleoperaion Confirmaion Inerrupion Full Auonomy
26 Muli-Robo Team size Communicaion range Communicaion opology Communicaion bandwidh Processing abiliy Team Reconfigurabiliy Team Composiion
27 Localizaion Tracking: Known iniial posiion Global Localizaion: Unknown iniial posiion Re-Localizaion: Incorrec known posiion (kidnapped robo problem
28 Graphical Models, Bayes Rule and he Markov Assumpion Acions a Beliefs b T( j a i, i b 2 Observable Hidden Observaions O(z j i Z Z 2 Saes 2 p( y p( Bayes rule: p( y p( y Markov : p(, a, a0, z0, a, z,, z p(, a
29 Derivaion of he Bayesian Filer 0,..., (, ( ( ( d o a p a p o p Bel (, ( ( ( d Bel a p o p Bel Firs-order Markov assumpion shorens middle erm: Finally, subsiuing he definiion of Bel( - : The above is he probabiliy disribuion ha mus be esimaed from he robo s daa
30 Ieraing he Bayesian Filer Propagae he moion model: Bel ( P( a, Bel ( d Updae he sensor model: Compue he curren sae esimae before aking a sensor reading by inegraing over all possible previous sae esimaes and applying he moion model Bel ( P( o Bel ( Compue he curren sae esimae by aking a sensor reading and muliplying by he curren esimae based on he mos recen moion hisory
31 Differen Approaches Kalman filers (lae-60s? Gaussians approimaely linear models posiion racking Eended Kalman Filer Informaion Filer Unscened Kalman Filer Muli-hypohesis ( 00 Miure of Gaussians Muliple Kalman filers Global localizaion, recovery Discree approaches ( 95 Topological represenaion ( 95 uncerainy handling (POMDPs occas. global localizaion, recovery Grid-based, meric represenaion ( 96 global localizaion, recovery Paricle filers ( 98 Condensaion (Isard and Blake 98 Sample-based represenaion Global localizaion, recovery Rao-Blackwellized Paricle Filer
32 Mone-Carlo Sae Esimaion (Paricle Filering Employing a Bayesian Mone-Carlo simulaion echnique for pose esimaion. A paricle filer uses N samples as a discree represenaion of he probabiliy disribuion funcion (pdf of he variable of ineres: S [, w : i N] i where i is a copy of he variable of ineres and w i is a weigh signifying he qualiy of ha sample. In our case, each paricle can be regarded as an alernaive hypohesis for he robo pose. i CS-47 Inroducion o Roboics and Inelligen Sysems 32
33 Paricle Filer (con. The paricle filer operaes in wo sages: Predicion: Afer a moion (a he se of paricles S is modified according o he acion model S where (n is he added noise. f ( S, a, n The resuling pdf is he prior esimae before collecing any addiional sensory informaion. CS-47 Inroducion o Roboics and Inelligen Sysems 33
34 Paricle Filer (con. Updae: When a sensor measuremen (z becomes available, he weighs of he paricles are updaed based on he likelihood of (z given he paricle i w P( z w i The updaed paricles represen he poserior disribuion of he moving robo. i i CS-47 Inroducion o Roboics and Inelligen Sysems 34
35 Resampling For finie paricle populaions, we mus focus populaion mass where he PDF is subsanive. Failure o do his correcly can lead o divergence. Resampling needlessly also has disadvanages. One way is o esimae he need for resampling based on he variance of he paricle weigh disribuion, in paricular he coefficien of variance: M 2 var( w ( i cv ( Mw ( 2 i E ( w ( i M ESS M cv 2 CS-47 Inroducion o Roboics and Inelligen Sysems i 35 2
36 The Kalman Filer Moion model is Gaussian Sensor model is Gaussian Each belief funcion is uniquely characerized by is mean m and covariance mari Compuing he poserior means compuing a new mean m and covariance from old daa using acions and sensor readings Wha are he key limiaions? Unimodal disribuion 2 Linear assumpions
37 Wha we know Wha we don know We know wha he conrol inpus of our process are We know wha we ve old he sysem o do and have a model for wha he epeced oupu should be if everyhing works righ We don know wha he noise in he sysem ruly is We can only esimae wha he noise migh be and ry o pu some sor of upper bound on i When esimaing he sae of a sysem, we ry o find a se of values ha comes as close o he ruh as possible There will always be some mismach beween our esimae of he sysem and he rue sae of he sysem iself. We jus ry o figure ou how much mismach here is and ry o ge he bes esimae possible
38 Kalman Filer Componens (also known as: Way Too Many Variables Linear discree ime dynamic sysem (moion model Sae Conrol inpu Process noise F B u G w Sae ransiion funcion Conrol inpu funcion Noise inpu funcion wih covariance Q Measuremen equaion (sensor model Sensor reading Sae Sensor noise wih covariance R z H n Sensor funcion Noe:Wrie hese down!!!
39 Compuing he MMSE Esimae of he Sae and Covariance Wha is he minimum mean square error esimae of he sysem sae and covariance? F B u z H P S Esimae of he sae variables Esimae of he sensor reading T T F P F GQ G Covariance mari for he sae T H P H R Covariance mari for he sensors
40 The Kalman Filer Propagaion (moion model: T T G Q G F F P P B u F / / / / Updae (sensor model: T T T P H S H P P P r K S H P K R H P H S z z r H z / / / / / / / / /
41 bu wha does ha mean in English?!? Propagaion (moion model: Updae (sensor model: - Sae esimae is updaed from sysem dynamics - Uncerainy esimae GROWS - Compue epeced value of sensor reading - Compue he difference beween epeced and rue - Compue covariance of sensor reading - Compue he Kalman Gain (how much o correc es. - Muliply residual imes gain o correc sae esimae - Uncerainy esimae SHRINKS T T G Q G F F P P B u F / / / / T T T P H S H P P P r K S H P K R H P H S z z r H z / / / / / / / / /
42 Kalman Filer Block Diagram
43 w w ~ ( ~ ~ Calculaion of CS-47 Inroducion o Roboics and Inelligen Sysems 43
44 Calculaion of ~ ~ y y y ~... ~ cos( sin( ~ ~ cos( cos( sin( ~ cos( ~ cos( cos( ] sin( ~ sin( cos( ~ [cos( ~ cos( cos( ~ cos( ~ cos( cos( cos( cos( ( cos( ~ v v v v v v y y y w v w v v v w v v w v v w v v w v v 44 CS-47 Inroducion o Roboics and Inelligen Sysems
45 Covariance Esimaion 45 CS-47 Inroducion o Roboics and Inelligen Sysems 2 2 / / 0 0 ] [ where ] [ ] ~ ~ [ ] ~ ( ~ [( ] ~ ~ [ v T T T T T T T T T w w E Q G Q G F F P G w w G E F X X F E G w F X G w F X E X X E P
46 Some observaions The larger he error, he smaller he effec on he final sae esimae If process uncerainy is larger, sensor updaes will dominae sae esimae If sensor uncerainy is larger, process propagaion will dominae sae esimae Improper esimaes of he sae and/or sensor covariance may resul in a rapidly diverging esimaor As a rule of humb, he residuals mus always be bounded wihin a ±3 region of uncerainy This measures he healh of he filer Many propagaion cycles can happen beween updaes
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