Autonomous Robotics CS 685
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Date Created: 09/28/15
08685 Lecture Notes Jana Ko eckd 1 Modelling Dynamical Systems The behavior of the system is best characterized in terms of state and its evolution over time Before we proceed we informally set up some terminology which will enable us to characterize behaviors of dynamical systems and design controllers for variety of systems The basic entities which describe the behavior of the dynamical system are o X set of states of the system and the environment 0 Y set of outputs Information available to the controller since the information about the entire state is often not available to the controller 0 U set of control actions The domains from which these entities can come from depends on what type of behavior we are trying to capture For example on case of mobile robot the state X z y 0 E 9 x 9 x 51 while if the subject of our control is washing machine the part of the state can be the water in the washing machine which can be X hot warm cold Example Consider your new digital camera De ned this system in terms of its state input and output ie specify the domains of each 11 System State Equations From the control theoretic standpoint we distinguish two entities the subject of our control which is in control literature often also referred to as plant and the controller The behavior of the system is described by time trajectories Xtytut We will focus for the moment on the trajectories obeying laws which can be described in terms of differential equations or in terms of difference equations Xt1 fXt7ut yt1 90475 0 VV For a special class of linear systems the system state equations have the following form X AxtBut y Cxlttgt lt5 where ABC are matrices of appropriate dimensions For linear systems there is an ample of techniques which give us guidelines how to characterize system s performance stability controlla bility observability and how to design control laws which are optimal with respect to some chosen objective The set of states inputs and output is nite the trajectories of the system can be for example described by nite state machine with inputs and outputs The goal of control is then do design a control policy which speci es what control actions should be done in every possible situation In the most general setting the control policy can be viewed as a mapping 7rHmHU or 7rHyHU from state or output histories Hm Hy to control actions Point Mass Here we demonstrate a simple instance of such system state equations for a point mass system and how to go from between dynamic equations and system state equations Consider a point mass in ideal environment with no friction under in uence of external forces Few The dynamic equations of this system are fully characterized by Newton s second law F51 The behavior of the system is at each instance characterized by its position and its velocity Hence the state of the system x m ilT The system state equation which captures the evolution of the system s state over time can then be described as differential equation m 8 glxlttgtl1fmlulttgt 12 Control strategies Lets have a look at some examples and different components of the control law In the following examples we will assume that the output of the system y is directly the state x Consider mass spring damper mechanism rst in the absence of any external forces miksxkdi0 In the homework we had a chance to observe how the behavior of the state x m MT depends on the choice of constants k9 kd and initial conditions Hence the open loop dynamics of the system is m l f i l W m m and the control input in this case is zero Suppose now that we will apply some external force of the following form Fen ikpm 7 kvz which is proportional to the current position and current velocity of the mass This would yield following dynamics equations mi ksx kdi ikpz 7 kvab 6 mi ks kpz kd km 0 7 Note that the second question describes the system of the same type but by adding the external force terms we effectively changed the coef cients of the system and hence change the system s behavior Example For mass spring damper system above7 suggest the formula for Fem such that the dynamic equations of the closed loop system will have the following have the following dynamics7 will behave as a simple point mass 92 F We did this example in class Fen will be some function on Example Consider again a simple point mass system with no damping and no friction i Fem We would like the point mass follow particular trajectory which was computed ahead of time id id md Suppose we rst apply external force Fez id If we simply use this control law the dynamics of the system would be i id In case we would like to compensate for the possible initial errors in m z let s consider the following control law Fem id 7 lave 7 kpe with e t9 7 0d This would yield the following system dynamics 92 id 7 kve 7 kpe 8 e kve kpe 0 9 This derivation can vary depending on how is the error de ned The above equations now described the error dynamics We can now investigate the behavior of the error as a function of time and choose the constants kp kw appropriately to yield the desired performance In the context of robotics this control law is also referred to as computed torque law The id part of the control law is also referred to as feed forward term and it would be sufficient if our model is perfect Proportional Derivative Control Even simplest control law which we can apply is to Fem 71w 7 kpe Compared to previous case this control law has no feed forward term In practice7 when the objective is to track very complex trajectories it is quite hard to achieve without the feed forward term Furthermore proportional derivative control law leaves some steady state error In order to compensate for steady state errors additional term integral term can be added to the system Fm 7mm kpe k edt o What is difference between closed loop an open loop system 7 o What is the role of feed back in the control system 7 Previously Reactive control architecture composition of behaviors feedback controllers based on different prioritization schemes Subsumption architecture one particular way of combining them one behavior can subsume another one behavior can inhibit another parallel execution of all behaviors fixed priority Characteristics of the Reactive Control Architectures no notion of representation of the environment Behavior Based Architecture We had previously examples of behaviors Feedback controllers taskachieving behaviors design motivated by potential field based technique How to composed them Some examples of composition superposition motivated by potential field techniques Behaviors Wha r are behaviors Behaviors are feedback confrollers Behaviors are execu red in parallel Achieve specific goals avoidobs racles gofogoal Can be combined to achieve more complex ne rworks make inpu rs of one behavior ou rpu rs of ano rher Behaviors can be designed ro lookahead build and mainfain represen ra rion of fhe world gsme U1 Representation of behaviors Func rional represen ra rion r 35 robo r schemas Feedback confrollers gradien r of some po ren rial func rion Lookup fable Sfimulus response diagrams Discre re andor confinuous represen ra rions differen rial equafions or iffhen rules gt wallfollowing example A r fhe level of inferac rion be rween behaviors Abs rrac r represen ra rion in terms of FSM39s Behavior Representations Previously we had following examples Continuous representation Potential field Techniques attractive repulsive potential fields Schema motivated by the potential fields more general vector fields goto goal follow corridor Arbib 81 perceptual schema embodies sensing object oriented constuct depending on situation different perceptual schema can be activated navigation in the dark vs during daylight motor schema used to represent action Toad behavior Rana Computatrix toad looking for flies when the fly Is detected attraction towards the goal modeled as vector field When two flies Are detected frog snaps in the middle both real frog and simulated Arbib 81 issues with superposition local minima maxima oscilatory behavior Superposition of different behaviors More on differenT behavior composiTions Compe l39i l39ive one acTive aT The Time Fixed prioriTizaTion subsumpTion Brooks example 39 Based on The highesT level of acTivaTion acTivaTion can depend on inTenTion Maes 80 example 39 VoTing archiTecTure example Coopera l39ive schemes some fusion 39 superposiTion of The vecTor fields or schemas Compose behaviors To form assemblages of more elemenTary behaviors examples move To pole wander Behavior based ArchiTecTure MoTivaTion 1 To keep all The advanTages of The ReacTive ConTrol 2 Allow represenTaTion of The environmenT 3 Allow bigger flexibiliTy and reconfiguraTion depending on The Task This is whaT subsumpTion archiTecTure was lacking Behavior Assemblages Power of absTracTion 39 ModulariTy 39 Reuse of elemenTary behaviors 39 reason over Them 39 Coarser level of granulariTy good for adapTaTion and learning AbsTracTion39s in Terms of FSM39s ComposiTion of The behaviors Examples FSM for navigaTion Pole finding roboT AAA compeTiTion Wander and avoid behavior assemblage empTy Wander 39 for Trash Full and empty Blue can 39 Move 0 quotrraShCan Example of rrash collec ring robo r each node is an assemblage of behaviors more de rails on The fransHions Behavior Composi39rion Programming language for behavior composi on ElemenTary behaviors FSM39s Composition operators Sequen al B1 B2 Condi onal B1 B2 Parallel B1 II B2 Disabling B1 B2 ITeraTive B1 B2 B1 B2 QFWNP Examples of more complex Tasks as neTworks of elemenTary behaviors behaviors can communicaTe via shared memory Example Classroom naviga on Clean Up Foraging EmergenT Behaviors Apparenle new behaviors can emerge from InTeracTions of rules InTeracTions of behaviors InTeracTions wiTh The environmenT Since The behavior is jusT inpuT oquuT mapping exTernally observed 39 Occasionally explicile un modeled inTeracTionsbehaviors can be observed NoTion of emergenT behavior inTuiTive noT well defined excepT for some simple scenarios Wall following example flocking dispersing foraging EmergenT behaviors Flocking example 1 Don39T run inTo anoTher roboT 2 Don39T geT Too far from oTher roboTs 3 Keep moving UnexpecTed vs emergenT depends on The observer subjecTive noTion Due To The un modeled uncerTainTies The behaviors are noT Exachy repeaTablepredicTable EmergenT behaviors can be achieved from parallel execuTion of many behaviors For The purpose of analysis undesirable phenomena Behavior ComposiTion So far all The examples of behaviors were jusT reacTive conTrollers represenTed in conTinuous or discreTe manner How To design more sophisTicaTed behaviors eg To achieve map building of The environmenT 39 Map building in The conTexT of Behavior Based ArchiTecTure WhaT Type of map represenTaTion would fiT RepresenTaTion needs To saTisfy The premises of The behavior based archiTecTure Example ToTo The map building roboT M MaTaric Learning a disTribuTed map represenTaTion of The environmenT based on navigaTion behaviors 1992 DisTribuTed Map Building Map cannoT be cenTralized CAD CAM model Idea disTribuTe differenT parTs of The map over differenT behaviors Individual behaviors will be responsible for The porTions of The map They will be connecTed in such a way ThaT The connecTions will reflecT The adjacency in The acTual model of The environmenT NoTion of a landmark parTicular parT of The environmenT which The roboT can easily recognize Design behaviors To deTecT parTicular landmarks The choice of landmarks depends on The Types of sensors used DisTr39ibuTed Map building Three Types of landmarks 1 Walls 2 Corridors 3 Irregular messy areas ToTo moves around safely and deTecTs landmarks Each landmark behavior remembered some informaTion abouT landmark Type heading and lengTh Map building conTinued 1 Every Time new landmark was discovered new behavior was Added and connecTed To The previous behaviors via communicaTion wires 2 If no behavior was acTivaTed landmark is new added if exisTing behavior was acTivaTed inhibiT oTher behaviors example 3 Was also used for paTh planning PaTh can be opTimized based on The lengTh of individual segmenTs LocalizaTion wiThin The Topological map DisTribuTed Map Building DifferenT layers 1 Move around safely 2 DeTecT new landmarks 3 Build a map and localize wiThin The map 4 Find a paTh Towards a goal Real Time reacTive behavior no cenTral place for keeping The represenTaTion of The environmenT as opposed To deliberaTive archiTecTure Behavior Based ArchiTecTure Summary Pros and cons 1 2 3 4 DisTribuTed operaTe aT larger Time scales MainTain The reacTiviTy Can form and mainTain represenTaTions NoTion of The emergenT behaviors someThing which is hard To model and hard To analyze can be achieved by superposiTion DifficulT To provide any guaranTees Behavior Based approached exploiT emergenT behaviors in some insTances They have To be avoided More examples of sysTems see book 61 62 MulTi agenT ArchiTecTures AdvanTages of having groups of roboTs achieving some Tasks exploraTion manipulaTion demining simpler less expensive roboTs cooperaTion can achieve more complex Tasks Taxonomy of The collecTives size communicaTion range communicaTion Topology reconfigurabiliTy of Topology or physical Homogeneous or HeTerogeneous Processing abiliTy cenTralized disTribuTed MoThership and Team concest Example MulTi roboT Collision Avoidance Air Traffic ConTrol Collision Avoidance DisTribuTed mulTi agenT moTion planning approach PoTenTial and VorTex field based moTion planning GeneraTion of The proToType maneuvers 2D planar conflicT resoluTion 212D conflicT resoluTion n Avoidance Vector field based approach on of participming veovon fields sz aka mnzuvzr Collision Avoidance 7 Prototype Maneuvers Rmmdabmn Wham27 Collision Avoidance ProToType Maneuvers D avoidance maneuver horizontal and verTical conflicT resoluTion Collision Avoidance Visualization CS685 Lecture Notes Jana Ko ecka 1 Combining Measurements An essential part of the control system is the capability of sensing the state of the system These notes brie y summarize the material covered in the class 0 we need sensors to sense the state of the system 0 sensors are inaccurate 0 how to combine measurements from sensors 0 model the uncertainties in the sensing explicitly Several means of combining static measurements without explicit characterization of their uncer tainties were discussed in the class Suppose now that we have knowledge about the performance of our sensor which is modeled by conditional probability distribution capturing the probability of true state x give the observation y Consider a simple example of a scalar linear systems of the following form xk 1 Uk 1 2M Mk wk 2 The rst equation in the system is so called system dynamics equation The state of the system z is the position of the point on the line which does not change over time This position is perturbed by a zero mean Gaussian noise process with variance 0 also denoted N0ag The variance of this noise re ect certainty about our dynamic model which in this case is that the position just uctuates slightly around its previous position The second equation is the measurement equation In this example it captures the state x can be measured directly as this measurement is denoted y Again since measurements are typically obtained with sensors there is certain amount of uncertainly which here is again modeled by zero mean Gaussian noise process with variance oi N0 030 We will introduce more complicated models of dynamics and measurements later Supposed that we would like to estimate the state of the system from the noise measurements The estimate of the state is denoted by i to distinguish it from the true state As discussed previously we consider Bayesian perspective and represent our knowledge about sensors measurements using conditional distribution This distribution in robotics can be often learned through repeated trials with some ground truth state and application of the Bayes rule pmy 0ltpylp In case this conditional distribution is Gaussian a suitable candidate of the estimate can be the mean of this conditional distribution Recall that the mean and variance completely characterize a Gaussian distribution 90 i W 1 7 535 7 V 27m p i 202 Lets consider the one dimensional system scalar system above and suppose that we have two sensors for measuring the position x along z axis each sensor returns the correct position corrupted by 2990
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