CIS 540 Principles of Embedded Computation Spring Instructor: Rajeev Alur

Stability of Hybrid Systems Is the dynamics in mode A stable? (s2 = -0.2s1) ? (s2 = 5s1) ? CIS 540 Spring 2015; Lecture April 29 A ds1 = -s1-100s2 s2>=-0.2s1 ds2 = 10s1-s2 B ds1 = -s1+10s2 s2>=5s1 ds2 = -100s1-s2 Is the dynamics in mode B stable? Both modes have stable dynamics, but switching causes instability!

Stability of Hybrid Systems CIS 540 Spring 2015; Lecture April 29

Design and Modeling of Hybrid Systems  Automated Guided Vehicle  Goal: Follow a track as closely as possible  Design of mode-switching controller  Obstacle Avoidance for Robotic System  Reach target while avoiding obstacles  Augment obstacle estimation via communication  Multi-hop control network  Maintain stability of multiple plants when information flows among sensors, actuators, and computing elements over shared network  Design control algorithm in conjunction with scheduling policy for the network CIS 540 Spring 2015; Lecture April 29

Model-Based Design and Analysis Requirements Specify correctness formally Modeling High-level design Analysis Debugging Design-space exploration Implementation And Testing CIS 540 Spring 2015; Lecture April 29

Multi-Robot Coordination  Autonomous mobile robots in a room  Goal of each robot:  Reach a target at a known location  Avoid obstacles (positions of obstacles not known in advance)  Minimize distance travelled  Cameras and vision processing algorithms allow each robot to estimate obstacle positions  Estimates are only approximate, and depend on relative position of obstacles with respect to a robot’s position  How often should robot update these estimates ?  Each robot can communicate with others using wireless links  How often and what information?  How does communication help ?  High-level motion control (path planning)  Decide on speed and direction CIS 540 Spring 2015; Lecture April 29

Path Planning with Obstacle Avoidance X Y Robot R (x,y) Direction  Robot R’ (x’,y’) ’ ’ Target(x f,y f ) Obstacle O 2 Obstacle O 1 Assumptions: Two dimensional world Point Robots Fixed speed v CIS 540 Spring 2015; Lecture April 29

Path Planning with Obstacle Avoidance X Y (x,y)  (x’,y’) ’ ’ Target(x f,y f ) O2O2 O1O1 State variables: (x,y); (x’,y’) Initialization: (x,y) := (x 0, y 0 ); (x’,y’) := (x’ 0, y’ 0 ) Dynamics: dx = v cos , dy = v sin  ; dx’ = v cos  ’, dy’ = v sin  ’ Safety requirement: (x,y) is not in O 1 U O 2 & (x’,y’) is not in O 1 U O 2 Liveness requirement: Eventually (x,y) = (x f, y f ) & Eventually (x’,y’) = (x f, y f ) Performance: Reduce distance travelled! CIS 540 Spring 2015; Lecture April 29

Modeling Obstacles  For modeling and analysis, in context of motion planning, need to simplify obstacle shapes and complexity of image processing algorithms  Simplicity and abstraction: Key to modeling  Assume each obstacle/estimate is a circle  Can be described by coordinates of center and radius  Assumption: Real obstacle is always contained in estimated circle  Alternative: ellipses (more accurate)  Consider an obstacle with center (x o,y o ) and radius r  Radius of smallest circle that envelopes the actual obstacle  Estimate of the obstacle as computed by a robot using image processing algorithms of a robot  A circle with center (x o,y o ) and radius e > r  Closer is the robot to the obstacle, better is the estimate  Decreases with distance of robot from obstacle, and converges to r CIS 540 Spring 2015; Lecture April 29

Obstacle Estimation Y Estimated radius e = r + a(d-r) 0 < a < 1 is a constant x o,y o r Estimate from distance d (x,y) Distance d e (x’,y’) Distance d’ Estimate from distance d’ CIS 540 Spring 2015; Lecture April 29

Rule for Obstacle Estimation X Y (x,y)  Target(x f,y f ) Robot R maintains radii e 1 and e 2 that are estimates of the obstacles e1e1 (x o1,y o1 ) e2e2 (x o2,y o2 ) Obstacle estimation is computationally expensive Every t e seconds, robot executes discrete update: e 1 := min( e 1, r 1 + a(dist((x,y), (x o1,y o1 )) –r 1 ); e 2 := min( e 2, r 2 + a(dist((x,y), (x o2,y o2 )) –r 2 ); Computation of robot R’ is symmetric CIS 540 Spring 2015; Lecture April 29

Path Planning X Y (x,y)  (x f,y f ) Shortest path: Straight line to target Preferred direction  0   If estimate of obstacle 1 intersects straight path, calculate two paths that are tangents to obstacle  If estimate of obstacle 2 intersects straight path, or obstacle 1, calculate tangent paths Path P4 Path P1 Plausible paths: P1 and P4 Calculate which one is shorter: Planning algorithm returns either  1 or  4 CIS 540 Spring 2015; Lecture April 29

Path Planning  Function plan with inputs  Current position of robot  Target position  Obstacle 1 position (center and radius estimate)  Obstacle 2 position (center and radius estimate)  Output: Direction for motion  Best possible path to target while avoiding obstacles assuming estimates are correct  Function plan written in C code (can be embedded in model)  Does it help to execute planning algorithm again as robot moves ?  Yes! Estimates may improve suggesting shorter paths  Invoke planning algorithm every t p seconds CIS 540 Spring 2015; Lecture April 29

Communication  Each robot has its own estimate of each obstacle  Robot 2’s estimates may be better than Robot 1’s own estimates  Strategy: Every t c seconds, send your own estimates to the other robot, and receive estimates from the other  If your own estimates are e 1 and e 2, and receive estimates e’ 1 and e’ 2, execute e 1 := min (e 1, e’ 1 ); e 2 := min (e 2, e’ 2 ) CIS 540 Spring 2015; Lecture April 29

Effect of Coordination X Y (x,y) (x f,y f ) Suppose Path P1 was preferred Path P4 Path P1 Communication with Robot 2 gives a better estimate of obstacle 2, but not for obstacle 1 P2 Path P2 is now viable. Running planner again could choose path P2 CIS 540 Spring 2015; Lecture April 29

System of Robots (real x real) in (real x real) out Hybrid process Robot (real x real) Robot2Robot1 CIS 540 Spring 2015; Lecture April 29

Robot Model clock z p, z e, z c :=0 cont x := x 0, y := y 0 real e 1, e 2 := e 0 ;  :=  0 dx = v cos  dy = v sin  z p <= t p & z c <= t c & z e <= t e ~ (x = x f & y = y f ) x = x f & y = y f dx = dy = 0 z c = t c  out ! (e1, e2); z c := 0 z p = t p   := plan(x,y,e1,e2); z p := 0 z e = t e  z e := 0; e 1 := min( e 1, r 1 + a(dist((x,y), (x o1,y o1 )) –r 1 ); e 2 := min( e 2, r 2 + a(dist((x,y), (x o2,y o2 )) –r 2 ); in? (e1’, e2’)  e1 := min(e1,e1’); e2:=min(e2,e2’) CIS 540 Spring 2015; Lecture April 29

Analysis  Key system parameters  How often should a robot communicate?  How often should a robot execute planning algorithm  How often should a robot execute image processing algorithm to update obstacle estimates?  Design-space exploration: Choose values of t c, t p, t e  Reduce distance travelled, but also account for costs of communication/computation  Symbolic analysis beyond the scope of current tools, so need to run multiple simulations CIS 540 Spring 2015; Lecture April 29

Illustrative Execution: No Communication CIS 540 Spring 2015; Lecture April 29 Distance travelled by R = 8.81

Illustrative Execution: No Communication CIS 540 Spring 2015; Lecture April 29 Distance travelled by R = 8.64

Hybrid Systems Wrap-up  Integrated modeling of control, communication, and computation  See notes for modeling of multi-hop control networks  Not covered: Linear Hybrid Automata (section 9.3)  Restricted continuous dynamics and discrete tests/updates  Symbolic reachability analysis by representing regions as polyhedra CIS 540 Spring 2015; Lecture April 29

Course Topics  Goal: Introduction to principles of design, specification, analysis and implementation of CPS  Disciplines  Model-based design  Concurrency theory  Distributed algorithms  Formal specification  Verification techniques and tools  Control theory  Real-time systems  Hybrid systems  Emphasis on mathematical concepts and logical reasoning CIS 540 Spring 2015; Lecture April 29

Logistics  Project phase 2: due Friday  Final: Wed, May 6, Noon Moore 216  Topics: all covered during lectures during the entire semester  Open book: Can consult textbook and your notes during exam  Extended office hours (next week): Mon 3 - 4, Tues 4 – 6  Review: Friday’s recitation CIS 540 Spring 2015; Lecture April 29