Determining the Nonexistence of Evasive Trajectories for Collision - PowerPoint PPT Presentation

Determining the Nonexistence of Evasive Trajectories for Collision Avoidance Systems Matthias Althoff, Sebastian S¨ ontges Technische Universit¨ at M¨ unchen September 28, 2015 Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 1 / 43

Motivation Assessing the Risk of Traffic Situations When to initiate an emergency maneuver? Collision “almost” unavoidable Approach: Test if there is none or only “very few” evasive trajectories Figure: Existence of evasive trajectories Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 2 / 43

Motivation Uncountably Many Possible Situations How “many” future trajectories of other traffic participants are possible? How “many” evasive trajectories are there? Answer for both: Infinitely many. Figure: Set of evasive trajectories Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 3 / 43

Motivation Overview of Our Previous Approach ➁ trajectory planning ➀ occupancy prediction controller ➃ trajectory tracking ➂ collision checking Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 4 / 43

Motivation Outline Model of the ego vehicle and other traffic participants 1 Set-based behavior prediction 2 Determining the nonexistence of evasive trajectories 3 Set representation for reachability analysis Computational tricks Test results 4 Further use of reachability analysis: Verification of single trajectories 5 Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 5 / 43

Models of the Ego Vehicle and Other Traffic Participants Constraints for Traffic Participants Initially the following constraints are considered: C 1: positive longitudinal acceleration is stopped when a parameterized speed v max is reached. C 2: driving backwards in a lane is not allowed. C 3: positive longitudinal acceleration is inversely proportional with speed above a parameterized speed v S (modeling a maximum engine power). C 4: maximum absolute acceleration is limited by a max . C 5: actions that cause leaving the road/lane boundary are forbidden. When a violation of a constraint of a traffic participant is sensed, it is no longer considered in future predictions for that particular traffic participant. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 6 / 43

Models of the Ego Vehicle and Other Traffic Participants Physical Modeling (1) Traffic participants are a point mass with � ( a lat ) 2 + ( a long ) 2 ≤ a max (Kamm’s circle) . The lateral acceleration is provided by a normalized steering input u 1 : a lat = a max u 1 . Due to the maximum absolute acceleration ( C 4), the longitudinal acceleration is bounded by � ( a max ) 2 − ( a lat ) 2 . a long = c1 Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 7 / 43

Models of the Ego Vehicle and Other Traffic Participants Physical Modeling (2) Constraints C 1 (max. velocity), C 2 (no driving backwards), and C 3 (max. engine power) are considered in another bound:  v S a max v , v S < v < v max ∧ u 2 > 0   a long = a max , (0 < v ≤ v S ∨ ( v > v S ∧ u 2 ≤ 0)) c2  0 , v ≤ 0 ∨ ( v ≥ v max ∧ u 2 > 0)  The longitudinal acceleration combining both bounds is for a normalized acceleration input u 2 ∈ [ − 1 , 1] � a long a long | u 2 | ≤ a long u 2 , c1 , a long = c2 c2 a long a long | u 2 | > a long sgn ( u 2 ) , c1 . c1 c2 Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 8 / 43

Set-Based Behavior Prediction Reachability Analysis jump exact reachable set initial set x 1 possible steady state x 2 trajectory Informal Definition A reachable set is the set of states that can be reached by a dynamical system in finite or infinite time for a set of initial states, uncertain inputs, and uncertain parameters. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 9 / 43

Set-Based Behavior Prediction Overapproximative Reachable Sets overapproximative reachable set exact invariant set reachable set initial set x 1 x 2 unsafe set Exact reachable set only for special classes computable → overapproximation computed for consecutive time intervals. Overapproximation might lead to spurious counterexamples. Simulation cannot prove correctness. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 10 / 43

Set-Based Behavior Prediction Abstraction Technique for Other Traffic Participants Overapproximative Occupancy Given are models M i , i = 1 . . . m which are abstractions of model M 0 , i.e., reach ( M 0 ) ⊆ reach ( M i ). The occupancy of the model M 0 can be overapproximated by m � � � � � reach ( M 0 ) ⊆ reach ( M i ) . proj proj � i =1 Two models: Longitudinal dynamics along road boundaries (upper bound), lateral dynamics towards road boundaries (left/right bound). lower left bound initial occupancy upper bound lower right bound Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 11 / 43

Set-Based Behavior Prediction Occupancy Along Road Boundaries The dynamics becomes monotone when following a lane center. Definition (Monotone dynamics) For the initial state x (0) ∈ R (0) and inputs u ( t ) ∈ U the dynamics is monotone when the following holds for the solution χ ( t , x (0) , u ( · )): if ∀ i , j , t ≥ 0 : x i (0) ≤ ¯ u j ( t ) ≤ ¯ x i (0) , u j ( t ) then ∀ i , t ≥ 0 : χ i ( t , x (0) , u ( · )) ≤ χ i ( t , ¯ x (0) , ¯ u ( · )) . � From this follows that e.g. the upper bound is provided by max. position, max. velocity, and max. acceleration: occupancy set for some time interval path s y s x s Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 12 / 43

Set-Based Behavior Prediction Occupancy Towards Road Boundaries For lateral dynamics there exists no single combination of an initial state and an input trajectory determining the boundary. a y a Given the vehicle-fixed angle of the acceleration φ a x vector a , possible trajectories are: 2 const. acceleration ( φ = 90 ◦ ) w const. acceleration 0 s y ( φ = 110 ◦ ) const. acceleration ( φ = 130 ◦ ) − 2 0 10 20 s x Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 13 / 43

Set-Based Behavior Prediction Occupancy Towards Road Boundaries: Method A Using limit of absolute acceleration (constraint C 4): Occupancies are circles with center c ( t ) and radius r ( t ): � s x (0) � � v x (0) � r ( t ) = 1 2 a max t 2 . c ( t ) = + t , s y (0) v y (0) From this follows the boundary of occupation: � � 2 b x ( t ) = v 0 t − a 2 max t 3 1 � a 2 max t 3 max t 4 − 4 a 2 , b y ( t ) = . 2 v 0 2 v 0 O ( t k − 2 ) r ( t k +1 ) 4 [ b x ( t ) , b y ( t )] T 0 s y c ( t k +1 ) [ b x ( t ) , − b y ( t )] T − 4 O ([ t k , t k +1 ]) 0 20 10 s x Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 14 / 43

Set-Based Behavior Prediction Occupancy Towards Road Boundaries: Method B and C Method B: Assume independence of lateral and longitudinal acceleration → Solution of time t s for switching the steering angle to avoid road departure: � a max w + 1 2 v 2 0 − v 0 t s = a max Method C: Combination of method A and B. const. acceleration ( φ = 90 ◦ ) const. acceleration 2 ( φ = 110 ◦ ) w const. acceleration 0 s y ( φ = 130 ◦ ) method A − 2 method B method C 0 10 20 s x Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 15 / 43

Set-Based Behavior Prediction Examples: Lane 1 Step 1: × 10 6 5.5915 5.5914 5.5914 5.5914 5.5914 5.5914 5.5913 5.5913 M1 M2 5.5913 M3 5.5913 3.865 3.8652 3.8654 3.8656 3.8658 3.866 3.8662 3.8664 3.8666 3.8668 × 10 5 M1: restricted absolute acceleration. M2: restricted acceleration and velocity in longitudinal direction. M3: staying within road boundaries. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 16 / 43

Set-Based Behavior Prediction Examples: Lane 1 Step 2: × 10 6 5.5915 5.5914 5.5914 5.5914 5.5914 5.5914 5.5913 5.5913 M1 M2 5.5913 M3 5.5913 3.865 3.8652 3.8654 3.8656 3.8658 3.866 3.8662 3.8664 3.8666 3.8668 × 10 5 M1: restricted absolute acceleration. M2: restricted acceleration and velocity in longitudinal direction. M3: staying within road boundaries. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 17 / 43

Set-Based Behavior Prediction Examples: Lane 1 Step 3: × 10 6 5.5915 5.5914 5.5914 5.5914 5.5914 5.5914 5.5913 5.5913 M1 M2 5.5913 M3 5.5913 3.865 3.8652 3.8654 3.8656 3.8658 3.866 3.8662 3.8664 3.8666 3.8668 × 10 5 M1: restricted absolute acceleration. M2: restricted acceleration and velocity in longitudinal direction. M3: staying within road boundaries. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 18 / 43

Set-Based Behavior Prediction Examples: Lane 1 Step 4: × 10 6 5.5915 5.5914 5.5914 5.5914 5.5914 5.5914 5.5913 5.5913 M1 M2 5.5913 M3 5.5913 3.865 3.8652 3.8654 3.8656 3.8658 3.866 3.8662 3.8664 3.8666 3.8668 × 10 5 M1: restricted absolute acceleration. M2: restricted acceleration and velocity in longitudinal direction. M3: staying within road boundaries. Matthias Althoff Nonexistence of Evasive Trajectories September 28, 2015 19 / 43