prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What is the effect of accounting for path preference on the robot's belief update?

Read the following text and answer briefly.

Context 1: Paper Info

Title: Incorporating Human Path Preferences in Robot Navigation with Minimal Interventions
Publish Date: 16 Mar 2023
Author List: Oriana Peltzer, Dylan Asmar, Mac Schwager, Mykel Kochenderfer

Figure

Hyperplane arrangement of a twodimensional space containing two obstacles (colored in gray) The robot is located inside the pink polytope, surrounded by three adjacent obstacle-free polytopes Each hyperplane on the boundary of the robot's polytope corresponds to one of the nonredundant constraints in eq (4) (b)Graph derived from the hyperplane arrangement The nodes on the graph designate polytopes, and edges designate transitions to adjacent polytopes To estimate the human's preference, the robot updates a posterior over the goal and over which of the graph transitions φ 1 , φ 2 and φ 3 is preferred by the human (c)Example preference defined over the graph The location of the goal is indicated in yellow in the lower right polytope For each node, the outgoing pink arrow designates the edge on the graph corresponding to the preferred transition between polytopes 
Simple, 10 × 10, 8 polytopes (b) Map 2: Office, 10 × 10, 56 polytopes (c) Map 3: Classroom, 20 × 20, 73 polytopes (d) Sampled observations and robot's executed trajectories 
Fig 5: Maps used for simulating the robot navigation problem with path preferences In (d), the heading angles observed are indicated with arrows The goal is indicated with a pink circle, and the orange robot corresponds to the starting location The blue robot follows a policy that accounts for path preference, while the green robot does not The opacity of the robots increases with time 
Map 1 problem setup and example realizations for goal-only (green) and path preference (blue) solution methods The robot starts at the lower left corner of the environment, and the goal of the task (pink circle) is in the upper left area The robot does not know which goal, among 10 options (shown in light blue squares), is the correct goal The human provides noisy observations, indicated by arrows, at each iteration The green robot selects actions according to the goal-only baseline, and the blue robot uses our proposed method to infer path preferences The polytopes composing G are drawn in blue Probability of correct goal WLPHVWHS +J (c) Entropy of goal distribution g 
Fig  6: Probability of the correct goal, fig 6b, and entropy of the goal belief distribution P (g), fig 6c, for the same problem setup, fig 6a In this problem instance, the human's preference is to go to the goal by passing on the right side of the obstacle Results are averaged over 50 runs and the area filled represents one standard deviation above and below the mean value The goal-only baseline shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference 
Success rates in the simple environment (Map 1) The results are averaged over 6 randomly sampled problem instances (start location, goal location, and goal possibilities), and over 50 runs per problem instance ∆T is the number of time steps separating two consecutive human inputs The robot's mission time is Tmax = 30 time steps We selected γ h = 1 5, corresponding to relatively noisy human inputs and making the problem more difficult to solve for the robot 
Computation times for Goal Only and Path Preference methods on Map 1 (fig 5a),Map 2 (fig 5b), and Map 3 (fig 5c),averaged over 100 runs with randomly sampled problem instances The 95 % confidence interval is provided with the mean We evaluate computation time at the first iteration of each run (where the search depth takes on its highest value Tmax) 

abstract

Robots that can effectively understand human intentions from actions are crucial for successful human-robot collaboration  In this work, we address the challenge of a robot navigating towards an unknown goal while also accounting for a human's preference for a particular path in the presence of obstacles 
This problem is particularly challenging when both the goal and path preference are unknown a priori  To overcome this challenge, we propose a method for encoding and inferring path preference online using a partitioning of the space into polytopes  Our approach enables joint inference over the goal and path preference using a stochastic observation model for the human 
We evaluate our method on an unknown-goal navigation problem with sparse human interventions, and find that it outperforms baseline approaches as the human's inputs become increasingly sparse  We find that the time required to update the robot's belief does not increase with the complexity of the environment, which makes our method suitable for online applications 

INTRODUCTION

Collaboration between humans and robots has become increasingly important and one key aspect of this collaboration is the ability for robots to adapt to human decisions  In many scenarios, such as a robot navigating through a busy room to deliver an item, it is important for the robot to take into account human preferences 
For instance, humans may prefer a specific path that would allow their colleagues to notice the item being delivered, but this preference may change dynamically based on various factors such as changes in the environment or unforeseen circumstances  While some preferences can be incorporated into the path-planning process, accommodating dynamic user preferences in real-time remains challenging 
In this paper, we propose a way to enable robots to adapt to human preferences dynamically by leveraging real-time feedback to inform decision-making  In this work, we tackle the problem of robot navigation in which the robot cannot observe the goal or the preferred path to the goal, but must make navigation decisions that are influenced by humans through recommended actions 
Prior work has explored how to adapt to a human's preference through feedback, but such approaches often require a high level of intervention, which can be time-consuming and impractical in real-world scenarios  To optimize the use of human input and quickly infer the human's preference, Fig  : An autonomous robot navigates in a simulated classroom towards a goal location (pink circle) 
At the start of its mission, it receives direction indications (arrows) from a human that indicate which path it should take to get to the goal  In this scenario, the human wants the robot to go around the desks on the right side of the classroom  A robot that does not reason over path preferences (green) will take the shortest path to the goal regardless of the human's input 
Our method (blue) infers the human's path preference from these indications and adapts to their recommendations  we propose an approach that leverages probabilistic representations of human preference and incorporates real-time feedback  Previous research by Bajcsy et al  considered an online adaptation problem in a manipulation task, where the person can apply forces to the robot to indicate their preferences 
By allowing the robot to continue its task while taking into account a probabilistic representation of human preference, their approach does not require frequent inputs  Building on this idea, we adopt a similar approach to adapt to a human's preference in the context of a robot autonomously navigating through a known environment, such as a cluttered office space 
Specifically, we focus on allowing the human to influence the robot's trajectory with respect to obstacles, by providing guidance on preferred routes or paths, while the robot continues to execute its task  Paths can be represented using homotopy classes   However, homotopies can pose computational challenges when used to encode and infer human preferences 
When the robot maintains a belief over homotopy classes, the inference problem can become exponentially complex with the number of obstacles in the space  Additionally, when the goal is unknown, the number of variables increases with the number of candidate destinations  This complexity can render the decision-making problem intractable 
Our solution is to encode path preference based on a partitioning of the environment into polytopes   This representation allows path preferences to be expressed as sets of preferred transitions between adjacent polytopes  Paths belonging to different homotopy classes correspond to different sequences of transitions 
By leveraging conditional independence assumptions, we can make the Bayesian inference problem tractable  These assumptions exploit the fact that human actions provide information about the path in a piece-wise manner  For example, indicating a preference for navigating around a particular obstacle only provides information about the local area and not the entire path 
Finally, after updating its belief representation over the human's preference, the robot can adapt to indications by replanning online  Our contributions are as follows  • We formulate the human-robot collaboration problem as a Partially Observable Markov Decision Process (POMDP) where both the goal of the task and the human's path preference are unknown random variables 
• We propose an encoding of a human's path preference using a partitioning of the environment into polytopes, along with conditional independence assumptions that make the Bayesian inference problem tractable to infer the task goal and path preference online  • Through simulations in two environments of different sizes and complexity, we show that our method is effective for solving problems where the robot must reach a goal that is unknown a-priori while simultaneously adapting to a human's indications 
Our method shows higher success rates compared to baseline approaches when the human inputs are sparse  Our approach enables a robot to make effective navigation decisions in collaboration with a human, even when the goal and path preference are not known in advance, and with minimal human input  In recent years, there has been a growing interest in shared autonomy and interactive systems, where humans and robots work together to accomplish tasks 
Several approaches have been proposed to address the challenge of enabling effective collaboration between human and robot agents while still achieving high task performance  Losey et al  and Jeon, Losey, and Sadigh propose a framework where a human operator is given control of a task-relevant latent action space while an autonomous system handles the rest 
Dragan and Srinivasa present a formalism for arbitrating between a user's input and a robot's policy when both human and robot share control of the same action space  Cognetti et al  [7] provide a method for real-time modifications of a path,       Fig  : We model the intent inference problem with the above diagram 
At each step in time, the robot receives an observation ot from the human conditioned on its current location st, the intended goal g, and the human's path preference θ  The robot updates its belief over g and θ and transitions to a next location st+1  while Hagenow et al  present a method that allows an outside agent to modify key robot state variables and blends the changes with the original control 
However, a common challenge of these approaches is the high level of intervention required from humans  Best and Fitch propose a method for predicting an agent's intended trajectory from observations  Rather than maintaining a belief over the agent's future path, they infer the agent's intended goal among a set of candidate locations at the boundary of the space 
This approach provides information on where the agent is heading and generates a distribution of candidate future trajectories for the agent  Inferring the goal of the task among a discrete set of candidates is also relevant to the area of shared autonomy  Javdani, Srinivasa, and Bagnell propose a formalism for shared control of a robotic arm, where the robot must assist the human in picking up an object but needs to infer which object the human has chosen from joystick inputs 
Planning with homotopy class constraints is useful in problems where the robot's requirements are given with respect to obstacles, and Yi, Goodrich, and Seppi consider topological constraints provided by human operators  Bhattacharya propose an efficient algorithm for solving pathplanning problems under homotopic constraints 
However, the number of homotopy classes for a given problem can be infinite, and as the robot changes location and updates its representation of the world, carrying out inference over homotopy classes in a dynamic environment requires recomputing the set of homotopies at every iteration, making the belief update challenging 
Prior work has addressed the challenge of shared autonomy by considering how robots can infer a human's intended goal, or how they can infer the preferred path to a goal  However, we argue that inferring the goal and the path as separate problems can lead to over-confidence in incorrect beliefs about the user's preferences 
To illustrate this point, consider the following scenario: a robot and a human are collaborating to move an object from one end of a room to Fig  : Using the hyperplanes composing the H-representation of each obstacle, we construct a hyperplane arrangement of the obstacle-free space (a)  We define the human's preference for the robot's one step action choices as the posterior distribution (given all human input up to that point) over transitions from the current to the neighboring polytopes, i e  edges on the graph 
Each time the robot transitions to a new polytope, the set of neighbor polytopes and the distribution over human preferences are updated  another, but there is an obstacle in the way  The human would like the robot to take a path around the obstacle on the left, even though the goal is on the right  If the robot only infers the goal from the human's inputs, it may incorrectly assume that the goal is on the right, and become over-confident in this belief 
On the other hand, if the robot only infers the preferred path, it may mistakenly assume that the goal is on the left, leading to a failure in completing the task  To overcome these challenges, our work proposes a joint inference approach that considers both the human's intended goal and their preferred path to that goal 
Specifically, we model the human's preference over different homotopy classes and leverage a conditional independence assumption to provide a tractable solution  In our approach, we assume that the human's inputs are noisily rational conditioned on both the goal and the preference  By jointly inferring the goal and path preference, we can avoid over-confidence in incorrect beliefs about the user's preferences, leading to improved system performance 
We consider the problem of robot navigation in a known environment to an unknown destination, where a human can intervene and provide a heading direction to the robot using a joystick or force cues  The human also has a preference on which path the robot should take with respect to obstacles, and our objective is for the robot to understand the human's intentions and execute the task with minimal interventions 
Let g be a discrete random variable denoting the goal of the task, belonging to a set of candidates Ω g , and let θ be a discrete-valued random variable representing the human's path preference, belonging to a set of possible preferences Θ  The physical location of the robot at time index t is denoted by s t ∈ R 2 , and the robot's action at time index t, belonging to some action space A, is denoted by a t  
The transition model T (s t+1 | s t , a t ) is deterministic, meaning the robot has full control over its future location  At any time step, the human may provide an observation to the robot  When the human intervenes, the robot receives a direction (heading angle) that can be mapped to a future location in space 
More specifically, we map the direction to an intended location, which is the resulting robot location after advancing in the indicated direction for one time step  For simplicity, we consider that the robot directly makes an observation o t of the location indicated by the human  We assume that the robot has a stochastic observation model for the human P (o t | s t , g, θ) that is conditioned on both the goal of the task g and the human's preferred path θ 
We further assume that having chosen a goal and path preference, the human takes actions to noisily minimize a cost function C g,θ that measures the cost of moving from the robot's current location to the goal along the preferred path  For example, C g,θ (s t , o t ) can be the length of the shortest path from location s t to the goal g after taking a first step to o t , and constrained by path preference θ 
We use C g,θ to induce a probability distribution over observations, given by: where γ h is a hyperparameter that designates the rationality coefficient  This model assumes the human will pick the lowest cost action with the highest probability and the likelihood of an action decreases exponentially with the increase in cost  
Our inclusion of the path preference θ sets our approach apart from   The model is shown in fig  represented as a Bayesian Network 

Inference

At each time step where the human provides an observation, the posterior P (g, θ) is given through the Bayesian update We note that the number of Bayesian updates required at each iteration to update the belief is equal to the cardinality of Ω g × Θ  In addition, each Bayesian update involves computing C g,θ (  
,   ) in eq  ( ), which involves solving an optimization problem (such as a shortest path problem)  In section IV, we propose a specific encoding of preference θ for resolving eq  ( ), while ensuring the number of computations of the cost C g,θ ( ,  ) per update does not grow exponentially with the number of obstacles 

Decision Making

We consider a navigation problem where the robot receives reward according to the model R(s t , g, θ, a t )  We wish to find the optimal policy π that maximizes the expected discounted sum of future rewards, with discount factor γ  The above problem is a Partially Observable Markov Decision Process (POMDP)  
In this section, we propose an encoding of human's path preference θ for computing the posterior in eq  ( )  Devifrom the concept of homotopy classes, we define the preference according to a partitioning of the environment into polytopes, as shown in fig  , creating a hyperplane arrangement of the space 
Hyperplane arrangements have been used by Vincent and Schwager in the context of Neural Network verification  In our setting, we leverage this representation to define path preferences as preferred transitions between adjacent regions of the space 

Hyperplane Arrangement

We assume a two-dimensional environment composed of m polytopic obstacles, each defined by their half-space representation (H-representation) where A i ∈ R di×2 and b i ∈ R di , and where d i is the number of edges (hyperplanes) composing polytope i  Let n = i d i be the total number of hyperplanes  We leverage each obstacle's H-representation to construct a hyperplane arrangement of the environment as shown in fig 
 e  a partitioning of the space into polytopes  More specifically, each location in space belongs to a polytope j for which we can write an H-representation of the form where α j i ∈ {−1, 1} di is a vector specific to polytope j and obstacle i corresponding to the relative position of any point in the set with respect to each hyperplane in O i  
Fig  : Intent inference model in a hyperplane arrangement of the obstacle free space  We spatially decompose the preference θ into a set of preferred neighboring polytopes per region of the space  Within each polytope j, the human preference pj is a discrete distribution over the preferred neighbor in N (j) 
We assume that for a location st belonging to polytope j, and given goal g and preference pj, the observation ot and any other preference p i,i =j are conditionally independent  Concatenating elements from each obstacle's Hrepresentation, we can write polytope j's H-representation as where Some of the constraints in eq  ( ) (corresponding to rows of A, b and α j ) are redundant, i e  the set P j does not change upon their removal 
We can further reduce the Hrepresentation of a polytope to include only non-redundant constraints  By removing the rows corresponding to redundant constraints, we obtain new matrices A j e , b j e and α j e such that we can write the polytope's reduced H-representation as The non-redundant constraints correspond to edges of the polytope 
In other words, as the robot continually moves in space, the first hyperplane that it will cross upon exiting the polytope will correspond to one of the polytope's nonredundant constraints  Vincent and Schwager outline an iterative method for removing redundant constraints by solving n linear programs 
We use this method in practice for computing α j e for each polytope  We can now characterize each polytope by a vector α j e ∈ {−1, 1} n j e , where n j e ≤ n is the number of essential constraints of the polytope  The polytopes P j partition the environment into a hyperplane arrangement 

Path Preference

In this section, we provide a definition of preference θ according to a graphical representation of the environment based on the hyperplane arrangement  Under this representation, a path preference corresponds to a set of preferred transitions  In other words, for each polytope in the space, the human will have a preference to which neighboring polytope they wish to transition 
Let G := (V, E) be an undirected graph, where vertices are obstacle-free polytopes, and edges connect two adjacent polytopes  Each polytope is described by a unique vector α j as defined in eq  ( )  Two polytopes are adjacent if they share non-redundant constraints (rows in eq  ( )) corresponding to the same hyperplane (i e  they are on opposite sides of the hyperplane) 
Let N (v) be the set of neighbors of a vertex v  For each vertex, we denote p v the discrete-valued random variable describing which edge in N (v) the human intends to transition to  Using this formalism, we define a path preference as the set of preferred transitions over all nodes in the graph, Let m θ = v∈V |N (v)| be the cardinality of Θ, and m g = |Ω g | the number of possible goals 
A priori, the number of Bayesian updates required to update the belief at every iteration should be m θ × m g   Now, let us assume the conditional independence relationships described by the new problem diagram in fig    More specifically, we introduce the assumption that conditioned on a robot location s t , the goal g, and the preference for the corresponding vertex p v in the graph, the observation o t and the preference for any other vertex are conditionally independent 
In other words, the observations the human provides can be defined conditioned only on the robot location, the goal, and the human's preference for its current vertex p v   By introducing this assumption, each update step only requires updating the joint (p v , g), reducing the number of cost computations to |N (v)| × m g  
We can notice that by introducing this assumption, we removed the direct relationship between the number of polytopes in the environment and the complexity of the Bayesian update in eq  ( )  In practice, components of θ are not mutually independent  For example, if the human preference at a vertex v 1 is
, it is unlikely that the human will also prefer p v2 = (v 2 , v 1 ) (turning back)  We can improve our model by assuming a dependent relationship between preferences for adjacent edges, which does not significantly increase the complexity of the inference problem  An interesting property of our encoding is that any two paths that belong to different homotopy classes will cross different sequences of polytopes, i e  they correspond to a different sequence of edges on G 
This can be proved by contradiction  Let us suppose that two continuous trajectories ξ 1 and ξ 2 , with the same start and end points and that do not intersect any obstacle, traverse the same regions in G in the same order  From the construction of the hyperplane arrangement, each polytope that the paths traverse through is obstacle-free 
Therefore, within each polytope, there is no obstacle in the area located in between the portions of ξ 1 and ξ 2 that belong to the region  A smooth transformation of ξ 1 into ξ 2 can be obtained by transforming each portion of ξ 1 belonging to the polytopes it intersects into the corresponding portion of ξ 2 for the same polytopes, where the extremities of the trajectory portions are connected to one another along the polytope's edges (where the same edge is crossed by both paths) 
Along this transformation, the paths do not intersect any obstacle, and therefore ξ 1 and ξ 2 belong to the same homotopy class 

EXPERIMENTS

We evaluate our model on a simulated navigation task where the robot must reach a goal that is unknown a priori while respecting the path preferences indicated by a human  The robot navigates in a grid world containing obstacles  The transition model is deterministic: the robot selects an adjacent location on the grid to reach at the next time step 
The robot is also allowed to take diagonal actions  Each location s t in the map can be mapped to a vertex v t ∈ G  Therefore, the actions leading to locations mapped to different vertices correspond to edges on the graph  We note f (s t , a t ) the edge crossed by taking action a t from location s t  
The robot is given a mission time limit T max for reaching the goal  In this problem, we assume that the human selects actions to noisily minimize a cost function C g,θ , where θ is defined as per eq  ( ), corresponding to the length of the shortest path to the goal constrained by the preference (where the robot is only allowed to make transitions on G along preferred edges) 
More specifically, where δ(s t , g | o t , p vt ) designates the length of the shortest path from s t to g passing by o t and constrained by preference p vt   This is a slight variant of the cost function proposed by Best and Fitch , where we add in a conditioning on the path preference  We compute costs by running the A path planning algorithm on the environment maps (grid worlds with diagonal actions) and impose preference constraints by pruning invalid transitions from the search tree 
Reward model  At each step in time, the robot receives a reward which is a sum of three components: a goal-specific reward a preference-specific reward or penalty We compute solutions to the POMDP defined in section III-B with the online solver POMCP , and with the particularity that within the rollouts, the robot does not expect to collect human inputs 
Each time a solution is computed, the robot takes an action and may receive an observation  If it does, it updates its belief distribution over the unknown problem variables and resolves the POMDP over a receding horizon 

Baselines

• Goal only  The robot solves the POMDP while ignoring the effects of path preference  Similarly to , we assume the human is taking action to minimize a goaldependent cost C g (s t , o t ) = δ(s t , g | o t ), where the conditioning on the preference is removed  We also omit the path preference's contribution to the reward R pref  
• Compliant  The robot complies with the human input, but does not take an initiative  If the user stops providing information, the robot continues in the last direction indicated for 5 time steps (conserving its momentum), then stops  • Blended  We designed an arbitration function to decide between our proposed policy (accounting for path preferences) and the user's recommendation when the robot receives inputs 
Our metric to evaluate confidence in the robot's prediction for the purpose of arbitration is the entropy of the intention distribution H(g, p i ), where p i denotes the preferred neighbor for the current region  Because our representation of the world is discrete, the arbitration is given by a step function 
Denoted by U , the action corresponding to the human's input, and P , the robot's prediction for the optimal action, we write the policy where we chose h = 1 6 as the confidence threshold 

Results

When evaluating the algorithm, we consider that a run is successful if the robot reached the goal within its allocated mission time T max and only made transitions between graph vertices corresponding to the human's preferences  We vary the time delay between human inputs, from constant guidance (∆ T = 1) to only a single observation (∆ T ≥ T max ) 
Success rates  Table I reports the success rates for experiments conducted over six randomly sampled problem instances and 50 runs per instance in Map 1 (fig  )  When the human provides inputs at every iteration, the compliant policy shows the highest success rates  However, as ∆ T increases, the compliant robot is not able to accomplish the task within the allotted time as it does not receive sufficient inputs to do so, and performance decreases compared to the autonomous baselines 
We find that in these runs, accounting for path preference consistently improves performance compared with the goal-only baseline  Results also show that blending the user's input with the robot's policy (Path Preference + Blend) when the human provides information leads to improved performance  Belief entropy 
Figure shows a challenging problem instance where the directions the human provides do not align directly with the shortest path to the goal  By ignoring the effects of preferences in the problem model (goal only), the robot quickly infers from observations that the upper left goal is less likely than others (P (g) drops) 
The strong decrease in entropy shows that the robot becomes overconfident in this prediction  Overconfidence in an incorrect goal will prevent the agent from finding the correct goal once the human's indications directly align with it, as it needs to correct for the wrong predictions, as shown in the path realization (fig 
)  In this realization, the goal-only method (green robot) fails to search the upper left area within the allotted time  By accounting for path preferences in its model, the blue robot's entropy over the goal distribution decreases more steadily, allowing for it to leverage the human's latest observations and reach the goal successfully 
shows an over-confident prediction (shown by the strong reduction in belief entropy) that the correct goal is less likely, making it more difficult to reach the correct goal compared to a method that accounts for path preference  Computation time  In table II we provide the time required to solve the POMDP, and the time required to update the robot's belief as it receives new observations 
We compute solutions on three maps: a simple 10 × 10 grid world with 8 polytopes (fig  ), a 10 × 10 grid world with 56 polytopes (fig  ), and a 20×20 grid world with 73 polytopes (fig  )  The latter environment being larger, we increase the mission time and the depth of the search tree in POMCP from T max = 30 (Map 1 and Map 2) to T max = 60 (Map 3) 
We do not notice an increase in the time required to update the robot's belief with an increase in problem complexity, which is consistent with our observation that the complexity of the Bayesian update should not increase with the number of obstacles or polytopes  On the contrary, the belief update time on Map 2 and Map 3, containing more obstacles, is reduced compared to the first map 
More obstacles result in fewer iterations when solving the constrained shortest path problem with A   Adding constraints due to the obstacles and polytopes reduces the size of the A search tree  C  Limitations Simulation environments  In our simulations, we hardcoded the preference policy over the maps (e g  in Map 1, go around the table counter-clockwise) 
We randomly sampled problem instances (start and goal locations, and goal options) to reduce the bias introduced by these preference choices  To best evaluate and compare the different approaches, it would be best to sample preferences among a distribution of preferences chosen by a human (for example, from benchmarks resulting from a collection of data) 
Creating such a benchmark is an interesting direction for future work  Hyperplane arrangement construction  The main limitation of our approach is that the size and geometry of each polytope depends strongly on the geometry of the obstacles, as seen in fig    Because of this, the robot can make predictions over preferences that are too refined compared with the topology of the environment 
A direct consequence is that when the size of the polytopes is small, the information provided by the human can be incorrectly interpreted as a preference on the robot's immediate action  Our method can be improved by changing the structure of the hyperplane arrangement so that it relies on the topology of the environment, but does not vary strongly with the geometry of the features in the environment 
For this purpose, topometric maps and region construction algorithms are promising directions  We presented an approach for encoding and inferring a human's path preference in an environment with obstacles  By leveraging a partitioning of the space into polytopes and a stochastic observation model, our method allows for joint inference over the goal and path preference even when both are unknown a-priori 
Our experiments on an unknown-goal navigation problem with sparse human interventions demonstrate the effectiveness of our approach and its suitability for online applications  The time required to update the robot's belief does not increase with the complexity of the environment, which further highlights the practicality of our method 
Context 2: Paper Info

Title: Bistability between π-diradical open-shell and closed-shell states in indeno[1,2-a]fluorene
Publish Date: Unkown
Author List: Shantanu Mishra (from IBM Research Europe -Zurich), Manuel Vilas-Varela (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leonard-Alexander Lieske (from IBM Research Europe -Zurich), Ricardo Ortiz (from Donostia International Physics Center (DIPC)), Igor Rončević (from Department of Chemistry, University of Oxford), Florian Albrecht (from IBM Research Europe -Zurich), Diego Peña (from Department of Organic Chemistry, Center for Research in Biological Chemistry and Molecular Materials (CiQUS), University of Santiago de Compostela), Leo Gross (from IBM Research Europe -Zurich)

Figure

Fig  1 | Non-benzenoid non-alternant polycyclic conjugated hydrocarbons a, Classical nonbenzenoid non-alternant polycyclic conjugated hydrocarbons: pentalene, azulene and heptalene b, Generation of indacenes and indenoindenes through benzinterposition and benzannelation of pentalene, respectively Gray filled rings represent Clar sextets c, Closed-shell Kekulé (left) and openshell non-Kekulé (right) resonance structures of QDMs Note that meta-QDM is a non-Kekulé molecule All indenofluorene isomers, being derived through benzannelation of indacenes, contain a central QDM moiety d, Closed-shell Kekulé (top) and open-shell non-Kekulé (bottom) resonance structures of indenofluorenes Compared to their closed-shell structures, 1 and 5 gain two Clar sextets in the openshell structure, while 2-4 gain only one Clar sextet in the open-shell structure Colored bonds in d highlight the ortho-and para-QDM moieties in the two closed-shell Kekulé structures of 5  e, Scheme of on-surface generation of 5 by voltage pulse-induced dehydrogenation of 6 (C20H14) Structures 7 and 8 represent the two monoradical species (C20H13) 
Fig  2 | Characterization of open-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of 5OS in the triplet configuration for the spin up (occupied) level (isovalue: 0 002 e -Å -3 ) Blue and red colors represent opposite phases of the wave function b, Corresponding DFT-calculated spin density of 5OS (isovalue: 0 01 e -Å -3) Blue and orange colors represent spin up and spin down densities, respectively c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) d, DFT-calculated bond lengths of 5OS e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig 7 f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible Scanning parameters: I = 0 3 pA (V = -1 2V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively Scale bars: 10 Å (f) and 5 Å (g) 
Fig  3 | Characterization of closed-shell indeno[1,2-a]fluorene on bilayer NaCl/Au(111) a, DFTcalculated wave functions of the frontier orbitals of closed-shell 5 0 (isovalue: 0 002 e -Å -3 ) The wave functions shown here are calculated for the 5para geometry b, DFT-calculated bond lengths of 5ortho (top) and 5para (bottom) c, Constant-height I(V) spectra acquired on a species of 5 assigned as 5para, along with the corresponding dI/dV(V) spectra Open feedback parameters: V = -2 V, I = 0 15 pA (negative bias side) and V = 2 2 V, I = 0 15 pA (positive bias side) Acquisition position of the spectra is shown in Supplementary Fig  7  d, Scheme of many-body transitions associated to the measured ionic resonances of 5para Also shown are STM images of assigned 5para at biases where the corresponding transitions become accessible Scanning parameters: I = 0 15 pA (V = -1 5 V) and 0 2 pA (V = 1 7 V)  e, Laplace-filtered AFM image of assigned 5para STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 7 Å  f, Selected bonds labeled for highlighting bond order differences between 5para and 5ortho For the bond pairs a/b, c/d and e/f, the bonds labeled in bold exhibit a higher bond order than their neighboring labeled bonds in 5para g, Laplace-filtered AFM images of 5 on bilayer NaCl/Cu(111) showing switching between 5OS and 5para as the molecule changes its adsorption position The faint protrusion adjacent to 5 is a defect that stabilizes the adsorption of 5  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3Å  STM and STS data in c and d are acquired on the same species, while the AFM data in e is acquired on a different species Scale bars: 10 Å (d) and 5 Å (e,g) 
NMR (300 MHz, CDCl3) δ: 7 51 (m, 2H), 7 40 -7 28 (m, 5H), 7 27 -7 20 (m, 2H), 7 13 (d, J = 7 7 Hz, 1H), 2 07 (s, 3H), 1 77 (s, 3H) ppm  13C NMR-DEPT (75 MHz, CDCl3, 1:1 mixture of atropisomers) δ: 141 2 (C), 141 1 (C), 140 0 (C), 139 4 (2C), 137 5 (C), 137 4 (C), 136 0 (3C), 134 8 (C), 134 5 (C), 134 1 (C), 134 0 (C), 133 7 (C), 133 6 (C), 131 6 (CH), 131 2 (CH), 131 1 (CH), 130 7 (CH), 129 8 (CH), 129 7 (CH), 129 5 (CH), 129 4 (CH), 129 0 (CH), 128 9 (CH), 128 7 (2CH), 128 6 (2CH), 127 2 (CH), 127 1 (CH), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 20 6 (CH3), 20 5 (CH3), 17 7 (CH3), 17 5 (CH3) ppm MS (APCI) m/z (%): 327 (M+1, 100) HRMS: C20H16Cl2; calculated: 327 0702, found: 327 0709 
NMR (500 MHz, CDCl3) δ: 7 93 (d, J = 7 6 Hz, 1H), 7 85 (d, J = 7 5 Hz, 1H), 7 78 (d, J = 7 7 Hz, 1H), 7 65 (d, J = 7 4 Hz, 1H), 7 61 (d, J = 7 5 Hz, 1H), 7 59 (d, J = 7 7 Hz, 1H), 7 47 (ddd, J = 8 4, 7 2, 1 1 Hz, 1H), 7 42 (dd, J = 8 1, 7 0 Hz, 1H), 7 35 (m, 2H), 4 22 (s, 3H), 4 02 (s, 3H) ppm  13C NMR-DEPT (125 MHz, CDCl3) δ: 144 1 (C), 143 3 (C), 142 3 (C), 141 9 (C), 141 8 (C), 141 2 (C), 138 2 (C), 136 5 (C), 127 0 (CH), 126 9 (CH), 126 7 (CH), 126 6 (CH), 125 3 (CH), 125 2 (CH), 123 6 (CH), 122 2 (CH), 119 9 (CH), 118 4 (CH), 37 4 (CH2), 36 3 (CH2) ppm MS (APCI) m/z (%): 254 (M+, 88) HRMS: C20H14; calculated: 254 1090, found: 254 1090 

abstract

Indenofluorenes are non-benzenoid conjugated hydrocarbons that have received great interest owing to their unusual electronic structure and potential applications in nonlinear optics and photovoltaics  Here, we report the generation of unsubstituted indeno[1,2-a]fluorene, the final and yet unreported parent indenofluorene regioisomer, on various surfaces by cleavage of two C-H bonds in 7,12-dihydro indeno[1,2-a]fluorene through voltage pulses applied by the tip of a combined scanning tunneling microscope and atomic force microscope 
On bilayer NaCl on Au(111), indeno[1,2a]fluorene is in the neutral charge state, while it exhibits charge bistability between neutral and anionic states on the lower work function surfaces of bilayer NaCl on Ag(111) and Cu(111)  In the neutral state, indeno[1,2-a]fluorene exhibits either of two ground states: an open-shell π-diradical state, predicted to be a triplet by density functional and multireference many-body perturbation theory calculations, or a closedshell state with a para-quinodimethane moiety in the as-indacene core 
Switching between open-and closed-shell states of a single molecule is observed by changing its adsorption site on NaCl  The inclusion of non-benzenoid carbocyclic rings is a viable route to tune the physicochemical properties of polycyclic conjugated hydrocarbons (PCHs)   Non-benzenoid polycycles may lead to local changes in strain, conjugation, aromaticity, and, relevant to the context of the present work, induce an open-shell ground state of the corresponding PCHs  
Many nonbenzenoid PCHs are also non-alternant, where the presence of odd-membered polycycles breaks the bipartite symmetry of the molecular network   Figure shows classical examples of non-benzenoid non-alternant PCHs, namely, pentalene, azulene and heptalene  Whereas azulene is a stable PCH exhibiting Hückel aromaticity ([4n+2] π-electrons, n = 2), pentalene and heptalene are unstable Hückel antiaromatic compounds with [4n] π-electrons, n = 2 (pentalene) and n = 3 (heptalene) 
Benzinterposition of pentalene generates indacenes, consisting of two isomers s-indacene and as-indacene (Fig  )  Apart from being antiaromatic, indacenes also contain proaromatic quinodimethane (QDM) moieties (Fig  ) , which endows them with potential open-shell character  While the parent s-indacene and asindacene have never been isolated, stable derivatives of s-indacene bearing bulky substituents have been synthesized  
A feasible strategy to isolate congeners of otherwise unstable non-benzenoid non-alternant PCHs is through fusion of benzenoid rings at the ends of the π-system, that is, benzannelation  For example, while the parent pentalene is unstable, the benzannelated congener indeno[2,1-a]indene is stable under ambient conditions (Fig  )  
However, the position of benzannelation is crucial for stability: although indeno[2,1a]indene is stable, its regioisomer indeno[1,2-a]indene (Fig  ) oxidizes under ambient conditions   Similarly, benzannelation of indacenes gives rise to the family of PCHs known as indenofluorenes (Fig  ), which constitute the topic of the present work 
Depending on the benzannelation position and the indacene core, five regioisomers can be constructed, namely, indeno [ Practical interest in indenofluorenes stems from their low frontier orbital gap and excellent electrochemical characteristics that render them as useful components in organic electronic devices  
The potential open-shell character of indenofluorenes has led to several theoretical studies on their use as non-linear optical materials and as candidates for singlet fission in organic photovoltaics   Recent theoretical work has also shown that indenofluorene-based ladder polymers may exhibit fractionalized excitations 
Fundamentally, indenofluorenes represent model systems to study the interplay between aromaticity and magnetism at the molecular scale   Motivated by many of these prospects, the last decade has witnessed intensive synthetic efforts toward the realization of indenofluorenes  Derivatives of 1-4 have been realized in solution , while 1-3 have also been synthesized on surfaces and characterized using scanning tunneling microscopy (STM) and atomic force microscopy (AFM), which provide information on molecular orbital densities , molecular structure and oxidation state  
With regards to the open-shell character of indenofluorenes, 2-4 are theoretically and experimentally interpreted to be closed-shell, while calculations indicate that 1 and 5 should exhibit open-shell ground states   Bulk characterization of mesitylsubstituted 1, including X-ray crystallography, temperature-dependent NMR, and electron spin resonance spectroscopy, provided indications of its open-shell ground state  
Electronic characterization of 1 on Au(111) surface using scanning tunneling spectroscopy (STS) revealed a low electronic gap of 0 4 eV (ref  )  However, no experimental proof of an openshell ground state of 1 on Au(111), such as detection of singly occupied molecular orbitals (SOMOs) or spin excitations and correlations due to unpaired electrons , was shown 
In this work, we report the generation and characterization of unsubstituted 5  Our research is motivated by theoretical calculations that indicate 5 to exhibit the largest diradical character among all indenofluorene isomers   The same calculations also predict that 5 should possess a triplet ground state 
Therefore, 5 would qualify as a Kekulé triplet, of which only a handful of examples exist   However, definitive synthesis of 5 has never been reported so far  Previously, Dressler et al  reported transient isolation of mesityl-substituted 5, where it decomposed both in the solution and in solid state , and only the structural proof of the corresponding dianion was obtained 
On-surface generation of a derivative of 5, starting from truxene as a precursor, was recently reported   STM data on this compound, containing the indeno[1,2-a]fluorene moiety as part of a larger PCH, was interpreted to indicate its open-shell ground state  However, the results did not imply the ground state of unsubstituted 5  Here, we show that on insulating surfaces 5 can exhibit either of two ground states: an open-shell or a closed-shell 
We infer the existence of these two ground states based on high-resolution AFM imaging with bond-order discrimination and STM imaging of molecular orbital densities   AFM imaging reveals molecules with two different geometries  Characteristic bond-order differences in the two geometries concur with the geometry of either an open-or a closed-shell state 
Concurrently, STM images at ionic resonances show molecular orbital densities corresponding to SOMOs for the open-shell geometry, but orbital densities of the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) for the closed-shell geometry  Our experimental results are in good agreement with density functional theory (DFT) and multireference perturbation theory calculations 
Finally, we observe switching between open-and closed-shell states of a single molecule by changing its adsorption site on the surface  Synthetic strategy toward indeno[1,2-a]fluorene  The generation of 5 relies on the solution-phase synthesis of the precursor 7,12-dihydro indeno[1,2-a]fluorene (6)  Details on synthesis and characterization of 6 are reported in Supplementary Figs 
  Single molecules of 6 are deposited on coinage metal (Au(111), Ag(111) and Cu(111)) or insulator surfaces  In our work, insulating surfaces correspond to two monolayer-thick (denoted as bilayer) NaCl on coinage metal surfaces
Context 3: Paper Info

Title: SLAS: Speed and Lane Advisory System for Highway Navigation
Publish Date: Unkown
Author List: Faizan Tariq, David Isele, John Baras, Sangjae Bae

Figure

Fig  1 Motivational Example With a slow moving vehicle ahead, the ego vehicle (in blue) may decide to either change lane to the fast moving lane (left) to minimize travel time or adjust its speed without changing lanes to preserve safety but it would be unwise for it to switch to the slow moving lane (right) as that would not benefit travel time or safety 
Fig  3  Simulation Setup Scenario Runner sets up the scenario for the CARLA Simulator, which then communicates with the SLAS and the Planning and Control ROS (Robot Operating System) nodes through the ROS bridge node 
Fig  4  Testing scenario with three lanes: lane 0 (left), lane 1 (center) and lane 2 (right) The expected motion of the ego vehicle, over the course of the simulation, is shown with numbered frames The right most lane (lane 3) is reserved for merging traffic so it is not utilized in our simulation 
Fig  5  Left: Travel time comparison Center: Lane choice (lateral position) comparison The center lines of lanes 0 (left), 1 (center) and 2 (right) have fixed lateral displacements of 0m, 3 5m and 7m respectively Right: Headway comparison With no leading vehicle, the headway is restricted by the visibility range of 50m 

abstract

This paper proposes a hierarchical autonomous vehicle navigation architecture, composed of a high-level speed and lane advisory system (SLAS) coupled with low-level trajectory generation and trajectory following modules  Specifically, we target a multi-lane highway driving scenario where an autonomous ego vehicle navigates in traffic 
We propose a novel receding horizon mixed-integer optimization based method for SLAS with the objective to minimize travel time while accounting for passenger comfort  We further incorporate various modifications in the proposed approach to improve the overall computational efficiency and achieve real-time performance 
We demonstrate the efficacy of the proposed approach in contrast to the existing methods, when applied in conjunction with state-of-the-art trajectory generation and trajectory following frameworks, in a CARLA simulation environment 

INTRODUCTION

Lane changing is considered to be one of the most risky driving behaviors since it is highly contingent upon multimodal trajectory predictions of neighboring vehicles and requires timely decision making   It is further influenced by a number of uncertainty factors such as road conditions, measurement accuracy, and a long tail of behavioral uncertainty of on-road agents 
However, if executed efficiently, lane changing coupled with speed adjustment can yield significant improvement in minimizing overall travel time while ensuring passenger comfort   To elaborate further, consider the scenario presented in Fig    Based on the predicted motion (shown in a lighter shade) of the neighboring vehicles (shown in orange), the ego vehicle (shown in blue) may decide to either change lane left in an attempt to minimize its travel time or slow down in the current lane to maintain safety 
However, it would be imprudent for the ego vehicle to risk changing lane right and consequently get stuck behind a slow moving vehicle even though there is presently a greater headway  This simple scenario highlights the importance of foresight and long planning-horizon in strategic decision making for autonomous vehicles 
Existing methods like MOBIL give us the ability to change lanes but behave greedily (prioritizing immediate rewards) oftentimes, which can lead to sub-optimal performance  It was shown in that the lane changing performance can be improved with an A inspired approach, but the formulation was limited to constant speed 
Such an approach is unable to assess the benefits of speed adjustment 1 University of Maryland, College Park, MD, USA  Email: {mftariq,baras}@umd edu  2 Honda Research Institute, San Jose, CA, USA  Email: {disele,sbae}@honda-ri com  Research supported by Honda Research Institute, USA   in minimizing overall travel time 
As will become apparent in Section IV, it may be necessary at times to sacrifice on shortterm benefits to gain long-term performance improvements  In such a scenario, an approach with speed adjustment coupled with long planning horizon has the foresight to deliver significantly better results  Moreover, the inclusion of speed adjustment in the decision making process inhibits the risk of incurring trajectory infeasibility as the environment conditions may prevent the ego vehicle from traveling at a constant reference speed and the low-level planner may be unable to handle such a discrepancy 
Therefore, in this work, we propose a low complexity receding horizon optimization based approach that outputs the lane change maneuvers coupled with speed adjustments for long planning horizons (> 15s) while guaranteeing safety  The long horizon strategic decision making gives ego vehicle the ability to proactively anticipate and handle challenging driving situations 
Literature review: In the literature, speed and lane changing decisions are generally considered from a motion planner's perspective , which allows for a simultaneous determination of target lanes and waypoints to perform the maneuver  The motion planning methods present in the literature can broadly be categorized into sampling-based, learning-based and optimization-based approaches 
In regards to the sampling-based approaches, single-query methods, in particular the different variants of RRT, are preferred over multi-query methods, like roadmap-based methods, due to the faster execution time and their ability to incorporate non-holonomic constraints   Even though these methods are able to incorporate safety guarantees by sampling feasible trajectories from a reachable safe set , the overall driving experience is often rather uncomfortable due to the concatenation of individual trajectories 
Moreover, the asymptotic optimality guarantees availed by arXiv:2303 00861v1 [cs RO] 1 Mar 2023 these methods do not help with real-world implementation in complex driving scenarios since they tend to have high sample complexity   In terms of the learning-based methods, the preferred approach seems to be the variations of Reinforcement Learning techniques applied in a simulated environment , , , ,  
These approaches, although seeming to work well in simulation, have concerns regarding real-world implementation due to the large amount of training data that they require, the exploration of unsafe behaviors during training, and a general inability to handle edge cases  They mainly utilize neural networks as function approximators which yields low computational complexity but also results in a lack of explainability and safety guarantees 
Lastly, the optimization-based approaches, especially the derivatives of optimal control methods, are abundant in the literature  In contrast to the potential-field based approaches that yield decent collision avoidance performance but are unable to accommodate vehicle dynamics, the optimal control methods , especially the derivatives of Model Predictive Control (MPC) approach , , , yield excellent collision avoidance performance while accommodating vehicle dynamics 
However, this performance comes at a cost of high computational complexity, arising mainly from the non-convex collision avoidance and the non-linear dynamics constraints  This, in turn, restricts the planning horizon to merely a few seconds  The key requirements for the algorithmic design of an autonomous vehicle include real-time operation, safety guarantees, optimality with respect to some metric(s), and accounting for the behavior variability of on-road agents 
Considering these requirements, we propose an optimization-based behavioral planning framework that enables autonomous vehicle maneuvering on multilane highways  While having the benefits of optimizationbased approaches, our method achieves a low computational complexity by employing a binary representation of the decoupled lane indicator dynamics in lieu of lateral dynamics, and utilizing algorithmic modifications to aid numerical computations 
Specifically, our method provides: • optimality with respect to travel time and comfort; • safety and feasibility guarantees; • real-time applicability for a long planning horizon; and • modularity in design, which enables the integration of external trajectory prediction modules  The proposed method fills in the research gap by meeting all the key algorithmic requirements while simultaneously gaining the foresight to make strategic decisions that yield long-term performance benefits, as verified in Section IV 
In this section, we present the algorithmic pipeline and formalize the road, observation and vehicle dynamics models that will be utilized in the subsequent sections 

Algorithmic Pipeline

Fig  illustrates the algorithmic pipeline of the proposed navigation architecture, in reference to the various existing   Algorithmic pipeline of the proposed navigation architecture  The raw sensory input data is processed by the Perception, and Simultaneous Localization and Mapping (SLAM) modules to place the autonomous vehicle relative to the various environmental entities in a unified frame of reference 
This information is then passed on to the navigation stack, composed of the behavioral planning, motion planning, and vehicle control modules  The output of the navigation module is passed down further in terms of actuation commands (brake, throttle and steering) to the actuators  algorithmic modules deployed on an autonomous vehicle 
The taxonomy of the various components of the navigation stack (highlighted by the dotted rectangle) is borrowed from   This pipeline essentially improves the pipeline introduced in by adding a speed advisory system  Our main focus is the development of the behavior planning module, highlighted as SLAS in Fig    SLAS outputs the target lane and reference speed which are utilized by the motion planning module to generate a reference trajectory for the ego vehicle 
The vehicle controllers compute the throttle and steering commands to track the trajectory accordingly  For the motion planning module, we adopt the Neural Networks integrated Model Predictive Control (NNMPC) due to its ability to accommodate the behaviors of neighboring vehicles in the trajectory generation process 
In our approach, we assume that the perception (of other vehicles) and the localization (of ego vehicle) are known without any uncertainty, for simplicity, but the modular architecture avails us the ability to integrate any perception or SLAM module in the overall framework



ground_truth: The belief entropy decreases more steadily.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the two ground states observed for indeno[1,2-a]fluorene on NaCl surfaces?

Read the following text and answer briefly.

Context 2:  Voltage pulses ranging between 4-6 V are applied by the tip of a combined STM/AFM system, which result in cleavage of one C-H bond at each of the pentagonal apices of 6, thereby leading to the generation of 5 (Fig  ) 
In the main text, we focus on the generation and characterization of 5 on insulating surfaces  Generation and characterization of 5 on coinage metal surfaces is shown in Supplementary Fig    )  Blue and orange colors represent spin up and spin down densities, respectively  c, Probability density of the SOMOs of 5OS (isovalue: 0 001 e -Å -3 ) 
d, DFT-calculated bond lengths of 5OS  e, Constant-height I(V) spectra acquired on a species of 5 assigned as 5OS, along with the corresponding dI/dV(V) spectra  Open feedback parameters: V = -2 V, I = 0 17 pA (negative bias side) and V = 2 V, I = 0 17 pA (positive bias side)  Acquisition position of the spectra is shown in Supplementary Fig    f, Scheme of many-body transitions associated to the measured ionic resonances of 5OS 
Also shown are STM images of assigned 5OS at biases where the corresponding transitions become accessible  Scanning parameters: I = 0 3 pA (V = -1 2 V and -1 5 V) and 0 2 pA (V = 1 3 V and 1 6 V)  g, Laplace-filtered AFM image of assigned 5OS  STM set point: V = 0 2 V, I = 0 5 pA on bilayer NaCl, Δz = -0 3
Å  The tip-height offset Δz for each panel is provided with respect to the STM setpoint, and positive (negative) values of Δz denote tip approach (retraction) from the STM setpoint  f and g show the same molecule at the same adsorption site, which is next to a trilayer NaCl island  The bright and dark features in the trilayer NaCl island in g correspond to Cl -and Na + ions, respectively 
Scale bars: 10 Å (f) and 5 Å (g)  To experimentally explore the electronic structure of 5, we used bilayer NaCl films on coinage metal surfaces to electronically decouple the molecule from the metal surfaces  Before presenting the experimental findings, we summarize the results of our theoretical calculations performed on 5 in the neutral charge state (denoted as 5 0 ) 
We start by performing DFT calculations on 5 0 in the gas phase  Geometry optimization performed at the spin-unrestricted UB3LYP/6-31G level of theory leads to one local minimum, 5OS, the geometry of which corresponds to the open-shell resonance structure of 5 (Fig  , the label OS denotes open-shell) 
The triplet electronic configuration of 5OS is the lowest-energy state, with the openshell singlet configuration 90 meV higher in energy  Geometry optimization performed at the restricted closed-shell RB3LYP/6-31G level reveals two local minima, 5para and 5ortho, the geometries of which (Fig  ) exhibit bond length alternations in line with the presence of a para-or an ortho-QDM moiety, respectively, in the as-indacene core of the closed-shell resonance structures of 5 (Fig  )  
Relative to 5OS in the triplet configuration, 5para and 5ortho are 0 40 and 0 43 eV higher in energy, respectively  Additional DFT results are shown in Supplementary Fig    To gain more accurate insights into the theoretical electronic structure of 5, we performed multireference perturbation theory calculations (Supplementary Fig  ) based on quasi-degenerate second-order n-electron valence state perturbation theory (QD-NEVPT2) 
In so far as the order of the ground and excited states are concerned, the results of QD-NEVPT2 calculations qualitatively match with DFT calculations  For 5OS, the triplet configuration remains the lowest-energy state, with the open-shell singlet configuration 60 meV higher in energy  The energy differences between the open-and closed-shell states are substantially reduced in QD-NEVPT2 calculations, with 5para and 5ortho only 0 11 and 0 21 eV higher in energy, respectively, compared to 5OS in the triplet configuration 
We also performed nucleus-independent chemical shift calculations to probe local aromaticity of 5 in the openand closed-shell states  While 5OS in the triplet configuration exhibits local aromaticity at the terminal benzenoid rings, 5OS in the open-shell singlet configuration, 5para and 5ortho all display antiaromaticity (Supplementary Fig  ) 
The choice of the insulating surface determines the charge state of 5: while 5 adopts neutral charge state on the high work function bilayer NaCl/Au(111) surface (irrespective of its openor closed-shell state, Supplementary Fig  ), 5 exhibits charge bistability between 5 0 and the anionic state 5 -1 on the lower work function bilayer NaCl/Ag(111) and Cu(111) surfaces (Supplementary Figs  ) 
In the main text, we focus on the characterization of 5 on bilayer NaCl/Au(111)  Characterization of charge bistable 5 is reported in Supplementary Figs    We first describe experiments on 5 on bilayer NaCl/Au(111), where 5 exhibits a geometry corresponding to the calculated 5OS geometry, and an open-shell electronic configuration 
We compare the experimental data on this species to calculations on 5OS with a triplet configuration, as theory predicts a triplet ground state for 5OS  For 5OS, the calculated frontier orbitals correspond to the SOMOs ψ1 and ψ2 (Fig  ), whose spin up levels are occupied and the spin down levels are empty 
Figure shows the DFT-calculated bond lengths of 5OS, where the two salient features, namely, the small difference in the bond lengths within each ring and the notably longer bond lengths in the pentagonal rings, agree with the open-shell resonance structure of 5 (Fig  )  Figure shows an AFM image of 5 adsorbed on bilayer NaCl/Au(111) that we assign as 5OS, where the bond-order differences qualitatively correspond to the calculated 5OS geometry (discussed and compared to the closed-shell state below) 
Differential conductance spectra (dI/dV(V), where I and V denote the tunneling current and bias voltage, respectively) acquired on assigned 5OS exhibit two peaks centered at -1 5 V and 1 6 V (Fig  ), which we assign to the positive and negative ion resonances (PIR and NIR), respectively  Figure shows the corresponding STM images acquired at the onset (V = -1 2
V/1 3 V) and the peak (V = -1 5 V/1 6 V) of the ionic resonances  To draw a correspondence between the STM images and the molecular orbital densities, we consider tunneling events as many-body electronic transitions between different charge states of 5OS (Fig  )  Within this framework, the PIR corresponds to transitions between 5 0 and the cationic state 5  
At the onset of the PIR at -1 2 V, an electron can only be detached from the SOMO ψ1 and the corresponding STM image at -1 2 V shows the orbital density of ψ1  Increasing the bias to the peak of the PIR at -1 5 V, it becomes possible to also empty the SOMO ψ2, such that the corresponding STM image shows the superposition of ψ1 and ψ2, that is, |ψ1| 2 + |ψ2| 2 (ref 
)  Similarly, the NIR corresponds to transitions between 5 0 and 5 -1   At the NIR onset of 1 3 V, only electron attachment to ψ2 is energetically possible  At 1 6 V, electron attachment to ψ1 also becomes possible, and the corresponding STM image shows the superposition of ψ1 and ψ2  The observation of the orbital densities of SOMOs, and not the hybridized HOMO and LUMO, proves the open-shell ground state of assigned 5OS 
Measurements of the monoradical species with a doublet ground state are shown in Supplementary Fig    Unexpectedly, another species of 5 was also experimentally observed that exhibited a closedshell ground state  In contrast to 5OS, where the frontier orbitals correspond to the SOMOs ψ1 and ψ2, DFT calculations predict orbitals of different shapes and symmetries for 5para and 5ortho, denoted as α and β and shown in Fig   
For 5ortho, α and β correspond to HOMO and LUMO, respectively  The orbitals are inverted in energy and occupation for 5para, where β is the HOMO and α is the LUMO  Fig  shows an AFM image of 5 that we assign as 5para  We experimentally infer its closed-shell state first by using qualitative bond order discrimination by AFM 
In high-resolution AFM imaging, chemical bonds with higher bond order are imaged brighter (that is, with higher frequency shift Δf) due to stronger repulsive forces, and they appear shorter   In Fig  , we label seven bonds whose bond orders show significant qualitative differences in the calculated 5ortho, 5para (Fig  ) and 5OS (Fig  ) geometries 
In 5para, the bonds b and d exhibit a higher bond order than a and c, respectively  This pattern is reversed for 5ortho, while the bond orders of the bonds a-d are all similar and small for 5OS  Furthermore, in 5para bond f exhibits a higher bond order than e, while in 5ortho and 5OS bonds e and f exhibit similar bond order (because they belong to Clar sextets) 
Finally, the bond labeled g shows a higher bond order in 5para than in 5ortho and 5OS  The AFM image of assigned 5para shown in Fig  indicates higher bond orders of the bonds b, d and f compared to a, c and e, respectively  In addition, the bond g appears almost point-like and with enhanced Δf contrast compared to its neighboring bonds, indicative of a high bond order (see Supplementary Fig  for height-dependent measurements) 
These observations concur with the calculated 5para geometry (Fig  )  Importantly, all these distinguishing bond-order differences are distinctly different in the AFM image of 5OS shown in Fig  , which is consistent with the calculated 5OS geometry (Fig  )  In the AFM images of 5OS (Fig  and Supplementary Fig  ), the bonds a-d at the pentagon apices appear with similar contrast and apparent bond length 
The bonds e and f at one of the terminal benzenoid rings also exhibit similar contrast and apparent bond length, while the central bond g appears longer compared to assigned 5para  Further compelling evidence for the closed-shell state of assigned 5para is obtained by STM and STS  dI/dV(V) spectra acquired on an assigned 5para species exhibit two peaks centered at -1 4 V (PIR) and 1 6 V (NIR) (Fig  ) 
STM images acquired at these biases (Fig  ) show the orbital densities of β (-1 4 V) and α (1 6 V)  First, the observation of α and β as the frontier orbitals of this species, and not the SOMOs, strongly indicates its closed-shell state  Second, consistent with AFM measurements that indicate good correspondence to the calculated 5para geometry, we observe β as the HOMO and α as the LUMO 
For 5ortho, α should be observed as the HOMO and β as the LUMO  We did not observe molecules with the signatures of 5ortho in our experiments  We observed molecules in open-(5OS, Fig  ) and closed-shell (5para, Fig  ) states in similar occurrence after their generation from 6 on the surface  We could also switch individual molecules between open-and closed-shell states as shown in Fig  and Supplementary Fig   
To this end, a change in the adsorption site of a molecule was induced by STM imaging at ionic resonances, which often resulted in movement of the molecule  The example presented in Fig  shows a molecule that was switched from 5para to 5OS and back to 5para  The switching is not directed, that is, we cannot choose which of the two species will be formed when changing the adsorption site, and we observed 5OS and 5para in approximately equal yields upon changing the adsorption site 
The molecule in Fig  is adsorbed on top of a defect that stabilizes its adsorption geometry on bilayer NaCl  At defect-free adsorption sites on bilayer NaCl, that is, without a third layer NaCl island or atomic defects in the vicinity of the molecule, 5 could be stably imaged neither by AFM nor by STM at ionic resonances (Supplementary Fig  ) 
Without changing the adsorption site, the state of 5 (open-or closedshell) never changed, including the experiments on bilayer NaCl/Ag(111) and Cu(111), on which the charge state of 5 could be switched (Supplementary Figs  )  Also on these lower work function surfaces, both open-and closed-shell species were observed for 5 0 and both showed charge bistability between 5 0 (5OS or 5para) and 5 -1 (Supplementary Figs  ) 
The geometrical structure of 5 -1 probed by AFM, and its electronic structure probed by STM imaging at the NIR (corresponding to transitions between 5 -1 and the dianionic state 5 -2 ), are identical within the measurement accuracy for the charged species of both 5OS and 5para  When cycling the charge state of 5 between 5 0 and 5 -1 several times, we always observed the same state (5OS or 5para) when returning to 5 0 , provided the molecule did not move during the charging/discharging process 
Based on our experimental observations we conclude that indeno[1,2-a]fluorene (5), the last unknown indenofluorene isomer, can be stabilized in and switched between an open-shell (5OS) and a closed-shell (5para) state on NaCl  For the former, both DFT and QD-NEVPT2 calculations predict a triplet electronic configuration 
Therefore, 5 can be considered to exhibit the spin-crossover effect, involving magnetic switching between high-spin (5OS) and low-spin (5para) states, coupled with a reversible structural transformation  So far, the spin-crossover effect has mainly only been observed in transition-metal-based coordination compounds with a near-octahedral geometry  
The observation that the switching between open-and closedshell states is related to changes in the adsorption site but is not achieved by charge-state cycling alone, indicates that the NaCl surface and local defects facilitate different electronic configurations of 5 depending on the adsorption site 
Gas-phase QD-NEVPT2 calculations predict that 5OS is the ground state, and the closed-shell 5para and 5ortho states are 0 11 and 0 21 eV higher in energy  The experiments, showing bidirectional switching between 5OS and 5para, indicate that a change in the adsorption site can induce sufficient change in the geometry of 5 (leading to a corresponding change in the ground state electronic configuration) and thus induce switching 
Switching between open-and closed-shell states in 5 does not require the breaking or formation of covalent bonds , but a change of adsorption site on NaCl where the molecule is physisorbed  Our results should have implications for single-molecule devices, capitalizing on the altered electronic and chemical properties of a system in π-diradical open-shell and closed-shell states such as frontier orbital and singlet-triplet gaps, and chemical reactivity 
For possible future applications as a single-molecule switch, it might be possible to also switch between open-and closed-shell states by changing the local electric field, such as by using chargeable adsorbates   Scanning probe microscopy measurements and sample preparation  STM and AFM measurements were performed in a home-built system operating at base pressures below 1×10 -10 mbar and a base temperature of 5 K  Bias voltages are provided with respect to the sample 
All STM, AFM and spectroscopy measurements were performed with carbon monoxide (CO) functionalized tips  AFM measurements were performed in non-contact mode with a qPlus sensor   The sensor was operated in frequency modulation mode with a constant oscillation amplitude of 0 5 Å  STM measurements were performed in constantcurrent mode, AFM measurements were performed in constant-height mode with V = 0 V, and I(V) and Δf(V) spectra were acquired in constant-height mode 
Positive (negative) values of the tip-height offset Δz represent tip approach (retraction) from the STM setpoint  All dI/dV(V) spectra are obtained by numerical differentiation of the corresponding I(V) spectra  STM and AFM images, and spectroscopy curves, were post-processed using Gaussian low-pass filters 
Au(111), Ag(111) and Cu(111) surfaces were cleaned by iterative cycles of sputtering with Ne + ions and annealing up to 800 K  NaCl was thermally evaporated on Au(111), Ag(111) and Cu(111) surfaces held at 323 K, 303 K and 283 K, respectively  This protocol results in the growth of predominantly bilayer (100)-terminated islands, with a minority of trilayer islands 
Sub-monolayer coverage of 6 on surfaces was obtained by flashing an oxidized silicon wafer containing the precursor molecules in front of the cold sample in the microscope  CO molecules for tip functionalization were dosed from the gas phase on the cold sample  Density functional theory calculations  DFT was employed using the PSI4 program package  
All molecules with different charge (neutral and anionic) and electronic (open-and closed-shell) states were independently investigated in the gas phase  The B3LYP exchangecorrelation functional with 6-31G basis set was employed for structural relaxation and singlepoint energy calculations  The convergence criteria were set to 10 −4 eV Å −1 for the total forces and 10 −6 eV for the total energies 
Multireference calculations  Multireference calculations were performed on the DFToptimized geometries using the QD-NEVPT2 level of theory , with three singlet roots and one triplet root included in the state-averaged calculation  A (10,10) active space (that is, 10 electrons in 10 orbitals) was used along with the def2-TZVP basis set  
Increasing either the active space size or expanding the basis set resulted in changes of about 50 meV for relative energies of the singlet and triplet states  These calculations were performed using the ORCA program package   Nucleus-independent chemical shift (NICS) calculations  Isotropic nucleus-independent chemical shift values were evaluated at the centre of each ring using the B3LYP exchangecorrelation functional with def2-TZVP basis set using the Gaussian 16 software package  
Starting materials (reagent grade) were purchased from TCI and Sigma-Aldrich and used without further purification  Reactions were carried out in flame-dried glassware and under an inert atmosphere of purified Ar using Schlenk techniques  Thin-layer chromatography (TLC) was performed on Silica Gel 60 F-254 plates (Merck) 
Column chromatography was performed on silica gel (40-60 µm)  Nuclear magnetic resonance (NMR) spectra were recorded on a Bruker Varian Mercury 300 or Bruker Varian Inova 500 spectrometers  Mass spectrometry (MS) data were recorded in a Bruker Micro-TOF spectrometer  The synthesis of compound 6 was developed following the two-step synthetic route shown in Supplementary Fig  , which is based on the preparation of methylene-bridge polyarenes by means of Pd-catalyzed activation of benzylic C-H bonds  
Supplementary Figure | Synthetic route to obtain compound 6  The complex Pd2(dba)3 (20 mg, 0 02 mmol) was added over a deoxygenated mixture of 1,3-dibromo-2,4-dimethylbenzene (9, 100 mg, 0 38 mmol), boronic acid 10 (178 mg, 1 14 mmol), K2CO3 (314 mg, 2 28 mmol) and XPhos (35 mg, 0 08 mmol) in toluene (1:1, 10 mL), and the resulting mixture was heated at 90 °C for 2 h 
After cooling to room temperature, the solvents were evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording 11 (94 mg, 76%) as a colorless oil  The complex Pd(OAc)2 (7 mg, 0 03 mmol) was added over a deoxygenated mixture of terphenyl 11 (90 mg, 0 27 mmol), K2CO3 (114 mg, 0 83 mmol) and ligand L (26 mg, 0 06 mmol) in NMP (2 mL) 
The resulting mixture was heated at 160 °C for 4 h  After cooling to room temperature, H2O (30 mL) was added, and the mixture was extracted with EtOAc (3x15 mL)  The combined organic extracts were dried over anhydrous Na2SO4, filtered, and evaporated under reduced pressure  The reaction crude was purified by column chromatography (SiO2; hexane:CH2Cl2 9:1) affording compound 6 (8 mg, 11%) as a white solid  in AFM imaging due to their reduced adsorption height compared to the rest of the carbon atoms 
We attribute this observation to the significantly different lattice parameter of Cu(111) (2 57 Å) compared to Au(111) and Ag(111) (2 95 Å and 2 94 Å, respectively) , such that the apical carbon atoms of the pentagonal rings of 5 adsorb on the on-top atomic sites on Au(111) and Ag(111), but not on Cu(111) 
Our speculation is based on a previous study of polymers of 1 on Au(111) by Di Giovannantonio et al  , where both tilted and planar individual units of 1 were observed depending on whether the apical carbon atoms of the pentagonal rings in 1 adsorbed on the on-top or hollow sites of the surface, respectively 
Given the strong molecule-metal interaction, we found no electronic state signatures of 5 on all three metal surfaces  STM set point for AFM images: V = 0  e, Frontier orbital spectrum of 5 -1   In the anionic state, ψ2 becomes doubly occupied and ψ1 is the SOMO  Filled and empty circles denote occupied and empty orbitals, respectively 
For each panel, zero of the energy axis has been aligned to the respective highest-energy occupied orbital 
Context 3:  Throughout the manuscript, Z will denote the set of integers and R the set of real numbers 
For some a, c ∈ Z and a < c, we will write For some e, g ∈ R and e < g, we will write

Road Model

The physical road structure is modeled as a continuous multi-lane highway with negligible curvature and unidirectional traffic flow  The lanes on the highway are clearly demarcated and at any given time k, the number of available lanes for the vehicles to travel on is denoted by N l (k) while the road speed limit is denoted by V l  
Therefore, the set of lanes available for traveling at a given time instant k is denoted by   We work with the Frenet coordinate system where the distance along the road is denoted by the longitudinal displacement (s) and the distance perpendicular to the road is defined by the lateral displacement (d) 
Each lane is assigned a lane indicator variable l  The leftmost lane, with respect to the direction of traffic flow, is assigned a value of l = 0 while each subsequent lane is assigned an increasing integer value for l, as depicted in Fig   

Vehicle Model

Since we aim to have real-time computations for a long planning horizon (> 15s), we model the vehicle dynamics with a linearized decoupled dynamical system  For the highway driving scenario, where the road curvature is typically small, it is reasonable to assume a decoupling between the lateral and the longitudinal dynamics , especially for the behavior planning layer 
Therefore, we utilize a linear constant acceleration model for the longitudinal dynamics and abstract out the lateral dynamics with a lane indicator variable  For the lane change dynamics, we use a moving average filter coupled with a rounding function to model the time required by the ego vehicle to change lanes 
This is compactly represented as: where s 0 (k), v 0 (k), l 0 (k) and L(k) denote the ego vehicle's longitudinal displacement, speed, lane indicator and target lane, respectively, at time instant k; the subscript i indexes the vehicles on the road with 0 being reserved for the ego vehicle; T s denotes the discretization time step; and N corresponds to the number of time steps required to change lane 
The state (x 0 (k)) and control input (u 0 (k)) to the system at time instant k are defined as: where V m denotes the maximum speed of the ego vehicle 

Observation Model

For practical considerations, we restrict the ego vehicle's visibility range to the sensory perception limit, denoted by R v   Then, the set of vehicles in ego vehicle's visibility range at time instant k, represented by O(k), is defined as: where s i (k) corresponds to the longitudinal displacement of the observed vehicle 
Remark 1: For the multi-lane highway driving scenario, occlusion does not play a prominent role so we do not account for it in the existing formulation  However, the proposed framework can easily accommodate occlusion and measurement uncertainties since the receding horizon approach bases its decision on the most up-to-date information available at any given time, as demonstrated in  
In this section, we describe the prediction model to generate the predicted future trajectories of observed vehicles and present a discussion on the proposed receding horizon optimization-based behavioral planning module 

Trajectory Prediction

Reliable behavior and trajectory prediction of other traffic participants is crucial for safe maneuvering of autonomous vehicles  The algorithm proposed in Section III-B is able to incorporate any generic prediction module available in the literature as long as it can provide a deterministic predicted future trajectory for a given vehicle 
In this work, we formulate a low-complexity prediction model that highlights the flexibility and efficiency of our proposed approach  For an observed vehicle i ∈ O(k), the future speed profile is predicted using a piece-wise linear function while the lane profile is assumed to stay constant for the duration of the prediction horizon 
At a given time step k, the estimated acceleration (ā k i ) and the estimated speed (v k i ) parameters are obtained through linear regression with mean-squared error on the past o k i > 1 speed observations  Based on the estimated parameters, we predict the future speed and longitudinal displacement as follows:
Here, H a corresponds to the acceleration horizon while vk i (j) and ŝk i (j) respectively represent the predicted speed and longitudinal displacement for vehicle i, j time steps into the future starting from the current time instant k  Remark 2: Due to the modular nature of the proposed framework, the behavior planning module detailed in Section III-B can work with advanced maneuver-based (e g 
Markov Chain ) and interaction-based (e g  Social Generative Adversarial Networks ) trajectory prediction modules, allowing for interactive maneuvering behaviors 

Speed and Lane Advisory System

The goal of our behavior planning module, Speed and Lane Advisory System or in short, SLAS, is to determine a sequence of speed and lane change commands that would enable the ego vehicle to maximize its speed, thus minimizing the travel time, while accounting for driver comfort and abiding by its dynamical, actuator, and safety limits 
The output of this module is a relatively smooth speed and lane change profile which is then passed on to a motion planner  It is necessary to incorporate the dynamical and actuator limits in the behavioral planning module so as not to provide the motion planner with goals that are not reachable, and jeopardize the safety of the overall system as a result 
In the subsequent discussion, we provide a formulation of the optimization problem for SLAS; highlight the modifications necessary to improve the computational complexity; and, present safety and feasibility analysis  1) Optimization Problem with Integer Constraints: SLAS is posed as an optimization problem, with the objective to maximize speed while minimizing frequent lane changes and abrupt changes in speed 
The output of SLAS, at time instant k, is the control input u 0 (k + 1), as defined in   The optimization problem is formulated as follows: Objective Function: In the formulation above, the optimization variables are the ego vehicle's speed (v k (j)) and target lane (L k (j)), j step into the future, starting from time instant k 
Here, H corresponds to the planning horizon  The scalarization parameters γ 1 , γ 2 and γ 3 in the objective function account for a relative tradeoff between maximizing speed, minimizing lane changes and minimizing abrupt changes in speed respectively  Increasing γ 1 yields a more aggressive behavior with the priority placed on maximizing speed while γ 2 and γ 3 combine to place an emphasis on maximizing passenger comfort by reducing lane and speed changes respectively 
Dynamical Constraints: These constraints are put in place to ensure the dynamical feasibility of the solution  The constraints ( ), and serve to initialize the longitudinal displacement, speed and target lane respectively for the optimizer, based on the values observed at time instant k  The constraints and ( ) bound the ego vehicle's speed by the speed limit and the acceleration limits of the vehicle respectively 
The ego vehicle's speed is then used to calculate the projected longitudinal displacement in   The target lane values at any planning step (j) are restricted to the set of reachable values by ( ), ( ) and   Here, restricts the target lane to the set of available lanes (L(k)), ensures that the lane change, if needed, is made to the adjacent lane only and (17) models the time steps (N ) required for a lane change 
The flooring function can easily be transformed into a couple of linear constraints by the introduction of an auxiliary integer variable, as shown in the Appendix  Finally, l k (j) is merely the internal representation of the lane the ego vehicle is projected to travel on at planning step j  Safety Constraint: The safety constraint ensures that the ego vehicle maintains a minimum safe distance (L s i (j)) to the nearest vehicle i, in its projected lane of travel (l k (j)), at planning instant j 
We borrow the definition of this safe distance from , where the authors provide a formalization, based on the clause from Vienna Convention on Road Traffic that states that "A vehicle [   ] shall keep at a sufficient distance [   ] to avoid collision if the vehicle in front should suddenly slow down or stop "
Furthermore, the absolute value constraint can be decomposed into linear constraints by the application of big-M method and the introduction of an auxiliary variable, as shown in the Appendix  Remark 3: The proposed formulation can accommodate arbitrary number of lanes at any given time instant k  This means that if at any given time, the number of available lanes for traveling either increases or decreases, the proposed formulation will still continue to hold 
This is an important consideration since many a times on highways, some lanes are blocked due to various unanticipated situations such as road accidents, roadwork, narrowing of road etc  2) Computational Complexity Reduction: This section details the optimization problem reformulation with binary variables, optimization warm start technique and lazy constraint implementation, all of which combine to improve the computational complexity of our SLAS module 
Binary Variables: The proposed formulation in Section III-B has relatively high computation complexity (computation time of ∼ 2s in the worst case scenario -slow moving traffic blocking all the lanes) due to the integer decision variables yielding a mixed-integer optimization problem   To circumvent the computational overload, we reformulate the problem with binary variables that replace the integer variables, as follows:
where the Lk (i, j) represents the modified target lane variable, indexed by the lane (i) as well as the planning step (j) and Lk (a, b) = 1 represents the choice of lane a ∈ L as the target lane at planning step b ∈ Z   Then, some of the constraints from the SLAS formulation in Section III-B are modified as follows:
Here, initializes the target lane, (21) restricts the target lane at any planning step to the set of available lanes, restricts the lane change between consecutive planning steps to the adjacent lanes, and ( ) represents the augmented safety constraint  The implication ( =⇒ ) in ( ) can easily be transformed into a linear constraint (see Appendix) 
The augmented minimum safety distance ( Ls i (j)) incorporates the time required to execute the lane change maneuver (N ) from ( ) into the following unified safety constraint: where L l is the width of the lanes (see Fig  ), δ(k) is the signed lateral deviation of the ego vehicle from the previous target lane's boundary at time step k, and γ d (δ k (j)) is the dynamic cost of deviation from the previous target lane (L(k − 1)) 
Moreover, in the cost function , we take L k (0) = L(k − 1)  These costs are introduced to prevent the swerving (canceling of lane switch before completion) behavior, unless absolutely necessary (for safety purposes)  Remark 4: Since the ego vehicle is considered to have changed lane once it crosses a lane boundary, the deviation δ k (j) is considered from the lane boundary instead of the center of the target lane to maintain the continuity of γ d (δ k (j)) with respect to the lateral displacement of the ego vehicle 
Specifically, δ k (j) > 0 if the ego vehicle has crossed the previous target lane boundary and 0 otherwise  This is an important consideration since a discontinuity in γ d (δ k (j)), upon completion of lane change, may lead to infeasibility  Remark 5: The swerving behavior is suppressed but not completely eliminated with a hard constraint since such a behavior is necessary at times to react to the environment's unpredictability 
This reactive strategy, which is a distinctive feature of our approach, avails the algorithm the ability to proactively 'change its mind' in case something unanticipated happens in the environment that can jeopardize safety  Optimization Warm Start: To aid the optimizer in finding an initially feasible solution, we provide the solution from the previous time step as a reference 
Formally, This doesn't imply that the solution from time step k − 1 will hold exactly at time step k, owing to the unmodeled disturbances, but providing this reference aids the optimizer in finding an initially feasible solution in the vicinity of the reference solution  This observation is rooted in the premise that the solution for the long planning horizon is not expected to change significantly between time steps, given the sampling time is not too large, and the predicted behavior of on-road agents does not alter significantly 
It is also worth pointing out that the priority here is quickly finding a feasible solution that obeys the safety constraints and actuator limits, and recursively improving it rather than excessively iterating to reach at a global optimum  In our experiments, it was observed that a suboptimal solution was qualitatively not significantly different from the optimal one 
Therefore, we utilize the cutting planes method for optimization , which first looks for a feasible solution, using our provided reference, and then recursively updates it until either the globally minimal solution is found or the time limit is reached  Lazy Constraints: To further enhance the computational efficiency, we introduce a lazy implementation of the lane changing constraints  
It was observed in our experiments that a feasible solution without the lane changing constraints ( ) can be found several order of magnitude (∼ 10×) quicker than if we include these constraints so we decided to have a lazy implementation for them  With a lazy implementation , the solver finds a set of feasible solutions without the inclusion of these constraints and then determines the feasibility of those solutions from the reduced problem with respect to the lazy constraints 
3) Feasibility: By an argument similar to the one presented in , it is a relatively straightforward proof for recursive feasibility of the problem, i



ground_truth: Open-shell π-diradical state and closed-shell state with a para-quinodimethane moiety.

prompt_front: Now, answer the following question based on the text below, only give me the answer and do not output any other words.

Question: What are the benefits of using binary variables in the SLAS formulation?

Read the following text and answer briefly.

Context 3: e  the optimization problem will continue to stay feasible, if initially feasible, with the trivial solution being matching the speed of the leading vehicle and not changing lanes 
In this section, we detail our experimental setup, demonstrate the performance of SLAS, and report a qualitative as well as a quantitative comparative analysis  The baselines in our comparative analysis are set to: Extended-Astar (EA ) , MOBIL , and no lane-change model (No-Change) 

Experimental Setup

The implementation setup, depicted in Fig  , is composed of the CARLA simulator (Version 0 9 11) , SLAS module (Section III-B), and the planner and controller module   To solve the optimization problem for SLAS, we use Gurobi Optimizer (Version 9 1 1)   The simulations are performed on a computer equipped with an Intel Xeon(R) CPU E5-2643 v4 @ 3 40GHz × 12 and NVIDIA Titan XP, running Ubuntu 20 04 LTS 
On average, the time required for each optimization step is ∼ 0 096s, while the maximum time limit for the optimizer is set to 0 2s, indicating the strong potential for real-time applicability 

Case Study

Figure illustrates the test case scenario for our comparative analysis  The scenario is composed of a highway segment with four lanes and the rightmost lane reserved for merging vehicles  The ego vehicle is initialized to follow a slow moving vehicle in lane 1 and has even slower moving traffic to its right in lane 2  Thus, the only option for it, in order to minimize travel time, is to switch left to lane 0 with faster moving traffic and greater headway 
Once it moves to lane 0, and overtakes the slow moving vehicle in lane 1, it has two options: either to keep traveling in lane 0 without making any lane change decisions until getting close to the lead vehicle or proactively exploiting the gap in lane 2 to switch to lane 3 in anticipation of traffic buildup in lanes 1 and 2  A strategic decision maker with foresight will choose to take the later option and make the decision proactively for a greater overall benefit 
The evaluation metrics for the comparative analysis include: travel time, lateral displacement, headway and distance to the closest vehicle  As for the simulation parameters, the simulation step size is set to 0 05s (simulation frequency of 20Hz); the velocities of vehicles in lanes 0, 1 and 2 are set to 8, 5 and 2 m/s respectively while the speed limit V l is set to 15m/s; the length of the highway patch is set to 350m while the width between the lanes is set to 3 5m; and the sensor visibility range is set to R v = 50m 
The parameters for SLAS are set as follows: T s = 0 4s, H = 40, N = 3, A min = −5m/s 2 , A max = 3 5m/s 2 , γ 1 = 1, γ 2 = 0 1 and γ 3 = 0 01  The values of these parameters can be tuned to yield an aggressive or defensive behavior of the algorithm  1) Travel Time: The left plot in Fig  depicts the travel time as a function of longitudinal displacement for the four algorithms 
As seen in the plot, our method (SLAS) maintains a lower overall travel time as compared to the other methods  Quantitatively speaking, SLAS outperforms EA , MOBIL and No-change methods by 12 72%, 23 52% and 54 34% respectively in terms of the time required to complete the simulation scenario  This shows that our method's foresight compensates for its apparent conservativeness arising from the need to preserve passenger comfort 
2) Lateral Displacement: To identify the differences in lane changing behaviors between the four approaches, the relationship between lateral and longitudinal displacements over the course of the simulation is highlighted in the center plot of Fig    In the plot, the lateral displacement of 0 corresponds to the center of lane 0 while the center of each following lane is 3 5m away 
Comparing the performance of the four algorithms, we see SLAS and EA showing relatively similar performances, resulting from proactive decision making  In contrast, since MOBIL only assesses the advantage of switching to the adjacent lanes, it is unable to see the benefit of proactively switching to lane 2  This explains why EA and SLAS start outperforming MOBIL in terms of travel time (left plot) at around the 130 [m] mark for longitudinal displacement 
As for a direct comparison between SLAS and EA , the benefits of having speed advisory system become apparent in this center plot  Due to speed control, SLAS is able to constantly maintain a greater headway (right plot) without having to brake significantly upon getting too close to the lead vehicle  This results in a smooth lateral displacement profile which allows the vehicle to change lanes with minimal jerk (quantitative analysis to follow in Section IV-C) and deliver better overall timing performance (left plot) 
3) Headway: The right plot in Fig  shows the headway maintained by the ego vehicle over the course of the simulation  In accordance with our prior discussion, MOBIL cruises behind the front vehicle, maintaining a relatively low headway until a sufficient space in the adjacent lane is found to perform the lane-change maneuver 
On the other hand, EA and SLAS show a comparable headway trajectory, however, SLAS maintains a greater headway throughout and achieves the maximum headway prior to EA   Quantitatively, SLAS maintains on average 9 43%, 36 57% and 113 17% more headway than the EA , MOBIL and No-change approaches respectively 
This strong performance by SLAS can be attributed to its incorporation of safety guarantees coupled with its consideration for passenger comfort  4) Distance to closest vehicle: Finally, we compare the distance that ego vehicle maintains from the closest vehicle throughout the simulation  On average, SLAS maintains 9 28%, 32 01%, and 22 84% more distance in comparison to EA , MOBIL and No-change approaches respectively 
These numbers are a testament to the strength of our approach resulting from consideration of long planning horizon coupled with speed control 

Monte Carlo Simulations

To demonstrate the long-term performance of the three approaches (SLAS, EA and MOBIL), we run a series of Monte Carlo simulations on scenarios with randomized initial positions (within a range of 8m) and velocities (within    ranges of 8, 5 and 2 m/s assigned to each of the three lanes randomly) of traffic participants 
The result from 50 simulations is presented in Table   In this table, the columns represent the different evaluation metrics, the rows identify the three algorithms, and the values highlighted in green represent the best result with respect to each evaluation metric  The evaluation metrics, going from left to right in the table, are completion time (s), brake (R [−1,0] ), brake jerk (R [−1,0] ), throttle (R [0,1] ), throttle jerk (R [0,1] ), angular acceleration ( • /s 2 ) and angular jerk ( • /s 3 ) 
Apart from completion time, the remaining metrics, based on the commands passed to the vehicular actuators (Fig  ), are used to model passenger comfort  In terms of average performance, SLAS greatly outperforms the other methods when it comes to passenger comfort since it explicitly accounts for comfort in the formulation 
However, it does so at a cost of slightly reduced performance in regards to travel time, when compared to EA , since SLAS tries to strike a balance between minimizing travel time and maximizing passenger comfort  SLAS also secures the lowest standard deviation, for each of the evaluation metrics, when compared to the other methods, which points to the consistency in its long-term performance 

CONCLUSION

We propose a novel behavior planning module for the multi-lane highway maneuvering scenario, that outputs strategic target lane and reference speed commands, and incorporate it with a state-of-the-art motion planning and control framework  We formulate the approach as a receding horizon mixed integer optimization with the goal to minimize travel time while accounting for passenger comfort for a long planning horizon 
In order to reduce the computational overload, we reformulate the problem by replacing integer variables with binary ones and further incorporate various modifications to aid numerical computations  We also carry out a detailed comparative analysis to demonstrate the performance of our approach on the CARLA simulator 
Our future work includes incorporating various delays and uncertainty measures in the perception, localization and prediction modules to evaluate the robustness properties of our approach 

Flooring Constraint

For y ∈ Z and x ∈ R, the constraint y = x can be represented by the following linear constraints: y ≤ x, y + 1 ≥ x + where > 0 accounts for the feasibility tolerance  where M 0 (big-M) and > 0 accounts for numerical errors (chosen to be 0 1 in our implementations) 

Absolute Value Constraint

For ∆s, L s ∈ R, the constraint |∆s| − L s ≥ 0 can be represented as: This can further be generalized, as done in our implementation, to have different forward and rear safety margins as: ∆s ≥ L f s ∨ ∆s ≤ −L r s where L f s and L r s are forward and rear safety margins respectively  This can be represented with the following linear constraints:
s ≥ 0 where M 0 (big-M) and c ∈ {0, 1} is responsible for making a choice between the two constraints 



ground_truth: Reduced computational complexity.