I Introduction
Reachavoid problems are prevalent in many engineering applications, especially those involving strategic or safetycritical systems. In these situations, one aims to find a control strategy that guarantees reaching a desired set of states while satisfying certain state constraints, all while accounting for unknown disturbances, which may be used to model adversarial agents. Reachavoid sets capture the set of states from which the above task is guaranteed to be successful. Reachavoid problems are challenging to analyze due to the asymmetric goals of the control and disturbance, leading to nonconvex, maxmin cost functions [1, 2, 3]. Due to the complexity of the cost function, dynamic programmingbased methods for computing reachavoid sets on a grid representing a state space discretization, such as HamiltonJacobi (HJ) formulations, have been popular and successful [4, 1, 2].
One specific class of reachavoid problems is the reachavoid game, in which the system consists of two adversarial players or teams. The first player, the attacker, assumes the role of the controller, and aims to reach some goal. The other player, the defender, assumes the role of the disturbance, and tries to prevent the attacker from achieving its goal. In [5, 6], the authors analyzed the twoplayer game of capturetheflag by formulating it as a reachavoid game, and then obtaining optimal strategies and winning regions for each player. The authors in [7, 8] extended previous results by analyzing the multiplayer case in which each team has an arbitrary number of players. Other dynamic programmingbased methods for stochastic systems also exist [9, 10]. Other important applications of reachavoid problems include motion planning in the presence of moving obstacles [11, 12, 13, 14]. In particular, the multivehicle motion planning problem is analyzed from the perspective of reachavoid problems and solved using dynamic programming in [14].
Due to the difficulties of analyzing reachavoid problems, dynamic programming methods have enjoyed great success due to their optimality and generality when the state space is smaller than 6D. For larger state spaces, computational burden becomes the main challenge. To address this challenge, heuristics typically based on information pattern simplifications and multiagent structural assumptions have been proposed. In
[15], the authors consider an openloop formulation of the reachavoid game in which one of the players declares its strategy at the beginning of the game; in addition, the player’s dynamics are assumed to be single integrators in 2D. An openloop framework is also used in [16] for pursuitevasion. A semiopenloop approach for 2D single integrator players, based on the idea of “path defense,” has been proposed in [17]. In the context of multivehicle motion planning, prioritybased heuristics have been used [14]. More broadly, methods for analyzing reachavoid problems as well as related multiagent systems make trade offs among generality of system dynamics, generality of the problem set up, conservatism of solutions, and computational efficiency [18, 19, 20, 21, 22, 23, 24, 25].Since the action of an opposing agent is modeled as a disturbance in the joint system, reachavoid problems are closely related to robust planning, in which disturbance rejection is of primary concern. In this context, methods that produce value functions with Lyapunovlike properties have been very effective. When the system dynamics are nonlinear in general but have small state spaces, HJ methods are able to produce Lyapunovlike functions through dynamic programming [4, 2]. Computational complexity has been alleviated in specific scenarios using decomposition techniques [26, 27]; however, these are not applicable to general reachavoid problems.
When the system dynamics and functions representing sets are polynomial, the search for Lyapunovlike functions can be done more efficiently by leveraging sumofsquares (SOS) programs, which can be converted to semidefinite programs [28] and solved using standard optimization toolboxes [29]. SOS programs involve checking whether polynomial functions can be written as SOS, which is a sufficient condition for nonnegativity or positivity, and thereby establishing Lyapunovlike properties. In addition, complex problem statements involving sets and implications can be written as SOS constraints. SOS programs have been used extensively in robust planning with methods involving barrier certificates and robust funnels [30, 31, 32, 33]. Other methods that utilize nonlinear optimization also exist; for example, [34] and [35] utilize nonlinear optimization techniques for motion planning through dynamic environments for a single vehicle and a flock of vehicles, respectively.
Statement of contributions: In this paper, we propose a method for computing the reachavoid set and synthesizing a feedback controller that is guaranteed to drive the system into a target set while staying out of an avoid set. Our approach combines dynamic programming and SOS optimization: the reachavoid set, represented by a Lyapunovlike value function, is obtained backwards one time step at a time as in dynamic programming, and at each time step a SOS program is solved. Building on previous SOSbased work such as [32, 33], we explicitly encode the avoidance constraint so that a single value function guarantees both reaching and avoidance, as in HJ methods [15, 6, 8]. Compared to previous dynamic programmingbased work such as [15, 6, 8], we trade off optimality of solution for computational complexity: although our method produces conservative reachavoid sets, we are able to analyze systems with higherdimensional state spaces. Unlike analytic approaches such as [17, 20, 22], our approach applies much more broadly to different problem setups and system dynamics. We demonstrate our method in simulations.
The remainder of this paper is structured as follows:

In Section II, we formulate the reachavoid problem and provide some background about SOS programming.

In Section III, we derive the SOS constraints for computing the reachavoid set based on its basic properties.

In Section IV, we propose a dynamic programming approach for solving the SOS program more efficiently.

In Section V, we numerically validate our theory by comparing our method to the HJ method, and present, to the best of our knowledge, the first 6D reachavoid set computed using a general numerical method.

In Section VI, we conclude and provide suggestions for future directions.
Ii Preliminaries
Iia Problem Formulation
Consider a system which evolves according to its dynamics, given by the following ordinary differential equation:
(1) 
where is the system state, is its control, and is the disturbance. We assume that the control must be of a timevarying state feedback form, , and that and are measurable. Importantly, we make no assumption on the disturbance other than that it is bounded. We denote the function spaces from which and are drawn as and , respectively.
The system dynamics is assumed to be uniformly continuous, bounded, and Lipschitz continuous in for fixed and . So given , there exists a unique trajectory solving (1) [36].
We would like to compute the set of joint states from which the attacker wins the game of time horizon . This is captured by the reachavoid set, defined as follows:
(2)  
The avoid set is the set of states that the system must avoid while reaching the target. Note the following propety:
Property 1
Final condition. .
IiB SOS Programming Background
In this section, we provide a brief introduction to SOS programs. For a more detailed discussion, please refer to [28] and [33]. In this paper we will represent a set of states as : . This allows us to transform setbased constraints such as to constraints of the form in our proposed optimization problem. Note that such a constraint is generally nonconvex.
When is a polynomial in , checking nonpositivity over is still NPhard [28]. However, in this case the constraint can be relaxed to the SOS condition , where are polynomials. This condition is equivalent to , where
is a vector of polynomial basis functions up to less than or equal to half the degree of
, and is a semidefinite matrix of appropriate size. Note that this constraint is satisfied by matching coefficients on the left and righthand sides. A shorthand for the above constraint is “”.One is often interested in guaranteeing nonpositivity over a subset of state space. For example, one may desire over the set where . A constraint in the form of where can be written as .
Iii Solution via SOS Programming
We now formulate a SOS program whose solution characterizes the reachavoid set defined in Eq. (2), and provides a feedback controller that guarantees reaching and avoidance.
As in earlier literature [33], let be characterized by the sublevel set of some function :
(3) 
Iiia The Value Function
Our SOS formulation is motivated by basic properties of the reachavoid set. The first is Property 1 in Section II. Taking the convention that , this property can be stated as .
Property 2
Lyapunovlike property. By definition of , if and , then there exists such that for all , for any arbitrarily small .
In terms of the value function, and with the convention , Property 2 becomes
where
(4) 
This Lyapunovlike property states that if is not in and is on the boundary of , then there must some control , over which the SOS program will optimize, such that regardless of the chosen disturbance , will remain nonpositive. The boundary of is described by the condition , and the nonpositivity condition on ensures the continued nonpositivity of .
Property 3
Avoidance property. By the definition of , if and , then it cannot be in .
IiiB Control Parametrization and Bounds
Equation (4) depends on the control , which is bounded according to the system dynamics (1). By the definition of the reachavoid set in Eq. (2), we must take this into account.
In this paper, we aim to search for a feedback controller , so following the control bounds constraint in [32] and [33], we enforce the following constraint:
(5) 
As long as is linear, the set is semialgebraic for a given .
IiiC The SOS Program
Putting all of the above into consideration, we arrive at the following optimization problem in Eq. (6), which maximizes the volume of the reachavoid set while enforcing the constraints described above. We will describe how volume can be maximized in Section IVC.
(6)  
subject to  
As introduced in Section IIB, we can convert all the above constraints into SOS form. Plugging in Eq. (4) for , , the optimization program becomes (7):
(7)  
subject to  
where .
Iv Solving the SOS Program via Dynamic Programming
The optimization in (7) involves polynomials in continuous time and state space. In this section, we discretize time so that the problem can be solved in a dynamic programming fashion, one time step at a time, akin to what is done in HJ reachability. This way, we avoid optimizing over an entire time horizon, which is computationally expensive.
Iva Time Discretization
We define time samples . All quantities dependent on time are now indexed by : for example, and . In addition, we approximate^{1}^{1}1Discretization error can be reduced with higher order schemes. the derivatives of and with respect to :
(8) 
Now, the optimization problem becomes (9).
(9)  
subject to  
where .
IvB Dynamic Programming
The optimization in (9) involves decision variables in the entire time horizon. However, the structure of the optimization program allows us to break it down into smaller problems, each representing one time step. This allows the computational complexity of our proposed method to scale linearly with the number of time discretization points.
Given or with some arbitrary , the optimization program starts at and decrements after every optimization of the form
(10)  
subject to  
where .
IvC Volume Maximization
Maximizing the volume of a polynomial sublevel set is potentially intractable [37]. We therefore substitute the objective of the previously defined optimization problems with a heuristic. Using cost heuristics does not remove any guarantees from our approach.
Since represents , one heuristic for maximizing volume is to restrict to be SOS, minimize the integral of over a region of interest as in [37], and maximize as in [32]. We will later also use to slightly relax the SOS program in Section IVD. Similarly to what is described in [37], we write where represents the coefficients of and a monomial basis. This allows us to write the integral as a linear function of , making it amenable to SOS optimization:
(12) 
IvD Implementation Details
Since the optimization (10) is bilinear and thus nonconvex, we propose heuristics for obtaining useful solutions. This section describes the heuristics, provides the final optimization program (13), and presents Alg. 1 for solving it.
Value function invariance: One property that also arises from the definition of in Eq. (2) is as follows:
Property 4
Invariance property.
In terms of the value function, this implies that . Using a slack variable , we encoded this property as an additional soft constraint in Eq. (13e) to guide the optimization. The value of is minimized in the objective (13a) with weight . In our experience, without , the optimization does not reliably produce value functions that represent nonempty sets.
Region of interest: To facilitate the search for polynomials and constant , we relax the constraints so that they only apply in some region of interest , which can be chosen to either be large enough to contain the reachavoid set , or in general contain the region of the state space that one wishes to consider. This is done by adding the terms , , , in the constraints (13b), (13c), (13d), (13e), respectively.
Initialization: To obtain a feasible initial guess, we introduce a slack variable to allow to be initially nonnegative in (13b). Throughout alternations of the optimization, we minimize in the objective (13a). In addition, we increase the weight by some factor to drive the value of down, as described in Alg. 1. Once is below some threshold , we consider the solution to be numerically feasible and stop optimizing over . In practice, we are consistently able to obtain feasible solutions.
Final optimization problem: All of the above considerations lead to the final optimization program for each time step in (13). The bilinear terms are and in (13b), in (13d), and in (13e). Therefore, we can optimize the three sets of variables , , and in an alternating fashion, where . Note that as mentioned in Section IVC, cannot be optimized at the same time as due to the volume maximization heuristic. The optimization algorithm is outlined in Algorithm 1.
(13a)  
subject to  (13b)  
(13c)  
(13d)  
(13e)  
(13f)  
where is a constant vector of weights, and
V Numerical Example: the ReachAvoid Game
We now demonstrate our approach for computing reachavoid sets by analyzing the reachavoid game, which involves an attacker with state trying to reach a target while avoiding capture by the defender with state . Let the joint state be denoted . The joint dynamics are given in Eq. (1), where we use to model the control of the attacker, and to model the control of the defender.
For convenience, we will use to denote the positions of the attacker and the defender, respectively. In general will be subsets of the player states , respectively.
In the context of reachavoid problems, the target set is the set of joint states such that the attacker is at the target:
The avoid set is the set of joint states in which the attacker is captured by the defender. In this paper we will assume that the attacker is captured by the defender when the positions of the two players are within some capture radius:
(14) 
We now present reachavoid set computations for two examples of reachavoid games. The first example involves two single integrator players moving in 2D space; the joint state space dimension is 4D; we compare our computation results with those obtained from HJ reachability, which is the most general method that provides the optimal solution up to small numerical errors. The second example involves two kinematic car players moving in 2D; the joint state space dimension is 6D, and computation using HJ reachability is intractable.
Va Two single integrator players
Consider the following player dynamics:
(15)  
(16)  
(17) 
Traditionally, the reachavoid set for these player dynamics can be computed using HJ reachability [6, 8]. Under special scenarios such as those in which players have the same maximum speed, analytic methods may be employed. Like HJ reachability, our SOSbased approach is applicable in a general setting. In this section, by comparing our results with those of HJ reachability, we demonstrate that our numerical results are conservative approximations of the optimal reachavoid set, and therefore maintains reaching and avoidance guarantees.
For this example, we have chosen the maximum speeds to be , and the target set to be approximately a square of length centered at the origin, . The hyperparameters of the optimization are , , , , . The maximum degree of was set to , that of set to , maxIter set to , the region of interest set to , and the time discretizations set to .
To visualize the 4D reachavoid set , we fix the defender position and show 2D slices of over several values of in Figure 2. One can notice that the growth of over time is not uniform as expected. This can be attributed to the solution of the SOS program being suboptimal, since the problem is nonconvex. However, it is important to note that given the constraints of the SOS program, any feasible solution offers reaching and avoidance guarantees.
Figure 3 shows computations of sliced at various defender positions . The outer magenta boundary is the computation result from HJ reachability, and represents the true reachavoid set up to small numerical errors. The solid blue boundary is the computation result from our SOSbased method. HJ reachability is better suited for this smaller 4D system, as the optimal solution can be obtained. However, any state inside the set computed using SOS programming is inside the set computed using HJ reachability, which means our computation results, although conservative, maintain reaching and avoidance guarantees.
Computations for this example were done on a desktop computer with an Intel Core i7 2600K CPU and 16 GB of RAM. The SOS computations took approximately 17 minutes with the above parameters using the spotless toolbox [38] and Mosek [29], and the HJ computations took approximately 25 minutes on a grid with points using the level set toolbox [39] and the helperOC library [40]. Computational time varies greatly with the maximum degree of polynomials chosen in the case of the SOSbased method, and with the number of grid points in the HJ method.
VB Two kinematic car players
In this section, we demonstrate our method on a system involving two kinematic cars. The joint dynamics are
(18)  
(19)  
(20)  
(21) 
Here we apply our SOSbased approach to derive and certify a controller for one of the cars that enables it to reach a goal in state space no matter what control action the other car might perform. This is indeed useful in the context of, for example, designing a safe lanefollowing controller that makes no assumption on the policy of other drivers, except for control bounds. Even though our controller and reachavoid set are once again quite conservative, they provide formal guarantees on a system that is typically too highdimensional for treatment with state of the art approaches like HJbased methods.
For this example, the maximum speed and turning rates of the attacker are and . We assume that the other car, the defender, will not try to collide with us on purpose and is therefore limited to speeds of and for its velocity and turn rate. Such an assumption can be made when one assumes, for example, that the other driver is simply trying to stay inside his lane. The target set, representing our desire to have the attacker stay on the road without colliding with the other car, is a circle in the middle of the road () on the right side of the region of interest (a box from to in , and from to in ,). We also bound and to be inside the interval , which allows us to use a Chebyshev approximation with an accuracy of for and for , namely:
The approximation makes our dynamics polynomial, and therefore amenable to SOS optimization. The hyperparameters of the optimization are , , , , . The maximum degree of was set to , that of set to . The maximum number of iterations (maxIter) was set to , and the time discretizations set to .
The right column of Figure 4 shows slices of the reachavoid slices for the states corresponding to , , and the indicated defender position. The left column shows the result of applying the resulting controller in simulation. Even though the sets are conservative, none of the constraints are violated in simulation. Also note that the policy used by the defender is simply to maximize its velocity in some direction. Importantly though, the controller returned by our algorithm is robust to any policy the defender might use. A video of this example is available online^{2}^{2}2https://www.youtube.com/watch?v=gUytdFHkjYY.
Vi Conclusions and Future Work
We presented a novel method for computing reachavoid sets and synthesizing a feedback controller that guarantees reaching and avoidance when the system starts inside the set. Our method utilizes sumofsquares (SOS) optimization to tradeoff optimality of solution for computational complexity, allowing us to compute, for the first time to the best of our knowledge, 6D reachavoid sets, although our solution is conservative. Combining SOS optimization with dynamic programming, we also greatly reduce the computational complexity of solving the SOS program.
Future work includes investigating ways to reduce both the degree of conservatism in our solutions and the computation time by improving the optimization algorithm. In addition, since our method is applicable to polynomial system dynamics and general problem setups, there is much room for the exploration of potential applications of our work. Lastly, we would like to test our algorithm through hardware implementation.
Vii Acknowledgements
The authors would like to thank Anirudha Majumdar for his guidance on sumofsquares programming. This work was supported by the Office of Naval Research YIP program (Grant N000141712433) and by the Toyota Research Institute (TRI). This article solely reflects the opinions and conclusions of its authors and not ONR, TRI or any other Toyota entity.
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