Learning Performance-oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation
Authors: Lakshmideepakreddy Manda
Shaoru Chen
Mahyar Fazlyab
Venue: CoRL 2024 (2024)
Keywords: Control Barrier Functions
Safety
Hamilton-Jacobi Partial Differential Equation
TL;DR: We use Hamilton-Jacobi PDEs to learn Control Barrier Functions for complex environments

Abstract: Control Barrier Functions (CBFs) offer an elegant framework for constraining nonlinear control system dynamics to an invariant subset of a pre-specified safe set. However, finding a CBF that simultaneously promotes performance by maximizing the resulting control invariant set while accommodating complex safety constraints, especially in high relative degree systems with actuation constraints, remains a significant challenge. In this work, we propose a novel self-supervised learning framework that holistically addresses these hurdles. Given a Boolean composition of multiple state constraints defining the safe set, our approach begins by constructing a smooth function whose zero superlevel set provides an inner approximation of the safe set. This function is then used with a smooth neural network to parameterize the CBF candidate. Finally, we design a physics-informed training loss function based on a Hamilton-Jacobi Partial Differential Equation (PDE) to train the PINN-CBF and enlarge the volume of the induced control invariant set. We demonstrate the effectiveness of our approach on a 2D double integrator (DI) system and a 7D fixed-wing aircraft system (F16).

Learning Performance-oriented Control Barrier Functions Under Complex Safety Constraints and Limited Actuation
Lakshmideepakreeddy Manda
Shaoru Chen Microsoft Research
Department of Electrical and Computer Engineering Johns Hopkins University
New York, United States
United States
cnnoorucchn@@iirroofffe
1manda1 @juu. @euuueuuu edu
Mahyar Fazlyab
Department of Electrical and Computer Engineering Johns Hopkins University, United States mahyarfazlyabejhu.ce
Abstract: Control Barrier Functions (CBFs) provide an elegant framework for constraining nonlinear control system dynamics to remain within an invariant sub- set of designated safe set. However, identifying a CBF that balances perfor- mance-by by maximizing the control invariant set-and accommodates complex safety constraints, especially in systems with high relative degree and actuation limits, poses a significant challenge. In this work, we introduce a novel self- supervised learning framework to comprehensively address these challenges. Our method begins with Boolean composition of multiple state constraints that de- fine the safe set. We first construct smooth function whose zero superlevel set forms an inner approximation of this safe set. This function is then combined with a smooth neural network to parameterize the CBF candidate. To train the CBF and maximize the volume the resulting control invariant set, we design physics- informed loss function based on Hamilton-Jann (PDE). We validate the efficacy of our approach on 2D double integrator (DI) system and a 7D fxed-wing aireraft system (F16).
Partial Differential Equation
1 Introduction
CBFs are powerful tool to enforce safety constraints for nonlin- ear control systems [1], with many successful applications in au-
tonomous driving [2], UAV navigation [3], robot locomotion [4], and safe reinforcement learning [5]. For control-affine nonlinear systems, CBFs can be used to construct a convex quadratic pro- gramming (QP)-based safety filter deployed online to safeguard against potentially unsafe control commands. The induced safety filter, denoted as CBF-QP, corrects the reference controller to main in a safe control invariant set.
While Control Barrier Functions (CBFs) provide an efficient method to ensure safety, finding such functions can be challenging. Specifically, there is no guarantee that the resulting safety filter will remain feasible throughout operation. This primarily because the "model-free" construction of the filter only incorporates the con- straints. Complex constraints, high relative degree, and bounded actuation exacerbate the challenge of ensuring feasibility. Various techniques have been proposed to address these challenges such
Figure 1: Illustraiion of the learned CBF-QP filtering many initializations
PID reference control on the DI system. The CBF zero contour drawn its value heatmap bounds the learned control invariant set.
8th Conference Robot Learning (CoRL. 2024), Munich, Germany.

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as CBF composition for complex constraints [6. 7, 8. 91, higher-order CBFs for high relative de- gree [10] [11], and integral CBFs [12] for limited actuation. Despite significant progress, these approaches can make the filter overly restrictive, thus limiting performance.
Contributions We propose novel self-supervised learning framework for CBF synthesis that systematically addresses all the above challenges. First, we handle complex safety constraints and high relative degree in CBF synthesis by encoding the safety constraints into the CBF parameteriza- tion with minimal conservatism. Second, we design a physics-informed training loss function based on Hamilton-Jacohi (HJ) reachability analysis [13] to satisfy bounded actuation while maximizing the learned control invariant set volume. We evaluate our method on the double-integrator and the high-dimensional fixed-wing aircraft system and demonstrate that the proposed method effectively learns a performant CBF even with complex safety constraints. We call the proposed framework Physics-informed Neural Network (PINN)-CBF [14].
1.1 Related work
Complex safety constraints For safe set described by Boolean logical operations on multiple constraints, [6] composes multiple CBFs accordingly through the non-smooth min/max operators. introduces smooth composition of logical operations constraints, which was later extended to si- multancously handle actuation and state constraints [8][9] using integral CBFs [12]. [15] proposes an algorithmic way to create single smooth CBF arbitrarily composing both min and max opera-
tors. Such smooth bounds have been used in changing environments [16]. Notably, [17] ensures the input constrained feasibility of the CBF condition while composing multiple CBFs.
High-order CBF High-order CBF (HOCBF) [10] and exponential CBF [11] are systematicly ap- proach CBF construction when the safety constraints have high relative degree. However, control- ling the conservatism of these approaches is challenge. To reduce conservatism improve the performance of HOCBF, various learning frameworks have been proposed [18, 19, 20J that allow tuning of the class K. functions used in the condition.
Learning CBF with input constraints Motivated by the difficulty of hand-designing CBFs, learning- -bsed approaches building on the past [21] have emerged in recent years as an alterna- tive [22, 23, 24, 25]. Liu et al. [26] explicitly consider input constraints in learning a CBF by finding counterexamples on the 0-level set of the CBF for training, while Dai et al. [27] propose a data- efficient prioritized sampling method. [28] explores adaptive sampling for training HJB NN models favoring sharp gradient regions. Drawing tools from reachability analysis, the recent work [29] it- eratively expands the volume of the control invariant set by learning the policy value function and improving the performance through policy iteration. Similarly, Dai et al. [30] expand conservative hand-erafted CBFs by learning on unsafe and safe trajectories, and Qin et al. [31] applies actor-critic RL framework to learn a CBF value function while [32] learns a control policy and CBF together for black-box constraints and dynamics. Our method differs by being controller-indeeendent and bas- ing its learning objective on the HJ partial differential equation (PDE), which defines the maximal control invariant set, without requiring trajectory training data.
HJ reachability- methods The value functions in HJ reachability analysis have been ex- tended to construct control barrier-value functions (CBVF) [33] and control Lyapunov-value func- tions [34], which can be computed using existing toolboxes [35]. Tokens et al. [36] further apply such tools to refine existing CBF candidates, and Bansal et al. [37] learn neural networks solution to a HJ PDE for reachability analysis. Of particular interest to our work is the CBVF, which is close to the CBF formulation and provides a characterization of the viability kernel. Our work aims to learn a neural network CBF from data without using computational tools based on spatial discretization.
Notation An extended class function a function a: (-b, a) + R for some a, 0 that is strictly increasing and satisfies a<0) = O. We denote L, (h) = fo E R" h(z) 2 r) as the r- superlevel set of a continuous function l(x)). The positive and negative parts of number a E are denoted by (a)+ operations of conjunction, disjunction, and negation, respectively. For two statements A and B, we have
max (a, 0) and (a)
min(a, 0), respectively. Let A, V, denote the logical
have-((A -(A A B) = (-A) V (-B) and -(A v B) (-A) A (-B).
2

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2 Background and Problem Statement
Consider continuous time control-aftine system:
I [() + g(x)u, E,
where C In the domain of the system, I E D is the state, u E C R" is the control input, U S Rd denotes the control imput constraint. We assume that f: D + R", g: D + _ locally Lipschitz continuous and 24 is a convex polyhedron. We denote the solution of (1) at time t 2 0 by zz(t). Given a set X' C DD that represents safe subset of the state space, the general objective of safe control design to find a control law T(x) that renders X invariant under the closed-loop dynamics i f()) + (z)) i... if (xto) E X for some to 2 0, then x(t) E X for all / 2 4- A general approach to solving this problem is through control barrier functions.
2.1 Control Barrier Functions
Suppose the safe set is defined by the O-superlevel set of smooth function c(-) such that X = {or c(z) 2 0}. For X to be control invariant, the boundary function c(.) must satisfy max,eu ö(æ, u) 2 0 when c(x) = O by Nugomo's 's theorem [38]. However, since c(-) does not necessarily satisfy the condition, we settle with finding a control invariant set contained in through CBFs.
Definition 1 (Control barrier function). Let S == (h) S S D be the O-superlevel set of continuously differentiable function h: - R. Then h(-) is control barrier finction for system (1) if there exists an extended class fffuctio a such that
(up(Ly)(x) Lgg(x)) a(h(x))) 0, Vx ED, well
(2)
where Ljh(z)= = VI(x) f(x) and Lgh(z)= = Vh(x)] g(x) are the Lie derivatives of f(z).
Given CBF h(-), the non-empty set of point-wise safe control actions is given by any locally
Lipschitz continuous controller ((x) E (ebb(()) = (u Eu Lyh(r)= + (z)(())u + a(h(z) 2 0)
renders the S forward invariant for the closed-loop system, which enables the construction of
manimally-invasive safety filter:
(xx) = argmin I|u ur (x)]1 subject to Lyh(a)) + Mx)()) + (((())) 2 0, u uEu,
(3)
where uy (x) is any given reference but potentially unsafe controller. Under the assumption that 2 a polyhedron, (e))())) is also polyhedral set and problem (3) becomes convex QP.
2.2 Problem Statement
While the CBF-QP filter is minimally invasive and guarantees xxt) E (h) C X' for all time, small Lj (h) essentially limits the ability of the reference control execute task. To take the performance of the reference controller into account, we consider the following problem.
Problem Ceefoormance-riiente CBF), Given the inpul-constrained system (1) and safe set X defined by complex safety constraints (to be specified in Section 3.1), synthesize CBF h(.) with induced control invariant set S Li (h) such that (i) S C X and (ii) the volume of S is maximized. Formally, this problem can be cast as an infinite-dine dimiiieiddmnnionlll optimization problem:
maximize KE#
volume(L, (h)) (performance)
(4)
subject to
Lt (h) C (safety)
sup(Lyh(z)) + L,h(2) a(h(z))))2 Vr (control invoriance) well
where is the class of scalar-valued continuously differentiable functions.
3 Proposed Method
In this section, we present our three-step method of learning PINN-CBF:

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(S1) Composition of complex state constraints: Given multiple state constraints composed by Boolean logic deseribing the safe set X. we equivalentyy represent X as a zero super level set of single non-smooth function c(.), i.e., X = 1. = (x c(z) 2 0}.
(S2) Inner approximation of safe set: Given the constraint function c(-) obtained from the previ- ous step, we derive smooth minorizer c(-) of c(-), i.e., c(z) 2 c(x) for all E D, s.t. Lg (c) S X.
(S3) Learning performance-oriente CBF: To approximate the largest control invariant subset of Lg (c), we design training loss function based on control barrier-value functions and HJ PDE [33]. We propose parameterization of the CBF and a sampling strategy exploiting the structure of the PDE.
3.1 Composition of Complex State Constraints
Suppose we are given N sets S; := L = far s,(x) 2 0} where each s; R
continuously differentiable and the safe X' described by logical operations on { Since all Boolcan logical operations can be expressed as the composition of the three fundamental operations conjunction, disjunction, and negation [6, 15], it suffices to only demonstrate the set operations shown below:
1. Conjunction: x E S; AND x E Sj + E S = (x 8(x) = (i) sy(x)) 2 0}- sy 2 0}.
2. Disjunction: I E S, OR E Sj + S US; = {2 ((x) =
3. Negation: NOT x E S + x E S.. x ((x) == - (x) 2 20 (complement of S,).
The conjunction and disjunction of two constraints can exactly expressed as one constraint com- posed through the min and max operator, respectively. Furthermore, negating a constraint (z)z) 2 0 only requires flipping the sign s; These logical operations enable us capture complex geome- tries and logical constraints as illustrated in the following two examples.
Example 1 (Complex geometric sets). Consider = [I1 x2] E I2² and two rectangular obstacles
given by O == {i 8: I < [ with i = 1, The union of the two rectangular obstacles O, U. is nonconvex set. Define the following functions
(1) 2 + C1, (1) x (3))) 22 + 82(x) =2 bb;
85(x) = - + 02 80(x) = x1 - -22 (7(x) = -2 + d2, (8(2) = x2 bz:
and let c(x) max(min (I), s2(2), sa())) min(ss(x), s6(x), sT()), sz())))- Then, have T-g (c) = O U Oz.
Example 2 (Loogical constraints). Consider the three constraints s1(x) 2 0, 1, 2, 3, at least two of which must be satisfied. This specification is equivalent to the constraint c(x) 2 0, where
clx) = max(min(s) (z), 82(x)), minn(22 (x)) (x)). min(si (x), sa(x)))
In summary, by composing the min and max operators, we can construct level-set function Ra + RR such that x E X' + c(z) 2 0, i.e. X = Lg (c). Being an exact description of X,, however, e(-) is not smooth. Next, we find a smooth lower bound of c(-) that facilitates CBF design.
3.2 Inner Approximation of Safe Set
To find smooth lower bound for g(-), we utilize its compositional structure. We bound the max operators using the log-sum-exponentin function as follows [15] (min follows similarly):
log(M) < max(si, (MM) 1 log Σ BBL),
log
(5)
with 3 > As 3 > oo, these bounds can be made arbitrarily accurate. We note that both the lower and upper bounds in (5) are smooth and strictly increasing in each input 8;. Therefore, to obtain a lower bound e(x) of c(x), suffices only to compose the lower bounds on each min and max function. In Figure we show the effect of on the resulting inner approximation.

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Figure 2: Effect of the smoothing parameter 3 on the resulting inner approximation. 3 E {+ox, 2,2).
3.3
Learning PINN-CBF
The smooth function c(-). whose O-superlevel set provides an inner approximation of the safe set X, necessarily CBF. Thus, we aim find the "closest" CBF approximation of gf-). We learn our Neural Network (NN) model using a Hamilton-Jacobi (HJ) PDE from reachability analysis whose infinite-time horizon solution precisely characterizes the CBF maximizing the volume of L (h). Our neural network model is trained by minimizing the PDE residual, grounding the method in physics, and justifying its name (PINN)-CBF.
Hamilton-Jacobi PDE for reachability Consider the dynamics (1) in the time interval [t. 0), where & 0 and are the initial time and state, respectively. Define Ur.oo) as the set of Lebesgue measurable functions
U. Let v5s) 108; xr I, u(.)) [t. 0J R" denote the unique solution of (1) given x and u(-) E Ut.ooj. Given bounded Lipschitz continuous function f: D + R,
[4,
the viability kernel of L (E) { ((x) 0} is defined as
V(t) = {o E LØ (E) Bu(-) E Ui.oj s.t. Vs ((,) E (O))
which is the set of all initial states within Lo (e) from which there exists an admissible control signal u(-) that keeps the system trajectory within L (f) during the time interval [t, OJ. Solving for be posed an optimal control problem, where V(t) can be expressed the superlevel set of a value function called control barrier-value function (CBVF).
the viability kernel
Definition (CBVF [33]). Given a discount factor y 2 0. the control barrier-value function By
D (-00, 0] + R is defined as (x, t) ;= max min e el(s ((bb(s)) u(-)El(p..) eE[r.0]
For < O, we have V(I) fo (x, > unique Lipschitz continuous viscosity solution of the following HJ PDE with terminal condition (x, 0) = ((x) [33, Theorem 3]:
0} [33, Proposition 2]. Additionally, B, is the
min
Ot B, (x, max V ((())))))
(7)
well
Under mild assumptions Lipschitz continuous dynamics and ((x) being a signed distance func- tion, B, t) is differentiable almost everywhere [39]. Furthermore, taking -oo, the steady state solution (x) = (2, -00) gives us the maximal control invariant set contained in Lt (f) [33, Section II.B]:
N(B,, () == min ((x) (x)) max 1,
Vx E D
(8)
well
We will leverage the above PDE to learn a CBF h whose zero superlevel set approximates the maximal control invariant set.
CBF parameterization In the context of our problem, we are interested in the viability kernel of T (c(x)), where e the smoothed composition of the constraints. Thus, our goal to learn a CBF h that satisfies the PDE (h, c) 0. To this end, we parameterize the CBF candidate
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(x = c(x) (x)x),
where dg: D - E20 is a non-negative continuously differentiable function approximator with pa- rameters 0. In this paper, we parameterize 6g in the form of multi-layer perceptron:
60(z)= = 0+ bb). 2k+1 < + bu), k
=0=
(10)
where {We, by 1K are the learnable parameters, (-) is a smooth activation such as the ELU, Tanh, Swish functions [40], and 0+ (-) is smooth function with non-negative outputs, i.e., (+(r) 2
0, vr E R. We choose (-) as the Softplus function, a-(r)- = log(1 + exp(3r)), $ >>0 0.
The parameterization (9) offers several advantages over using a standard MLP model for ha- First, accelerates learning by enforcing the non-negativity constraint c(x:) > h(x) in (8) by design. Second, complex constraints are directly integrated into the system via g(x), simplifying the learning process. Importantly, ho (ar) is automatically non-positive in regions containing obstacles, allowing the learning process to focus elsewhere.
In summary, the parameterization in (9) ensures that he(-) is smooth and satisfies ha(x) S c(x) S c(z) for all E DD implying that L (he) S (c) S Lg (c) = X by construction.
Initialization We propose specialized initialization scheme by letting Wr, by. 0, initializing the candidate hi with This initialization favors finding the closest CBF to g(ar during training. Setting only the last layer zero allows training to break symmetry and retain gradient flow.
Training loss and sampling distribution Given the parameterization (9), we now propose to train he to approximately satisfy the steady-state HJ PDE. This leads the following physics-informed risk minimization
minimize
j(0) Exny (HHI(x;0,n) + AxCBF(2;0,7)) A 0.
(1la)
(=;(1;,7) min c(x)) ho(2), max LnxxL +
(11b)
CCBF(I;0,1) max
max (,h)(z) Lgho(z)u (2o())))-h((()000 Nell
(11c)
The first loss Liu 0, 7) is the square of the PDE residual to increase the volume of L (he), while the second loss LCBF( ( 0, 7) enforces the CBF condition. We note that the latter is implicitly present in the PDE but not enforced by the corresponding loss (2; 7)
Finally, represents sampling distribution over I.g (c), meaning no learning is required within obstacle regions. Efficient sampling in these regions can be achieved using intelligent sampling methods like envelope rejection [41] or the random walk Metropolis-Hastingg (RWMH) [42], both of which are well-suited for cluttered environments. Notably, the training process remains self- supervised, as only domain points are collected for learning.
In the following, demonstrate that jointly enforcing Lu and LCBF together with intelligent sam- pling (IS) improves the performance and safety of the learned PINN-CBF. We note that for poly- hedral actuation constraint sets, the supremum over is achieved one of the vertices, yielding closed-form expression. We summarize our self-supervised training method in Algorithm 1.
Algorithm Training PINN-CBFs
Input: Constraints {c;), smoothing parameter | 0, discount factor y 0, regularization parameter À 2 0
Output: Modules 6-(+) and c(-) s.t. he c(-) (-)-)
1. Following sections 3.1 and 3.2, compose constraints {ci} using 3 to form
2. Sample training data. X {æ; E L [
3. Initialize weights and biases of the model & randomly with Wz, by 0.
4. ERM: minimize
j(Ø) ΣN [Au (I () + ALcrr(2 8 7

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Experiments
We demonstrate the efficacy of PINN-CBF on a 2D double integrator system and a 7D fixed-wing aircraft system. Experiments are performed on Google Colab, where the PINN-CBF training is performed on T4 GPU with ISGB RAM and its validation on a CPU with 52GB RAM. The code is available at https: //tp::/ggtttt/tt// www//////////////////oooe/eeee// httph///..co/mooo//P/-/---- C/INN-CBF.
4.1 Experiment Setup
Double Integrator First, we consider the double integrator benchmark V 0 == u, with E I denoting the position and u R denoting the velocity. The action u R represents the acceleration of the system and directly operates on v.
We generate complex obstacle configuration consisting of rotated rectangles and unit walls bor- dering the state space D (zx -1 S x I 11, -6 S a I 6} shown in Fig. 2. Then, we parameterize PINN-CBF following Section 3.3 and train it with N = 10 uniform samples. At each state in the training data, we compute the values and gradient of using JAX [43] flexible automatic-diiferenttaai and hardware acceleration. The acceleration u. is bounded by E [-1. The nominal controller is given by a PID controller meant to stabilize the system to a target position (see Appendix A.
Fixed- Wing Aircraft The Dubins fixed-wing Plane system has 7D states and 3D limited actuation control. Its full dynamics are shown in Appendix A.2. Sampling and training are done as before except with N 10 compensate for higher dimensional space
{(n, e, d, o. sb, Vir -6 & n,e I 6, -1 < d S 11, 2n I 0,0, v/ I 2T, 0.5 S Vi I 2}. Although 106 (VI0f)7 = 10 can overcome the curse of dimensionality. The action u = {Ay. p. 9} is bounded magnitude by |u| < [10.5, 1, 1]. Rectangular obstacles occur in n, c, d. We chose a nominal trajectory resembling
suggested by naive dimensional scaling, we aim to demonstrate how learning
takeoff to evaluate the PINN-CBF filter against relevant baselines on collision avoidance.
4.2 Results
Double-Integrator We evaluate the combination of loss functions (11b) and (11c) with intelligent sampling (IS) from Algorithm 1, as shown in Table 1. All proposed ablations and baseline methods use the same architecture (9) for fair comparison, accommodating complex input-space constraints. The MLP for dg has layer widths of 50 50 1, while DeepReach [37] includes time as an additional input and is applied over the full horizon with
Using three metrics, We validate at 10a grid samples covering the effective obstacle-fri space (c). The residual error [W(h, c) denotes how close the learned CBF is to the maximal-invariant solution of the HJ PDE, LCBF measures safety violation, and Vol (g( (ho)))Vo(((3 (c)) represents the volume ratio between the learned invariant set and the smoothed safe set, which serves as an upper bound enforced by the architecture.
Method
Mean [W c) over L (c) Mean Lerr over L (c)
Vol(L* (he))/Vol(L4) (c))
LHU LIU + X
0.026
0.016
0.566
0.026
0.015
0.547
Lu + XC (IS)
0.022
0.011
0.619
L2 LB [22]
2.002
8.057 10-5
0.0
NCBF [26]
0.178
0.177
1.0
ReepReach [37]
0.206
0.195
0.984
Table 1: Comparison of Lu methods with our ablations above baselines for the 2D Double-Integrane system.
In the ablations, scalarization with À 0.2 reduces safety violations but decreases volume. con- trast, intelligent sampling lowers residuals and safety violations while increasing volume, suggesting mitigates tradeoff scalarization. Among the baselines, directly enforcing the CBF condition
(2) with LE results in a highly safe but ineffective filter with zero volume. NCBF and ReepReach achieve high volume but with significantly higher safety errors. Next, we will examine how these methods perform on more complex fixed-wing plane system.
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Fixed- Wing Plane The same comparisons conducted for the Double-Integrale are also performed MLP architecture. DeepReach
with the following adjustments: all methods utiliize 701700 100 includes an additional input for time. Additionally, validation is carried out on 10" uniformly random sampled states instead of grid samples.
Model
Mean (W c) over L (c)
Mean LCBF over L (c)
Volume of L.j (ha)) Lo (c)
LIu
0.345
0.088
0.707
LHI + >C
0.345
0.082
0.702
LHU + XB (IS)
0.339
0.081
0.700
CE [22]
5.794
2.731 x 10-1
8.400 x 10-6
NCBF [26]
0.741
0.240
0,579
DeepReach [37]
0.543
0.500
0.951
Table 2: Comparison of Lu methods with our ablations above baselines for the 7D Fixed- Wing plane system.
Training with fCrF is ineffective, yielding nearly zero volume. The performance of NCBF severely declines due to the increased difficulty of sampling for counterexamples in higher dimensions. In contrast, DeepReach achieves higher volume at the cost of increased safety violations. Our method, PINN-CBF, achieves both low conservatism and safety. To illustrate this, we visualize the perfor- mance of PINN-CBF, NCBF, and DeepReach in filtering a nominal trajectory representing takeoff.
Fixed-Wiie Wing Aircraft Example In Fig. 3, the fixed- wing aireraft starting from the given position collides with 3D obstacle, while in Fig. 4, obstacles are avoided with the PINN-CBF safety filter.
x
Figure 3: Nominal reference trajectory illustrating crash. The red arrow is the direction of velocity.
0k
Dy
Figure 4: Agile takeoff avoiding obstacles under actuation limits: NCBF is conservative, DeepReach collides, PINN-CBF balances both. A vidco is linked here.
5
Conclusion and Limitations
We introduced self-supervised framework for learning control barrier functions (CBFs) for limited- actuation systems with complex safety constraints. Our approach maximizes the control invariant set volume through physics-ine -nnfome- learning on a Hamilton- (HJ) PDE characterizing the viability kernel. Additionally, we proposed neural CBF parameterization that leverages the PDE structure.
A key limitation of learning- based CBFs is the lack of guarantees on the control invariant set, as well as the need for many samples, similar to ihysics-informed methods. Our intelligent sampling (IS) strategy mitigates this to some extent, but the approach remains restricted to specific domains and obstacle configurations. Future work will explore domain-adaptive CBFs that generalize across different obstacle settings, building recent advances in physics-informed techniques and further investigating links between CBF learning, HJ reachability analysis, and Control Lyapunov Value Functions [44].
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A Appendix
A.1 Reference Control
The reference control for the (DI) obeys the PID control law
Unom (t) Kye(t) + Ki J. e(T) + Kd de(t) dt
where e(l) is the error between the current state and the target state at time l. For our experiments, we use the gains КØ = 1.2, K = -220, K = 0.1, Kÿ 0.1 with KT K =.
For (F16), we hand-design reference control policy resembling takeoff. To test the PINN-CBF filtering abilities, we obstruct the trajectory with rectangular obstacles and place walls restrict the safe spatial coordinates to box. In Figure 4, filtering demonstrates agile planning and obstacle avoidance. The following is an example control policy for takeoff while banking.
(t) t + P(t)
P sin(2t), Q(t) cos(t), =(0) 10,1,2,0,0.
A.2 3D Dubins Fixed-Wing Aireraft System Dynamics
Following [45], the dynamics of the Dubins fixed-wing aircraft (F16) system is given by:
= VT cos ei cos O,
é = VT sin & cos 0,
d Vr sin A
à P + sin ø tan 0 Q + cos ø tan 0 R.
0 = cos & Q -sin の R,
sin Φ
cos 0 Q + cos Φ R.
(12)
cos 0
Vr = AT,
gD
sin @ cos
VT
where [n e d ab Vy ]T are the states and [Ar P Q are the controls. In particular, n, e, d are the position coordinates, VT is the tangential velocity of the aircraft, Φ, y the Euler coordinates orienting the plane, Ar is the tangential acceleration control input, P and Q are rotational control inputs, and gn is gravitational acceleration.
12


Reviews:
[{"review_id": "Ln2C2MiCTc", "reviewer": "Reviewer_LoGn", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 3, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "The paper introduces a novel approach towards learning control barrier functions focused on improving the performance of the system by maximizing the volume of the control invariant set. They utilize HJ-PDE inspired training loss to achieve this. Their formulation handles multiple safety constraints by utilizing composition of safety constraints followed by smoothening to get an inner approximation of the safe set.", "summary_of_recommendation": "Please read the review above.", "technical_quality": 4, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 3, "quality_of_limitations": 1, "questions_for_rebuttal": "1. Since the paper utilizes a neural network based control barrier functions, 100% safety cannot be guaranteed. It would be great if the paper compares the safety rate of the different formulations listed in table 1. Does performance oriented CBF have lesser or more safety rate compared to other methods?\n\n2. What effect the does the discount factor $\\gamma$ have on the learned CBF. what value was chosen for implementation and was that kept the same throughout the training?\n\n3. In section 3.2 what does each input $y_i$ correspond to. This is a bit confusing. They should clearly specify it and relate it to terms in section 3.1.\n\n4. In Table 1, the volume of ${L_0}^{+}(h_{\\theta})$ is higher for NCBF as compared to $\\mathcal{L_{HJ}}$. Could you provide a reason for that?\n\n5. The limitations of the proposed method are not mentioned. Please add that as well.\n\n6. There are multiple typos in the paper which should be addressed.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "8MLV0lyLg6", "reviewer": "Reviewer_6drv", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 2, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "Synthesizing CBFs that maximize the resulting control invariant set for nonlinear high-dimensional systems remains an open challenge. The authors propose several useful modifications to existing learning-based reachability frameworks to stabilize and improve the training of a specially parameterized neural CBF. The efficacy of the proposed framework is demonstrated on a 2D double integrator system and a 7D fixed-wing aircraft system.", "summary_of_recommendation": "Although each proposed modification to the learning-based synthesis of the CBF is modest, the combined framework will likely serve as an illustrative baseline for future works. The paper is also greatly strengthened by high-quality writing, presentation, and context. For these reasons, I weakly recommend that the paper is accepted.", "technical_quality": 3, "clarity_of_presentation": 4, "potential_impact": 3, "robotics_focus": 3, "quality_of_limitations": 1, "questions_for_rebuttal": "- How does using the proposed smooth minorizer of the constraint function compare to using, e.g., the minimum signed distance function commonly used in HJ reachability problems?\n - The reviewer strongly suggests including a comparison of the proposed framework with an existing baseline to demonstrate the improvements achieved by the proposed parameterization of the neural CBF and proposed training scheme.\n - What key advantages does the CBVF formulation have compared against the HJ reachability value function formulation?\n - How are the algorithm hyperparameters $\\lambda, \\beta, \\gamma$ selected?", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}, {"review_id": "NlB6cdGGPd", "reviewer": "Reviewer_EPGh", "summary": "", "questions": "", "limitations": "", "rating": 3, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": 2, "strengths_and_weaknesses": "", "was_revised": false, "final_justification": "", "rebuttal": "", "claims_and_evidence": "", "methods_and_evaluation_criteria": "", "theoretical_claims": "", "experimental_designs_or_analyses": "", "relation_to_broader_scientific_literature": "", "essential_references_not_discussed": "", "other_strengths_and_weaknesses": "", "other_comments_or_suggestions": "", "questions_for_authors": "", "summary_of_paper": "This paper proposes to learn a neural network control barrier function to achieve maximal performance by solving the HJ PDE using CBVF. To find the inner approximation of safe set, logical operators are adopted to construct state constraints. Experiments are conducted on two simulation examples.", "summary_of_recommendation": "Incremental contribution, unconvincing experiment, but more convincing after rebuttal", "technical_quality": 3, "clarity_of_presentation": 3, "potential_impact": 2, "robotics_focus": 3, "quality_of_limitations": 1, "questions_for_rebuttal": "See weakness above.", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "extra_scores": {}}]

Author rebuttal:

Meta-review: The paper introduces an approach to learning control barrier functions focused on improving the performance of the system by maximizing the volume of the control invariant set. They utilize HJ-PDE inspired training loss to achieve this. The proposed formulation handles multiple safety constraints by utilizing a composition of safety constraints followed by smoothening to get an inner approximation of the safe set.

Strengths
- The paper is well-written and easy to follow. The technical content is clearly presented.
- The proposed framework is well-motivated by theoretical analysis of the structure of the CBVF.

Limitations

- Several reviewers have raised concerns regarding the novelty and contribution of the proposed approach over existing methods. 

- The experiment results do not demonstrate a clear advantage achieved by the proposed modifications over existing approaches. Further quantitative comparison is required to establish the utility of the proposed approach.

- The experiment results presented as it is somewhat confusing. It will help to separate out the baseline comparisons and ablation studies, clearly marking the winning approach.

- The authors should discuss the limitations of the proposed approach, such as the effect of using a learning-based approach for safety.

**Comments post rebuttal**

The authors have successfully addressed reviewers' concerns. The new experiments conducted by the authors were particularly useful in demonstrating the utility of the proposed method. The authors should incorporate the additional experiment results into the revised paper to make it more solid and convincing. In addition, the authors should include a metric for comparison that is more interpretable than a loss term (e.g., some sort of % safety rate?) in the final version.
Decision: Accept