Safe Control Under Input Limits with Neural Control Barrier Functions
Authors: Simin Liu
Changliu Liu
John Dolan
Venue: CoRL 2022 Poster (2022)
Keywords: safe control
input limits

Abstract: We propose new methods to synthesize control barrier function (CBF) based safe controllers that avoid input saturation, which can cause safety violations. In particular, our method is created for high-dimensional, general nonlinear systems, for which such tools are scarce. We leverage techniques from machine learning, like neural networks and deep learning, to simplify this challenging problem in nonlinear control design. The method consists of a learner-critic architecture, in which the critic gives counterexamples of input saturation and the learner optimizes a neural CBF to eliminate those counterexamples. We provide empirical results on a 10D state, 4D input quadcopter-pendulum system. Our learned CBF avoids input saturation and maintains safety over nearly 100% of trials. 

Safe control under input limits with neural control barrier functions
Simin Liu, Changliu Liu, John Dolan
Robotics Institute
Carnegie Mellon University
(siminliu, cliu6, jdolan) Candrew. cmu.
Abstract: We propose new methods to synthesize control barrier function (CBF)- based safe controllers that avoid input saturation, which can cause safety viola- tions. In particular, our method is created for high-dimensional, general nonlinear systems, for which such tools are scarce. We leverage techniques from machine learning, like neural networks and deep learning, to simplify this challenging prob- lem in nonlinear control design. The method consists of learner-critic critic architec- ture, in which the critic gives counterexamples of input saturation and the learner optimizes neural CBF to eliminate those counterexamples We provide empir- ical results on a 10D state, 4D input quadcopter-pendun system. Our learned CBF avoids input saturation and maintains safety over nearly 100% of trials.
Keywords: safe control, input limits
1 Introduction
In theory, control barrier functions are an appealing tool for safe control. However, it is difficult to make the derived controllers respect input limits, which reduces their usage in practice. CBFs target the set invariance class of safety problems, in which safety is defined keeping system's 'ss state to prescribed region. A large part of their appeal is that they offer mathematical guarantees of safety. Such assurances are essential for safety-critcaal robotics applications, like collision-free drone flight [1, 2], manipulators that work safely around humans [3], and stable bipedal walking [4]. However, these safety guarantees break down when input saturation occurs, since that means the system cannot exert the force required for evasive maneuver. The system then becomes endan- gered (or dangerous), with the possibility of expensive equipment failure or people getting harmed. It is therefore imperative that we account for input limits when designing CBFs.
CBFs are energy functions which map states to an energy value, with safe states having lower en- ergy. In energy function methods, an energy function is found and then a controller is crafted that dissipates the energy. The core problem of these methods is constructing the energy function. A valid energy function has to meet complex constraints that depend on the input limits, system dy- namics, and safety specification. So far, this problem of designing CBFs around input limits has been studied to a limited degree, mostly for small or simple systems. To our knowledge, nothing has been proposed which handles the nonlinear and high-dimensional systems that are more realistic in robotics. Currently, many state-of-the-att approaches rely on hand-designing CBFs. This works well for certain simple systems, like the kinematic bicycle system [5, 6, 7, 8, 9, 10, 11, 12]. Some of these hand-design methods more systematic, deriving non-saturating CBF for special classes of systems, like polynomial systems [13, 14]. Yet another variation of hand-design is to hand-select parametric function for the CBF and optimize the parameters to avoid saturation [15, 16, 10]. The problem of designing set, well-known problem in the controls community [17]. This area has long history and has been studied under different viewpoints and names, including viability kermel computation [18] and infinite-time reachable set computation [19]. For condensed survey, see [20]. The most generic framework for computing control invariant sets is HJ Reachability [21]. Although it can handle non- linear systems and gives formal guarantees against saturation, it cannot typically scale past systems
saturaauua CBF also equivalent to computing a control invariant
6th Conference on Robot Learning (CoRI. 2022), Auckland, New Zealand.

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of 6 or 7 dimensions in the state. This paper takes a different approach from HJ Reachability and abandons formal guarantees for better scalability.
For the most part, existing approaches for synthesizing non-saturating CBF apply to narrow classes of systems and can involve considerable human effort. In contrast, we envision framework for automating CBF synthesis that has a wide range of applicability. To achieve this, we borrow ideas from machine learning (ML). We take inspiration from the related field of Lyapunov function (LF) synthesis, which has incorporated ML with success. LFs certify stabilization, rather than safety. However, finding a non-saturating CBF essentially finding function that satisfies a constraint on set of inputs, which is the same problem as in LF synthesis. Recent works in LF synthesis represent the LF as NN and then train it to satisfy the function constraints [22, 23, 24, 25J. We borrow this paradigm for the unique problem of input saturation of CBFs. For additional related work, please see Appendix Sec. 6.1.
There are several advantages to framing the problem as NN training. Firstly, it allows us to handle synthesis for nonlinear systems. Good non-saturating CBFs for nonlinear systems tend be nonlin- car functions, and NN are a richly expressive class of nonlinear functions. Secondly, it allows us to handle synthesis for systems with large state dimensions (2 10D). Neural networks can be trained quickly for inputs (here, the system state) of this size.
We synthesize CBFs that respect input limits by posing this as a problem of training neural function to satisfy limit-related constraints. Our contributions are as follows:
1. A novel way frame the synthesis of non- non-saturating CBF.
2. The design of a training framework, including neural CBF design, loss function and training algorithm design.
3. Experimental validation on 10D state, 4D input nonlinear system.
The rest of this paper is laid out as follows: Sec. explains how CBFs work and carefully define the input saturation problem. Then, Sec. 3 details our approach, including the design of the neural CBF, training losses, and training algorithm. Finally, Sec. 4 describes how we test our method on a challenging guancuuter-pendulum system against several baselines.
2 Preliminaries
In this section, provide review of CBFs, mathematically define non-saturating CBF, and explain the premise of CBF synthesis. First, some notation: for a function R" R, let C & <o) be its zero-sublevel set, ÔC = {c}= the boundary of this set, and Int(C) (c) <o the interior. Now, assume the following is given: (1) control-affine system 1 = f(x: g(x)u, where x E D C R"" u E U C Thm and f R - TR.", I -> Raxm are locally Lipschitz continuous on IR"" (2) imput set bounded convex polytope, and (3) safety specification p R + R. which implicitly defines the allowable set A _ {p) Further, assume p(x:) is continuous and smooth. Given p(z), we can define r E = as the relative degree from p(x) to (i.e. the first derivative of p(x) where u appears).
We now walk through the process of producing a safe controller via CBF methods. In safe control, the goal is to keep some subset of the allowable set forward invariant, which means keeping any trajectory starting within the subset inside of it for all time. We call this subset the safe set and will be defined by function, the control barrier function, which we design. The CBF will also be used to define the safe controller that ensures forward invariance.
In the absence of input limits, we would just form CBF known function of the safety specifi- cation p(x):
2 at } 1 +
I
(limit-blind CBF)
where < O. See [5, 26] for an explanation. The safe set S C A defined by this limit-blind CBF is elaborated in Appendix Sec. 6.2. In the presence of input limits, we will have to modify this design, but we will come back to this. Next, to define safe controller using CBF is straightforward. A

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safe controller simply needs to repel the system back into the safe set whenever it reaches the set boundary. The following theorem formalizes this idea:
Theorem (Taken from [5]). Given CBF á and safe set S, any feedback controller k(x) R IRmh satisfying
6(z, k(x))
Volz) Vo()) 9() 0 Vz E as
(1)
L2((2)
(x)(2))
renders the system forward invariant over S.
Note that you can also require CBF to satisfy stricter inequality Ø(x) < -a(((x)) for all r E DD where ox-)) is a class-k function. We elaborate on this alternative in Appendix Sec. 6.2. The theorem above requires the controller to repel the system (decrease Φ) from the boundary (x: E 0S). We are allowed to use any nominal controller, knom, as long as we modify its inputs to satisfy Eqn. the boundary. Thus, CBF-based safe controller simply filters (modifies) nominal controller online by applying the QP below at every step of control execution:
k(x) = arg min uel
||u
Knom (2)
(CBF-QP)
2
10
if as
s.t. Lj((x) + Lg0(x)u
(2)
00 O.W.
The issue with using a limit-blind CBF for this controller that it can cause controller saturation. Specifically, saturation occurs when no E U exists that satisfies Eqn. 1 (constraint 2), causing the loss of safety guarantees." What we need is to synthesize a non-saturating CBF, which mathe- matically defined as:
Definition 1 (Non-saturating CBF). A function o R" + R is a non-saturating CBF over a set S for all x OS:
inf d(r u) S 0 well
Intuitively, Def. just requires that there exist feasible control input to decrease & (push the system to the interior of S) at every state on the boundary, 0s. We approach this problem by modifying the limit-blind CBF:
cr od)
6- II
1 1
p P p
(modified CBF)
where (²r)) IR" + R such that (x))) S {o(x)) This modification acts to shrink the associated safe set from S St With proper choice of function p (æ)) we can exclude irrecover- able states (states where no fecasible input preserves safety) from the safe set boundary. In the rest of the paper, we focus on learning the function p² (x) to produce a non-saturating CBF.
Problem scope: to review, our proposed method applies to nonlinear, control-affine systems and safety problems that can be described by smooth safety specification function p(x)). We also assume the system is deterministic and fully known. The method also applies to systems of high relative degree, since we based the modified CBF off of higher-order CBF.
Learning Non-Saturating Control Barrier Functions
In this section, we present neural CBF design (Sec. 3.1) and then discuss the training framework which optimizes it with respect to control limits 3.2, 3.3). Learning is formulated as min-max optimization problem of the following form, where 9 denotes the parameters of o":
min max C(0,z) x)) 0 xEaS'
(4)
In practice, to avoid an unsolvable QP when saturation occurs, we add slack variable to Eqn. 2. However, will still violate safety guarantees.
3

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angle
aeggle
anglle
anggle
(a) Iter. 0
(b) Iter. 70
(c) Itor. 140
(d) Iter. 750
Figure 1: Learned safe sets for the toy cartpole problem four iterations during training. Candidate counterexamples are marked in black. It is observed that the critic correctly identifies that the states with severe saturation are those with angle and angular velocity the same sign (angular velocity swinging the pendulum out from the vertical). In (d), green denotes the larges/ non-saturating safe set, computed using MPC. Note that our volume enlargement is so effective that the learned safe set attains 95% of the largest volume.
We call this loss [(0, x) the saluration risk. This min-max problem is solved using a learner-critic algorithm (Alg. 1). where the critic and learner repeatedly find where the worst saturation occurs and then update the CBF to reduce saturation there. We also propose some strategies for training stably, boosting critic efficiency, and for increasing the volume of the safe set. For visualization, plot the critic's counterexamples for toy cartpole problem and intuitively justify their correctness (Fig. 1).
3.1 Neural CBF Design
Building upon the previous work discussed in Sec. 2, to design a non-saturating CBF, we only need to consider the design of the function p² (2) from the modified CBF, Let nn IR - R be multilayer perceptron with tanh activations. Then, we define
p'(x) = (n(z))) + p(z)
-
(5)
where Ig is state, identified by the user, that should belong to any reasonable learned safe set. Specifically, Te should belong to the allowable set AA and should satisfy (x.) < O for i [0, r 1] (this constrains any possible higher-order components in Ze; for example, velocities should be directed away from unsafe zones). For safety problems that limit the system's 's distance from an equilibrium, the equilibrium itself should lie any reasonable safe set. For anti-collision problems, re can be point far from the ego robot. For additional recommendations, see the Appendix. We have designed p² to obey three constraints: Constraint I: p is smooth. This is required preserve make existence and uniqueness arguments for the closed-
the smoothness of d", which allows loop system. Our p² satisfies this because nn has smooth tanh activations and also is assumed smooth. Constraint 2: The 0-sublevel set of p* is contained within the 0-sublevel set of p: {p" [p'}<o S (p)s0- The allowable set can be shrunk but not enlarged; otherwise, dangerous states may be incorporated. Our design meets this criterion because p" p. Constraint 3: S* is nonempty, where S* is defined by p" With our design, Ee E S by the definition of S* (see Appendix) and our assumptions on Te.
3.2 Training Framework
In the following sections, we present a loss function that encourages satisfaction of Eqn. and the algorithm for solving the min-max problem on this loss. First, define 0 as the parameters of o*, which include the weights nn and the e; coefficients. Our loss function, called saturation risk, is defined as:
L(8,x)
A inf (a u)
(saturation risk)
weld

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Algorithm Learning non-saturating CBF
1: function LEARN(X 0)
2: Set learning rate aj
3: -
0 V E
R(0)]
From Eqn. 6,
4: return 0
5:
end function
6: function COMPUTECE())
7: Set learning rate ae. number of gradient steps N
8: X uniformly sample set on the boundary
b See Alg.
9:
for i in [O..... N] do
10:
G f- VxL(0,
Batch gradient
11:
P <- project G along boundary
12:
X + X + a. P
> Batch update
13:
X project X to boundary
>> See Alg.
14:
end for
15: X f- worst saturating states in X
16:
return A
17:
end function
18:
function MAIN()
Input: dynamical system it, safety specification P
20:
Randomly initialize neural CBF parameters
21:
Repeat:
22:
Xcee + COMPUTECE(()
D Xace is set of counterexamples
0 LEARN(X 0)
24: Until convergence
return /
26: end function
It is measure of the best-case saturation at given state When L(0, 0, then saturation occurs at x; when [(A, x) > 0, measures how severe the saturation is. Thus, our min-max problem (Eqn. 4) is to minimize the worst best-case saturation the boundary. When the worst best-case negative, i.e.
max C(O, x)) 0
(training goal)
#ESS
then we have successfully found a non-saturating CBF.
To solve the min-max problem on L(O, .r) (Eqn. 4), we propose a leamer-critic algorithm (Alg. 1). Essentially, the algorithm alternates between the critic computing counterexamples (maximization with respect to and the learner updating the CBF (minimization with respect to 0). The critic uses projected gradient descent to produce an approximate maximizer, 4", and then the learner uses gradient descent to minimize the saturation loss at 6" Since both learner and critic perform gradient useful to re-express it as an analytic function, rather than continuous minimization. To find the analytic expression, observe that this C(O, x) is minimization over where the objective affine (from Eqn. 1) and the constraint set a convex polyhedron Ut by assumption. This means the minimizing u" is one of the vertices v E V(2() of the constraint set. Thus, [(8, .)) can be computed as a discrete minimization, which analytic:
descent on L(O, xr)
L( xx A min 22(,, vEV(L))
(analytic risk)
3.3 Practical Training Methods
Batch optimization: in practice, training works better if both learner and critic batch of counterexamples, rather than just a single one. This means the critic optimizes a set of differently initialized counterexamples once and the learner takes weighted loss over subset of the best counterexamples:
a Vo
Σ
softmax((((,z))
(6)
Ex
5

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15
10
5
20
10
-5
0
-10
-10
-10
155
-15_1
-2
0
20
(b) 7 v. y
(c) ( v. 0
(d) 0 v. (pend, quad pitch vel)
(a)
(quad roll, roll vel)
(pend pitch, pitch vel)
Figure 2: (Left) quadcopter prenuttur-ppnuuumm system (image from [30].. (Others) Axis-aligned 2D slices of the 10D safe set (blue is ours, purple is hand-designed CBE, green is safe MPC). For each slice, unvisualized states have been set to
This has the advantage of stabilizing convergence without adding much overhead. It stabilizes con- vergence by preventing (1) deadlock and (2) inaccurate gradients throughout training. Deadlock when improvement at counterexample causes saturation another; averaging the learner's loss on a group of counterexamples avoids this by encouraging progress on many counterexam- ples at once. Inaccurate gradients refers to the fact that the learner should be using the gradient at the optimal counterexample x² (V, x²) z²)),, but since the critic is suboptimal, it uses differ- ent gradient, VoC(O, Clearly, this could derail training. However, we find that with batch- ing, the critic produces better counterexamples, giving us closer estimate of the true gradient (VoL(8, .²) = V0 ())). Plus, with averaging, the learner averages out the effect of inaccurate gradients. We show in the Appendix that the algorithm requires large batches of counterexamples to perform well (Table 2). All in all, these techniques are cheap and effective way to avoid the kind of training instability found in other counterexample-based methods, like adversarial training for image classifiers [27, 28,29]. 29].
Enlarging the safe set: in this section, we introduce a regularization term that we add to the training objective help enlarge the safe set. One measure quality for safe sets size. Since CBFs will allow a system to move freely inside a safe set, a larger safe set provides greater freedom towards accomplishing control objectives. Thus, larger safe set enables better task performance. To this end, we add regularization term to the training objective to encourage large safe set. For context, recall that our CBF og defines safe set S* Specifically. there is function
of øg that implicitly defines S* as its O-sublevel set: S = {ar|ss ss" (1)) S 0}); ss" (ar) is defined in the Appendix. To clearly indicate its dependence on the CBF and its learned parameters 0, we write it as ss here. Next, we evaluate the regularization term at some sampled states Xreg- We have:
R(0)
sigmoid (x))
xEXrep
The idea behind the sigmoid is to encourage states near the boundary (z)())(z)) = 0) to move (further) inside the safe set (ss(o)(z)) << 0). Sigmoid gives a gradient which encourages values near zero to become (more) negative. We demonstrate in the Appendix that including this term can increase the volume by several factors.
4 Experiments
In this section, train neural CBF challenging nonlinear, high-dimensional, and input- limited robotic system. We pose two experimental questions:
Q1. How well do we achieve our training objective?
Q2. Does the Cas-d safe controller ensure FI?

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Saturation the boundary
Safety of rollouts w/ diff fsom
Safe set volume
% non-sat. states
meun, dev
sat.
Rlyr
ktor-ass
(as fraction of ours)
Ours Hand- Hend-designed CBF
99.00
1.75 + 2.40
15.19
99.62
99.62
99.02
1,00
78.68
4.99 3.78
29.50
78.68
80.28
80.28
53.87
Safe MPC
99.06
99.54
99.44
0.08
Table I: Comparison of our method against baselines. The "mean, std dev of sat." is E[C(0, æ)] +
of [( x>] and "worst sat." is [(0, x.))
Our system is a pendulum on top of a quadcopter (Fig. 2). This is a coupled system, with the dynam- ics of both components found by the Guler-Lagrangian method [31]. The states are quadcopter posi- tion and roll- roll-pitch-yawe satch-yawwa orientation (I, y. and 7 a, .9) and roll-pitch pendulum orientation (Øp. 8p), as well as the first derivatives of these states. The inputs are thrust and torque (F, Tr, TB, Ta), which are limited to a bounded convex polytope set. See the Appendix for the system dynamics. The safety specification is to prevent the pendulum from tipping and the quadcopter from rolling:
p= p= + 8 57 (s/4)²
(8)
where 6, = arccos(cos((p) (0p) cos()p)) the pendulum's 's angle from the vertical. Since the quadcopter
position does not impact safety and the position dynamics can be decoupled, we exclude position and consider the resulting 10D state, 4D input system. Finally, in the design of p (ar), we let J. = Ö. which is the system's 's equilibrium.
Next, we propose metrics to answer each of our experimental questions:
Q1 metries. % of non- satusaatratinn states ôs* we measure how well we satisfy Eqn. 3 by uni- formly sampling states on 0s* and calculating what percentage of them are non-saturating. We also use these samples to compute the mean and variance of the severity of saturation, [E] and o [L, 2.). To approximate the severity of the worst saturation, [(0, xx",, we apply our critic and allow it to more samples and computation time than during CBF training.
Q2 metries. Q2 considers the in-the-loop control performance of our learned CBF. We measure % of simulated rollouts that are Fl by initializing rollouts randomly inside S and simulating their trajectories under the safe controller (CBF-QP) until just after they reach the boundary. Then, we record whether the system exited or remained inside S after arriving at the boundary. The value of this metric depends on the choice of nominal controller, knom. We try ko(z) = O, which yields an unactuated system, and higr and kigr-agg which are linear quadratic regulator (LQR) stabilizing controllers, with kigr-agg tuned to be more aggressive.
We also choose two
alternatives as our baselines:
Hand-designed CBF: for CBFs, this is a typical alternative. We hand-pick parametric CBF and then optimize the parameters for non-saturation.
Safe MPC: MPC is often used for safe planning and control and it can take input limits into account. The safety specification (p I 0) becomes a nonlinear state constraint and we also have to set the terminal constraint to be a smaller, known invariant set for the MPC solution to be forward invariant.
For details on any of these baselines, see the Appendix Sec 6.4. Note that safe MPC defines its safe set implicitly. This means the boundary of the safe set is not defined by function. Hence, it would be too time-consuming to sample states on the boundary and compute the metrics for Q1. For safe MPC, we only report the results for Q2.
Code. Our code can be found at https::/giitee http://ggghtuu...cms//////00//// cow hll htmmm//sl22111111nnn http://ccccossaa20/////tpm...tt.
Training details. The learning framework and metrics were implemented using Python and Py- Torch [32]. Training took about 2 hours on single NVIDIA GeForce RTX 2080 Ti GPU.
Discussion. As we can see in Table for our learned CBF, almost 100% of the boundary states are non-saturating and almost 100% of the rollouts are FI across the nominal policies. This shows that our neural CBF and learning framework are an effective combination and capable of handling systems of high complexity. We are not able to attain 100% non-saturating states and Fl rollouts, which is either due to suboptimality of training or limitations of the function class, as NNs are only universal approximators when their size is taken to the infinite. For our learned CBF, the severity of saturation at the saturating states is not negligible, but still low, and the worst saturation is large, but rarely encountered. The hand-designed CBF does poorly across all metrics (only 80%
7

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of the boundary states are non saturating and around 80% of rollouts are FI), Safe MPC is equally good as the learned CBF at ensuring safety (almost 100% rollout safety)². However, its significant disadvantage is that its safe sets are just fraction of the size of our own (8.4% respectively). Size is an important measure of the quality of safe set; these small sizes imply that this baseline can only ensure safety from relatively few states. The root of the issue is that safe MPC constructs a safe set by effectively expanding a smaller, "seed" invariant set provided by the user. The size of the safe set therefore depends greatly the size of the seed invariant set. As we have already established, it can be quite difficult for users to design invariant sets (equivalently non non-saturating CBFs) of any reasonable size.
Visualizing slices of the safe set can provide deeper insight into the results of training (Fig. 2). Recall that a safe set contains only states that are recoverable from danger, given our input limits. We observe that our CBF has diamond-shaped safe sets in slices B and C, which makes sense because small angles can still be recoverable at larger speeds. In slice B, safe MPC's set largely captures states where the signs of 7: differ. This makes sense too, since these are states where the angular velocity acts to return the angle to O. Safe MPC's set in slice B is also larger than ours. Since it is a non-saturating safe set, just like ours, we have to conclude that our algorithm could have found a larger non-saturating safe set. The reason for this suboptimality is that our volume regularization strategy is imperfect. It encourages expansion only at scattered points on the boundary, which means that some areas of the boundary may be unaffected. However, our regularization strategy seems to work well overall, since the volume of our safe set in 10D is much larger than that of safe MPC. In slices C and D, get glimpse of why that is. In both of these slices, safe MPC's set is tiny. It very tightly constraints the pendulum's angular velocity. While it makes sense for safe MPC to be more conservative towards the pendulum than the quadcopter (the pendulum is not directly actuated, while the quadcopter is), it is unnecessarily conservative.
Next, we analyze the shapes of the sets in slice D. Our safe set indicates that there should be maximum safe angular speed for quadcopter and pendulum. This makes sense, since higher angular velocities are certainly harder to pull back from. We also observe larger range for the quadcopter's pitch velocity than the pendulum's 's which makes sense because again, the quadcopter is directly actuated and the pendulum is not. On the other hand, the hand-designed CBF does not restrict Ô or in slice D. Due to the functional form of the hand-designed CBF (see Appendix Section 6.4), and / are only restricted when 60 > 0 or 33 > 0 (that is, when angular velocity is strictly acting to destabilize). We've assumed 0, B in slice D. This safety criterion is clearly too lenient, since large angular velocities that are currently swinging the system to the origin can cause overshooting and toppling shortly after. Overall, function form. In general, it can be hard to guess what the right form might be for a system of this complexity. However, our algorithmic approach, combined with the NN functional representation, removes the need for guessing.
can see that the nen-neural CBF does not have the right
Limitations and future work: in the future, we intend to loosen assumptions (see last paragraph of Sec. 2). particularly the assumption of a known and deterministic model. We would also like to test this method on more high-dimensiioal,, nonlinear systems. One limitation of our work is that we had to perform a change of variables on the states input to the neural CBF before it would train successfully (see Appendix for details). While the dynamies on the new states were still nonlinear, this indicates that a simple feed-forward network might not be the best neural function class.
5 Conclusion
In summary, we proposed framework for facilitating CBF synthesis under input limits. Thanks to our neural CBF representation and our effective and efficient learning framework, our method scales to nigher-dimensional nonlinear systems. We learned a virtually non- non- naturraan CBF on one such system, the quadcopter-pendulun We hope that this safe control tool will makes CBFs more accessible and of practical value to roboticists.
"In theory, MPC should attain perfect safety. However, nonlinear MPC solvers are not "complete" in the sense that even if there exists a solution to their nonlinear program, they may not find it. Hence, sometimes they may falsely consider the safety problem infeasible and provide an unsafe input.
8

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Acknowledgments
This material is supported by the United States Air Force and DARPA under Contract No. FA8750- 18-C- 18-C-0092 and the National Science Foundation under Grant No. 2144489. We thank Tianhao Wei, Weiye Zhao, Yiwei Lyu, and Qin Lin for useful conversations. We are also very grateful to Kate Shih, Qin Lin, Harrison Zheng, Raffaele Romagnoli, Tianhao Wei, Soumith Udatha, Tamas Molnar, and Jason Choi for reviewing drafts of this work.
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17


Reviews:
[{"review_id": "XRq9TDwtBQ", "reviewer": "Reviewer_8LBY", "summary": "This paper proposes a method for synthesizing control barrier functions (CBF) that avoids action saturation for safe controllers. The main contribution is the proposal of a limit-aware CBF, and the usage of a min-max learning framework that uses a critic to find worst-case saturation points and a learner for updating the CBF. They also propose two neural CBF designs, an efficient method for optimizing the learner as a discrete minimizing problem, and introduce a regularization term to encourage larger safety sets. The paper compares their method to a non-neural CBF baseline for a pendulum-balancing task on a quadcopter, and found that their method enables nearly 100% of the boundary states of the safe set to avoid input saturation and nearly 100% of the rollouts are forward invariant inside the safe set, which is a significant improvement over the baseline.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths\n* The introduction of a non-saturating CBF to avoid input-saturation for safe controllers, which is an important consideration when synthesizing CBFs.\n* The two proposed neural CBF designs are mathematically well motivated by the three constraints they need to satisfy to be valid safety specifications.\n* The two strategies of taking the best of a batch and warmstarting the optimizing process for finding the states along the safe set boundary where saturation occurs is a useful inclusion to the process for improving the critic.\n* The re-expression of the saturation risk as an analytical function and framing the learner update rule as a discrete minimization problem over counterexamples is both efficient and well-thought out.\n* The regularization term that encourages states near the safe set boundary to move inside the safe set (or further in) is clever and well-designed in this paper.\n\nWeaknesses:\n* This paper does not implement their method on robot hardware, which makes it hard to determine how effective it is in practice due to the limiting assumptions on known and deterministic dynamics. In lieu of this, it would be worthwhile and more convincing to at least see simulated experiments with noise added to the dynamics to see how the learned neural CBF performs. \n", "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": "", "summary_of_recommendation": "This paper addresses the important problem of avoiding input saturation when synthesizing control barrier functions (CBF) for safety specifications. It introduces a novel CBF formulation and a learning framework for learning to identify worst-case saturation states at the boundary of the safety set and improving the CBF. In addition, the paper includes many additional components to improve the efficiency and performance of the learning method, and demonstrates the efficacy of the proposed solution on a challenging pendulum-balancing domain with a simulated quadcopter. The paper does a sound job of mathematically motivating the design choices in this paper, and for these reasons I suggest this paper be accepted.\n", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "reviewer_expertise": "2: The reviewer is willing to defend the evaluation, but it is quite likely that the reviewer did not understand central parts of the paper"}, "extra_text": {"issues": "* Additional experiments that include stochasticity to the dynamics of the simulated experiments would be more convincing for understanding whether the existing technique is potentially applicable to robot hardware. \n* A discussion on how challenging and important hyperparameter optimization is in this method for learning (e.g: including the regularization term, how many samples are warmstarted vs. taking from best of a batch, etc.)\n\nMinor issues include:\n* Replacing related works that are cited as being on arxiv with the peer reviewed versions (e.g: Clark, Andrew. \"Verification and synthesis of control barrier functions.\" 2021 60th IEEE Conference on Decision and Control (CDC). IEEE, 2021.)\n* The x and y axis titles on Figure 2 (b), © and (d) are quite small and enlarging them would improve readability\n"}}, {"review_id": "UY4gyjwRwV", "reviewer": "Reviewer_V3nm", "summary": "this paper proposes a method to construct Control Barrier Functions using Neural Networks (NN) to avoid input saturations. The keypoint is to ensure the condition satisfaction on the boundary of the safe set, hence they formulate a min-max optimization problem with each step some counter-examples generated from the sampling process. As for the experiment, the authors design a 10d quadrotor-pendulum example and achieve better “satisfaction rate” and “forward invariant rollout rate” than non-NN CBF baseline approach.\n", "questions": "", "limitations": "", "rating": 2, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "Strengths: \n1. The paper is easy to read - it provides detailed thoughts on how to guide the training more efficiently and more stably.\n2. Unlike papers working on only low-dimension systems, they work on a challenging 10d quadrotor-pendulum dynamic.\n\nWeaknesses:\n1. The approach is only testified on one experiment\n2. The approach is only compared with one baseline\n3. No hardware experiments or ablation studies (for different design choices and improvements)\n", "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": "", "summary_of_recommendation": "I recommend rejecting this paper. Although the authors provide an interesting algorithm to construct CBF for input saturations, they lack thorough and solid experiments to persuade the readers about this approach’s benefit. The baseline is weak (only a CBF constructed in a non-NN fashion) and the experiment is only considering the case where an equilibrium point is available. The experiment cannot fully convince us why we need this new approach.\n", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Good: Minor revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "quality_of_limitations": "Additional details required", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed.", "reviewer_expertise": "5: The reviewer is absolutely certain that the evaluation is correct and very familiar with the relevant literature"}, "extra_text": {"issues": "1. Weak baseline: I doubt the strength of this baseline. There are lots of related works [1-4] in using Neural Networks to construct CBF and have witnessed great success. Since NN is not a novelty of this paper, it is unfair to only compare a non-NN CBF. Besides, I would also suggest comparing methods such as reinforcement learning. \n2. Lack of experiments: The system used in the paper has an equilibrium point so the CBF is constructed following design-1. But the paper also introduces the design-2 which is not implemented in the experiment. More experiments are expected to also support that design choice.\n3. Lack of ablation studies: We would like to see how each of the components / tricks in the paper leads to the performance gain.\n\n[1] Wang, Chuanzheng, et al. \"Learning control barrier functions with high relative degree for safety-critical control.\" 2021 European Control Conference (ECC). IEEE, 2021.\n[2] Qin, Zengyi, et al. \"Learning safe multi-agent control with decentralized neural barrier certificates.\" arXiv preprint arXiv:2101.05436 (2021).\n[3] Meng, Yue, Zengyi Qin, and Chuchu Fan. \"Reactive and safe road user simulations using neural barrier certificates.\" 2021 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS). IEEE, 2021.\n[4] Xiao, Wei, et al. \"Barriernet: A safety-guaranteed layer for neural networks.\" arXiv preprint arXiv:2111.11277 (2021).\n\n"}}, {"review_id": "OTN_Bf4QGd-a", "reviewer": "Reviewer_A9jW", "summary": "This paper presents a framework for learning the Input Constrained Control Barrier Function (CBF) for high-dimensional nonlinear systems. The authors formulate a min-max problem to learn the non-saturating CBF. The maximization problem focuses on generating worst-case data points, while the minimization problem attempts to learn the CBF. The authors validate their approach in simulations and demonstrate that the proposed method achieves a higher success rate ensuring safety.", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "1. Insufficient baselines. There are several works focusing on the input saturation problem using CBF, e.g. [1, 2]. The authors only compare with a baseline that does not consider input constraints. This would be difficult to evaluate the actual contribution. More comparisons with proper baselines would be desirable.\n2. The interpretation of the CBF design is vague. There are several questions that are not well discussed. Why was the Design 2 introduced and how is this design used? How does it perform in terms of training and ensuring safety?\n3. Some limitations are not discussed. 1. how to handle systems with higher relative order? 2. how to tackle problems with multiple safety specifications? Input constraints can also be considered as safety specifications.\n4. In the proposed approach, the generation of counterexamples is crucial. However, it is difficult to assess whether the proposed generation technique can be generalized to different tasks, e.g., where the boundary is non-convex. Perhaps the authors could try another task with more complex safety boundaries, such as a robot manipulation task.\n5. Several sections are not clearly explained. Some notations are used without explanation. \n6. Ablation studies are also beneficial to strengthen the paper.", "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": "", "summary_of_recommendation": "Given the discussion in the strengths and weaknesses, I suggest that the authors should make a major revision. Improve the clarity of this paper, compare it with more appropriate baselines, and try different tasks to verify the generalizability of the proposed method.", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Fair: Major revision needed to improve organization or clarity.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are not well addressed", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "3: The work is incremental and unlikely to have much impact, even though it may be technically correct and well executed.", "reviewer_expertise": "4: The reviewer is confident but not absolutely certain that the evaluation is correct"}, "extra_text": {"issues": "1. What is $x_e$ in line 116? Why author introduces Design 2 which is not used in the experiment? \n2. In Figure 1, there is a noticeable difference between the learned set and the MPC set. \n3. The explanation of the analytic saturation risk is not very clear. What is the difference between $\\mathcal{V}(\\mathcal{U})$ and $\\mathcal{U}$?\n4. “Another challenge to speed is systems with large state spaces…” \n5. “We project the saturation risk’s gradient along the boundary, use this to update x, and finally project x back to the boundary.” How is the projection done? It would be beneficial to visualize the procedure to improve the understanding.\n6. What is $s_{\\theta}^*$, and how is it computed? What are $\\mathbb{X}_{reg}$ and $\\mathcal{D}$?\n\n[1] Agrawal, Devansh R., and Dimitra Panagou. \"Safe control synthesis via input constrained control barrier functions.\" *2021 60th IEEE Conference on Decision and Control (CDC)*. IEEE, 2021.\n\n[2] Garg, Kunal, and Dimitra Panagou. \"Control-lyapunov and control-barrier functions based quadratic program for spatio-temporal specifications.\" *2019 IEEE 58th Conference on Decision and Control (CDC)*. IEEE, 2019.\n\n[3] Dalal, G., Dvijotham, K., Vecerik, M., Hester, T., Paduraru, C., & Tassa, Y. (2018). Safe exploration in continuous action spaces. *arXiv preprint arXiv:1801.08757*.\n\n[4] Liu, P., Tateo, D., Ammar, H. B., & Peters, J. (2021). Robot reinforcement learning on the constraint manifold. In *Conference on Robot Learning* (pp. 1357-1366). PMLR.\n\n[5] Akametalu, Anayo K., et al. \"Reachability-based safe learning with Gaussian processes.\" *53rd IEEE Conference on Decision and Control*. IEEE, 2014.\n\n[6] Bansal, Somil, et al. \"Hamilton-jacobi reachability: A brief overview and recent advances.\" *2017 IEEE 56th Annual Conference on Decision and Control (CDC)*. IEEE, 2017."}}, {"review_id": "HZ_60skMRwuW", "reviewer": "Reviewer_wTA7", "summary": "This paper proposes a modification to the Control barrier function (CBF) formulation to allow for non-saturating controls, i.e. controls that do not exceed the actuation limits of the controller. The authors go about this by first characterizing a non-saturating invariant set, and then propose a learning method to learn the parameters of a control barrier function that will ensure a non-empty safe set and a resulting control law to keep the system within the safe set limits. Note that the authors assume that the dynamics of the system are known, and therefore this work is more in line with optimization based control rather than learning about the system dynamics. ", "questions": "", "limitations": "", "rating": 3, "confidence": null, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": null, "clarity": null, "significance": null, "originality": null, "strengths_and_weaknesses": "**Strengths:**\n* Strong theoretical derivation and proofs ensure soundness of the work. While I am not aware of the full background of control barrier functions, I have checked the derivations and proofs within the paper. Prior knowledge was well referenced in the paper.\n* Useful research direction: Controller saturation, and related simplistic modeling of actuation systems is a key limitation for a lot of simulation based robotics research and it is nice to see work addressing the issue.\n* Challenging kinodynamic environment.\n\n**Drawbacks:**\n* Assumptions of knowledge of the system dynamics: As with any planning based approach, the requirement of knowing the system dynamics requires intensive efforts in system identification, and even then often the dynamics are only known with uncertainty. How senstive would the system be to misidentification of the dynamics? Perhaps the authors might be able to include an experiment on that.\n* Affine control law: How frequently is the assumption of an affine control violated in practise? What are the typical ways in which this is addressed? \n\nComments:\n* The authors seem to have adopted an approach where the system starts with a saturating safe-set and works inwards from there. Would the authors comment on a reversed approach where they grow the safe set from a smaller starting safe set? I raise this question as even though the authors present results where the learned FI set was almost entirely safe, there were still saturated control inputs. \n", "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": "", "summary_of_recommendation": "I found the presentation of the paper to be quite clear, the problem is well motivated and the research question is important. The prior works have been referenced, and the novelty of the paper is well established in the writing. I would recommend an accept, but as a non-expert in the field.", "technical_quality": "Excellent: Very convincing and thorough.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Highly relevant to robotics but no hardware experiments", "quality_of_limitations": "Limitations are addressed clearly", "questions_for_rebuttal": "", "reasons_to_accept": "", "reasons_to_reject": "", "questions_to_authors": "", "main_review": "", "summary_of_the_paper": "", "summary_of_the_review": "", "strength_and_weaknesses": "", "limitations_and_societal_impact": "", "review_text": "", "extra_scores": {"impact": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "issues": "Please refer to the main review. ", "reviewer_expertise": "3: The reviewer is fairly confident that the evaluation is correct"}, "extra_text": {}}]

Author rebuttal:

Decision: Accept (poster)