Verified Path Following Using Neural Control Lyapunov Functions
Authors: Alec Reed
Guillaume O Berger
Sriram Sankaranarayanan
Chris Heckman
Venue: CoRL 2022 Poster (2022)
Keywords: Path Following
Trajectory Tracking
Control Lyapunov Functions
Plan Execution
Verified Autonomy

Abstract: We present a framework that uses control Lyapunov functions (CLFs) to implement provably stable path-following controllers for autonomous mobile platforms. Our approach is based on learning a guaranteed CLF for path following by using recent approaches --- combining machine learning with automated theorem proving --- to train a neural network feedback law along with a CLF that guarantees stabilization for driving along low-curvature reference paths. We discuss how key properties of the CLF can be exploited to extend the range of  the curvatures for which the stability guarantees remain valid. We then demonstrate that our approach yields a controller that obeys theoretical guarantees in simulation, but also performs well in practice. We show our method is both a verified method of control and better than a common MPC implementation in computation time. Additionally, we implement the controller on-board on a $\frac18$-scale autonomous vehicle testing platform and present results for various robust path following scenarios.

Verified Path Following Using Neural Control Lyapunov Functions
Alec Reed
Guillaume Berger Sriram Sankaranarayanan Department of Computer Science University of Colorado Boulder, USA. first. Lastname@colorado edu
Christoffer Heckman
Abstract: We present a framework that uses control Lyapunov functions (CLFs) to implement provably stable polh-following controllers for autonomous mobile platforms. Our approach is based on learning a guaranteed CLF for path follow- ing by using recent approaches theorem proving to train a neural network feedback law along with a CLF that guarantees stabilization for driving along low-curvature reference paths. We dis- cuss how key properties of the CLF can be exploited to extend the range of the curvatures for which the stability guarantees remain valid. We then demonstrate that our approach yields a controller that obeys theoretical guarantees in simula- tion, but also performs well in practice. We show our method is both verified method of control and better than a common MPC implementation in compu- tation time. Additionally, we implement the controller on-board autonomous vehicle testing platform and present results for various robust path following scenarios.
combining machine learning with automated
g -scale
Keywords: Path Following, Trajectory Tracking, Control Lyapunov Functions, Plan Execution, Verified Autonomy.
Loop
1 Introduction
Makiu Sumple
Path following is a fundamental primitive for autonomous robotic platforms, wherein the goal is to steer the platform along a desired path in the workspace. This is important for mobile platforms since many approaches to platform design rely on the separation between high- level planners that search over paths without considering platform dynamics and lower-level plan execution which implements path follow- ing [1, 2]. However, path following is a chal- lenging problem since platform models of in- terest nonlinear. Popular approaches such as the "Stanley" controller (originally proposed by Hoffman al. for the 2005 DARPA grand challenge [3]) can become unstable depending on the choice of the controller's parameters and the curvature of the path [4]. Similarly, feed- back controllers based on linearization can also be unstable if the deviation is too large [5].
Vertler
Theary Vertied path
Vinilie V/
Bounde devitiioon
Gusranteed Control
Oontri Estmetion CLP Propectiee
Figure 1: Learning CLFs. Motion samples consist of state-control pairs used to train the neural net- work V/ with parameter 0. The Verifier produces counterexamples (C.E.) until Vo satisfies the spec- ifications. The green box contains our contribu- tions: 1) extraction of control inputs from Ve8 that ensure bounded path deviation, 2) constructing advanced maneuvers using single CLF, and 3) demonstration on a real platform.
Control Lyapunov functions (CLFs) ing a control system towards an equilibrium point [6, 7]. A CLF is a positive definite function such that for any state within well-defined basin of attraction, there exists control input that causes its derivative to be negative. Thus, CLFs naturally define stabilizing controllers [8, 9]. However, the generation of CLFs has remained a challenging problem for researchers, and various CLF gen-
general approach for controller design, typically for steer-
6th Conference on Robot Learning (CoRI. 2022), Auckland, New Zealand.

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eration methods have been developed. Neural networks have recently become a popular method for designing CLFs given their ability to converge even on systems with sophisticated dynamics. The idea is to use machine learning algorithms to infer candidate CL.s by minimizing special loss function called "Lyyapunvv risk" in combination with automated theorem-provie algorithms to verify the results of the learning algorithm. As result, if successful, the approach yields CLF as well as a stabilizing controller. Thus, neural CLF generation a promising approach for designing controllers for nonlinear systems. Nevertheless, its use has been limited to stabilization to fixed equilibria, whereas our application seeks to stabilize along a path. Secondly, empirical validation on an actual robotic test platform has not been publicly performed.
In this paper, we address the auth-following problem for a path in the workspace whose curvature k(s) varies with the distance s along the path. For K(s) = 0, which corresponds to a straight- line path, the problem is amenable to existing neural CLF synthesis tools, but it becomes progressively harder as the path curvature increases and becomes dependent on
To address this, we exploit key properties of the system and the generated CLF to design a formalism wherein the controller derived for a straight-line path can be used for paths with non-zero, varying curvature. We derive bounds on the maximum path curvature for which the controller is valid and we show that we can construct complex maneuvers using this approach.
Next, we demonstrate our approach on \/2 -scale vehicle platform. Our demonstrations show that (a) the model- design approach transfers to a real robotic platform even though the model used for learning the CLF is simpler model; and (b) the resulting approach is favorable compared to other verified and unverified path-following methods in both curved and straight path-following scenarios.
1.1 Related Work
Numerous methods have been studied for generating CLFs. Certain nonlinear control design meth- ods such as backstepping automatically synthesize CLFs but are not widely applicable. Tan and Packard studied approach using bilinear constraints that are hard to solve in practice [10]. Ma- jumdar et al. propose a similar approach for path following but use a control feedback law derived a heuristic initialization to solve bilinear constraints efficiently in prac- tice [11]. Data-driven approaches for CLFs were first studied by Khansarizadeh et al. [12]. How- ever, lot of data is potentially needed to generate CLFs and the final result needs to be verified against the system dynamics. Previous work by Ravanbakhsh et al. [13] showed an active approach for deriving CLFs for nonlinear systems that was adapted to path following using the idea of control funnel functions [14, 15). Furthermore, their approach was shown to be effective on a autonomous vehicle testbed in the lab. However, the techniques described so far focus mostly deriving poly- nomial CLFs. It is well known that CLFs can be non-polynomial even for polynomial dynamical systems. Recently, neural networks have been used to generate CLFs for systems with complex, non-polynomial dynamics. Introduced by Richards et al. [16] and Chang et al. [17], neural networks have been shown be an effective method for generating CLFs for a variety of systems.
through linearization
Majumdar et al. [18] provide the framework of "funnel libraries"' in which a set of funnels F and associated feedback controllers generated. This library of funnels can be sequenced together to form complex vehicle maneuvers. Bouyer et al. [19] showed that this approach can reduce the reach-avoid problem to that of timed automata by switching between appropriate controllers based on state measurements. Funnel libraries are an interesting approach towards a more rigorous plan execution that supports motion planner which can restrict itself maneuvers available in the library. However, maintaining such a library can be cumbersome. In particular, a large number of controllers may be needed to cover the space of possible maneuvers. The approach presented in this paper can simplify funnel libraries by covering a larger set of trajectories with fewer controllers.
Preliminaries: Problem Definition and CLF Learning
Consider the problem of stabilizing a dynamical system = f(x u). wherein x E D C R and E 24 C R.",, to the equilibrium point = 0 without loss of generality. This is a typical control problem, e.g., where a mobile platform has command inputs that can lead it to follow a specific trajectory which may be parametrized by a point in time. Let V D - IR be a control Lyapunov function (CLF) for f on D. Formally, a CLF is defined in Definition 1.

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Definition 1. A CLF is continuousty differentiable function V D - R that is positive definite, i.e, V(0) and (zx) > 0 for all # and satisfies the following decrease condition
E \ (0), 3u eu: VV(x) fxz u) < O.
(1)
Rather than requiring that the system reaches equilibrium, bility, wherein we define a neighborhood I of O, and require that the Lyapunov condition Eq. (1) is satisfied in the region D I, called the Lyapunov (satisfiability) region. Using the CLF V. control inputs can be generated for any state x that belongs to the largest level set of V contained completely within D to drive the system to the smallest level set of V that contains I, in finite time [19].
often interested in region sta-
The synthesis CLFs is known to be very challenging, and the complexity increases rapidly with that of the system dynamics and the control task. The core idea of this work is to synthesize CLFs for simple mancuvres for the robot platform (namely, straight-pati following) and then extend these to more complex maneuvres. To synthesize the "simple-task" CLFs, we used the approach discussed in Chang et al. [17], which is a neural network based learning solution for generating CLFs. In this framework potential neural CLFs are trained using state-control samples from the defined Lya- punov region D I and then verified by an SMT solver to ensure that the candidate CLF meets the conditions in Definition 1. Details of the CLF generation process are provided in appendix A.
Region of Attraction While the target of CLF training is to generate a CLF that is valid inside the Lyapunov region, important metric is the region of attraction (ROA) [20].
Definition 2. Let V be CLF with domain D for a system f(x, u). Any level set of V completely contained 'D defines a region of attraction. That is, for any B > 0, if (V(x) S B} C D, then (V(x) < 3} is an ROA for the system. The maximal region of attraction is the maximum level set of V contained within D.
Note that it is important to consider the largest level set contained in D. Lyapunov conditions do not directly ensure safety (i.e., infinite-time horizon containment) for any states the verified Lyapunov region since large level sets of V may exit D while descending the gradient. Upon exiting D there no guarantee that actions will be available reduce V. ensure safety, system's initial state must be contained within the maximal region of attraction.
2.0.1 Vehicle Model
To train the system we use a path-following model with a fixed velocity. The model consists of two states: de and l.. where de is the distance from the line as measured from the rear axle and 0g is the theta offset from nearest point on the target line (see Figure 2). The dynamics of this system are:
de = u sin(Oe)
6. = e tan(u)
e k(s) cos(fe)
L
dek(s)
Diagram of the states of the wheeled vehicle described in Equation 2, being the parameterized curve.
Figure 2:
where 0, 0 0,, and n(s) is the curvature of the refer- ence path at the nearest point. Note that k(s) is 0 for the straight-ling line path shown Figure 2.
To learn a neural CLF using the approach in Appendix A, state-control samples are generated for the learning algorithm. To calculate at any given sample (de, (d), an LQR controller.
2.1 Straight Path to Curved Path Following
Finding CLFs for the vehicle model described Eq. (2) for paths of various curvature would (a) be expensive and potentially infeasible (as the neural CLF training algorithm offers no guarantee of finding a CLF and the problem becomes more difficult as the curvature increases); and (b) lead to only finite set of curves that the vehicle execute. Instead, our approach consists of finding
3

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single CLF for the straight-ling path, and compute the maximum curvature for which the CLF is valid, in order extend the CLF paths with nonzero, possibly varying, curvature.
If the minimal required steering input for the straight-path CLF is smaller than the maximum allowed steering input Umaxx we can derive a curved-path CLF. More precisely, given CLF V for straight- line path, i.c., for x(s) = O in Eq. (2), with ROA contained in the region de E [-dmax: dmax], we define the minimal required input Ureq for V as the smallest value of satisfying that for all state in the Lyapunov region, there exists a steering input u with |u] S Uyeq that decreases the value of V, i... (x) f(x, u) < O. If Ureq < Umax then one can exploit the gap tan (umax) tan(Ureq) to follow path with curvature k(s) bounded as follows.
Proposition 1. [+(ss) S
(an(Mmaax) -tan(ura)) then V is a CLF Eq. (2). Lods
Proof. Note that for all de E [-dmaxx dmax], M(a) coo(fe))
I-der((x) I-dmax [x(3]] Thus, if I dmaa [x()
[e(aa]]
[e(a)]
tan (anmar) )- tan( one can use the steering input u(s) arctan(tan(u) + [-dnax<<(s)) ). where
[[((s)]
L
u E Ureqs Ureq] satisfies V (xc) f(x u) < O. holds that |u())] S Umax- Moreover, with this input the second equation of Eq. (2) becomes @ = stan(a) Since u is decreasing input for V with x(s) = 0, it is clear that is a CLF for curved-path following with input us)).
Similar to a library of fumnels [18], the CLF generated for straight-path following can be used for tasi (imax) -tau(urer) per Proposition 1. However, while previous L+ddmax
any path with curvature |s())] <
works require switching controllers [19, 18), our solution relies single CLF for the duration of the maneuver.
2.2 Control Extraction
Algorithm Vehicle Rollout CLF Minimization
import CLF, set vehicle model parameters, generate target path, discretize steering action space, set dl (time step), set K,, (control cost)
while sim == true do for steer in steering/Anless do
ler de + caw.errrr www.rrruuuuetttetteurrueeeeeenn steer))
ur Rollouts torr/oloooos.ooppeddccclll// x(0,,,
end for
< http://hhwn.anommmosssosorrrreed K2 uar Rolloutts)
end while
The definition of a CLF (Definition 1) guarantees the existence, at any state in the Lyapunov do- main, of a control input causing the CLF to decrease over infinitesimal time horizon (negative Lie derivative). A simple control strategy is then to simply execute the control that minimizes the Lie derivative at the current state. One can plot a theoretically executable path to descend the CLF as quickly possible using this method. However this approach usually results in saturation of the control surface. In our system, this holds true as the Lie derivative is a monotonic function of u, so that the control that minimizes the Lie derivative of the CLF is always the maximum or minimum u E U. Such a bang- bang controller [21] is not desirable in practice because abrupt changes in the input can damage the platform, and requires high actuation frequency.
A better practical method for control extraction is to simulate some actions and select the action that causes the largest reduction of V(x). Given Eq. (1) and provided the time horizon is small enough, there always exists a control action to reduce the CLF if the vehicle in the Lyapunov region of the CLE. The addition of a small control cost can be included for smoother operation. This approach allows for stabilization to the origin, as well as more realistic models to be introduced to compute realistic maneuvers. A simple, functional implementation of this idea is outlined in Algorithm 1. This algorithm can be expanded to operate like a traditional MPC [22] by increasing the look-ahead, and minimizing cost function balancing future V (zz) value, control cost, and terminal V(x) value.

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0.5
0.5
y (m)
0.5
-0.4
-2
0.8
-0.4
0.4
0.8
d,
2 (m)
(a)
(b)
Figure 4: Results of CLF minimization on the testing platform. (a) Colored points are sampled plat- form trajectories given various starting positions and orientations, black lines define the Lyapunov region and the red line the maximal ROA. Surface colors show the value of V(x), orange being larger, blue smaller. (b) Vehicle paths at starting states (de.0e). (0.4, 0.4), -0.4, (0.0, 0.4), (0.0, -0.4), (-0.4, 0.4). (-0.4, -0.4). The cyan area contains the ROA in the direction of de. the green area is the lower bound of the Lyapunov region of de and the red line is the target path and black lines are platform paths.
3
Implementation and Experimental Evaluation
3.1 CLF Generation
Our neural Lyapunov control framework was successful in generating CLFs for various target paths and vehicle parameters. Figure 4(a) shows a generated CLF for straight-line path. The CLF verified inside the Lyapunov region defined by inner and outer radius of 0.2 and 0.8 respectively. Note that while the CLF is valid in those bounds, the maximal ROA is the only region in which we guarantee safety, although in practice observe success, in both converging to the minimum and remaining in the Lyapunov region, outside the ROA.
3.2 Controller Implementation and Evaluation
Platform To test our approach in the real world we use --sclle vehicle platform in motion capture space. The wheel base the vehicle is 34 em and the velocity for all tests is fixed at 0.5 m/s. The vehicle includes an board computer for real-time computation and receives pose estimates from the motion capture system through WiFi connection. After computation, the control action transmitted control unit which controls the steer- ing servos and motor of the vehicle. Remarkably, each iteration of Algorithm is executed in less than 150 ps.
Figure 3: Vehicle Testing Platform
Control Extraction in Real Time The test objective is to use the generated CLF to stabilize the platform to straight-line path from I m at constant velocity of 0.5 m/s. On the real-world platform the Lie derivative minimization control extraction method is unable to stabilize the platform to a straight-line path: since the control surface always saturated, the platform continuously overshoots the distance to the target path, and therefore is unable stabilize.
-2.5 to
Using Algorithm we simulate potential rollouts for the platform and execute the lowest-cost action. We discretize the continuous action space [- ] to 31 equally-spaced steering values. We use the kinematic bicycle model to project our current state single time-step into the future, setting the time-step to be equal to system average update rate dt 150 μs.
5

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0.5
(m)
00.5
0.5
-055
0.3
x: (m)
(a)
(b)
Figure 5: (a) Reduced invariant region based on starting point (0, -0.4) where the solid red line is the maximal ROA and the dashed red line the reduced invariant region (0, 00.4). (b) Computed de boundary when platform starts at de = 0, 0g -044.
Figure 4(a) shows the convergence to the minimum of the CLF using Algorithm 1, starting from various positions in the Lyapunov region. In practice, while not guaranteed to converge, we that even positions starting outside the maximal ROA converge to V(x) =
Given any starting point in the maximal ROA we can plot the guaranteed maximum de and 0g by computing the level-set (Jrstar)) and taking max 0, and max d, the computed set. Figure 5(a) shows a reduced invariant region based on we can guarantee platform will deviate no further than [-48, 44] cm from the target path (a 27% de reduction when compared to the maximal ROA maximum de of 0.66).
(0, -044. For this particular starting position
Maximum Curvature Computation In Subsection 2.1, we showed that a CLF for traighh-path following can be extended to curved- path following if the range of input required for straight- following is smaller than the maximum available steering input Umax 1 The maximum allowed curvature depends gap tan (mmx)) tan (Mreq), where Uroq is the smallest range required for straight-path following. Given the CLF V computed Subsection 3.1, Ureq can be computed as:
OV (de, 0. sin( + aV (de, 0.) U vaannu) 0dg
o).
Uroq
max min dg
|u]
(3)
00,
L
where the maximum is taken for de, 0₂ in the Lyapunov region. Eq. (3) can be rewritten as a single maximization problem:
@V
(de, 0,) sin(
I tan (Ureq) = max
Ød.
(4)
d.
@v
(de. 0e)
20,
Eq. (4) has only two variables (de, (c),02), so that it can be solved easily using for instance grid search, nonlinear solvers like !popt [23]. We compute the value 7, tan ttan(rre)) 0.496. With L 0.3, we get that gap = 0.8512. By Proposition with dmax 0.8, it follows that V can be extended to surved-path following for any path with curvature H(s) S L+dmax 0.774, i.e. with radius of curvature R 2 1.3
Using the approach discussed in Section 2.1 we can execute complex object avoidance maneuver. The maneuver is curved path with varying curvature, bounded by 0.4 m (the minimal radius of curvature 2.5 m); see Figure 6. The figure shows platform paths of various starting positions and orientations after executing Algorithm on the constructed path.
3.3 Side-by-side comparison with alternative techniques
We compare our control method against two MPC implementations [22] (a non-verified controller) well as the "deep learned" controller trained when computing the neural CLF (DLC) (a verified

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2.5
3 (m)
0.5
0.5
-6
-4
2
(m)
Figure 6: Experimental results where colors are as in Figure 4a. It is shown that if the platform's 's initial position is contained in the maximal ROA the platform will never exit the teal region.
controller as discussed in Chang et al. [17]). One MPC has low cost of applying controls (MPCL) while the other one has a high cost of applying controls (MPCH). We compare our control method in straight- and curved-path -ered-ptthhhhh following problems. The straight target path shown in Figure 7(a) is straight line from 0 to 30 m, with an obstacle in the initial direction of the vehicle, just outside of the vehicle control funnel (as defined by the computed CLF in Figures 4 and 5). The initial position of the vehicle is on the target path, but oriented yaw by the largest 0g contained in the maximal ROA in Figure 4(a). This is an adversarial condition for non-verified controllers as we are unable to calculate the maximum path deviation. Rather, the amount of deviation from the target path will depend on design parameters such as control costs, MPC horizon and tuning constants. The curved target path is circle with a radius of 4 m. The vehicle begins at (0, 4) and completes one full lap of the path. Figure 7(b) shows the vehicle's 's deviation from the target path as time series.
1.2
0.8
(m)
(m)
MPCL MPCH
0.6
-0.5
0.4
DLC
Our Method
0.2
20
0
100
[m)
Simulation time
(a)
(b)
Figure 7: Comparison of control methods for (a) straight and (b) curved paths. (a) The red box represents an obstacle. (b) The y-axis represents the average deviation from the target path.
For each controller, run 50 simulations with Gaussian noise (μ+ 0, 0.1) added to the initial conditions 0. and de. Figure shows the test setup and simulated runs. Figure 7(a) shows the target paths with each colored area being the empirical average path (solid line in the middle) plus/minus one standard deviation.
As shown in Figure 7 and Table 1, our method is the only method that is both verified method of control (guaranteed to avoid the obstacle outside the control funnel) and able to stabilize to the target path (small steady state error). In particular, one will note that the other verified control method (DLC) oscillates around the target path, resulting in a large mean steady state error. Addi- tionally method provides the fastest convergence to the target, with small maximum deviation, and is more than 2 orders of magnitude faster than the MPC implementation, making it a reasonable control choice even when verified control is not mandatory.

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Table 1: Comparing performance and safety of verified and non-verified controllers over 50 simula- tions. "Average X" refers to the empirical average value of X over the 50 simulations, while "max X" and "mean X" refer to the maximum and mean value of X over given simulation. The steady state error is calculated as the Lı error upon the vehicle reaching steady state (after approximately 10 as shown in Figure 7).
Average mean
Average max de (meters)
Maximum Average max de (meters)
controller computation time (ms)
Safe paths
mean steady state error
Straight Path (Figure 7(a))
Our Method
0.75
50/50
0.39
0.56
0.0003
DLC [17]
0.08
50/50
0.42
0,49
0.0171
MPCL [22]
102
50/50
0.43
0,54
0.0046
MPCH [22]
99
12/50
0.74
0.98
0.0569
Curved Path (Figure 7(b))
Our Method
1.25
0.16
0.29
0.16
DLC [17]
0.09
0.42
0.68
0.62
MPCL [22]
120
0.31
0.31
0.31
MPCH [22]
110
1.00
1.04
0.85
4 Limitations
The primary limitation of this work is the method to generate verified CLFs. Some desirable CLFs do not converge solution (such CLFs with larger Lyapunov regions). The limiting factor is likely the verification phase, specifically the verification of Eq. (1), Rather than verifying 3u E U for which (z) f(x, u) < 0, we write u terms of our LQR controller (a function of the system state) to provide a single u for each state. Of course this greatly constrains the problem and means that the LQR controller constants must be trained simultaneously theory SMT solvers should be able to directly check Eq. (1) it is first-order logic inequality, however practice the control spaces are often too large. In future work one may be able to reduce the space by bounding WI to some neighborhood of a sampled control. Additionally these verification techniques are conditioned on how well-represented a system is by the model f(z, u). Namely, a model which is developed to represent certain behaviors of our system limit the extent of maneuvers that are verified and may be executed by the platform. This crucial limitation exists at the interface of model validation. wherein the adequacy of the model is evaluated. Possible next steps to address this limitation may be to consider online model synthesis and the extension of verification to these new models.
the CLF parameters [17]. In
5 Conclusion
In this work we show the ability to generate CLFs using neural networks [17], extract executable controls to stabilize the system, and construct safe trajectories using single CLF. Future work will seek improvements to CLF generation and verification. Additionally, CLFs can be used inside receding horizon formulation in Algorithm I provide potentially smoother control experience and faster stabilization to the target. Finally the feasibility of non-smooth maneuvering in this framework would provide another layer of robustness when operating in less ideal conditions.
Acknowledgments
This work was funded by NSF award #1932189 and the Belgian American Education Foundation (BAEF). All opinions expressed are those of the authors and not necessarily of the NSF or BAEF.

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References
[1] E. Gat. On three-layer architectures. In Artificial Intelligence and Mobile Robots. MIT Press, 1998.
[2] I.. Claussmann, M. Revilloud, D. Gruyer, and S. Glaser A Review of Motion Planning for Highway Autonomous Driving, IEEE Trans. Intell. Transp. Syst., 21(5):1826-1885 May 2019. ISSN 1558-0016. doi:10.1
[3] G. M. Hoffmann, C. J. Tomlin, M. Montemerio, and S. Thrun. Autonomous automobile trajectory tracking for off-road driving: Controller design, experimental validation and rac- ing. In 2007 American Control Conference, pages 4282788.
2007. doi:10.1 1t0.1199//CC.00000
[4] J. M. Snider. Automatic steering methods for autonomous automobile path tracking. Robotics Institute, Pittsburgh, PA, Tech. Rep. CMU-RITR-09-08, 2009.
[5] H. Ravanbakhsh, S. Sankaranarayanan, and S. A. Seshia. Formal policy learning from demon- strations. In International Conference Robotics and Automation (ICRA), pages 6037-6043. IEEE Press, 2019.
[6] E. Sontag. A lyapunov-lik characterization of asymptotic controllability. SIAM Journal on Control and Optimization, 21:462 471, 1983.
[7] Z. Arsttein. Stabilization with relaxed controls. Nonlinear Analysis: Theory, Methods & Ap- plications, 7(11):1163 1173, 1983, ISSN 0362-546X. doi: 06011000016003225400663330000000
[8] E. Sontag. A 'universal' construction of artstein's 's theorem on nonlinear stabilization. Systems and Control Letters, 13(2):117-223 Aug. 1989. ISSN 0167-6911. doi: 111.1111/1166669111900 90028-5. Funding Information: * Research supported in part by US Air Force Grant 88-0235.
[9] Y. Lin and E. D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Letters, 16(6):393 397, 1991. ISSN 0167-6911. dbt:10000160016--911199) 90111-Q.
[10] W. Tan and A. Packard. Searching for control lyapunov functions using sums of squares pro- gramming. sibi, 1(1).
[11] A. Majumdar, A. Ahmadi, and R. Tedrake. Control design along trajectories with sums of squares programming. In IEEE International Conference on Robotics and Automation, ICRA 2013, pages 4054-4061 2013.
[12] Mohammad Zhansari-Zadeh and Aude Billard. Learning control Yyарunov function to en- sure stability of dynamical system-based robot reaching motions. Robotics and Aulonomous Systems, 62(6):752- 765, 2014. ISSN 0921-8890. doi: 10. 101
[13] H. Ravanbakhsh and S. Sankaranarayanan. Learning control Iyаpunov functions from coun- terexamples and demonstrations. Autonomous Robots, 43(2):275-307, aug 2018. doi: 10.10777100114401--999911 URL https:///oot org/ 111.:00000001111111------1---110 9791-9.
[14] M. Mason. The mechanies of manipulation. In Proceedings. 1985 IEEE International Con- ference on Robotics and Automation, volume 2, pages 544-548, 1985. 10010.001// 188.10.
[15] H. Ravanbakhsh, S. Aghli, C. Heckman, and S. Sankaranarayanan. Path-following through control funnel functions. In 2018 JEEE/RSJ International Conference Intelligent Robots and Systems (IROS), pages 401-408, 2018. 100.1100//RO.2201.......
[16] S. M. Richards, F. Berkenkamp, and A. Krause. The Iyapunov neural network: Adaptive stability certification for safe learning of dynamical systems. In A. Billard, A. Dragan, J. Peters, and J. Morimoto, editors, Proceedingy of The 2nd Conference on Robol Learning, volume 87 of Proceedings of Machine Learning Research, pages 466-476. PMIR, 29-31 Oct 2018. URI. https: htaps::/pproceeee http:///eccceeee..cee/ mlr, press/v81 wates/vv//riiii http///chharddde. html.

---

[17] Y.-C. Chang, N. Roohi, and S. Gao. Neural Iyapunov control. Advances in neural information processing systems, 32, 2019.
[18] A.
Majumdar and Tedrake. Funnel libraries for real-time robust feedback motion planning. 00011
CoRR, abs/1601.04000 2016. URL http::/arxxi..orrg///
[19] Bouyer, N. Markey, N. Perrin, and P. schlehuber caissier. Timed-automata abstraction of switched dynamical systems using control invariants. Real-Time Systems, 53, 05 2017. doi: 10.1007/112241-016-99000
[20] Slotine and W. Li. Applied Nonlinear Control. Prentice-Hall, Englewood Cliffs, NJ, 1991.
[21] H. Hermes al. Functional analysis and time optimal control. 1969.
[22] R. Rajamani. Vehicle Dynamics and Control. Mechanical Engineering Series. Springer
US. 2011. ISBN 9781461414339,
URL https://book:: google. cotp/ooksmmmbookk
c2JFDox4KuUC.
[23] A. Wächter and L. T. Biegler. On the implementation of an interior poriot filter line-search algorithm for large-scale nonlinear programming. Mathematical Programming. 106:25-57, 2006.
10


Reviews:
[{"review_id": "xg3o8Tj7m4", "reviewer": "Reviewer_U6aE", "summary": "This paper considers the problem of tracking a curve path for a vehicle model. The authors apply the neural Lyapunov function approach to a straight line tracking problem. Given the learned Lyapunov function, the authors compute the largest curvature that the Lyapunov function can tolerate. The idea of this paper is straightforward, but the authors evaluated it on hardware.", "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:\n1. Experiments on hardware.\n\nWeakness:\n1. Theoretical contribution is minor.\n2. Some presentation issues.", "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": "Although the idea is simple, the experiments on hardware look promising.", "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": "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": "3: The reviewer is fairly confident that the evaluation is correct"}, "extra_text": {"issues": "1. The definition of u_{req} in Line 172 is confusing. Maybe move Eq. (5) here?\n2. Caption of figure 2 is too close to the text. Maybe remove the blank space on the top and bottom of the figure?\n3. L124: verified -> be verified.\n4. Explain the function in Algorithm 1."}}, {"review_id": "uPmG-Mdkdvp", "reviewer": "Reviewer_mRSR", "summary": "This paper proposes a framework to search for the best control for path tracking problems using neural control Lyapunov functions. The authors show the proposed method can ensure a bounded deviation of a target path and demonstrate the effectiveness of their algorithm on a real robot.", "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:\n\n• This paper is well-written and easy to read. \n\n• The framework provides a general and simple way to extract a stabilizing feedback controller. \n  \nWeaknesses:\n\n• In the experiment section, there is lacking comparison between the proposed method and existing path tracking methods. It would be also interesting to see the performance comparison in controllers extracted by polynomial CLFs and neural CLFs.\n\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": "Overall, this paper contributes to the current research field of search for a reliable controller with a theoretical guarantee and shows the process is practical in real robotic systems. However, it’s unclear to see how effective the proposed method is because of an absence of comparison. ", "technical_quality": "Good: Generally convincing, but some additional experiments or analyses needed.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "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": "4: The work contains interesting new ideas that will potentially have major impact in robotics or machine learning.", "issues": "Does the process encounter issues of scalability? ", "reviewer_expertise": "5: The reviewer is absolutely certain that the evaluation is correct and very familiar with the relevant literature"}, "extra_text": {}}, {"review_id": "pXuHidhVLe", "reviewer": "Reviewer_wkL6", "summary": "This paper applies existing techniques on learning Control Lyapunov Function (CLF) to the application of path following of ground vehicles. The main contribution of the paper is the ability to safely use a CLF designed for low curvature paths to paths with higher curvatures without losing the correctness guarantees of the CLF. The paper provides sufficient experimental evidence to support the proposed approach.", "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:\n- The paper applies the proposed framework to a real-world example with interesting experimental results.\n\nWeaknesses:\n- The paper's contribution is limited to applying existing algorithms (i.e., neural-based CLF learning and verification) with minor modifications.\n\n- Control Lyapunov functions for ground vehicles have been extensively studied, and several manually designed CLFs have been discussed in the literature. The paper lacks a clear justification of the motivation behind using a neural-based CLF for this setup. \n\n- The paper lacks a comparison against other techniques to understand the advantages and disadvantages of using the proposed framework.\n\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": "While the paper provides interesting experimental results on a real platform, the novelty of the paper is limited to applying existing algorithms of neural-based CLF with minor modifications.", "technical_quality": "Very Good:  Convincing.", "clarity_of_presentation": "Very Good:  Well organized and clear.", "potential_impact": "", "robotics_focus": "Sufficient demonstration on hardware", "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": "2: Unsure, difficult to assess.", "reviewer_expertise": "4: The reviewer is confident but not absolutely certain that the evaluation is correct"}, "extra_text": {"issues": "- The main technical problem the paper aims to solve is unclear. Does the paper aims to answer the question of \"how to obtain a CLF for ground vehicles?\". But such a problem can be analyzed manually, and a learning-based approach is not justified. Or does the paper aims to answer the question of \"Is neural-based CLF practical and can it be used for real-world example?\" for which the use of ground vehicles is just an example to show these limitations? However, to answer this question one would expect the use of neural-CLF on a diverse set of problems to understand its limitations. Or does the paper aims to answer \"how to use CLFs designed for paths with low curvatures to obtain guarantees on paths with higher curvature?\" and the neural-based CLF happens to be just an example of a CLF? \n\n- It is unclear how the proposed framework handles the mismatch between the dynamical model (equation 4) and the actual robotic platform. The numerical results section does not describe the effect of this mismatch and applies the CLF designed for the nominal model directly to the platform. \n\n- How does the proposed approach compare against other non-learning approaches like using MPC with CLF constraints?"}}, {"review_id": "VARvQTqc4aG", "reviewer": "Reviewer_VUEm", "summary": "This paper proposes a new way to do path following using neural network based control Lyapunov functions (CLF). For a given vehicle dynamic with straight-line reference paths, the authors construct the neural CLF to satisfy the Lyapunov conditions. Next they derive the curvature bounds to ensure the CLF controller stay valid. Hardware experiments are conducted to showcase the proposed approach's real-time performance.", "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: theoretical guarantees for the CLF converging behavior and tolerant reference curvature bound ; hardware implementation to showcase the proposed controller can work in real-time.\n\nWeaknesses: this work is incremental since there are already so many papers in using NN to construct CLF, and no baselines used in the experiments to demonstrate what is the benefit of this new NN CLF method. And they only use this CLF controller in a relative low-dimension experiment. The verification process can only converge within a small RoA.", "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": "My decision is weak reject. In the past few years we have witnessed so many papers in using neural networks to construct the Control Lyapunov Function, and I don't think this paper brings enough contribution to this field. The experiment is so simple and there aren't enough baselines or different setups to demonstrate the benefit of the proposed approach.\n", "technical_quality": "Fair: Significant additional evidence required to support the claims.", "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": "The only novelty I can sense from the paper is the way to derive a bound for the reference curvature so that one CLF can fit for different reference paths. This is good, but how much can we benefit from this? So my suggestion is the following:\n\n(1) For the vehicle experiment, first compare this method to the classical controllers (PID, LQR, MPC...) and to show that this experiment is indeed a hard problem that previous methods cannot really solve it elegantly. And then compare this method with reinforcement learning (I suggest pick some candidates from DDPG, SAC, TRPO, PPO) to show that this is not just another gym environment that can be easily tackled by RL.\n\n(2) Next, add different experiment settings. Compare with vanilla version of the CLF. Can your method converge under more initial conditions? What about the converging speed? How about the training time and model size? What happened if you try the curvature out of the bounded region?\n\n(3) Finally, please try more experiments. One vehicle example in such low-dimension is not persuasive at all. You definitely want to consider more challenging cases (more complicated vehicle models, drones...)\n"}}]

Author rebuttal:

Decision: Accept (poster)