Path-Following through Control Funnel...

Path-Following through Control Funnel Functions

Hadi Ravanbakhsh, Sina Aghli, Christoffer Heckman, and Sriram Sankaranarayanan∗

Abstract— We present an approach to path following usingso-called control funnel functions. Synthesizing controllers to“robustly” follow a reference trajectory is a fundamentalproblem for autonomous vehicles. Robustness, in this context,requires our controllers to handle a specified amount ofdeviation from the desired trajectory. Our approach considersa timing law that describes how fast to move along a givenreference trajectory and a control feedback law for reducingdeviations from the reference. We synthesize both feedback lawsusing “control funnel functions” that jointly encode the controllaw as well as its correctness argument over a mathematicalmodel of the vehicle dynamics. We adapt a previously describeddemonstration-based learning algorithm to synthesize a controlfunnel function as well as the associated feedback law. Weimplement this law on top of a 1/8th scale autonomous vehiclecalled the Parkour car. We compare the performance of ourpath following approach against a trajectory tracking approachby specifying trajectories of varying lengths and curvatures.Our experiments demonstrate the improved robustness ob-tained from the use of control funnel functions.

I. INTRODUCTION

Recent advances in motion planning have brought trulyautonomous systems closer to becoming a reality. However,one of the main challenges is their safety. Plan execu-tion may fail because of considerable uncertainties such asdisturbances, imprecise measurements, and modeling. Suchfailures can lead to safety violation with catastrophic conse-quences. Given a reference trajectory with associated timing,a straightforward solution is to design a feedback controlwhich tracks the reference trajectory and reduces trackingerrors. Another idea is to use “path-following” where thegoal is to track a path without timing constraints. Path-following techniques provide smooth convergence to thereference trajectory and avoiding input saturation [1], [2].In this paper, we investigate path-following techniques toimprove robustness and safety for plan execution.

First, we reformulate the problem as a path followingproblem that adds an extra control input to the system todecide how fast to move along the reference trajectory.This naturally allows our approach to speed up/slow downprogress along the reference trajectory. Furthermore, ourmethod is based on control funnel functions inspired bythe concepts of control funnels [3], [4] and control barrierfunctions [5]. Given a reference trajectory segment and adesired “safe region” around the trajectory, a control funnelfunction guarantees that the system moves along one end

This work was supported by NSF award no. 1646556.All authors are with the Department of Computer Science, University of

Colorado, Boulder, CO 80309 USA*Corresponding author. E-mail address: srirams at

colorado.edu

Fig. 1. Safe tracking with control funnels: the black box depicts an obstacle,solid red line shows the reference trajectory generated by the planner. Theprojection of the funnel on x-y plane is shown in green. The green circle otthe right is the head of the funnel and the one to the left is the tail. The blueregion enlarges the funnel to account for the non-zero size of the vehicle.

of the trajectory to another while remaining inside thesafe region. We use a previously-developed framework tosynthesize control funnels automatically given the vehicledynamics, the reference trajectory, and the surrounding saferegion [6].

We have implemented our method using a standard single-track model for capturing the ground vehicle dynamics. Theresulting controllers are then tested on a 1

8

th scale vehicleplatform called the Parkour car developed at CU-Boulder.Using our approach, we successfully synthesize controllersfor numerous reference trajectories and demonstrate the abil-ity of the controller synthesized on a toy model to drive anactual vehicle in the lab. We also show that, when contrastedwith trajectory tracking, path following approaches providea higher level of robustness as shown by some of ourexperiments that involved large disturbances applied to thecar during motion.

A. Related Work

Stability for autonomous vehicles is a challenging prob-lem. Brockett [7] showed that even a simple unicycle modelcannot be stabilized using continuous feedback laws. How-ever, continuous feedback laws exist for stabilization tonon-stationary trajectories; these feedback laws are usuallyobtained through linearization [8]. While trajectory trackinghas been widely used to solve plan execution, it has severalshortcomings which are addressed using path-following. Inpioneering work [9], [10], [11], [12], [13] the velocity of thevehicle tracked a desired reference velocity and the controlleris designed to steer the vehicle to the path. These path-following methods have been shown to yield a smootherconvergence to the trajectory while avoiding input saturation.

Beside these works, a wide diversity of approaches are usedto study the path-following problem. One line of work isbased on designing vector fields surrounding the path toguarantee reaching and following the path [14], [15]. Anotherapproach is to use model predictive control [16], [17]. In thisarticle, we consider a line of effort distinct from these others.

Hauser et al. [1] proposed the conversion of the trajec-tory tracking strategy to the so-called maneuver regulationstrategy. The main idea is to decrease the distance betweenthe state and the reference trajectory, not a specific point onthe trajectory. The reference trajectory xr(·) is parameterizedusing a variable θ (instead of time) and distance is defined asa function of x−xr(θ). θ and treated as a variable. An updatelaw (timing law) is then applied to ensure proper change of θ.Hauser and Hindman showed that this maneuver regulationtrick would yield a system that avoids input saturation.Similarly, Pappas [2] showed that by re-parameterizing thetrajectory, one could avoid input saturation. Subsequently,Encarnacao et al. [18] extended the technique for the outputmaneuvering problem on a restricted set of dynamics.

Following Hauser et al. [1], others have divided the taskinto two parts. The first task is to reach and follow thereference trajectory using the variable θ (instead of time),and the second task is to improve the solution using an extracontrol input θ. For example, in Skjetne et al. [19], firstthe system output is stabilized, and then a control law forθ is used to adjust the velocity. In this work, we use theextra freedom to control θ for increasing robustness. Morespecifically, this extra degree of freedom allows us to designmore robust control Lyapunov functions (CLFs) from whichwe extract the feedback as well as the timing law.

Control Lyapunov functions were originally introduced byArtstein and further developed by Sontag [20], [21]. Synthe-sis of CLFs is hard, involving bilinear matrix inequalities(BMIs) [22], [4]. Standard approaches such as alternatingminimization result often do not converge to a solution.To combat this, Majumdar et al. use LQR controllers andtheir associated Lyapunov functions for the linearization ofthe dynamics as good initial seed solutions [4]. In contrast,recent work by some of the authors remove the bilinearity byusing a demonstrator in the form of a MPC controller [6].Furthermore, this approach avoids local saddle points andhas a fast convergence guarantee.

Aguiar et al. [23] argue that there are performance limi-tations for systems with unstable zero dynamics if one usestrajectory tracking. However, using an extra control inputθ, this restriction vanishes. The timing law in this work isdesigned as a function of θ and its higher derivatives.

Recently, Faulwasser et al. [24] combined the idea of path-following with control funnels. They proposed designing thetiming law as a function of θ, the distance from the path(e.g., x − x(θ)), and their higher derivatives. For example,one can design a timing law which slows down the progressof θ when the distance between the state and the referenceis large. Similarly, we use control funnels to provide formalguarantees. However, the funnel is constructed using a CLF.Besides this, the timing law in our work is a function of the

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Fig. 2. A Schematic Diagram of the Bicycle Model.

state x and depends on the structure of the CLF.

II. BACKGROUND

This paper investigates CLF-based path following focusingon applications to ground vehicles. We will use the well-known bicycle model, whose state consists of its position(x and y), its orientation (α), and velocity (v). The rearaxle is perpendicular to the bicycle’s axis, the front wheel’sorientation can be adjusted to steer the vehicle (see Figure 2).Let γ be the angle between the front axle and the bicycleaxis (Fig. 2). We will assume that γ ∈ [−π4 , π4 ] is a controlinput to the model. Also, the thrust applied to the vehicle isT ∈ [−4, 4]. The model has the following dynamics:

.x = v sin(−α),

.y = v cos(α).

α = vl tan(γ),

.v = T , (1)

wherein l is the distance between the wheels and the controlinputs to the model (γ, T ) are shown in blue.

We will use this model to analyze the behavior of groundvehicles. In this paper, we will study the design of controlinputs to solve the problem of controlling the vehicle tofollow a given trajectory. Such trajectories are generatedusing planning algorithms such as RRTs and are oftendesigned to avoid obstacles in the workspace [25].

As an example, consider a scenario where the vehiclemoving with speed 2m/s needs to circumnavigate an obstacleas shown in Fig. 1. First, the planner generates a referencetrajectory shown with the solid red line. Note that, bydesign, the reference trajectory keeps some distance fromthe obstacle.

To guarantee safety and trajectory tracking at the sametime, we use a control funnel [3] (the green region in Fig. 1)which contains the reference trajectory. The correspondingcontrol law for a funnel guarantees that once the state isinside the funnel, it remains in the funnel until it reaches thedesired target set of states. In other words, the system safetyin the tracking process is formally guaranteed. In this work,we wish to improve the process of funnel design to increaserobustness. Our funnel design technique is based on stabilityanalysis which is discussed first.

A. Stability Analysis

The dynamics for autonomous vehicles can be modeledwith Euler equations to study the behavior of these systems.In a continuous time setting, the state of the system x ∈ Rnupdates w.r.t. an ordinary differential equation. Formally, .

x =f(x,u), where .

x is the derivative of x w.r.t. time and u ∈Rm is the control input. Stability is a fundamental propertyof dynamical systems. Numerous control problems can beviewed as controlling a given system to stabilize to a givenequilibrium state xr or an “equilibrium” reference trajectoryxr(t). Control Lyapunov functions (CLF) are a powerful toolfor designing such stabilizing controls [20]. We first describeCLFs for stabilizing to an equilibrium state.

Definition 1 (Control Lyapunov Functions): A CLF V isa smooth and radially unbounded function that maps eachstate to a real non-negative value, such that (a) V (xr) =0 and V (x) > 0 for all x 6= xr, and (b) (∀ x 6=xr) (∃u)

.V (x,u) < 0, wherein

.V is the Lie-derivative of

V wrt f :.V (x,u) = ∇V · f(x,u).

Condition (a) ensures that the value of V is zero at theequilibrium and strictly positive everywhere else. Condition(b) ensures that for any state, we can find a control inputthat can achieve an instantaneous decrease to the value ofV . In this sense, CLFs are equivalent to artificial potentialfunctions over the state space [26]. Having a CLF, one candesign a feedback law which always decreases the value of Vand therefore stabilizes the system to the equilibrium point.For instance, Sontag provides a simple means to extract afeedback law from a given CLF [21].

B. Trajectory Tracking vs. Path Following

Another form of stability appears in trajectory tracking,wherein the goal is to stabilize to a reference trajectory xr(t).Formally, let xd(t) : x(t) − xr(t) describe the “deviation”from the reference trajectory state at time t. The goal isto stabilize xd to the equilibrium 0 under the time-varyingreference frame that places xr(t) as the origin at time t. Thedynamics for xd is defined as: .

xd = f(x,u)− r(t), wherein.xr = dxr

dt = r(t).One of the key drawbacks of trajectory tracking is that

it specifies the reference trajectory xr(t) along with thereference timing, wherein the state xr(t) must ideally beachieved at time t. This poses a challenge for control designunless the timing is designed very carefully. Imagine, areference trajectory that traverses a winding hilly road atconstant speeds. This compels the control to constantlyaccelerate the vehicle on upslopes only to “slam the brakes”on downhill sections [27].

Path following, on the other hand, separates these concernsby allowing the user to specify a reference (feasible) pathparameterized with a scalar, θ (instead of time), xr(θ) yieldsa state for each θ and dxr

dθ = r(θ). As proposed by Hauseret al., one could define Π as a function that maps a state xto the closest state on the reference trajectory xr(·), usingan auxiliary map π [1], [28]:

π(x) : argminθ||x− xr(θ)||2P , Π(x) : xr(π(x))

timing law

feedback law

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Fig. 3. Schematic View of a System along the Parameterized Path.

where ||x||2P : xtPx is a Lyapunov function for thelinearized dynamics around the reference trajectory. In orderto stabilize the system to the reference path, Hauser et al.propose to decrease the value of ||x−Π(x)||2P . However, asthe projection function π can get complicated, they use localapproximations of π.

Following this, others have proposed to design a controllaw for a virtual input u0 that controls θ as a function oftime (called the timing feedback law), or in other words, theprogress (or sometimes regress) along the reference [19],[23]. Therefore, the deviation is now defined as xd(t) :x(t)−xr(θ(t)) wherein dθ

dt = u0. As depicted in Fig. 3, θ ismapped to a state on the path xr(θ). For example Faulwasseret al. [24] design the timing law as a function of θ, thedeviation (xd for state feedback systems), and their higherderivatives:

g(θ(k),x(k)d , . . . , θ,xd, u0) = 0 ,

wherein θ(k) is the kth derivative of θ. However, definingthe function g is a nontrivial problem.

III. PATH-FOLLOWING USING CLFWe now present the design of a path following scheme

by specifying a timing law as well as control for deviationfrom the reference trajectory based on a control Lyapunovfunction. Let us define a new coordinate system, in whichthe state of the system is zt : [θ,xtd], wherein x(t) = xd(t)+xr(θ(t)) is the original state of the system. Also, the controlinputs are vt : [u0,u

t]. We assume θ is directly controllableusing u0:

.θ = u0. Therefore, .

xr = dxr

dθ

.θ = r(θ)u0, and thus

.xd = f(xd + xr(θ),u)− r(θ)u0 .

First, our goal is to design a control that seeks to stabilizexd = 0. We define a CLF as a function V over xd, butindependent of θ, that respects the following constraints:

(A) : V (0) = 0 and (∀xd 6= 0) V (xd) > 0

(B) : (∀ θ, xd 6= 0) (∃u, u0).V (θ,x, u0,u) < 0 .

(2)

Note that although V is a function of xd, its derivative is afunction of xd, θ, u0,u:

.V (θ,xd, u0,u) = ∇V (xd) · .

xd

= ∇V (xd) · (f(xd + xr(θ),u)− r(θ)u0).

Conditions (A) and (B) are identical to those in Defini-tion 1, requiring the function V to be positive definite, and

the ability to choose controls to decrease V . Furthermore,note that the (∀ θ) quantifer in (B) guarantees that thisdecrease is achieved no matter where the current referencestate x(θ) lies relative to the current state xd. Also, we notethat the value of u0 (rate of change for θ) is obtained throughthe feedback law derived from the CLF V .

Theorem 1: If there exists a function V which respectsEq. (2), there exists a feedback function that is smooth almosteverywhere, and stabilizes to the reference trajectory.

All proofs are available in the extended version from theauthors’ web page1.

Thus far, we have only demonstrated how CLFs can beused to decide on a timing law as well as a control to nullifythe deviation to 0. However, this does not address a keyrequirement of progress: we need to ensure that

.θ > 0 so

that we make progress along the reference from one end toanother. Other limitations include that saturation limits on uare not enforced, and that the avoidance of obstacles is notconsidered. Finally, the stability is required to be global inthe entire state-space, which is very restricting. All of theserestrictions will be removed in the next section.

IV. PATH SEGMENT FOLLOWING PROBLEM

In the previous section, we discussed path following in aglobal sense using CLFs, but noted several limitations. Wewill refine our approach to address them in this section.

Finite Trajectories: Consider a reference trajectory seg-ment defined by xr(θ) for θ ∈ Θ : [0, T ]. However, definingstability for such segments is cumbersome. Therefore, wediscuss reachability in finite time along with safety (reach-while-stay).

We augment the given reference trajectory by addingan initial set I 3 xr(0), a goal set G 3 xr(T ), and aparameterized family of safe sets S(θ) 3 xr(θ) for θ ∈ Θ,such that S(0) ⊇ I and S(T ) ⊇ G. Let S :

⋃θ∈Θ S(θ)

denote the entirety of the safe set. For convenience, wehave defined S, I, G in the original state space x. We willnow define them in terms of the deviation xd to define thefollowing sets:

I : {xd | xd + xr(0) ∈ I}G : {xd | xd + xr(T ) ∈ G}

S(θ) : {xd | xd + xr(θ) ∈ S(θ)}.Finally, let us denote S :

⋃θ∈Θ S(θ).

Input saturation: Unlike infinite trajectories, here weassume initially θ = 0 and in the end θ = T . In thissetting, we must ensure progress for θ at each time-step.In other words, we wish to make sure

.θ > 0. Recall that θ is

controlled by a virtual input u0 (.θ = u0). Therefore, we need

input saturation for u0 and we assume u0 ∈ [u0, u0], whereu0 > 0 and u0 <∞. We also assume the reference trajectoryis feasible with respect to the dynamics. To ensure that thereference trajectory remains feasible for the path following

1http://www.cs.colorado.edu/˜srirams/drafts/iros-submission-2018-with-proofs.pdf

problem, we enforce that u0 ≤ 1 ≤ u0 as the timing law issimply

.θ = 1 for the original reference trajectory (θ(t) = t).

Besides u0, we will add saturation limits to the inputs in u,as well. Formally we restrict v ∈ V for a polytope V .

Definition 2 (Path Segment Following Problem): Thepath segment following problem, given (S(θ), I, G) andsaturation constraints V , is to derive a control feedback lawu = u(θ,xd) and timing law u0 = u0(θ,xd) such that forall initial states xd(0) ∈ I, θ(0) = 0, the resulting state-control trajectory of the closed loop (θ(t),xd(t), u0(t),u(t))satisfies the following conditions: (a) saturation constraintsare satisfied, (u0(t),u(t)) ∈ V for all times t, (b) thereexists t∗ > 0 such that xd(t∗) ∈ G, i.e, the goal is reached,and (c) for all t ∈ [0, t∗), xd(t) ∈ S(θ(t)), while stayinginside the safe set for all times until the goal is reached.Finally, note that condition (a) guarantees progress is madetowards θ = T starting from θ = 0.Proof Rules: The control Lyapunov function argumentcan get extended to control funnels for formally satisfyingthe reach-while-stay property [29]. We will define a controlfunnel as the sublevel sets of a smooth function V (θ,xd).For a smooth function V , and a relational operator ./∈ {<,≤,=,≥, >}, let us define the following families of setsthat are parameterized by θ: V ./β(θ) : {x | V (θ,x) ./ β}.Furthermore, let V ./β : ∪θ∈Θ V ./βθ .

Definition 3 (Control Funnel Function): A smooth func-tion V (θ,xd) is called a control funnel function iff thefollowing conditions hold:

(a) (∀xd ∈ I) V (0,xd) < β(b) (∀xd 6∈ int(G)) V (T,xd) > β(c) (∀θ ∈ Θ,xd 6∈ int(S(θ))) V (θ,xd) > β

(d) (∀θ ∈ Θ,xd ∈ S(θ) ∩ V =βθ )

(∃v ∈ V).V (θ,xd,v) < 0.

(3)

The idea, depicted in Fig. 4, is as follows. Initially(condition(a) in Eq. (3)), V < β (x ∈ V <β). Condition(d) guarantees that for all the states in a neighborhood ofset V =β , there exist a feedback which decreases the valueof V . Therefore, by providing a proper feedback, the statenever reaches boundary of V ≤β (V =β) because the value ofV can be decreased just before reaching V =β . As a result, Vremains < β. This means the state stays inside V <β as longas θ ∈ Θ. Also the state remains in S(θ) as value of V forother states is ≥ β (condition (c)). Since θ is increasing atminimum rate u0 at some point θ reaches T . Then, accordingto condition (b), the state must be in the interior of G (topgreen ellipse), because otherwise value of V would be ≥ β.

Theorem 2: Given a control funnel function V , thereexists a control strategy for reaching G such that for anyinitial state xd(0) ∈ I , the goal state is eventually reached atsome time t∗ satisfying T/u0 ≥ t∗ ≥ T/u0, while stayingin set S for 0 ≤ t ≤ t∗.

In practice, we replace V =β in condition (d) of Eq. (3)with V ≥β for some β ≤ β. Using this trick we make sure thevalue of V can be decreased in a larger region to improverobustness. Also, any set V ≤β (for β ≥ β) would be aninvariant until θ reaches T .

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Fig. 4. Schematic View of a Funnel for reach while stay. S(θ1) definesthe safe set when θ = θ1. V (θ) ≤ β is an invariant.

Increasing Robustness To improve robustness, one couldsimply maximize λ s.t.

Eq. (3)∧(∀θ ∈ Θ,xd ∈ S(θ) ∩ V =β)

(∃v ∈ V).V (θ,xd, u0,u) < −λ .

The bigger is the λ, the faster the value of V can bedecreased, and therefore the resulting control law would bemore robust.

V. EXPERIMENTS

In this section, we discuss the process of designing controlfunnels for a bicycle model, followed by implementation anddiscussions.

Synthesizing Funnels: We adapted a recently developeddemonstrator-based learning framework to synthesize controlfunnel functions for given sets I , G, and S [6]. Followingthat approach, the control funnel function V is parameterizedas a linear combination of some basis functions V (z) :∑rj=1 cjgj(z). Next, we provide an MPC-based demon-

strator that given a concrete state (θ,xd), demonstrates anoptimal control input (u0,u) by minimizing a cost function(distance to reference trajectory) for a given finite timehorizon. Furthermore, the conditions in Eq. (3) are checkedfor a given instantiation of parameters (c1, . . . , cr) using averifier that uses a LMI-based relaxation.

We use the bicycle model presented in Eq. (1), but usea “body fixed frame”, wherein the state of the vehicle isgiven by zt : [θ,xtR], and xR : [αR, xR, yR, vR]t. The statevariables in the inertial frame x(t) : [α(t), x(t), y(t), v(t)]t

are written in terms of xR as follows:αR(t) + αr(θ(t))

cos(αr(θ(t)))xR(t)− sin(αr(θ(t)))yR(t) + xr(θ(t))sin(αr(θ(t)))xR(t) + cos(αr(θ(t)))yR(t) + yr(θ(t))

vR(t) + vr(θ(t))

.

Fig. 5. Parkour car platform used for experiments.

In this frame, yR axis is always aligned to axis of the vehiclein the reference trajectory.

We observe that the change of coordinates allows foraccurate low order polynomial approximations. Also, ourexperimental results suggest that the learning frameworksucceeds in finding a control funnel function of lower degreeover the new coordinates when compared to the inertialframe. For all the experiments, we use the following param-eterization of V : V (θ,xR) : xtRCxR + c0θ, where c0 andC are the parameters to be synthesized. For demonstration,we use an off-the-shelf offline MPC with a simple quadraticcost function. Please refer to [6] for more details. As analternative to Path-Following based Control Funnel (PF-CF)we compare with Trajectory Tracking based Control Funnel(TT-CF) obtained by setting

.θ = 1, and eliminating the

control input u0.

Control Law Extraction: For running the experiments, weneed to extract control laws from control funnels. For a TT-CF, we merely use Sontag formula [21] with input saturation.Moreover, for a PF-CF, the controller stores and tracks valueof θ (as a non-physical variable). The control law is extractedfor both u and u0, and in addition to providing the feedbacku, the controller updates the value of θ according to controlinput u0.

Parkour Car In order to verify functionality of proposedmethod we perform experiments on a 1

8

th scale, four wheeldrive vehicle platform known as Parkour car (Fig.5) in alab environment equipped with a motion capture system.Parkour car has a wheel base of l = 34cm and includesan on-board computer to perform all computation on thevehicle. While in action, the main computer receives a poseupdate from motion capture system through WiFi connection,after which a new control action is calculated based on thesynthesized control law which then gets transmitted to anECU (Electronic Control Unit). The ECU handles signalconditioning for acceleration and steering motors on Parkourcar. One iteration of this control action calculation can be

performed in less than 300µs on a single CPU core runningat 3.5GHz. The low computation cost makes this methodattractive for real-time applications aboard platforms withlow computation capabilities.Straight Path: In the first experiment, we consider astraight path from x = −2 to x = 2 with the referencetrajectory xr(t) : [−π/2,−2 + 2t, 0, 2]ᵀ. The sets are

S(θ) : [−1, 1]3 × [−3, 3] , I : B0.5 , G : B0.5 ,

where Br is a ball of radius r centered at the origin.The learning framework then successfully finds a PF-CF.However, the learning framework fails to find a TT-CF. Thisdoes not rule out the existence of a TT-CF, however. Next,we increase the length of the path to 8m (from x = −4to x = 4). In this case, the learning framework can find aTT-CF. Fig. 6 shows simulation trajectories corresponding tothe PF-CF and the TT-CF. For comparison, starting from thesame initial conditions, the simulation is performed until xreaches x(0)+12. Fig. 6(a) shows the results for initial stateswhere the initial state is near I . The simulations suggest thatboth methods perform similarly and all trajectories convergeto the path (y converges to zero). The simulation time forall cases are similar and around 6s. Also, the velocity ofthe vehicle is almost constant for both methods. Fig. 6(b)shows the results for cases when the initial states are furtheraway from G (it needs more forces/time to reach G). In thiscase, the path-following method takes a longer time to reachx = 4 as the speed increases smoothly. Fig. 6(c) considersinitial states that are closer to G. For these case, the path-following method takes a shorter time to reach x = 12 asthe speed decreases smoothly. The results demonstrate thatthe path-following method yields a faster convergence tothe reference path. Moreover, the velocity changes smoothlywhile the trajectory tracking method settles the target veloc-ity immediately.

We also investigate the same problem (straight path fromx = −4 to x = 4) where the velocity is more restricted:

S(θ) :[−1, 1]3 × [−0.5, 0.5]

I : G :{x|4α2 + 4x2 + 4y2 + 16v2 ≤ 1} .Again, under these circumstances, learning TT-CF fails whilefinding PF-CF is feasible. In other words, in trajectorytracking the change of velocity is crucial for reducing thetracking error.Circular Path To carry out experiments on Parkour car andexamine the behavior over long trajectories, we consider acircular path with radius 1.5m. The vehicle moves with aconstant velocity π

2m/s and the reference trajectory wouldbe xr(t) : [π3 t, 1.5 cos(t), 1.5 sin(t), π2 ]t. For the learningprocess, we consider a finite trajectory (once around thecircle) where t ∈ [0, 6]. The sets are:

S(θ) : [−1, 1]× [−3, 3], I : B0.5, G : B0.5 .

Figure 7 shows the trajectories when the controller runs onParkour car. Despite the uncertainties in the measurementsand simple modeling, both controllers do a good job of

following the reference path. Fig. 8 shows the trajectoriesfor different initial states. Fig. 8(b) suggest that the tra-jectory tracking method may take shortcuts to satisfy timeconstraints.

We also investigated the same problem with higher refer-ence velocity. When the reference velocity is increased toπ (from π/2), we could not find a TT-CF. Nevertheless,increasing reference velocity does not seem to affect theprocess of finding PF-CF, and we can discover solutions evenif the reference velocity is 10π.

Oval Path: Following a circular path is easy as the curva-ture remains fixed. However, the problem is more challengingwhen the path is an oval. The goal is to follow an oval pathP : {y212 + x2

22 = 1}. First, a reference trajectory is generatedto follow this path closely. As polynomial approximationsof the reference path become more challenging, we dividethe reference path into two similar parts. Then, we find afunnel for each part and make sure we can concatenate thesetwo funnels. For the first part, the goal is to reach fromB0.5([2, 0]) to B0.5([−2, 0]) going in a CCW direction andthen reach from B0.5([−2, 0]) to B0.5([2, 0]) again in a CCWdirection. For both segments we use the following sets:

S(θ) : [−1, 1]× [−3, 3], I : B0.5, G : B0.5 .

Notice that since G for the first segments fits in I for thesecond segment, we can safely concatenate the funnels. Ifa trajectory tracking method is being used, the learningprocedure fails to find solutions. However, the path follow-ing method yields proper control funnels. Figure 9 showsthe trajectories generated from our experiments using theCF-based controller. The tracking is not precise when thecurvature is at its maximum. We believe the main reason isinput saturation for the steering, which occurs because of theimprecise model we use (Fig. 9).

Obstacle Avoidance: Going back to the scenario ofobstacle avoidance, we wish to find a control funnel toguarantee safety (avoiding the obstacle). Having a referencetrajectory, instead of defining S(θ), we simply define S as

S : {x | ([x, y]⊕ B0.25) ∩O = ∅} ,where O is the obstacle, ⊕ is the Minkowski sum, and0.25m is the distance between the center of the car andits corners (the body of Parkour car fits in B0.25). Thistrick allows to reason only about the center of the car, andsafety is guaranteed as long as the center of the car is inS. Next, we set I : B0.25 and G : B0.25. However, wecan not find a solution using the learning framework. Torelax the conditions, we allow G to be larger G : B0.5.In this case, we were able to find a solution (only if thepath-following method is being used). For the experiment,Parkour car moves toward the obstacle with different initialstates and the CF-based controller engages when the car is1.5m to the left or right of the obstacle’s x position. Fig. 10shows the projection of the funnel on x-y plain. We notethat if a trajectory starts from the head of the funnel, notonly its initial x and y, but also its initial v and α should

-4 -2 0 2 4 6 8-1

-0.5

0

0.5

-4 -2 0 2 4 6 81.95

2

2.05

-8 -6 -4 -2 0 2 40

0.5

1

1.5

2

2.5

-8 -6 -4 -2 0 2 4-2

-1

0

1

0 2 4 6 8 10 121.5

2

2.5

3

3.5

4

0 2 4 6 8 10 12-1

-0.5

0

0.5

1

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Solid (dashed) lines are simulation trajectories corresponding to PF-CF (TT-CF). Blue (red) trajectories start from the sameinitial condition.

Fig. 6. Simulation Results for a Straight Path

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Trajectory Tracking

Fig. 7. Projection of trajectories, generated on Parkour car, for the circularpath. Parkour car finishes five rounds around the circle. The referencetrajectory is shown in black.

also be inside the funnel. Fig. 10 (a) shows trajectorieswhere the initial state is inside the head of the funnel. Asshown, the trajectories remain inside the funnel and reachthe tail. However, as demonstrated in Fig. 10 (b), even ifthe trajectory starts outside of the funnel head, the wholebody of the car may remain in the guaranteed region (blueregion). Nevertheless, the safety is not guaranteed any longeras Fig. 10 (c) shows trajectories where Parkour car leaves theguaranteed region.

VI. CONCLUSIONS

In this work, we investigate the use of control Lyapunovfunctions for path following and provide a characterizationof control funnel functions for tracking a trajectory segment.Our approach lends itself to an efficient synthesis techniquepresented previously. We implement the resulting controlleron the Parkour car and show its effectiveness through a set oftracking problems. Future work will focus on integrating ourapproach more closely with planning approaches to augmentexisting approaches to the synthesis of control funnels [30].

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Fig. 8. projection of trajectories, generated on Parkour car, for the circularpath form different initial states. Blue (red) lines corresponds to the path-following (trajectory tracking) method. The reference trajectory is shown inblack. Initial state: (a) [−π/2, 0, 0, 0] and (b) [π, 2.25,−1.4, 0].

REFERENCES

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[2] G. J. Pappas, “Avoiding saturation by trajectory reparameterization,”in Proceedings of 35th IEEE Conference on Decision and Control,vol. 1, Dec 1996, pp. 76–81 vol.1.

[3] M. Mason, “The mechanics of manipulation,” in ICRA, vol. 2. IEEE,1985, pp. 544–548.

[4] A. Majumdar, A. A. Ahmadi, and R. Tedrake, “Control design alongtrajectories with sums of squares programming,” in ICRA. IEEE,2013, pp. 4054–4061.

[5] P. Wieland and F. Allgower, “Constructive safety using control barrierfunctions,” IFAC Proceedings Volumes, vol. 40, no. 12, pp. 462 – 467,2007.

[6] H. Ravanbakhsh and S. Sankaranarayanan, “Learning lyapunov (po-tential) functions from counterexamples and demonstrations,” in RSS,2017.

[7] R. W. Brockett et al., “Asymptotic stability and feedback stabilization,”Differential geometric control theory, vol. 27, no. 1, pp. 181–191,1983.

[8] G. Walsh, D. Tilbury, S. Sastry, R. Murray, and J. P. Laumond, “Sta-bilization of trajectories for systems with nonholonomic constraints,”IEEE Trans. on Automatic Control, vol. 39, no. 1, pp. 216–222, Jan1994.

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Fig. 9. Projection of the trajectory, generated on Parkour car, for the ovalpath. The reference trajectory is shown in black.

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Fig. 10. Projection of trajectories on x−−y plane generated by Parkourcar for the obstacle avoidance problem. Funnel boundary is shown in grayand as long as the center of car is in the funnel, the whole body of thecar remains in the blue region. (a) guaranteed traces, (b) not guaranteed butsafe traces, (c) not guaranteed and unsafe traces.

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