Signal temporal logic (STL) is a temporal logic formalism that involves logical predicates, denoted μ, whose truth values are evaluated over continuous signals. In this particular case, the continuous signals are the system’s state trajectories at time t, namely x(t). The predicates assume their logical valuates based on a (continuous) predicate function h : R n × R + → R as in μ : = T r u e if  h x , t ≤ 0 F a l s e if  h x , t > 0 . (3) Based on such predicates, an STL formula φ can be recursively defined as φ : : = T r u e | μ | ¬ φ | φ 1 ∧ φ 2 | φ 1 ∨ φ 2 | ♢ a , b φ | □ a , b φ | φ 1 U a , b φ 2 , where a , b ∈ R + with a ≤ b are timing bounds, ¬ represents negation, ∧ expresses conjunction, ∨ denotes disjunction, ♢ stands for eventually, □ stands for always and U denotes the until temporal operator (Maler and Nickovic, 2004).

If a solution x : R + → R n of (1) satisfies an STL specification φ at time t, then we write (x, t)⊧φ. The STL semantics are recursively given by the above (top of this page) rules.

A CBF enables controller synthesis for dynamic systems in a way that ensures that if the system starts inside a set, it will never leave that set, rendering the set forward invariant with respect to the dynamics of system. A CBF can characterize the set of allowable control inputs that guarantee forward invariance of certain regions for a dynamical system at hand. The required control input is picked from a set defined in terms of the CBF for example by solving an optimization problem in a sampled-data fashion (Ames et al., 2016).

Nonsmooth CBFs allow more flexibility in the encoding of state constraints and specifications compared to their smooth counterparts. The utilization of such functions typically leads to consideration of the dynamics of the system in the form (2), primarily due to the discontinuities introduced by the control law u when it depends on the gradient of a nonsmooth CBF In fact, since the latter are nonsmooth, their gradient cannot be defined everywhere in the usual way. At points of nondifferentiability, one can understand their gradient as a set, rather than a singleton vector, and express it using the concept of the generalized gradient, which in finite dimensional spaces enjoys the following concrete characterization as Theorem 1

THEOREM 1

[(Clarke, 1990, Theorem 2.5.1)]. Consider a locally Lipschitz function b : R n × [ t 0 , t 1 ] → R . Let S be any set of Lebesgue measure zero in R n + 1 and Ω b denote the zero-measure set where b is non-differentiable. Then, with c o ̄ { A } denoting the closure of the convex hull of set A , the generalized gradient ∂ b ( x , t ) of b at point (x, t) can be written in terms of the limits of sequences (x _i , t _i ) → (x, t) as follows ∂ b x , t = c o ̄ lim i → ∞ ∂ b ∂ x 1 … ∂ b ∂ x n ∂ b ∂ t ⊺ : S ∪ Ω b ∌ x i , t i → x , t . (4) Examples of nonsmooth functions include the point-wise minimum or maximum of a finite collection of locally Lipschitz functions. Indeed, these specific nonsmooth functions are of particular interest in the context of STL synthesis because they can capture the conjunction and disjunction of a number of predicates, when each of the latter is expressed by its own component CBF

In particular, suppose b ( x , t ) = max i ∈ { 1 , … , k } { b i ( x , t ) } , where each b i is Lipschitz near (x, t) (i.e., locally Lipschitz around (x, t)). Then b is Lipschitz near (x, t), and if one denotes I (x, t) the set of indices i for which b ( x , t ) = b i ( x , t ) then (Clarke, 1990, Proposition 2.3.12): ∂ max i ∈ 1 , … , k b i x , t ⊆ c o ∂ b i x , t | i ∈ I x , t , (5) with equality holding if b i is regular at x for all i ∈ I (x, t), in which case b is also regular. Similarly, if b ( x , t ) = min i ∈ { 1 , … , k } { b i ( x , t ) } and each b i is Lipschitz near (x, t), then again, b is Lipschitz near (x, t), and ∂ min i ∈ 1 , … , k b i x , t ⊆ c o ∂ b i x , t | i ∈ I x , t , (6) with equality holding now if − b i is regular at x for all i ∈ I (x, t), in which case − b is regular too.

Note that although the pairwise maximum of a finite set of continuously differentiable functions is regular, the pairwise minimum function may not be. Nonetheless, if all b i are differentiable at x, at least the generalized gradient can be computed in both cases in an expedient manner.

When either the dynamics of a system or the gradient of a function is set valued, the Lie (directional) derivative of the function along the solutions of the system will also be set-valued. Strong (Bacciotti and Ceragioli, 1999) and weak (Bacciotti and Ceragioli, 2006; Glotfelter et al., 2017) versions of set-valued Lie derivatives have been introduced depending on the regularity of the function being differentiated. In this paper we utilize the weak version, at the expense of more relaxed convergence conditions, because we need to consider generalized gradients of functions that may not be regular.

With ⟨⋅, ⋅⟩ denoting the inner product of two vectors, the weak set-valued Lie derivative of a scalar locally Lipschitz function b is now defined as L F W b x , t = η ∈ R ∣ ∃ ν ∈ F f + g u , ∃ ξ ∈ ∂ b x , t  s.t.  〈 ξ , [ ν 1 ] 〉 = η . (7) The following lemma (Lemma 1) links time derivative of function b along the solutions of (2) to the weak set-valued Lie derivative of b ( x , t ) :

LEMMA 1

[cf. (Glotfelter et al., 2019)]. Consider a Carathéodory solution x : [ 0 , t ′ ] → D ⊂ R n to the differential inclusion (2), and let b : R n × [ 0 , t ′ ] → R be a locally Lipschitz function. Then, b ̇ x , t ∈ L F W b x , t a.e. (8) For a locally Lipschitz scalar function b : D × [ 0 , t ′ ] → R with D ⊂ R n , consider the associated set C = x , t ∈ R n × R + ∣ x ∈ D , b x , t ≥ 0 . Now the notion of forward invariance can be linked to the concept of CBF through the following definition (Definitions 1,2):

DEFINITION 1

[cf. (Glotfelter et al., 2017, Def. 4) and (Lindemann and Dimarogonas, 2018, Def. 3)]. A continuous scalar function b : D × [ 0 , t ′ ] → R where D ⊂ R n is a candidate nonsmooth CBF if for all ( x , 0 ) ∈ C , there exists a Carathéodory solution to (2) such that ( x , t ) ∈ C for all t ∈ [0, t′].

DEFINITION 2

[cf. (Glotfelter et al., 2017, Def. 3) and (Lindemann and Dimarogonas, 2018, Def. 2)]. A continuous candidate nonsmooth CBF b : D × [ 0 , t ′ ] → R where D ⊂ R n is a valid nonsmooth CBF for (2), if for any ( x , 0 ) ∈ C there exists a class- KL function β : R + × R + → R + such that b x t , t ≥ β b x 0 , 0 , t ∀ t ∈ 0 , t ′ for all Carathéodory solutions of (2) starting from x (0).

The following theorem gives a useful equivalent condition for a valid CBF

THEOREM 2

[cf. (Glotfelter et al., 2017, Thm. 2)]. Let D ⊂ R n be an open and connected set, and b : D × [ 0 , t ′ ] → R a locally Lipschitz candidate nonsmooth CBF If there exists a locally Lipschitz extended class- K function α : R → R such that min L F W b x t , t ≥ − α b x , t , ∀ x , t ∈ D × 0 , t ′ , (9) then b is a valid non-smooth CBF for (2).

In the special case where the differential inclusion (2) reduces to a singleton and g(x)g(x)^⊺ is positive definite (so that a simple feedback transformation can bring (1) to the form x ̇ = u ), a straightforward application of Theorem 2 leads to Corollary 1

COROLLARY 1

A candidate non-smooth CBF b ( x , t ) is a valid non-smooth CBF for (1) if there exists a locally Lipschitz extended class- K function α : R → R such that sup u ∈ U b ̇ x t , t ≥ − α b x , t . In other words, for a valid non-smooth CBF there is always a control input u to make the set C ( t ) forward invariant.

This section borrows primarily from Sun and Tanner (2015), based on the foundation of sphere world navigation functions of Rimon and Koditschek (1992), to construct a time-varying CBF with navigation function properties. While Sun and Tanner (2015) allow for time-varying destination configurations, and Chen et al. (2020) consider time-varying obstacle locations, here the construction of the navigation function component of the CBF is itself time-invariant, just as in the original methodology (Rimon and Koditschek, 1992), although time-varying extensions appear plausible (Sun and Tanner, 2015; Chen et al., 2020).

An STL specification consists of logical predicates μ as in (3) that can be interpreted as different regions of interest in the state space of the dynamical system at hand, which need to be visited or avoided at particular time periods. Working in a sphere world, all regions of interest (those that a robot needs to approach or those it needs to avoid) are assumed to have spherical shapes. This assumption does not limit the generality of the approach since both Rimon and Koditschek (1992) for the time-invariant case, as well as Li and Tanner (2018) for the case of time-varying destinations, show that diffeomorphic transformations can extend navigation function properties from sphere to (forests of) star worlds.

The key feature of the construction of Sun and Tanner (2015) that is adopted here is the non-point destination. Specifically, instead of the target of navigation being the convergence to a single point, Sun and Tanner (2015) allow for a destination manifold in the shape of a spherical shell, which is the zero level set of the function h i x t = ‖ x t − x c i ‖ 2 − r i 2 , (11) which serves as the predicate function encoding logical predicate μ _i . In the above, one distinguishes the predicate function’s center x c i and its radius r _i . Consistent with STL semantics (3), μ _i is true when h _i (x) ≤ 0 and false otherwise. Regions of the robot’s workspace that always need to be avoided can be encoded as (static) obstacles and incorporated all together in a specific functional representation inspired by Rimon and Koditschek (1992). Specifically, assuming that the implicit representation of each one of those isolated (obstacle) regions is defined as a function ζ j x = ‖ x t − x ob j ‖ 2 − r ob j 2 j = 1 , … , M , where x ob j and r ob j denote the center and radius of each undesirable spherical (obstacle) region. Our understanding is that obstacles are being avoided as long as ζ _i (x) > 0. Similarly, the boundary of the workspace itself is captured by the function ζ 0 x = − ‖ x t − x ws ‖ 2 + r ws 2 , where x _ws and r _ws stand for the center and radius of the workspace, respectively. Given these constructs and the fact that all obstacles are assumed to be disjoint, the combined obstacle representation can take the form ζ x = ∏ j = 0 M ζ j x , and with that, a navigation function ϕ _i (x) can be explicitly constructed for predicate μ _i as ϕ i x = h i x h i x κ + ζ x 1 / κ , (12) with κ = 2n for n ∈ N in the role of a positive tuning constant which be set sufficiently high to guarantee navigation function properties for (12). Note that for all x that do not satisfy μ _i , it is 0 < ϕ _i (x) ≤ 1, and ∇ϕ(x) is non-zero almost everywhere (with the exception of a finite number M of isolated critical points).

The following examples illustrate how (12) can be used to construct a time-varying nonsmooth CBF b ( x , t ) that can encode a combination of predicates μ _i .

EXAMPLE 1

Consider the STL formula φ = ♢_[a,b] μ ₁ . If ϕ ₁(x) is defined as in (12) with h _i (x) being the predicate function for μ ₁ , then a CBF that captures φ as a specification can be constructed in the form b ( x , t ) = 1 − ϕ 1 ( x ) − c 1 ( t ) , where c 1 : R + → [ 0,1 ] is a nondecreasing function satisfying c ₁(0) = 0 and c ₁(t′) = 1 for some t′ ∈ [a, b]. Then for c ₁(t′) = 1, b x ( t ′ ) , t ′ ≥ 0 when ϕ 1 x ( t ′ ) ≤ 0 , which in turn happens only when h 1 x ( t ′ ) ≤ 0 , implying that μ ₁ is true.

EXAMPLE 2

Consider the STL formula φ = φ ₁ ∧ φ ₂ where φ 1 = ♢ [ a 1 , b 1 ] μ 1 and φ 2 = □ [ a 2 , b 2 ] μ 2 . Start off by constructing a separate CBF for each of the two component formulae: b 1 ( x , t ) = 1 − ϕ 1 ( x ) − c 1 ( t ) for φ ₁ , exactly as in Example 1 , and b 2 ( x , t ) = 1 − ϕ 2 ( x ) − c 2 ( t ) for φ ₂ , where c 2 ( t ) : R + → [ 0,1 ] is a nondecreasing function with c ₂(0) = 0 and c ₂(t′) = 1 for all t′ ∈ [a ₂, b ₂]. ¹ The CBF that expresses φ can now be formulated as a pointwise minimum of b 1 and b 2 , i.e., b ( x , t ) = min { b 1 , b 2 } .

EXAMPLE 3

Consider the STL formula φ = φ ₁ ∨ φ ₂ where φ ₁ and φ ₂ are as in Example 2 . CBF s b 1 ( x , t ) and b 2 ( x , t ) for φ ₁ and φ ₂ are constructed exactly as in Example 2 . However, this time the overall CBF is formulated as a pointwise maximum of b 1 and b 2 , i.e., b ( x , t ) = max { b 1 , b 2 } .

The construction of the CBF based on navigation function (12) provides number of advantages compared to existing CBF-based STL motion planning methods [e.g., (Lindemann and Dimarogonas, 2018)]; First note that since the navigation function can encode unsafe regions (obstacles) through ζ(x), it obviates the need for the explicit definition of additional logical predicates corresponds to such unsafe regions, thus reducing the size of the STL specification. This reduction in the size of STL is particularly useful if this method is used in conjunction with a reactive STL (event-based STL) motion planning methodology (Gundana and Kress-Gazit, 2021) that includes a prior higher-level automata synthesis step. Another advantage of the nonsmooth formulation is that not only paves the way for covering larger class of STL compared to those considered by Lindemann and Dimarogonas (2018), but also eliminates the conservatism associated with under-approximation of minimum operator for the sake of smoothness (see Section 5.2). Yet another advantage of control barrier navigation functions is related to a reduction of the computational load required for determining control inputs (see Section 4.2).

Based on the idea illustrated in Examples 1, 2 and 3, the following sections present the development of a three-step process to produce CBFs that encode general specifications in the STL fragment (10).

Section 4.1 primarily illustrated how the use of navigation functions and pointwise minimum functions can allow the construction of CBFs that tightly encode STL specifications in the fragment defined by (10). This section focuses on demonstrating that control design can also be facilitated due to the navigation function properties afforded by the proposed component CBFs.

Without loss of generality, let p be the total number of predicates and q be the total number of temporal operators appearing in the STL specification. For j ∈ {1, … , q}, and with the formulae inside the temporal operators written in CNF (Section 4.1.3), then temporal operator indexed k will be modelled by a CBF of the form b p + j x , t = min { max b k j x , t , … , b l j x , t , … , max b m j x , t , … , b n j x , t } (16) For some distinct k _j , l _j , m _j , n _j ∈ {1, … , p}. Then the temporal operators of the STL formula can themselves be arranged in CNF (Section 4.1.2), in which case the composite (total) CBF capturing the complete STL specification would take a similar compact form b x , t = min { max b p + k x , t , … , b p + l x , t , … , max b p + m x , t , … , b p + n x , t } (17) For some other distinct k, l, m, n ∈ {1, … , q}, with the understanding of each one of the b p + * component CBFs above is of the form (16). Note that all b i with i ∈ {1, … , p} are continuously differentiable functions, while all b p + j with j ∈ {1, … , q} are not, but they are still locally Lipschitz functions. In the rest of the paper we refer to those p continuously differentiable components of b as component CBFs. In view of (16), (17), define the index set of the component CBFs of the form (13) that simultaneously agree with the value of b in (17) as J x , t = i ∈ 1 , … , p | b x , t = b i x , t . Then the control input u that guarantees that b ( x , t ) ≥ 0 can be computed directly using the gradient of the CBF unless x is a point where the latter non-differentiable. At such points, resorting to QP for the determination of the control input u may be unavoidable, although there are still cases where such a computationally expensive procedure can be circumvented. The following sections illustrate different options, starting with the straightforward one where b is computed away from points of nondifferentiability.

Consider a robot in a 2D spherical workspace of radius 1, initially positioned at a configuration with coordinates x ₀ = (0.9,0.2)^⊺ and with dynamic x ̇ = u . The workspace contains a static spherical obstacle of radius 0.2236, centered at (0.5,0)^⊺. The obstacle region is obviously an area that the robot should always avoid.

In addition to avoiding obstacles, the robot has an array of mission objectives associated with different (spherical) regions of interest in its workspace. These mission objectives will naturally be expressed in STL In general, we will denote μ _i the predicate that is associated with the ith region of interest. Table 1 collects the topological information of the different regions of interest for the robot.

The STL task specification that the robot needs to satisfy is given in the following form: φ 1 = □ 3,7 μ 1 ∨ μ 2 ∨ ♢ 2,4 μ 3 ∧ ♢ 4,5 μ 2 ∧ μ 3 ∧ μ 4 U 6,10 μ 5 . (27) Using the construction process outlined in Sections 4.1.1–4.1.3, the barrier function for (27) is as follows: b = min max max b 1 , b 2 , b 3 , min b 2 ′ , b 3 ′ , min b 4 , b 5 . (28) Note that while b 2 (or b 3 ) and b 2 ′ (or b 3 ′ ) are constructed for the same region μ ₂ (or μ ₃), but due to the different time intervals associated with the temporal operators that contain μ ₂ (or μ ₃), they are in fact different barrier functions, as a result of using a different c(t) in their construction (see Section 4.1.1).

Figure 1 gives successive snapshots of the robot’s path through the workspace, as it is steered by the control law computed based on the process outlined in Section 4.2. The time instances associated with the snapshots showcased correspond to representative moments in relation to the temporal operators appearing in the STL specification φ ₁ in (27). First of all, as a result of maximum operator, visiting (μ ₁ ∨ μ ₂) at t = 3 is preferred over visiting μ ₃ since the former is much closer to the initial location of the robot. For the same reason, between μ ₁ and μ ₂, the former is selected to be visited at time t = 3. It should continue to remain inside the union of μ ₁ and μ ₂ (to ensure (μ ₁ ∨ μ ₂) remains true) from t = 3 until t = 7, which is verified in the sequence of subsequent snapshots at times t = 3, t = 5, t = 6, and t = 7. Meanwhile, however, and sometime in the [4, 5] time interval, the intersection of μ ₂ and μ ₃ must be visited (to make (μ ₂ ∧ μ ₃) true), a fact that is evident in the top left snapshot for t = 5 where the robot is shown to make a maneuver to the left to reach the intersection of μ ₂ and μ ₃. Then, the specification φ ₁ indicates that in the [6, 10] time interval predicate μ ₄ should first be satisfied before predicate μ ₅ becomes true. Indeed, the robot is shown at t = 6 to have touched the boundary of μ ₄; following that, at time instant t = 10 the robot is shown to have touched the boundary of μ ₅. While all these maneuvers take place, the robot always stays clear of the static obstacle, marked in Figure 1 with the solid red disk.

Figure 2 presents graphs that show the evolution of the two-dimensional control input u (x, t) that implements the STL task specification (27). As Figure 2 indicates, the control inputs experience discontinuities. Not surprisingly, several jumps occur at time instants coinciding with non-differentiable points of the barrier function.

To see better the computational savings of this method compared to approaches that required the repeated solution of the QP program for the determination of the control law, the time interval [0, 10] of the STL task (27) was discretized to 1,000 time steps. Among those, only 17 featured J (x, t) with cardinality two, while there were zero instances where |J (x, t)| > 2. Of those 17 time steps, which show as the non-differentiable points of the barrier function depicted in Figure 3, none of them marked a singular case; consequently, (23) applies to them all, and (18) used everywhere else. Therefore, in handling the STL task (27), the reported method never resorted to solving a QP problem.

This section includes an illustrative example that demonstrates how less conservative the presented solution can be [even for the smaller STL class considered by Lindemann and Dimarogonas (2018)] when satisfying STL specifications, specifically in cases where smooth formulations do not permit the satisfaction of these specifications.

Consider a robot in a 2D spherical workspace of radius 3, initially positioned at a configuration with coordinates x ₀ = (−2,1)^⊺. The STL specification of the robot’s mission in this workspace involves visiting two regions of interest whose topological information is presented in Table 2.

The STL task specification that the robot needs to satisfy is given in the following form: φ 2 = □ 1,3 μ 1 ∧ □ 2,4 μ 2 . (29) Within a smooth STL composition framework, planning for satisfaction of (29) proceeds as follows (Lindemann and Dimarogonas, 2018): 1) first one defines predicate functions h ₁ = 1 − ‖x‖ and h ₂ = 1 − ‖x − (1.5,0)^⊺‖ for regions μ ₁ and μ ₂ respectively; 2) then the barrier function for each sub-formula of (29) is constructed as b 1 = γ 1 ( t ) − ‖ x ‖ and b 2 = γ 2 ( t ) − ‖ x − ( 1.5 , 0 ) ⊺ ‖ ; 3) then one selects γ ₁(t) such that b 1 ≤ h 1 for t ∈ [1, 3], and γ ₂(t) such that b 2 ≤ h 2 for t ∈ [2, 4]; 4) finally, the composite barrier function is formed as b = − ln exp ( − b 1 ) + exp ( − b 2 ) .

This process renders (29) not satisfiable. To see this, focus on the time interval t ∈ [2, 3] when the robot needs to visit and remain in the intersection of μ ₁ and μ ₂. Note that in this time interval, there must be γ ₁(t) ≤ 1 and γ ₂(t) ≤ 1 to ensure b 1 ≤ h 1 and b 2 ≤ h 2 , respectively. However, since the robot needs to be located somewhere in the intersection of μ ₁ and μ ₂, it must be γ ₁(t) ≥ 0.5 and γ ₂(t) ≥ 0.5 to ensure that b 1 ≥ 0 and b 2 ≥ 0 . As a result, for a any legitimate choice of γ ₁(t) and γ ₂(t), there will be 0 ≤ b 1 ≤ 0.5 and 0 ≤ b 2 ≤ 0.5 . This results in a composite barrier function b < 0 in t ∈ [2, 3] for all legitimate choices of γ ₁(t) and γ ₂(t). While there exist solutions to satisfy the specification (keep each b 1 and b 2 non-negative), the conservatism introduced by the under-approximation of the minimum operator (corresponds to the conjunction in the STL formula) by the smooth exponential summation does not permit the satisfaction of (29). In contrast, the non-smooth formulation of this paper can handle (29) successfully. Figure 4 depicts snapshots of the robot’s path generated by the control design of Section 4.2, satisfying the STL task of (29). Having said that, it can be noted that there can be ways for this satisfiability gap to be reduced, as part of the QP problem—although, theoretically, it can never be completely eliminated.

This section demonstrates how the nonsmooth CBF theory can be applied in the context of early pediatric motor rehabilitation, to regulate play-based social interaction between infants and mobile robots. The primary clinical objective of these mobile robots is to encourage infant mobility through interactive gameplay. Figure 5 shows an (enriched, in terms of stimuli) robot-assisted motor rehabilitation environment for infants involving two robots engaged in free-play activities with an infant.

The robots shown Figure 5 have been remotely controlled during the studies conducted, with the operators following a pre-determined look-up table of appropriate robot responses to infant reactions. Similar to other instances of Human Robot Interaction (HRI) application reported in literature (McGhan et al., 2015; Zehfroosh et al., 2017), a Markovian model is used to model the interaction at the high level. The parameters of this Markovian model are learned through observations in sessions with human subjects (Zehfroosh et al., 2017). Synchronized video from a network of surrounding cameras provided input to action recognition machine learning algorithms capable of identifying certain infant behaviors of interest, such as walking, crawling, standing, sitting, etc., as well as transitions between them (Kokkoni et al., 2020). In an envisioned fully automated version of this robot-assisted rehabilitation environment, robots could receive direct feedback regarding the child’s reactions and adapt their gameplay behavior accordingly in order to further encourage infant mobility.

In specific gameplay scenarios involved in the HRI protocol followed, infants and robots were playing a game of tag, in which the robot had a small set of options with regards to its play-based interaction with the child: close the distance to the infant; increase the distance to the infant; stand still (Kokkoni et al., 2020). Analysis of session data from a small number of subjects seemed to point to a new hypothesis according to which the type of robot behavior that usually triggers infant motor responses rarely involves single atomic actions, but is rather more complex involving several actions in temporal succession. For instance, it looked as if the robot could convey a social non-verbal cue such as “follow me” if it initially approached the child within about 1 m in distance, stood still for a short time interval, then attempted to increase the distance slightly, before repeating in a back-and-forth moving pattern. Motivated by these observations, we have subsequently conjectured that robot responses modeled in an LTL framework may be more effective in triggering the desired subject responses (Zehfroosh and Tanner, 2019). The STL framework described in this paper allows us to bring this HRI method to a new level, including timing constrains.

To see how this could work in the context of the HRI scenario of Figure 5, consider a circular 2D robot workspace for Dash at the time when the “follow me” social cue is to be given (see Figure 6). Divide this workspace into three labeled regions R = {r ₁, r ₂, r ₃}, with the robot initialized in region r ₂. The desired back-and-forth moving pattern for the “follow me” task can be encoded by a random selection of some specific subregions of interest inside r ₁ and inside r ₂. Construct now an STL specification that requires the robot to visit those regions in order and with appropriate timing. For example, for three regions defined geometrically as in Table 3,the following STL behavior specification (in the form of formula φ ₃) for the robot can be defined as a way to signal “follow me” within 10 s to its human playmate: φ 3 = □ 2,4 μ 1 ∧ ♢ 5,6 μ 2 ∧ □ 8,10 μ 3 . (30) Figure 7 presents Dash’s trajectory for STL formula φ ₃ given in (30), realized through the nonsmooth control barrier navigation function methodology of Section 4. Just like the motion planning scenario of Section 5.1, the robot path is shown in terms of snapshots at important time instances that attempt to illustrate the satisfaction of the STL specification (30).