A Safety and Passivity Filter for Robot Teleoperation Systems
AA Safety and Passivity Filterfor Robot Teleoperation Systems
Gennaro Notomista and Xiaoyi Cai Georgia Institute of Technology, Atlanta, GA 30308, USA, [email protected] Massachusetts Institute of Technology, Cambridge, MA 02139, USA, [email protected]
Abstract.
In this paper, we present a way of enforcing safety and pas-sivity properties of robot teleoperation systems, where a human operatorinteracts with a dynamical system modeling the robot. The approachdoes so in a holistic fashion, by combining safety and passivity con-straints in a single optimization-based controller which effectively filters the desired control input before supplying it to the system. The resultis a safety and passivity filter implemented as a convex quadratic pro-gram which can be solved efficiently and employed in an online fashionin many robotic teleoperation applications. Simulation results show thebenefits of the approach developed in this paper applied to the humanteleoperation of a second-order dynamical system.
Keywords:
Robot teleoperation, safety of dynamical systems, passivityof dynamical systems, control barrier functions, integral control barrierfunctions
In robot teleoperation, the robot and a human operator can be seen as inter-connected systems that exchange inputs and outputs. The dynamics of thesesystems, as well as that of the communication channel between them, can leadto unpredictable behaviors of the compound system. Therefore, it is often con-venient analyzing robot teleoperation systems from an energetic point of view,which consists of keeping track of the energy the interconnected systems ex-change between each other.
Passivity -based approaches to the control of inter-connected systems [8] have demonstrated to be suitable in many application do-mains, ranging from telemanipulation [23,13,22] to teleoperation of multi-robotsystems [5].Passivity is an amenable property as it ensures the energy generated bythe system does not exceed the one injected through the input to the system.Energy injected from external sources or other interconnected systems can makea system become non-passive, as discussed in [4]. Passivity theory allows usto analyze dynamical systems from an energetic point of view, so it is a verysuitable design tool for dealing with interconnected systems and, therefore with a r X i v : . [ c s . R O ] F e b Gennaro Notomista and Xiaoyi Cai robot teleoperation. Additionally, energy considerations can be useful to accountfor delays in the communication between a human teleoperator and a roboticsystem that is remotely controlled [29], which can cause the performance of thealgorithms to degrade, in terms of both convergence rate and stability [18].Passivity-based approaches, as well as other energy-based methods, for thecontrol of robotic systems are considered in [4,12,26,28,15]. In [6], the authorsintroduce the concept of energy tanks , which is then extended in [20,21,19,7].These works consider additional dissipative forces on a teleoperation systemthat has to kept passive in order to prevent energy tanks from depleting—acondition that would introduce a singularity in their proposed approaches—soas to keep a positive passivity margin, intended as the energy dissipated by thesystem over time.Besides the passivity property, in many robotic applications, it is also desir-able to ensure the safety of the system, intended as the forward invariance of asubset of the system state space. This is particularly crucial when robotic sys-tems interact or collaborate with humans in order to perform a task. The safetyof human operators can be guaranteed by constraining the robot to operate insafe regions of the workspace. To this end, control barrier functions (CBFs) [2]are a control-theoretic tool which can be employed in order to ensure safety indynamical systems.Fig. 1: Passivity and safety filter: the human input u h the state feedback con-troller u fb are modified by a filter before being supplied to the dynamical system(e.g. a teleoperated robotic system) in order to ensure its safety and passivity.In this paper, we propose a way of dealing with safety and passivity objectivesin a holistic fashion. We do so by introducing a safety and passivity filter (seeFig. 1), whose goal is that of modifying the input to the robotic system in orderto render it safe and passive. The filter effectively modifies the system itselfin order to ensure that it remains safe and passive.
The proposed approach isable to seamlessly account for user-defined feedback control laws—which can beleveraged to endow the system with stability properties or to accomplish otherobjectives—combined with the inputs of a human interacting with the robot.The computational burden introduced by the designed filter is low, making theapproach amenable for the real-time implementation on many robotic platforms.
Safety and Passivity Filter for Robot Teleoperation Systems 3
Throughout this paper, we consider a robotic system modeled by the followingnonlinear control system: ® ˙ x = f ( x, u ) y = g ( x ) (1)where x ∈ R n , u ∈ R m and y ∈ R m are the state, input and output, respectively,of the system, f : R n × R m → R n is a Lipschitz continous vector fields and g : R n → R m .To account for both state feedback controllers and external human inputs,we explicitly consider the input u broken down as follows: u = u fb ( x ) + u h , (2)where u fb : R n → R m denotes the state feedback component of the input, and u h represents the input given by a human operator.As we are interested in provably guaranteeing passivity and safety propertiesof dynamical systems, in the following we briefly introduce this two concepts. Definition 1 (Passivity [10]).
The system (1) is passive if there exists a con-tinuously differentiable positive definite storage function V : R n → R such that,for all x and u , ˙ V = ∂V∂x f ( x, u ) ≤ u T y. (3) The system is called lossless if ˙ V = u T y . The definition of passivity is a statement about the system, which holds for allpossible values of the input u and the output y . Safety, on the other hand, is tiedto the definition of a safe set , i.e. a subset of the state space of the system wherewe want the state of the system to remain confined for all times. A techniquewhich proved to be applicable to a variety of robotic systems [3,27,11,24,17,16]and different scenarios consists of employing control barrier functions (CBFs).In the following, we introduce the definition of CBFs as in [2] and the mainresult which will be used in this paper to ensure controlled forward invariance,i.e. controlled safety. Definition 2 (Control Barrier Functions (CBFs) [2]).
Let
C ⊂ D ⊂ R n be the zero superlevel set of a continuously differentiable function h : D → R .Then h is a control barrier function (CBF) if there exists an extended class K ∞ function γ such that, for the system (1) , sup u ∈ R m { L f h ( x, u ) + γ ( h ( x )) } ≥ . (4) for all x ∈ D . An extended class K ∞ function is a continuous function γ : R → R that is strictlyincreasing and with γ (0) = 0. Gennaro Notomista and Xiaoyi Cai The notation L f h ( x ) denotes the Lie derivative of h along the vector field f .Given this definition of CBFs, the following theorem highlights how they can beused to ensure both set forward invariance (safety) and stability. Theorem 1 (Safety [2]).
Let
C ⊂ R n be a set defined as the zero superlevelset of a continuously differentiable function h : D ⊂ R n → R . If h is a CBFon D with a regular value, then any Lipschitz continuous controller u ( x ) ∈{ u ∈ R m : L f h ( x, u ) + γ ( h ( x )) ≥ } for the system (1) renders the set C forwardinvariant (safe). Additionally, the set C is asymptotically stable in D . Besides safety, in this paper, we are interested in enforcing passivity condi-tions onto a dynamical system, representing for example a robot interacting witha human in a teleoperation task. However, the condition of passivity, recalled in1, involves the input u . Recently introduced integral CBFs (I-CBFs) [1]—whichgeneralize control dependent CBFs [9]—can be leveraged to enforce passivityconditions. In order to take advantage of I-CBFs, the system (1) needs to be dynamically extended as follows: ˙ x = f ( x, u )˙ u = φ ( x, u, t ) + vy = g ( x ) (5)where v ∈ R m is the new control input and φ : R n × R m × R → R m will bedesigned to ensure that u = u fb ( x ) + u h , as in (2), as desired.We now have the necessary constructions to introduce integral CBFs. Definition 3 (Integral Control Barrier Functions (I-CBFs)[1]).
For thesystem (1) , with corresponding safe set S = { ( x, u ) ∈ R n × R m : h ( x, u ) ≥ } ⊂ R n × R m defined as the zero superlevel set of a function h : R n × R m → R with a regular value. Then, h is an integral control barrier function (I-CBF) if forany ( x, u ) ∈ R n × R m and t ≥ : ∂h∂u ( x, u ) = 0 = ⇒ ∂h∂x ( x, u ) f ( x, u ) + γ ( h ( x, u )) ≥ . (6)The implication in (6) guarantees that, by means of an I-CBF h , the inequal-ity˙ h ( x, u ) + γ ( h ( x, u )) = ∂h∂x ( x, u ) f ( x, u ) + ∂h∂u ( x, u ) φ ( x, u, t ) + γ ( h ( x, u )) ≥ S —can always be satisfied by a proper choice of φ ( x, u, t ). With thedefinition of I-CBFs, we concluded the introduction of the control-theoreticaltools employed in the next section to design an input filter to render a dynamicalsystem safe and passive. Safety and Passivity Filter for Robot Teleoperation Systems 5
In this section, we develop the safety and passivity filter. We brake down itsstructure into three components:1. Safety-preserving controller, described in Section 3.12. Passivity-preserving controller, described in Section 3.23. Tracking controller, described in Section 3.3These three components will be then combined in Section 3.4 where Proposition 2is stated, which ensures that the designed filter achieves the desired properties.
Define the safe set S x to be the zero superlevel set of a continuously differentiablefunction h x , i.e. S x = { x ∈ R n : h x ( x ) ≥ } . (8)We want the state x of the system (1) to be confined in S x for all times t . Thiscondition, corresponding to safety, can be enforced using Theorem 1.If h x has relative degree 1 with respect to the input u —i.e. the time derivative˙ h x explicitly depends on u —then it has relative degree 2 with respect to the input v , based on the dynamic extension described by (5). In order for Theorem 1 tobe applicable to the system (5), we need L f h x ( x, u ) to depend on the input v ,a condition that does not hold if the relative degree of h x with respect to v isgreater than 1.To circumvent this issue, following the idea in [16] or [1], let h (cid:48) x ( x, u ) := ˙ h x ( x, u ) + γ ( h x ( x )) . (9)Since h (cid:48) x depends on u , L f h (cid:48) x depends on v . Then, in order to ensure the safetyof S x , we may choose any control input v satisfying the following inequality: L f h (cid:48) x ( x, u, v ) + γ x ( h (cid:48) x ( x, u )) ≥ . (10)This way, by Theorem 1, h (cid:48) x ( x, u ) ≥ h x ( x, u ) + γ ( h x ( x )) ≥ h x ( x ) ≥
0, i.e. S x is safe (see also Example 8 in [16]). Remark 1. If h x has relative degree greater than 1 with respect to u , then re-cursive or exponential CBFs can be leveraged. See techniques developed in [11]and [16].To conclude this section, we notice that Theorem 1 and Remark 1 suggestthe definition of the following set of controllers: K x ( x, u ) = { v ∈ R m : L f h (cid:48) x ( x, u, v ) + γ x ( h (cid:48) x ( x, u )) ≥ } . (11)Theorem 1 can be then interpreted as: if v ∈ K x ( x, u ), the set S x is safe. Gennaro Notomista and Xiaoyi Cai
As pointed out before, as passivity is a condition on the control input u ratherthan the state x , in this paper, we employ integral CBFs (I-CBFs) to ensurepassivity conditions of a dynamical system. The following result—analogous toTheorem 1 for I-CBFs—will be leveraged. Theorem 2 ([1]).
Consider the control system (1) and suppose there is a cor-responding dynamically defined controller ˙ u = φ ( x, u, t ) . If the safe set S ⊂ R n × R m is defined by an integral control barrier function h : R n × R m → R ,then modifying the dynamically defined controller to be of the form ˙ u = φ ( x, u, t ) + v ∗ ( x, u, t ) (12) with v ∗ the solution of the quadratic program (QP) v ∗ ( x, u, t ) = argmin v ∈ R m (cid:107) v (cid:107) subject to ∂h∂u ( x, u ) v + ∂h∂x ( x, u ) f ( x, u )+ ∂h∂u ( x, u ) φ ( x, u, t ) + γ ( h ( x, u )) ≥ results in safety, i.e. the control system (5) with the dynamically defined con-troller (12) results in S being forward invariant: if ( x (0) , u (0)) ∈ S then ( x ( t ) , u ( t )) ∈S for all t ≥ . We now define an I-CBF which Lemma 1 shows to be suitable to ensure thepassivity of the system (1).Let V : R n → R be a continuously differentiable positive definite function,and define the following I-CBFs: h u ( x, u ) := g ( x ) T u − L f V ( x, u ) . (14)The corresponding safe set S u is defined as S u = { ( x, u ) ∈ R n × R m : g ( x ) T u − L f V ( x, u ) ≥ } . (15) Lemma 1 (Passivity as safety).
Safety of S u in (15) = ⇒ Passivity of (1) .Proof.
Assume S u is safe, i.e. ( x, u ) ∈ S u for all t . From (15), one has: g ( x ) T u − L f V ( x, u ) = y T u − ∂V∂x f ( x, u ) = y T u − ˙ V ≥ t . Thus, y T u ≥ ˙ V and, by Definition 1, the system is passive with storagefunction V and with respect to input u and output y . Remark 2.
The expression of h u in (14) represents the power dissipated by thesystem. In [20] and [15], methods to ensure the passivity of a system in terms ofenergy are proposed. While those approaches are more flexible, insofar as theyenforce conditions similar to h u ( x, u ) ≥ T max in [7]). Safety and Passivity Filter for Robot Teleoperation Systems 7
Similarly to what has been done before, the result in Theorem 2 suggests thedefinition of the following set of controllers: K u ( x, u ) = ß v ∈ R m : ∂h u ∂u ( x, u ) v + ∂h u ∂x ( x, u ) f ( x, u )+ ∂h u ∂u ( x, u ) φ ( x, u, t ) + γ ( h u ( x, u )) ≥ ™ . (17)By Theorem 2, the safety of S u —and, by Lemma 1, the passivity of (1)—isenforced using the I-CBF h u by picking a controller in K u ( x, u ).With this result in place, we are now ready to combine safety and passivity.Before presenting the safety and passivity filter, in the following section we showhow to ensure that the dynamically extended system (5) asymptotically behavesas the original system (1) when safety constraints are not violated. The dynamic extension of the system (1) proposed in (5) is required in order toenforce constraints on the input u —the passivity constraints—through a properchoice of v . On the other hand, due to this extension, we are not able to controlthe original system (1) using u anymore, but rather we have to design a suitablefunction φ in (5) in order to track the desired u using v . The objective of thissection is that of presenting a controller that serves this purpose .Assume we want u = u fb ( x ) + u h as in (2). As ˙ u = φ ( x, u ) + v , one could set˙ u = ˙ u fb ( x ) + ˙ u h + v ∗ = L f u fb ( x, u ) + ˙ u h (cid:124) (cid:123)(cid:122) (cid:125) =: φ ( x,u,t ) + v ∗ , (18)where v ∗ is given by (13) [14]. This choice, however, may cause u ( t ) to divergeover time more and more from its desired value u fb ( x ( t )) + u h ( t ), due to the factthat v ∗ from (13) is the minimizer of the difference between the time derivativesof the input functions. In fact, from (12), (cid:107) v ∗ (cid:107) = (cid:107) ˙ u − φ ( x, u, t ) (cid:107) , (19)is the norm of the difference between the derivative of u —rather than the inputfunction itself—and φ . The following theorem presents a dynamically definedcontrol law which results in u ( t ) converging to the desired value u fb ( x ( t )) + u h ( t )as t → ∞ whenever safety is not violated. Proposition 1.
Consider the system (1) and a desired nominal input signal (2) . Consider an I-CBF h : R n × R m → R defined to ensure the safety of the set It is worth noticing that there are cases in which a dynamically defined controller ˙ u is already available (see, for instance, [25]). Gennaro Notomista and Xiaoyi Cai S ⊂ R n × R m defined as its zero superlevel set. Then, the dynamically definedcontroller ˙ u = L f u fb ( x, u ) + ˙ u h + α u fb ( x ) + u h − u ) (cid:124) (cid:123)(cid:122) (cid:125) =: φ ( x,u,t ) + v ∗ , (20) where α > and v ∗ is given in (13) , will ensure the safety of the set S , as well asthe tracking of the nominal control signal (2) whenever the controller φ ( x, u, t ) is safe.Proof. First of all, by φ ( x, u, t ) being safe we mean that the constraints in (13)are inactive and, consequently, v ∗ ( x, u, t ) = 0. Then, by Theorem 2, the con-troller (20) results in the forward invariance, i.e. safety, of the set S . Therefore,we only need to confirm that, if the controller (20) is safe, then u will trackthe nominal controller (2). To this end, let us consider the following Lyapunovfunction candidate for the system in (5) with φ ( x, u, t ) given in (18): W ( u, x, u h ) = 12 (cid:107) u − u fb ( x ) − u h (cid:107) . (21)Its time derivative evaluates to:˙ W = ∂W∂u ˙ u + ∂W∂x ˙ x + ∂W∂u h ˙ u h = ( u − u fb ( x ) − u h ) T ( ˙ u − L f u fb ( x, u ) − ˙ u h ) (22)Substituting the proposed controller (20), we obtain˙ W = ( u − u fb ( x ) − u h ) T (cid:16) L f u fb ( x, u ) + ˙ u h + α u fb ( x ) + u h − u ) − L f u fb ( x, u ) − ˙ u h (cid:17) = α u − u fb ( x ) − u h ) T ( u fb ( x ) + u h − u )= − αW ( u, x, u h ) . (23)Thus, W ( t ) →
0, or equivalently u ( t ) → u fb ( x ( t )) + u h ( t ), as t → ∞ , i.e. theinput u will track the desired control signal (2). Remark 3.
The variable ˙ u only appears in the software implementation of thepassive and safe controller for the system (1). The actual input given to thesystem is its integral u ( t ). Therefore, the value α in the expression of the dy-namically defined controller (20) can be chosen arbitrarily large, being aware ofnot introducing rounding or numerical errors while solving the QP (13). As canbe noticed in the proof of Proposition 1, the larger the value of α is, the fasterthe convergence of u to the desired controller (2) when v ∗ = 0 (i.e. when nosafety-related modifications of ˙ u are required). Safety and Passivity Filter for Robot Teleoperation Systems 9
Remark 4.
Integrating the expression of the dynamically defined controller in(20) with respect to time, we get: u ( t ) = (cid:90) t ( L f u fb ( x ( τ ) , u ( τ )) + ˙ u h ( τ )) dτ + α (cid:90) t ( u fb ( x ( τ )) + u h ( τ ) − u ( τ )) dτ (cid:124) (cid:123)(cid:122) (cid:125) Integral control , (24)where we explicitly recognize the integral component of the dynamically definedcontroller which ensures the desired tracking properties [25]. In this section, we combine the results of the previous three subsection to designa safety and passivity input filter.
Proposition 2 (Main result).
Consider a dynamical system (1) , a set S x where we want the state x of the system to remain confined for all times (safety),and a continuously differentiable positive definite function V with respect towhich we want the system to be passive (passivity). If the controller v ∗ ( x, u, t ) = argmin v ∈ K xu ( x,u ) (cid:107) v (cid:107) , (25) where K xu ( x, u ) = K x ( x, u ) ∩ K u ( x, u ) ⊂ R m , exists for all times t , then thesystem (1) is safe and passive.Proof. The proof of this proposition is based on the combination of the resultsof Theorems 1 and 2 with Lemma 1.If the QP (25) has a solution for all t , then v ∗ ( x, u, t ) ∈ K xu ( x, u ) for all t .Then, by Theorem 1, as v ∗ ( x, u, t ) ∈ K x ( x, u ), S x defined in (8) using h x is for-ward invariant, i.e. safe. Moreover, as v ∗ ( x, u, t ) ∈ K u ( x, u ) for all t , Theorem 2ensures that S u defined in (15) using h u is safe. Thus, by Lemma 1, the system(1) is passive. Remark 5 (Safety and passivity filter).
Solving the QP (25) can be interpretedas filtering the desired control input given in (2)—comprised of a state feedbackcomponent, u fb , and a human input, u h —to obtain v ∗ . The filtered controller isthen integrated in software to obtain the actual control input u supplied to thesystem (1) to ensure its safety and passivity. See Fig. 2.The filtering, i.e. the synthesis of the safety- and passivity-preserving con-troller, is implemented as an optimization-based controller solution of a convexquadratic program. As such, it can be efficiently solved in online settings, evenunder real-time constraints, in many robotic applications.The following section shows the benefits of the safety and passivity filterdeveloped in this paper applied to the human teleoperation of a second-orderdynamical system, modeling a mechanical robotic platform. Fig. 2: Passivity and safety filter: the structure of the filter depicted in Fig. 1is specified using the results of the paper. The filter includes the computationof the function φ , whose expression is given in (20), the solution of the convexquadratic program (QP) (25), and an integration step, in order to compute thecontrol input u supplied to the system. In this section, we present the results of the application of the safety and passivityfilter developed in the previous section to the case of a second-order dynamicalsystem controlled both by a feedback controller and by an external control inputof an operator.The model of the system is the following: ˙ x = x ˙ x = − σx + uy = x , (26)where x , x , u, y ∈ R , σ >
0. Its dynamic extension (5) is: ˙ x = x ˙ x = − σx + u ˙ u = φ ( x, u, t ) + vy = x , (27)where v ∈ R . The desired input ˆ u is a PD controller aimed at driving the stateof the system to the origin:ˆ u = u fb ( x ) + u h = − k P x − k D x + u h , (28)where k P , k D > u h = [ − . , T . From the desired (28) the expressionof φ ( x, u, t ) can be obtained using (20): φ ( x, u, t ) = − k P x − k D ( − σx + u ) + k I (ˆ u − u ) + ˙ u h , (29) Safety and Passivity Filter for Robot Teleoperation Systems 11 where k I > α in (20), i.e. an integral gain, as noticed inRemark 4.To ensure the passivity of the system, the following storage function has beenemployed: V : R n → R : x (cid:55)→ (cid:107) x (cid:107) . (30)and the I-CBF h u (14) has been employed. For the system (26), the passivitycondition (16) becomes: A u v ≤ b u , (31)where A u ( x ) = x T (32) b u ( x, u, t ) = − (1 + 3 σ ) (cid:107) x (cid:107) + 2 σx T x − (cid:0) x T − σx T (cid:1) u (33) − x T φ ( x, u, t ) + γ u (cid:0) σ (cid:107) x (cid:107) − x T x − x T u (cid:1) . (34)Safety has been defined as the condition that x never enters the unit diskcentered at the origin. To this end, the following CBF has been defined: h x ( x ) = (cid:107) x (cid:107) − d , (35)where d = 1. As h x has relative degree 2 with respect to u , a recursive approachhas been employed, as discussed in Remark 1. Thus, the following two auxiliaryCBFs arise: h (cid:48) x = 2 x T x + (cid:107) x (cid:107) − d (36) h (cid:48)(cid:48) x = (cid:107) x (cid:107) + 2 (cid:107) x (cid:107) + 2(2 − σ ) x T x + 2 x T u − d , (37)and the safety condition (4) becomes: A x v ≤ b x , (38)where A x ( x ) = − x T (39) b x ( x, u, t ) =(2 x T + 2(2 − σ ) x T + 2 u T ) x + (4 x T + 2(2 − σ ) x T )( − σx + u )(40)+ 2 x T φ ( x, u, t ) + γ x ( h (cid:48)(cid:48) x ) . (41)Passivity and safety conditions are then combined in the following single QPequivalent to (25): v ∗ ( x, u, t ) = argmin v ∈ R (cid:107) v (cid:107) (42)subject to A x ( x ) v ≤ b x ( x, u, t ) (Safety constraint) (43) A u ( x ) v ≤ b u ( x, u, t ) (Passivity constraint) . (44) (b) Fig. 3: Trajectory (Fig. 3a) and passivity I-CBF (Fig. 3b) for the system (27)controlled using v ( t ) = 0 for all t . The state x converges to [ x , , x , ] = [ − . , h u and h x take negative values in Fig. 3b.The result of the implementation of the solution of (42) to control the system(27) are reported in the following.Figure 3 shows the trajectory of the state and of the I-CBF h u for the system(27) controlled using v ( t ) = 0 for all t . As no safety constraint is enforced, thesystem trajectory enter the red disk (unsafe region). Moreover, as no passivityconstraint is enforced, the value of h u becomes negative. From (14), this impliesthat y T u (cid:54)≥ ˙ V , i.e. energy is generated and the system is not passive.To mitigate this issue, we introduce the passivity I-CBF constraint A u ( x ) v ≤ b u ( x, u, t ). Figure 4 shows the results of the implementation of the controller v ∗ solution of the QP (42) without the safety constraint A x ( x ) v ≤ b x ( x, u, t ). Thetrajectory of the system still enters the unsafe red-shaded region, however thevalue of h u is always positive. For the same rationale discussed above, in thiscase energy is not generated and y T u ≥ ˙ V as desired, i.e. the system is passive.Finally, to show how safety and passivity constraints can be enforced in aholistic fashion, Fig. 5 shows the behavior of the system controlled by the solutionof the complete QP (42). The trajectory of the system is kept away from theunsafe region by the effect of the safety constraints and, at the same time, thevalue of h u remains positive for all times, i.e. the system is safe. In this paper, we introduced a safety and passivity filter which is able to guaran-tee that a dynamical system remains passive and a subset of its state space re-mains forward invariant. This technique is particularly suitable in robot teleoper-ation scenarios where a human is interconnected—by an input-output relation—to a robotic system and exchange energy with the system through the supplied
Safety and Passivity Filter for Robot Teleoperation Systems 13(a) (b)
Fig. 4: Trajectory (Fig. 4a) and passivity I-CBF (Fig. 4b) for the system (27)controlled using v ( t ) = v ∗ solution of the QP (42) without the safety constraint A x ( x ) v ≤ b x ( x, u, t ). The state x converges to [ x , , x , ] = [ − . ,
0] as expectedand, in addition to the simulation in Fig. 3, it does so while preserving passivityfor all times. In fact, in Fig. 3b, it can be seen how h u is always kept positive.Values of h x , on the other hand, become negative when the blue dot in Fig. 4ais inside the red-shaded disk. (a) (b) Fig. 5: Trajectory (Fig. 5a) and passivity I-CBF (Fig. 5b) for the system (27)controlled using v ( t ) = v ∗ solution of the QP (42) with safety and passivityconstraints. By enforcing the safety constraints, the blue dot in Fig. 5a is notallowed to enter the unsafe region (red disk). As a result, x does not converge to[ x , , x , ] = [ − . ,
0] as desired by the human input, but in this case it reachesthe value of x in the safe region closest to [ − . , − , h x and h u beingboth kept positive. control inputs. The passivity of the interconnected system guarantees that noenergy is generated by the interconnection, while the forward invariance prop-erty ensures the safety of the interaction between the human operator and therobotic system.Future work will be devoted to the feasibility analysis of the optimizationproblem which defines the safety and passivity filter, as well as to the intro-duction of estimation algorithms required to evaluate the input supplied by thehuman interacting with the robotic system. Moreover, in this paper, we shownthe approach applied to a simulated linear second-order system, representing asimple robotic system, controlled by a feedback controller as well as a human in-put. Future work will focus on applying this method to real manipulator robotsand multi-robot systems, which are commonly employed in robot teleoperationapplications. References
1. Ames, A., Notomista, G., Wardi, Y., Egerstedt, M.: Integral control barrier func-tions for dynamically defined control laws. Control Systems Letters (2020)2. Ames, A.D., Coogan, S., Egerstedt, M., Notomista, G., Sreenath, K., Tabuada, P.:Control barrier functions: Theory and applications. In: European Control Confer-ence, pp. 3420–3431 (2019). DOI 10.23919/ECC.2019.87960303. Ames, A.D., Grizzle, J.W., Tabuada, P.: Control barrier function based quadraticprograms with application to adaptive cruise control. In: Conference on Decisionand Control, pp. 6271–6278. IEEE (2014)4. Anderson, R.J., Spong, M.W.: Bilateral control of teleoperators with time delay.IEEE Transactions on Automatic control (5), 494–501 (1989)5. Chopra, N., Spong, M.W.: Passivity-based control of multi-agent systems. In:Advances in robot control, pp. 107–134. Springer (2006)6. Duindam, V., Stramigioli, S.: Port-based asymptotic curve tracking for mechanicalsystems. European Journal of Control (5), 411–420 (2004)7. Giordano, P.R., Franchi, A., Secchi, C., B¨ulthoff, H.H.: A passivity-based decentral-ized strategy for generalized connectivity maintenance. The International Journalof Robotics Research (3), 299–323 (2013)8. Hatanaka, T., Chopra, N., Spong, M.W.: Passivity-based control of robots: His-torical perspective and contemporary issues. In: 2015 54th IEEE Conference onDecision and Control (CDC), pp. 2450–2452. IEEE (2015)9. Huang, Y., Yong, S.Z., Chen, Y.: Guaranteed vehicle safety control using control-dependent barrier functions. In: American Control Conference, pp. 983–988. IEEE(2019)10. Khalil, H.K.: Nonlinear control. Pearson New York (2015)11. Nguyen, Q., Sreenath, K.: Exponential control barrier functions for enforcing highrelative-degree safety-critical constraints. In: American Control Conference, pp.322–328. IEEE (2016)12. Niemeyer, G., Slotine, J.J.: Stable adaptive teleoperation. IEEE Journal of oceanicengineering (1), 152–162 (1991)13. Niemeyer, G., Slotine, J.J.E.: Telemanipulation with time delays. The InternationalJournal of Robotics Research (5), 1186–1205 (2019)18. Olfati-Saber, R., Murray, R.M.: Consensus problems in networks of agents withswitching topology and time-delays. IEEE Transactions on automatic control (9), 1520–1533 (2004)19. Secchi, C., Franchi, A., B¨ulthoff, H.H., Giordano, P.R.: Bilateral teleoperationof a group of uavs with communication delays and switching topology. In: 2012IEEE International Conference on Robotics and Automation, pp. 4307–4314. IEEE(2012)20. Secchi, C., Stramigioli, S., Fantuzzi, C.: Position drift compensation in port-hamiltonian based telemanipulation. In: 2006 IEEE/RSJ International Conferenceon Intelligent Robots and Systems, pp. 4211–4216. IEEE (2006)21. Secchi, C., Stramigioli, S., Fantuzzi, C.: Control of interactive robotic interfaces:A port-Hamiltonian approach, vol. 29. Springer Science & Business Media (2007)22. Sieber, D., Hirche, S.: Human-guided multirobot cooperative manipulation. IEEETransactions on Control Systems Technology (4), 1492–1509 (2018)23. Stramigioli, S., Van Der Schaft, A., Maschke, B., Melchiorri, C.: Geometric scatter-ing in robotic telemanipulation. IEEE Transactions on Robotics and Automation (4), 588–596 (2002)24. Wang, L., Ames, A.D., Egerstedt, M.: Safety barrier certificates for collisions-freemultirobot systems. Transactions on Robotics (3), 661–674 (2017)25. Wardi, Y., Seatzu, C., Cortes, J., Egerstedt, M., Shivam, S., Buckley, I.: Track-ing control by the newton-raphson method with output prediction and controllerspeedup. arXiv preprint arXiv:1910.00693 (2019)26. Wohlers, M.R.: Lumped and distributed passive networks: a generalized and ad-vanced viewpoint. Academic press (2017)27. Wu, G., Sreenath, K.: Safety-critical control of a planar quadrotor. In: AmericanControl Conference, pp. 2252–2258. IEEE (2016)28. Yamauchi, J., Atman, M.W.S., Hatanaka, T., Chopra, N., Fujita, M.: Passivity-based control of human-robotic networks with inter-robot communication delaysand experimental verification. In: 2017 IEEE International Conference on Ad-vanced Intelligent Mechatronics (AIM), pp. 628–633. IEEE (2017)29. Zampieri, S.: Trends in networked control systems. IFAC Proceedings Volumes41