Towards manipulability of interactive Lagrangian systems

Automatica 119 (2020) 108913

Contents lists available at ScienceDirect

Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Towards manipulability of interactive Lagrangian systems✩ Hanlei Wang Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing 100094, China

article

info

Article history: Received 30 August 2018 Received in revised form 26 November 2019 Accepted 12 February 2020 Available online 3 July 2020 Keywords: Dynamic feedback Infinite manipulability Bilateral teleoperation Dynamic-cascade framework Switching topology Time-varying delay Lagrangian systems

a b s t r a c t This paper investigates manipulability of interactive Lagrangian systems with parametric uncertainty and communication/sensing constraints. Two standard applications are teaching operation of robots and teleoperation with a master–slave system. We systematically formulate the concept of infinite manipulability for general dynamical systems, and investigate how a unified motivation based on this concept yields a design paradigm towards guaranteeing the infinite manipulability of interactive dynamical systems and in particular facilitates the design and analysis of nonlinear adaptive controllers for interactive Lagrangian systems. Specifically, based on a new class of dynamic feedback, we propose adaptive controllers that achieve both the infinite manipulability of the controlled Lagrangian systems and robustness with respect to the communication/sensing constraints, mainly owing to the resultant dynamic-cascade framework. We also show that a special case of our main result provides a delayindependent solution to the problem of nonlinear bilateral teleoperation with arbitrary unknown time-varying delay. © 2020 Elsevier Ltd. All rights reserved.

1. Introduction One important trend of modern automatic machines is to facilitate the human–machine interaction. For instance, collaborative robots (the study of which has become particularly active in the robotics industry) are expected to be used in the scenario that the collaboration between the robots and human operators is frequently involved (e.g., the teaching operation in the standard ‘‘teach-by-showing’’ approach (Asada & Izumi, 1989; Craig, 2005; Ikeuchi & Suehiro, 1994; Weiss, Sanderson, & Neuman, 1987)). Another example is teleoperation with a master–slave system in which case the slave robot is kept to be synchronized with the master robot that is guided by a human operator (see, e.g., Hokayem & Spong, 2006; Sheridan, 1993). The fundamental issues behind these typical application scenarios are quite different from the common automatic control systems that emphasize stability with respect to an equilibrium; for instance, it is well known that the equilibrium of a teleoperator is implicitly specified by the human operator (typically unknown a priori), and that it is the similar case for a robot manipulator under the standard teaching operation. ✩ This work was supported by the National Natural Science Foundation of China under Grants 61374060 and 61333008, and the National Key Basic Research Program (973) of China under Grant 2013CB733100. The material in this paper was partially presented at the Chinese Automation Congress (CAC), November 30–December 2, 2018, Xi’an, China. This paper was recommended for publication in revised form by Associate Editor Warren E. Dixon under the direction of Editor Daniel Liberzon. E-mail address: [email protected]. https://doi.org/10.1016/j.automatica.2020.108913 0005-1098/© 2020 Elsevier Ltd. All rights reserved.

Historically, the control problems involved in teleoperation have received sustaining interest, which yields many significant results (see, e.g., the pioneering result in Anderson and Spong (1989) and the results discussed in Hokayem and Spong (2006) and Sheridan (1993)). But the connection between teleoperation and standard control theory might be still relatively weak, mainly due to the lack of fundamental concepts that may enhance this connection, though there are some exceptional ones. In particular, the exploitation of the passivity concept in bilateral teleoperation (see, e.g., Anderson & Spong, 1989; Niemeyer & Slotine, 1991) is, in certain sense, a constructive attempt to address the connection issue (for instance, passivity often implies the potential stability of the system (Anderson & Spong, 1989; Niemeyer & Slotine, 1991, 2004)), and the past decades have witnessed the wide applications of this concept in teleoperation (see, e.g., Hokayem & Spong, 2006; Niemeyer & Slotine, 2004; Nuño, Basañez, & Ortega, 2011). In recent years, benefiting from the extensive interest in control of multi-agent systems, many synchronizationbased controllers have been proposed for teleoperators with their nonlinear dynamics being taken into account (see, e.g., Chopra, Spong, & Lozano, 2008; Lee & Spong, 2006; Liu & Chopra, 2013; Nuño, Basañez, Ortega, & Spong, 2009; Wang & Xie, 2018) and the special case of the results in Abdessameud, Polushin, and Tayebi (2014), Nuño, Ortega, Basañez, and Hill (2011) and Wang (2014) (focusing on consensus of networked Lagrangian systems on directed topologies) can also be used in a teleoperator. A critical issue that spans the long history of teleoperation is the robustness with respect to the communication delay (see, e.g., Hokayem & Spong, 2006), especially if the delay is time-varying. Many

2

H. Wang / Automatica 119 (2020) 108913

results (e.g., Anderson & Spong, 1989; Chopra et al., 2008; Liu & Chopra, 2013; Nuño, Ortega, & Basañez, 2010) achieve the robustness with respect to arbitrary unknown constant delay (which can also be referred to as delay-independent), yet this becomes frustrating as the delay is time-varying and in fact no delay-independent result has been witnessed in the case of timevarying delay. For instance, the results in Nuño, Aldana, and Basañez (2017) and Nuño et al. (2009) rely on designing the damping gain based on the upper bound of the time-varying delay and the result in Abdessameud et al. (2014) also exploits some a priori information of the delay for specifying the controller gains. For networked Lagrangian systems, the issue associated with the coupling between the dynamics of each system and network interaction remains and is even much more severe, due to the fact that the topology might be directed and/or switching (see, e.g., Liu, 2015; Liu, Min, Wang, Liu, & Liao, 2014; Nuño, Ortega, et al., 2011; Wang, 2014). For linear identical integrator systems or those systems that can be transformed to integrator systems by feedback, some strong results have been achieved for the consensus/synchronization problem (see, e.g., Chopra & Spong, 2008; Münz, Papachristodoulou, & Allgöwer, 2011); in particular, both the time-varying delay and switching directed topologies are considered in Münz et al. (2011) in the context of multiple identical single-integrator systems. The consideration of uncertain highorder systems on undirected jointly-connected topologies, using dynamic feedback, occurs in Rezaee and Abdollahi (2017). The results for uncertain Lagrangian systems with switching topologies (and time delays) are presented in, e.g., Liu (2015) and Liu et al. (2014) and due to the use of static feedback, these results generally impose relatively strong requirements concerning the interaction topologies or time delays (for instance, the interaction topologies are required to be balanced or regular). An attempt based on dynamic feedback design for uncertain Lagrangian systems occurs in Wang (2020), which is mainly for realizing consensus of multiple Lagrangian systems with general switching directed topologies and arbitrary time-varying delays, and the obtained topology/delay-independent solutions are mainly attributed to the dynamic feedback design. However, most of these results do not systematically consider the interaction between the network and an external subject (for instance, a human operator) while this becomes typical in the previously discussed problems of teleoperation and teaching operation. The systems involving interaction with an external subject are typically referred to as interactive systems and in the specific context here as interactive Lagrangian systems. The investigation of such systems over the past mainly concentrates on the stability of the interactive systems; one common concept that is exploited is passivity since it is well recognized that passivity of the system typically implies stability as the system interacts with a human operator. On the other hand, achieving passivity shows some potential limitations as handling the system uncertainty and tough circumstances of the communication channel (see, e.g., Chopra, Spong, Hirche, & Buss, 2003; Nuño et al., 2010; Nuño, Ortega, et al., 2011). The attempts to resolve this issue along other perspectives (not based on passivity) occur in, e.g., Abdessameud et al. (2014), Liu and Chopra (2013), Nuño et al. (2017, 2010), Nuño, Ortega, et al. (2011) and Wang (2014). Yet the systematic and rigorous formulation concerning the interactive behaviors of the combined system (for instance, the system consisting of the networked system and human operator) is still rarely witnessed. The results in Liu and Chopra (2013) and Nuño et al. (2017) either consider some specific dynamics of human operators or present some particular ad hoc discussions concerning the human–robot interaction. Frequency-domain formulations for addressing the human– system interaction are presented in, e.g., Son, Franchi, Chuang,

Kim, Bülthoff, and Giordano (2013) and Yokokohji and Yoshikawa (1994), which rely on several particular frequency-domain performance indices for describing the response from the human operator’s input action to the master–slave position error (Yokokohji & Yoshikawa, 1994) or cross correlation of the position tracking error of the slave (Son et al., 2013); in particular, the result in Son et al. (2013) introduces the frequency-domain performance indices for quantifying the exerted effort of the human operator for the control of the slave system consisting of multiple unmanned aerial vehicles (UAVs), i.e., maneuverability. However, the specific formulations in Son et al. (2013) and Yokokohji and Yoshikawa (1994) are subjected to the well-recognized potential limitations of the frequency-domain approach and also exhibit their ad hoc feature as taking into account the system uncertainty/nonlinearity and time-varying communication/sensing constraints and as addressing the inherent issue concerning the adjustment of the equilibrium point of a teleoperator or general interactive Lagrangian systems. In summary, beyond the standard passivity concept that mainly focuses on stability concerning the human–system interaction, the rigorous and systematic formulation/framework towards addressing the general fundamental mechanism concerning the human–system interaction remains to be established, especially as involving nonlinear dynamics, uncertainty, and time-varying communication/sensing constraints. In this paper, we systematically formulate the concept of infinite (dynamical) manipulability to rigorously quantify the interactive behavior of Lagrangian systems under an external input action (force or torque), and the concept here extends/generalizes the one introduced in the specific context of consensus of networked robotic systems in Wang and Xie (2018) to general dynamical systems with mathematically rigorous formulation. Motivated in part by the result in Wang and Xie (2018) concerning the importance of existence of an infinite gain from the external force/torque to the consensus equilibrium increment, a design paradigm towards guaranteeing the infinite manipulability of general interactive dynamical systems (namely, the existence of an infinite gain from the external input action to the specified output) is formally proposed. Differing from the concept of passivity in the literature that mainly addresses the stability issue of a human-manipulator interactive system, the concept of infinite manipulability is mainly for addressing the required amount of effort associated with the dynamic maneuvering of interactive (Lagrangian) systems. Specifically, based on a new class of dynamic feedback, we develop adaptive controllers to systematically address the issue of manipulability of a single Lagrangian system and that of networked Lagrangian systems with switching topology and unknown time-varying communication delay; the resultant closed-loop system is a dynamic-cascade one, which is in contrast to the system in Wang (2020) and also to the standard cascade system. The new feature of the proposed adaptive controllers lies in the dynamic feedback design of the reference velocity and acceleration while the basic adaptive structure is the same as the standard one in Slotine and Li (1987). Our result covers two practically important applications, i.e., teaching operation of a robot manipulator and bilateral teleoperation with unknown time-varying delay. In particular, we demonstrate how the motivation of studying the manipulability of bilateral teleoperators leads to the first delay-independent solution to the problem of bilateral control of nonlinear teleoperators with arbitrary unknown time-varying delay (to the best of our knowledge). A preliminary version of the paper was presented in Wang (2018) and it mainly focuses on the concept of infinite manipulability and the exploitation of this concept for developing an adaptive controller for teleoperators with timevarying delay. In the present paper, we expand Wang (2018) to further address the case of a single Lagrangian system and that of networked Lagrangian systems with switching topology and time-varying delay.

H. Wang / Automatica 119 (2020) 108913

2. Equations of motion of Lagrangian systems The equations of motion of a m-DOF (degree-of-freedom) Lagrangian system can be written as (Slotine & Li, 1991; Spong, Hutchinson, & Vidyasagar, 2006) M(q)q¨ + C (q, q˙ )q˙ + g(q) = τ

(1)

m

where q ∈ R is the generalized position (or configuration), M(q) ∈ Rm×m is the inertia matrix, C (q, q˙ ) ∈ Rm×m is the Coriolis and centrifugal matrix, g(q) ∈ Rm is the gravitational torque, and τ ∈ Rm is the exerted control torque. Three typical properties concerning the dynamics (1) are listed as follows. Property 1 (Slotine & Li, 1991; Spong et al., 2006). The inertia matrix M(q) is symmetric and uniformly positive definite. Property 2 (Slotine & Li, 1991; Spong et al., 2006). The Coriolis and centrifugal matrix C (q, q˙ ) can be appropriately chosen so that the ˙ matrix M(q) − 2C (q, q˙ ) is skew-symmetric. Property 3 (Slotine & Li, 1991; Spong et al., 2006). The dynamics (1) depend linearly on a constant parameter vector ϑ and this leads to M(q)ζ˙ + C (q, q˙ )ζ + g(q) = Y (q, q˙ , ζ , ζ˙ )ϑ

(2)

where ζ ∈ Rm is a differentiable vector, ζ˙ is the derivative of ζ , and Y (q, q˙ , ζ , ζ˙ ) is the regressor matrix. 3. Manipulability of dynamical systems We start by considering a standard simple example, namely the motion of a point mass governed by mx¨ = u + f

(3)

where x ∈ R is the position of the point mass, m ∈ R is the mass, u ∈ R is the control input, and f ∈ R is the external force from a subject (e.g., a human operator). Let us now consider the problem of the degree of the adjustability of the position x under the action of f . Suppose that u takes the standard damping action as u = −bx˙ with b being a positive design constant, and we have that mx¨ = −bx˙ + f .

(4)

Following the standard practice, we obtain the transfer function from f to x as G(p) =

1 mp2 + bp

(5)

with p denoting the Laplace variable, √ and further the H∞ norm of G(p) as supω |G(jω)| = supω [1/(|ω| m2 ω2 + b2 )] = ∞ where |·| denotes the modulus of a complex number. As is well recognized, the H∞ norm (which is well known to be equal to the L2 -gain for linear time-invariant systems) describes the energy-like relation between the input and output, i.e., the relation between the L2 norm of the output and that of the input. Specifically, for the example above, we have that (see, e.g., Desoer & Vidyasagar, 1975; Ioannou & Sun, 1996) sup f ̸ =0

∥x∥2 = sup |G(jω)| = ∞ ∥f ∥2 ω

(6)

where ∥ · ∥2 denotes the standard L2 norm of a function. This would imply that an input with a finite L2 norm holds the possibility of producing an output with an infinite L2 norm [intuitively imitating a standard case in the calculus that 0 · ∞ (in the sense of limit) can reach an arbitrary finite quantity], and consequently,

3

it would be possible for a human operator to maneuver the position of the point mass to an arbitrary value with finite energy consumption (in the sense that the L2 norm of the exerted force is finite). This potentially reduces the required amount of effort from the human operator. In particular, consider an external force f (t) = 1/(t + 1) (which is well known to have a finite L2 norm), and the output x, in accordance with (6), holds the possibility of having an infinite L2 norm. The actual consequence can be illustrated by considering the output corresponding to this specific input, and by integrating (4) with respect to time, it can be shown that [suppose that x(0) = 0 and x˙ (0) = 0] mx˙ = −bx + ln(t + 1) and this implies that the output x is the response of a standard stable filter with an unbounded input ln(t + 1). It is well recognized that the output x (i.e., the position of the point mass) converges to infinity as t → ∞ (x has an infinite L2 norm), in comparison with the fact that f (t) is squareintegrable and converges to zero as t → ∞ [f (t) has a finite L2 norm]. We now formally introduce the concept of infinite manipulability or infinite dynamical manipulability for general dynamical systems, which generalizes the one introduced in the specific context of consensus of networked robotic systems in Wang and Xie (2018) to consider general dynamical systems with mathematically rigorous formulation. Manipulability of a dynamical system in terms of its specified output basically describes the degree of adjustability of the output corresponding to an external input action, and it is essentially equivalent to the standard concept of reachability/controllability or output reachability/controllability for dynamical systems. The distinctive point concerning (dynamical) manipulability may lie in its emphasis on the (physical) interactive maneuvering behavior of (controlled) dynamical systems acted upon by an external subject, in contrast to the concept of reachability/controllability or output reachability/controllability that is typically associated with stability or stabilizability of a dynamical system itself. What is of particular interest, as is shown in our later result, is the infinite manipulability and it is typically associated with marginally stable dynamical systems. Definition 1. Consider a dynamical system y = G∗ (f ) with G∗ denoting a mapping, f the external input action, and y the output. (1) The dynamical system is said to be infinitely manipulable if it is manipulable (controllable) and if the gain of the mapping from the external input action to each component of the output is infinite. (2) The dynamical system is said to be infinitely manipulable with degree k, k = 1, 2, . . . if it is infinitely manipulable and if the mapping contains k pure integral operations with the infinite portion of the gain being only due to the k pure integral operations. In most practical cases, the property of the gain of the mapping from the external input action to each component of the output is or can generally be ensured to be equivalent to that of the mapping from the external input action to the output. Hence, the gains used in the standard input–output analysis (see, e.g., Desoer & Vidyasagar, 1975; Ioannou & Sun, 1996; van der Schaft, 2000) can be directly adopted for quantifying the system manipulability (as also illustrated in the above simple example), and for facilitating the formulation later, the quantification of manipulability of a dynamical system over the interval [0, t ] is denoted by Mtf ↦→y , and M∞ f ↦ →y is typically denoted by Mf ↦ →y for conciseness. In many applications, it is desirable to maintain the infinite manipulability of the system concerning the specific output (e.g., for reducing the amount of effort exerted by a human operator in the course of adjusting the system equilibrium). On the other hand, for a system with overly high manipulability, it might

4

H. Wang / Automatica 119 (2020) 108913

be difficult for a human operator to accurately adjust the system equilibrium. For instance, the mapping given by (5) contains one pure integral operation [i.e., there is only one factor 1/p in G(p)], and thus the system (4) is said to be infinitely manipulable with degree one. In contrast, if the damping parameter b is set to be zero, the manipulability degree of the system becomes two. It would typically be more difficult to manipulate an infinitely manipulable system with degree two than an infinitely manipulable system with degree one. Intuitively, we can consider the manipulation of a point mass on a frictionless horizontal plane and that of a point mass on a horizontal plane with viscous friction. The accurate positioning for the case without any friction is expected to be much more difficult than that for the case with viscous friction. For conciseness of subsequent formulations and demonstrations, the manipulability analysis based on input–output gains in the sequel, if not particularly mentioned, follows the standard practice; see also the relevant results in, e.g., Angeli, Sontag, and Wang (2000), Desoer and Vidyasagar (1975), Ioannou and Sun (1996), Khalil (2002), Rotea (1993), van der Schaft (2000), Sontag (1998) and Vidyasagar (1993).

Remark 1. The damping control with gravitational torque compensation as well as the stability of the closed-loop system with τ ∗ = 0 is well recognized, and the result here is for revisiting this standard problem in the context of rigorously analyzing the manipulability of the system and for showing that the system is actually infinitely manipulable with degree one.

4. Manipulability of Lagrangian systems

where α and λM are positive design constants, and z(0) can be specified as an arbitrary constant vector. Define

For the Lagrangian system given by (1), we investigate the manipulability of the system with its generalized position or velocity as the output. We expect to realize the infinite manipulability of the system in terms of the output. In particular, the infinite manipulability in terms of its generalized position has important applications in teaching operation of robot manipulators.

s = q˙ − z .

4.1. Damping control with gravitational torque compensation Consider the standard damping control with the gravitational torque compensation

τ = −α q˙ + g(q)

(7)

where α is a positive design constant, and under an external input action τ ∗ , this control yields M(q)q¨ + C (q, q˙ )q˙ = −α q˙ + τ ∗ .

(8)

The above system defines a mapping from τ ∗ to q, and as is previously discussed, the gain of this mapping quantifies the manipulability of the system. As is typically done, consider the Lyapunov function candidate V = (1/2)q˙ T M(q)q˙ , and we have that (using Property 2) V˙ = −α q˙ T q˙ + q˙ T τ ∗ . Using the result from the standard basic inequalities q˙ T τ ∗ ≤ (α/2)q˙ T q˙ + [1/(2α )]τ ∗T τ ∗ , we obtain that V˙ ≤ −(α/2)q˙ T q˙ + [1/(2α )]τ ∗T τ ∗ . Following the typical practice (see, e.g., Khalil, 2002), we obtain that the L2 -gain from τ ∗ to q˙ is less than or equal to 1/α . On the other hand, the L2 -gain from q˙ to the position increment q − q(0) is that of a pure integral operation, which is well known to be supω (1/|ω|) = ∞. Therefore, the L2 -gain from τ ∗ to q − q(0) satisfies the property that Mτ ∗ ↦→q−q(0) ≤ (1/α ) supω (1/|ω|) = ∞ and in addition Mτ ∗ ↦→q−q(0) has the same order as the upper bound (as in the typical practice of calculating the gains for nonlinear systems), and thus the infinite manipulability of the system with degree one is ensured (since the mapping only contains one pure integral operation). Furthermore, in the case that τ ∗∫ ∈ L2 ∩ L∞ , there t exists a positive constant ℓM such that ℓM − 0 τ ∗T (σ )τ ∗ (σ )dσ ≥ 0, ∀t ≥ 0. Consider the ∫quasi-Lyapunov function candidate t V ∗ = V + [1/(2α )][ℓM − 0 τ ∗T (σ )τ ∗ (σ )dσ ] (the adoption of the second term follows Lozano, Brogliato, Egeland, and Maschke (2000, p. 118)) whose derivative can be directly demonstrated to satisfy the property that (using the previous result concerning V˙ ) V˙ ∗ ≤ −(α/2)q˙ T q˙ ≤ 0, and we obtain by the typical practice that q˙ → 0 as t → ∞.

4.2. Adaptive control With parametric uncertainty, the gravitational torque compensation is no longer accurate, which would possibly result in reduction of manipulability. More importantly, we expect to rigorously address the quantitative performance of the system (e.g., ensuring the efficiency of teaching operation) in addition to the manipulability even if we do not exactly know the system model or the system model is subjected to a variation. This can be accommodated in part by the flexibility of adaptive control (see, e.g., Slotine & Li, 1989; Wang, 2017). We first introduce a vector z ∈ Rm by z˙ = −α q˙ + λM (q˙ − z)

(9)

(10)

The adaptive controller is given as

τ = − Ks + Y (q, q˙ , z , z˙ )ϑˆ ϑ˙ˆ = − Γ Y T (q, q˙ , z , z˙ )s

(11) (12)

where K and Γ are symmetric positive definite matrices, and ϑˆ is the estimate of ϑ . The dynamics of the system can be described by

⎧ ⎪ ⎨q¨ = −α q˙ + λM s + s˙ M(q)s˙ + C (q, q˙ )s = −Ks + Y (q, q˙ , z , z˙ )∆ϑ + τ ∗ ⎪ ⎩ϑ˙ˆ = −Γ Y T (q, q˙ , z , z˙ )s

(13)

where ∆ϑ = ϑˆ − ϑ . Eq. (13) defines a system referred to as a dynamic-cascade system since the cascade component λM s+˙s involves both the vector s [generated by the lower two subsystems of (13)] and its derivative s˙, in contrast to the system in Wang (2020) and also to the standard cascade systems. As is standard (see, e.g., Ortega & Spong, 1989; Slotine & Li, 1987), consider the Lyapunov-like function candidate V = (1/2)sT M(q)s + (1/2)∆ϑ T Γ −1 ∆ϑ , and its derivative along the trajectories of the system can be written as (using Property 2) V˙ = −sT Ks + sT τ ∗ . Theorem 1. The adaptive controller given by (11) and (12) for the Lagrangian system given by (1) ensures that q˙ → 0 in the case that the external input action τ ∗ ∈ L2 ∩ L∞ . In addition, the system with q as the output is infinitely manipulable with degree one if the gravitational torque is a priori compensated. Proof. In the case that τ ∗ ∈ L2 ∩∫L∞ , there exists a positive t constant ℓ∗M such that ℓ∗M − (1/2) 0 τ ∗T (σ )K −1 τ ∗ (σ )dσ ≥ 0, ∀t ≥ 0. Using similar procedures as in the previous analysis, we directly obtain V˙ ∗ ≤ −(1/2)sT Ks ≤ 0 with V ∗ = ∫ t that ∗ ∗T V + [ℓM − (1/2) 0 τ (σ )K −1 τ ∗ (σ )dσ ], which implies that s ∈ ∫t L2 ∩ L∞ and ϑˆ ∈ L∞ . This yields the result that 0 s˙(σ )dσ = ∫t s − s(0) ∈ L∞ and 0 s˙(σ )dσ + s(0) = s ∈ L2 . Consider the first subsystem of (13) with q˙ as the output, and in accordance with the standard linear system theory, the output can be considered as the superposition of the output under the input s˙ and that

H. Wang / Automatica 119 (2020) 108913

under the input λM s. The system given by q¨ = −α q˙ + λM s + s˙ with λM s + s˙ = 0 and with q˙ as the output is exponentially stable and strictly proper by the standard linear system theory. Hence, from the input–output properties of exponentially stable and strictly proper linear systems (Desoer & Vidyasagar, 1975, p. 59), we obtain that the output of the system corresponding to the input λM s is square-integrable and bounded. From Wang (2020, Proposition 2), we obtain that the output corresponding to the input s˙ is also square-integrable and bounded. This implies that q˙ ∈ L2 ∩ L∞ in accordance with the standard superposition principle for linear systems. We then obtain that z = q˙ − s ∈ L2 ∩ L∞ , and this leads us to obtain from (9) that z˙ ∈ L2 ∩ L∞ . From the second subsystem of (13) and using Property 1, we obtain that s˙ ∈ L∞ . Therefore, q¨ ∈ L∞ , implying that q˙ is uniformly continuous. From the properties of square-integrable and uniformly continuous functions (Desoer & Vidyasagar, 1975, p. 232), we obtain that q˙ → 0 as t → ∞. We now consider the manipulability of the system with q as the output. First, consider the mapping from τ ∗ to s, and as is well recognized in adaptive control, this mapping is obviously not L∞ -gain bounded yet it is L2 -gain bounded and also L2 ↦ → L∞ gain (see, e.g., Angeli et al., 2000; Rotea, 1993; Sontag, 1998 for the detail) bounded. In fact, the previous analysis implies that the L2 -gain from τ ∗ to s is less than or equal to 1/λmin {K } with λmin {·} denoting the minimum eigenvalue of a matrix and that if τ ∗ ∈ L2 , then s ∈ L2 ∩ L∞ . From the first subsystem of (13), we can investigate the mapping from s to q, and the output q˙ of this subsystem, due to its linear nature, can be considered to be the superposition of two outputs corresponding to the inputs s˙ and λM s, respectively in accordance with the standard superposition principle for linear systems. From Wang (2020, Proposition 2), the ∫ t input s˙ yields a square-integrable ∫ t and bounded output since s˙(σ )dσ + s(0) = s ∈ L2 and 0 s(σ )dσ = s − s(0) ∈ L∞ in 0 the case that s ∈ L2 ∩ L∞ . In addition, the input s˙ yields the first portion of q (denoted by χ1 ), the boundedness of which can be observed by the integral operation of χ¨ 1 = −α χ˙ 1 + s˙ with respect to time. In fact, we have that

χ˙ 1 = −αχ1 + αχ1 (0) + χ˙ 1 (0) − s(0) + s

(14)

which means that χ1 in (14) is the output of an exponentially stable linear system with bounded input and is thus bounded by the standard linear system theory. The input λM s, on the other hand, only yields a square-integrable and bounded output (second portion of q˙ ) and it does not lead to the boundedness of the second portion of q (denoted by χ2 ) since the integral operation of χ¨ 2 = −α χ˙ 2 + λM s with respect to time gives

χ˙ 2 = −αχ2 + αχ2 (0) + χ˙ 2 (0) + λM

t

∫

s(σ )dσ

5

Remark 2. The action λM (q˙ − z) introduced in (9) is for ensuring the infinite manipulability of the system in terms of its generalized position under the external input action τ ∗ . The basic structure of the adaptive controller given by (11) and (12) follows the fundamental result in Slotine and Li (1987) (this adaptive structure is also exploited in the sequel in the context of networked Lagrangian systems and teleoperators), and new reference velocity and acceleration [i.e., z and z˙ given by (9)] are introduced to address the manipulability issue of a single Lagrangian system with parametric uncertainty. The design given by (9), (11), and (12) is one specific but also representative case with s being square-integrable and bounded. The design guided by the infinite manipulability can be quite general. For instance, we can specify the z-dynamics as z˙ = −α q˙ + λM column{(q˙ (k) − z (k) )2ρ+1 , k = 1, . . . , m}, and the control torque is given as τ = 1

¯ −kcolumn {sgn(s(k) )|s(k) | ρ ∗ −1 , k = 1, . . . , m}+ Y (q, q˙ , z , z˙ )ϑˆ where ρ ∗ > 1, ρ = 0, 1, . . . and ρ is also subjected to the condition that 2ρ + 1 < ρ ∗ /(ρ ∗ − 1), k¯ is a positive design constant, and q˙ (k) , z (k) , and s(k) are the kth components of q˙ , z, and s, respectively. It can be directly shown by the typical practice and similar procedures as in the proof of Theorem 1 that this control ensures the stability of the system with τ ∗ ∈ Lρ ∗ ∩ L∞ . It is similar for the case of networked Lagrangian systems and teleoperators. 4.3. Specifying velocity as the output The controllers above ensure the infinite manipulability of the system with the generalized position as the output, and on the other hand, we note that the gain of the mapping from the external input action to the velocity q˙ is actually finite, yielding finite manipulability of the system with respect to the velocity. This would imply that for adjusting the velocity of the system to a desired value, the external input has to hold a torque constantly. To achieve the infinite manipulability of the system in terms of the velocity, we can simply set α = 0 and then the gain of the mapping from τ ∗ to q˙ becomes infinite and in addition this mapping contains one pure integral operation. Hence, the infinite manipulability with degree one can be guaranteed. 5. Manipulability of networked Lagrangian systems Consider n Lagrangian systems with the dynamics of the ith system being governed by (Slotine & Li, 1991; Spong et al., 2006) Mi (qi )q¨ i + Ci (qi , q˙ i )q˙ i + gi (qi ) = τi m

(15)

0

and s ∈ L2 ∩ L∞ .1 We now explicitly calculate the L2 -gain of the mapping from s to q, and to this ∫ t end, we combine (14) and (15) as q˙ = −α[q − ψ0 ] + λM 0 s(σ )dσ + s with ψ0 = [α q(0) + q˙ (0) − s(0) ∫ t ]/α . By following the standard practice, the L2 -gain from λM 0 s(σ )dσ + s to q − ψ0 can be derived as 1/α (i.e., the H∞ norm of the transfer ∫ t function), and the upper bound of the L2 -gain from s to λM 0 s(σ )dσ + s can be derived as √

λ2M supω (1/|ω|2 ) + 1. Therefore, the L2 -gain from τ ∗ to q − ψ0 √ satisfies Mτ ∗ ↦→q−ψ0 ≤ [1/(αλmin {K })] λ2M supω (1/|ω|2 ) + 1 = ∞. As in the typical practice of calculating the gains for nonlinear dynamical systems, this shows that the system is infinitely manipulable with degree one. ■

1 It is well known that square-integrability of a function does not imply that its integral is bounded and this function holds the possibility of being integral unbounded.

(16)

where qi ∈ R is the generalized position or configuration, Mi (qi ) ∈ Rm×m is the inertia matrix, Ci (qi , q˙ i ) ∈ Rm×m is the Coriolis and centrifugal matrix, gi (qi ) ∈ Rm is the gravitational torque, and τi ∈ Rm is the exerted control torque. We briefly introduce the graph theory in the context involving the n Lagrangian systems by following Olfati-Saber and Murray (2004) and Ren and Beard (2005, 2008). As now standard, we adopt a directed graph G = (V , E ) for describing the interaction topology among the n systems where V = {1, . . . , n} is the vertex set that denotes the collection of the n systems and E ⊆ V × V is the edge set that denotes the information interaction among the n systems. A graph is said to contain a directed spanning tree if there exists a vertex k∗ ∈ V so that any other vertex of the graph has a directed path to k∗ , where the vertex k∗ is referred to as the root of the graph. Denote by Ni = {j|(i, j) ∈ E } the set of neighbors of the ith system. The weighted adjacency matrix W = [wij ] associated with G is defined in accordance with the rule that wij > 0 in the case that j ∈ Ni , and wij = 0 otherwise. The standard assumption regarding the diagonal entries of W that wii = 0, ∀i = 1, . . . , n is adopted. In the case that the interaction topology switches,

6

H. Wang / Automatica 119 (2020) 108913

the interaction graph among the systems becomes time-varying. Denote by GS = {G1 , . . . , Gns } the set of the interaction graphs among the n systems, and these graphs share the same vertex set V yet their edge sets are typically different. The union of a collection of graphs Gi1 , . . . , Gis with is ≤ ns is a graph with the vertex set given by V and the edge set given by the union of the edge sets of Gi1 , . . . , Gis . Denote by t0 , t1 , t2 , . . . with t0 = 0 an infinite sequence of times at which the interaction graph switches, and this sequence is assumed to satisfy the standard property that TD ≤ tκ+1 − tκ < TM , ∀κ = 0, 1, . . . with TD and TM being two positive constants where the assumption concerning the dwell time that tκ+1 − tκ ≥ TD , ∀κ = 0, 1, . . . is standard for the case of switching topology (refer to Ren & Beard, 2008 for the details). In the following, we design an adaptive controller to realize consensus of the n Lagrangian systems with switching topologies and time-varying delays, and simultaneously ensure the infinite manipulability of the system. The delays are assumed to be piecewise uniformly continuous and uniformly bounded. We define a vector zi ∈ Rm as z˙i = − α q˙ i − Σj∈Ni (t) wij (t)[ξi − ξj (t − Tij )] + λM (q˙ i − zi )

(17)

where α and λM are positive design constants, zi (0) is an arbitrarily specified constant vector, ξi is defined as ξi = q˙ i + α qi (see Chopra & Spong, 2006), Tij is the time-varying delay from the jth system to the ith system, and as in the typical practice ξj (t − Tij ) is set as zero if 0 ≤ t < Tij . Define si = q˙ i − zi .

(18)

The adaptive controller is given as

τi = − Ki si + Yi (qi , q˙ i , zi , z˙i )ϑˆ i ϑ˙ˆ i = − Γi YiT (qi , q˙ i , zi , z˙i )si

(19) (20)

where Ki and Γi are symmetric positive definite matrices, ϑˆ i is the estimate of a parameter vector ϑi , and the regressor matrix Yi (qi , q˙ i , zi , z˙i ) and the vector ϑi are defined in accordance with the linearity-in-parameter property of the Lagrangian system (Slotine & Li, 1991; Spong et al., 2006), i.e., Mi (qi )z˙i + Ci (qi , q˙ i )zi + gi (qi ) = Yi (qi , q˙ i , zi , z˙i )ϑi . The dynamics of the ith system can be described by

⎧ ⎪ ξ˙ = −Σj∈Ni (t) wij (t)[ξi − ξj (t − Tij )] + λM si + s˙i ⎪ ⎨ i Mi (qi )s˙i + Ci (qi , q˙ i )si = −Ki si + Yi (qi , q˙ i , zi , z˙i )∆ϑi ⎪ ⎪ ⎩ϑ˙ˆ = −Γ Y T (q , q˙ , z , z˙ )s i

i i

i

i

i

i

(21)

i

where ∆ϑi = ϑˆ i − ϑi . The above system, similar to the previous case, is also dynamic-cascaded in the sense that the cascade component λM si +˙si involves both the vector si and its derivative s˙i . Theorem 2. Suppose that there exist an infinite number of uniformly bounded intervals [tκν , tκν+1 ), ν = 1, 2, . . . with tκ1 = t0 satisfying the property that the union of the interaction graphs in each interval contains a directed spanning tree and that the time-varying delays are piecewise uniformly continuous and uniformly bounded. Then, the adaptive controller given by (19) and (20) with zi being given by (17) ensures (1) consensus of the n systems without external physical interaction, i.e., qi − qj → 0 and q˙ i → 0 as t → ∞, ∀i, j = 1, . . . , n, and (2) the infinite manipulability of the system with degree one in terms of an external physical input action at the torque level and the consensus equilibrium increment if the gravitational torques are a priori compensated.

Proof. We first follow the standard practice to analyze the lower two subsystems of (21) (see, e.g., Ortega & Spong, 1989; Slotine & Li, 1987). Specifically, consider the Lyapunov-like function candidate Vi = (1/2)sTi Mi (qi )si + (1/2)∆ϑiT Γi−1 ∆ϑi and its derivative along the trajectories of the system can be written as V˙ i = ˙ i (qi ) − 2Ci (qi , q˙ i ) (see, −sTi Ki si ≤ 0 with the skew-symmetry of M e.g., Slotine & Li, 1991; Spong et al., 2006) being utilized, ∀i. This leads us to immediately obtain that si ∈ L2 ∩ L∞ and ϑˆ i ∈ L∞ , ∀i. All the equations expressed as the first subsystem of (21) can be written compactly as

ξ˙ = FD (ξ ) + λM s∗ + s˙∗

(22)

with ξ = [ξ , . . . , ξ ] s = [ , . . . , ] and FD (·) denoting a linear mapping that involves the delay operation. Following the standard practice (see, e.g., Lee & Spong, 2007; Ren & Beard, 2008), we introduce a vector ξE = [ξ1T − ξ2T , . . . , ξnT−1 − ξnT ]T . In accordance with the result in Münz et al. (2011), the output ξE of the linear system given by (22) with s∗ = 0 and s˙∗ = 0 uniformly asymptotically converges to zero and furthermore ξ˙ also uniformly asymptotically converges to zero (in the case that s∗ = 0 and s˙∗ = 0). Therefore, from the standard linear system theory, the system (22) with ξ as the state and with s∗ = 0 and s˙∗ = 0 is uniformly marginally stable with the state uniformly converging to certain constant vector. For the system given by (22), using Wang (2020, Proposition 3) and the standard superposition principle (with λM s∗ and s˙∗ , respectively, as the inputs), we obtain that FD (ξ ) ∈ L∞ . Eq. (17) can be rewritten as [using (18)] z˙i = −α zi + (λM − α )si − Σj∈Ni (t) wij (t)[ξi − ξj (t − Tij )], which yields the result that zi ∈ L∞ and z˙i ∈ L∞ in accordance with the input–output properties of exponentially stable and strictly proper linear systems (Desoer & Vidyasagar, 1975, p. 59), ∀i. Thus, q˙ i ∈ L∞ , ∀i. We then obtain that s˙i ∈ L∞ from the second subsystem of (21) and by using the property that Mi (qi ) is uniformly positive definite (see, e.g., Slotine & Li, 1991; Spong et al., 2006), ∀i. This implies that q¨ i ∈ L∞ , ∀i. Hence, q˙ i and si are uniformly continuous, ∀i. Using the properties of square-integrable and uniformly continuous functions (Desoer & Vidyasagar, 1975, p. 232), we obtain that si → 0 as t → ∞, T 1

T T , n

∗

sT1

sTn T ,

∀i. From (20), we obtain that ϑ˙ˆ i ∈ L∞ , ∀i. The result that q˙ i ∈ ˙ L∞ , z˙i ∈ L∞ , and ϑˆ i ∈ L∞ implies that qi , zi , and ϑˆ i are all uniformly continuous, ∀i. Then, we obtain from (17) that z˙i is

piecewise uniformly continuous by additionally considering the assumption that the time-varying delays are piecewise uniformly continuous and uniformly bounded and the standard assumption concerning the dwell time (i.e., tκ+1 − tκ ≥ TD , ∀κ = 0, 1, . . . ), ∀i. From the second subsystem of (21) and using the aforementioned property that Mi (qi ) is uniformly positive definite, we obtain that s˙i is piecewise uniformly continuous, ∀i. The application of the standard generalized Barbalat’s lemma (see, e.g., Jiang, 2009; Wang, 2020) immediately yields the result that s˙i → 0 as t → ∞, ∀i. For the system given by (22) with λM s∗ + s˙∗ as the input and ξE as the output, we obtain from the standard input–output properties of linear time-varying systems that ξE → 0 as t → ∞, and we also obtain from Wang (2020, Proposition 3) that ξ˙ → 0 as t → ∞. From the definition of ξi , we obtain that q¨ i = −α q˙ i + ξ˙i , upon which using the input–output properties of exponentially stable and strictly property linear systems (Desoer & Vidyasagar, 1975, p. 59) yields q˙ i → 0 as t → ∞, ∀i. Therefore, qi − qj → 0 as t → ∞, ∀i, j. We next demonstrate that the manipulability of the system is infinite with degree one if an external physical input action is exerted at the torque level on a system that acts as the root of the interaction graph (in the sense that there exist an infinite number of uniformly bounded intervals such that the system acts as the root of the union of the interaction graphs

H. Wang / Automatica 119 (2020) 108913

in each interval). Without loss of generality, suppose that the κ ∗ th system, 1 ≤ κ ∗ ≤ n acts as the root and is subjected to an external physical input action τκ∗∗ at the torque level. The derivative of Vκ ∗ along the trajectories of the system now satisfies the property that (by using the standard basic inequalities) V˙ κ ∗ ≤ −(1/2)sTκ ∗ Kκ ∗ sκ ∗ + (1/2)τκ∗∗T Kκ−∗1 τκ∗∗ , and this implies that the L2 -gain from τκ∗∗ to sκ ∗ is less than or equal to 1/λmin {Kκ ∗ }. The upper bound of the L2 -gain from s∗ to λM s∗ + s˙∗ , similar to the previous case, can be directly obtained √ by calculating the H∞ norm of the transfer function as λ2M + (supω |ω|)2 ∗ ∗ ˙ (due to the fact that s and s are not purely related by the differential operation). For the system (22), it can be directly shown from Wang (2020, Proposition 3) that the L2 -gain from λM s∗ + s˙∗ to ξ˙ is finite, and by letting ξc = (1/n)Σkn=1 ξk , we obtain that the L2 -gain from λM s∗ + s˙∗ to ξ˙c is less than or equal to a positive constant h∗ (i.e., the L2 -gain is finite). The L2 -gain from ξ˙c to ξc − ξc (0) is that of a pure integral operation, ∗ i.e., supω (1/|ω|). Then, we obtain the L √2 -gain from τκ ∗ to ξc −

ξc (0) as Mτκ∗∗ ↦→ξc −ξc (0) ≤ (h∗ /λmin {Kκ ∗ }) λ2M supω (1/|ω|2 ) + 1 = ∞. Let qc = (1/n)Σkn=1 qk , and by exploiting the result that ξc = q˙ c + α qc (from the definition of ξi , i = 1, . . . , n) with the L2 -gain from ξc − ξc (0) to qc − [qc (0) + (1/α )q˙ c (0)] being 1/α (which is obtained by calculating the H∞ norm of the transfer function), the L2 -gain from τκ∗∗ to qc − [qc (0) + (1/α )q˙ c (0)] can be shown to satisfy Mτ ∗∗ ↦→qc −[qc (0)+(1/α )q˙ c (0)] ≤ κ √ [h∗ /(αλmin {Kκ ∗ })] λ2M supω (1/|ω|2 ) + 1 = ∞. This implies that the system is infinitely manipulable with degree one.

■

Remark 3. The definition of zi given by (17) is motivated by but differs from Wang (2020) in the sense that zi is no longer the pure integral concerning the system state. This is reflected in the newly introduced term λM (q˙ i − zi ) = λM si , which is shown to be crucial for ensuring the infinite manipulability of the system. In the case with only switching topologies [a special case of Theorem 2 with Tij = 0, ∀j ∈ Ni (t), ∀i], our result does not necessarily require the communication among the systems, and the availability of relative position and velocity measurement among the neighboring systems and that of position and velocity measurement for each system would be adequate for implementation. Remark 4. The proposed controller can also ensure the position consensus of networked Lagrangian systems in the case that the external input action is square-integrable, bounded, and piecewise uniformly continuous, which can be shown using similar procedures as in the proof of Theorem 1; in the scenario of manipulation of a single Lagrangian system (Section 4), the convergence of the velocity of the system is ensured with a square-integrable and bounded external input. This fundamentally ensures that the external subject (e.g., a human operator) can easily manipulate the controlled Lagrangian systems and simultaneously that the asymptotic consensus among the systems or convergence of the velocity of the system be maintained. An intuitive interpretation concerning the possibility of simultaneously achieving the two objectives is associated with the properties of functions that are ‘‘square-integrable yet not integral bounded’’. A well-known function of this category is f (t) = 1/(t + 1) and its integral can ∫bet directly shown to satisfy the well-recognized property that f (σ )dσ = ln(t + 1) → ∞ as t → ∞. In particular, due to 0 the square-integrability (and boundedness and piecewise uniform continuity) of the external input action, the asymptotic consensus among the systems or convergence of the velocity of the system is maintained even under the external input action, and due to the possibility of integral unboundedness of the external input action, manipulating the system to an arbitrary equilibrium without using so much effort (i.e., with finite energy consumption) becomes possible.

7

6. Application to bilateral teleoperation with time-varying delay Bilateral teleoperation with arbitrary unknown time-varying delay is a long-standing problem, and to the best of our knowledge, no delay-independent solution has been systematically developed. The standard scattering/wave-variable-based approach (Anderson & Spong, 1989; Niemeyer & Slotine, 1991) can typically handle arbitrary unknown constant time delay. The modifications to the original scattering/wave-variable-based approach appear in, e.g., Ching and Book (2006), Chopra et al. (2003), Munir and Book (2002), Niemeyer and Slotine (1998) and Yokokohji, Imaida, and Yoshikawa (1999) for handling timevarying delay or position drift. It is typically recognized that the scattering/wave-variable-based approach exhibits potential limitations as handling the problem of position drift (see, e.g., Chopra, Spong, Ortega, & Barabanov, 2006; Lee & Spong, 2006). For resolving this problem, numerous synchronization-based results under constant or time-varying delay are presented (see, e.g., Chopra et al., 2008, 2006; Hua, Yang, Yan, & Guan, 2017; Lee & Huang, 2010; Lee & Spong, 2006; Liu & Khong, 2015; Nuño, ArteagaPérez, & Espinosa-Pérez, 2018; Nuño et al., 2009, 2010; Polushin, Liu, & Lung, 2006; Polushin, Takhmar, & Patel, 2015). However, most of these results, as handling the case that the delay is time-varying, are generally delay-dependent. Here we provide a solution to this long-standing problem and the solution is independent of arbitrary time-varying delays in the sense that the delays are only required to be piecewise uniformly continuous and uniformly bounded, which can be considered as a special case of the result in Section 5. The dynamics of a teleoperator can be written as (Hokayem & Spong, 2006; Spong et al., 2006) M1 (q1 )q¨ 1 + C1 (q1 , q˙ 1 )q˙ 1 + g1 (q1 ) = τ1 + τ1∗

(23)

M2 (q2 )q¨ 2 + C2 (q2 , q˙ 2 )q˙ 2 + g2 (q2 ) = τ2 − τ2∗

(24)

with τ1 being the torque exerted by the human operator on the master robot (the first robot) and τ2∗ the torque exerted by the slave robot (the second robot) on the environment. The adaptive controller is given as ∗

τ1 = − Ks1 + Y1 (q1 , q˙ 1 , z1 , z˙1 )ϑˆ 1 ϑ˙ˆ 1 = − Γ1 Y1T (q1 , q˙ 1 , z1 , z˙1 )s1 τ2 = − Ks2 + Y2 (q2 , q˙ 2 , z2 , z˙2 )ϑˆ 2 ϑ˙ˆ 2 = − Γ2 Y2T (q2 , q˙ 2 , z2 , z˙2 )s2

(25) (26) (27) (28)

with z1 and z2 being defined as z˙1 = − α q˙ 1 − λ[ξ1 − ξ2 (t − T2 )] + λM (q˙ 1 − z1 )

(29)

z˙2 = − α q˙ 2 − λ[ξ2 − ξ1 (t − T1 )] + λM (q˙ 2 − z2 )

(30)

where K is a symmetric positive definite matrix, λ is a positive design constant, and T1 and T2 are the time-varying delays. Theorem 3. Suppose that the time-varying delays are piecewise uniformly continuous and uniformly bounded. Then, the adaptive controller given by (25), (26), (27), and (28) with z1 and z2 being respectively given by (29) and (30) for the teleoperator given by (23) and (24) ensures position synchronization of the master and slave robots provided that τi∗ ∈ L2 ∩ L∞ and τi∗ is piecewise uniformly continuous, ∀i = 1, 2. If the gravitational torques are compensated a priori, the controller ensures static torque reflection and the manipulability of the teleoperator is infinite with degree one. The proof of Theorem 3 can be performed by following similar procedures as in that of Theorem 2 and Theorem 1. The special

8

H. Wang / Automatica 119 (2020) 108913

issue that needs to be further demonstrated for bilateral teleoperation is that of force/torque reflection, and this can be completed by following the standard practice (see, e.g., Lee & Spong, 2006; Liu & Chopra, 2013; Liu & Khong, 2015). In particular, consider the scenario that q˙ i , q¨ i , and z˙i converge to zero, i = 1, 2, and it can be directly shown that τ1∗ → (λα K /λM )(q1 − q2 ) and τ2∗ → −(λα K /λM )(q2 − q1 ). This implies that τ1∗ → τ2∗ , i.e., the static torque reflection in the sense of Lee and Spong (2006) is achieved. Remark 5. Our result is in contrast to Abdessameud et al. (2014) and Nuño et al. (2009) which rely on the conditions involving the gains based on some a priori information of the time-varying delay, and in particular neither the upper bound of the delay nor that of the discontinuous change of the delay is required. The assumption concerning the derivative of the delay in, e.g., Chopra et al. (2003) and Niemeyer and Slotine (1998) is also no longer required. Another favorable point of our result is that the robustness with respect to piecewise uniformly continuous and uniformly bounded delays implies that our result also achieves robustness with respect to packet loss/switching in the communication channel, either for the case of bilateral teleoperation or for that of networked Lagrangian systems. 7. Conclusion In this paper, we have systematically formulated the concept of infinite (dynamical) manipulability for dynamical systems and then investigated how a unified motivation based on this concept yields a systematic design paradigm for general interactive dynamical systems and interactive Lagrangian systems with parametric uncertainty and communication/sensing constraints. Specifically, the proposed design paradigm guarantees the infinite manipulability of the controlled Lagrangian systems with particularly strong robustness with respect to the interaction topology and time-varying communication delay. In addition, our result provides a solution to the long-standing problem of nonlinear bilateral teleoperation with arbitrary unknown time-varying delay. We would like to further discuss the connection between the physics of human–system interaction and mathematical properties of general functions, which becomes particularly prominent in the present work and shows some interesting features that might arouse our sense/admiration of the delicate connection between the pure mathematics and physics (which has been historically witnessed for numerous times in various context). Our result can be considered as a contribution to this (probable) historical truth/fact from the perspective of systems and control. Specifically, what attracts our attention in this study are those functions that are square-integrable yet not integral bounded. As is shown in our main result, the incorporation of a controlled square-integrable (and bounded) function that is not integral bounded (this function is generated by the closed-loop system) is crucial for ensuring both the easy manipulation of the system (this yields the infinite manipulability of the closedloop system and consequently reduces the required amount of effort of the human operator) and the asymptotic position consensus (synchronization) among the Lagrangian systems. Squareintegrability of functions often leads to the consequence that they converge to zero (for instance, if the functions are further uniformly continuous), which is well recognized in the field of systems and control. On the other hand, it is also well known that some of the square-integrable functions hold the possibility that their integrals with respect to time are unbounded. Our study shows how such a property concerning general functions is systematically exploited in designing nonlinear controllers for interactive Lagrangian systems and associated with the gain properties of dynamical systems such as infinite manipulability and physical properties such as finite amount of energy.

Acknowledgment The author would like to thank Dr. Tiantian Jiang for the helpful discussions and suggestions. References Abdessameud, A., Polushin, I. G., & Tayebi, A. (2014). Synchronization of Lagrangian systems with irregular communication delays. IEEE Transactions on Automatic Control, 59(1), 187–193. Anderson, R. J., & Spong, M. W. (1989). Bilateral control of teleoperators with time delay. IEEE Transactions on Automatic Control, 34(5), 494–501. Angeli, D., Sontag, E. D., & Wang, Y. (2000). A characterization of integral input-to-state stability. IEEE Transactions on Automatic Control, 45(6), 1082–1097. Asada, H., & Izumi, H. (1989). Automatic program generation from teaching data for the hybrid control of robots. IEEE Transactions on Robotics and Automation, 5(2), 166–173. Ching, H., & Book, W. J. (2006). Internet-based bilateral teleoperation based on wave variable with adaptive predictor and direct drift control. Journal of Dynamic Systems, Measurement, and Control, 128(1), 86–93. Chopra, N., & Spong, M. W. (2006). Passivity-based control of multi-agent systems. In S. Kawamura, & M. Svinin (Eds.), Advances in robot control: From everyday physics to human-like movements (pp. 107–134). Berlin, Germany: Springer-Verlag. Chopra, N., & Spong, M. W. (2008). Output synchronization of nonlinear systems with relative degree one. In V. D. Blondel, S. P. Boyd, & H. Kimura (Eds.), Recent advances in learning and control (pp. 51–64). London, U.K.: Springer-Verlag. Chopra, N., Spong, M. W., Hirche, S., & Buss, M. (2003). Bilateral teleoperation over the Internet: The time varying delay problem. In Proceedings of the american control conference (pp. 155–160). Denver, CO, USA. Chopra, N., Spong, M. W., & Lozano, R. (2008). Synchronization of bilateral teleoperators with time delay. Automatica, 44(8), 2142–2148. Chopra, N., Spong, M. W., Ortega, R., & Barabanov, N. E. (2006). On tracking performance in bilateral teleoperation. IEEE Transactions on Robotics, 22(4), 861–866. Craig, J. J. (2005). Introduction to robotics: Mechanics and control (3rd ed.). Upper Saddle River, NJ: Prentice-Hall. Desoer, C. A., & Vidyasagar, M. (1975). Feedback systems: Input-output properties. New York: Academic Press. Hokayem, P. F., & Spong, M. W. (2006). Bilateral teleoperation: An historical survey. Automatica, 42(12), 2035–2057. Hua, C.-C., Yang, X., Yan, J., & Guan, X.-P. (2017). On exploring the domain of attraction for bilateral teleoperator subject to interval delay and saturated P+d control scheme. IEEE Transactions on Automatic Control, 62(6), 2923–2928. Ikeuchi, K., & Suehiro, T. (1994). Toward an assembly plan from observation Part I: Task recognition with polyhedral objects. IEEE Transactions on Robotics and Automation, 10(3), 368–385. Ioannou, P. A., & Sun, J. (1996). Robust adaptive control. Upper Saddle River, NJ: Prentice-Hall. Jiang, H. B. (2009). Hybrid adaptive and impulsive synchronisation of uncertain complex dynamical networks by the generalised Barbalat’s lemma. IET Control Theory & Applications, 3(10), 1330–1340. Khalil, H. K. (2002). Nonlinear Systems (3rd ed.). Upper Saddle River, NJ: Prentice-Hall. Lee, D., & Huang, K. (2010). Passive-set-position-modulation framework for interactive robotic systems. IEEE Transactions on Robotics, 26(2), 354–369. Lee, D., & Spong, M. W. (2006). Passive bilateral teleoperation with constant time delay. IEEE Transactions on Robotics, 22(2), 269–281. Lee, D., & Spong, M. W. (2007). Stable flocking of multiple inertial agents on balanced graphs. IEEE Transactions on Automatic Control, 52(8), 1469–1475. Liu, Y.-C. (2015). Distributed synchronization for heterogeneous robots with uncertain kinematics and dynamics under switching topologies. Journal of the Franklin Institute, 352(9), 3808–3826. Liu, Y.-C., & Chopra, N. (2013). Control of semi-autonomous teleoperation system with time delays. Automatica, 49(6), 1553–1565. Liu, Y.-C., & Khong, M.-H. (2015). Adaptive control for nonlinear teleoperators with uncertain kinematics and dynamics. IEEE/ASME Transactions on Mechatronics, 20(5), 2550–2562. Liu, Y., Min, H., Wang, S., Liu, Z., & Liao, S. (2014). Distributed adaptive consensus for multiple mechanical systems with switching topologies and time-varying delay. Systems & Control Letters, 64, 119–126. Lozano, R., Brogliato, B., Egeland, O., & Maschke, B. (2000). Dissipative systems analysis and control: Theory and applications. London, U.K.: Springer-Verlag. Munir, S., & Book, W. J. (2002). Internet-based teleoperation using wave variables with prediction. IEEE/ASME Transactions on Mechatronics, 7(2), 124–133. Münz, U., Papachristodoulou, A., & Allgöwer, F. (2011). Consensus in multi-agent systems with coupling delays and switching topology. IEEE Transactions on Automatic Control, 56(12), 2976–2982.

H. Wang / Automatica 119 (2020) 108913 Niemeyer, G., & Slotine, J.-J. E. (1991). Stable adaptive teleoperation. IEEE Journal of Oceanic Engineering, 16(1), 152–162. Niemeyer, G., & Slotine, J.-J. E. (1998). Towards force-reflecting teleoperation over the Internet. In Proceedings of the IEEE international conference on robotics and automation (pp. 1909–1915). Leuven, Belgium. Niemeyer, G., & Slotine, J.-J. E. (2004). Telemanipulation with time delays. International Journal of Robotics Research, 23(9), 873–890. Nuño, E., Aldana, C. I., & Basañez, L. (2017). Task space consensus in networks of heterogeneous and uncertain robotic systems with variable time-delays. International Journal of Adaptive Control and Signal Processing, 31(6), 917–937. Nuño, E., Arteaga-Pérez, M., & Espinosa-Pérez, G. (2018). Control of bilateral teleoperators with time delays using only position measurements. International Journal of Robust and Nonlinear Control, 28(3), 808–824. Nuño, E., Basañez, L., & Ortega, R. (2011). Passivity-based control for bilateral teleoperation: A tutorial. Automatica, 47(3), 485–495. Nuño, E., Basañez, L., Ortega, R., & Spong, M. W. (2009). Position tracking for nonlinear teleoperators with variable time delay. International Journal of Robotics Research, 28(7), 895–910. Nuño, E., Ortega, R., & Basañez, L. (2010). An adaptive controller for nonlinear teleoperators. Automatica, 46(1), 155–159. Nuño, E., Ortega, R., Basañez, L., & Hill, D. (2011). Synchronization of networks of nonidentical Euler–Lagrange systems with uncertain parameters and communication delays. IEEE Transactions on Automatic Control, 56(4), 935–941. Olfati-Saber, R., & Murray, R. M. (2004). Consensus problems in networks of agents with switching topology and time-delays. IEEE Transactions on Automatic Control, 49(9), 1520–1533. Ortega, R., & Spong, M. W. (1989). Adaptive motion control of rigid robots: A tutorial. Automatica, 25(6), 877–888. Polushin, I. G., Liu, P. X., & Lung, C.-H. (2006). A control scheme for stable forcereflecting teleoperation over IP networks. IEEE Transactions on Systems, Man, and Cybernetics–Part B: Cybernetics, 36(4), 930–939. Polushin, I. G., Takhmar, A., & Patel, R. V. (2015). Projection-based force-reflection algorithms with frequency separation for bilateral teleoperation. IEEE/ASME Transactions on Mechatronics, 20(1), 143–154. Ren, W., & Beard, R. W. (2005). Consensus seeking in multiagent systems under dynamically changing interaction topologies. IEEE Transactions on Automatic Control, 50(5), 655–661. Ren, W., & Beard, R. W. (2008). Distributed consensus in multi-vehicle cooperative control. London, U.K.: Springer-Verlag. Rezaee, H., & Abdollahi, F. (2017). Adaptive stationary consensus protocol for a class of high-order nonlinear multiagent systems with jointly connected topologies. International Journal of Robust and Nonlinear Control, 27(9), 1677–1689. Rotea, M. A. (1993). The generalized H2 control problem. Automatica, 29(2), 373–385. van der Schaft, A. (2000). L2 -gain and passivity techniques in nonlinear control (2nd ed.). London, U.K.: Springer-Verlag. Sheridan, T. B. (1993). Space teleoperation through time delay: Review and prognosis. IEEE Transactions on Robotics and Automation, 9(5), 592–606. Slotine, J.-J. E., & Li, W. (1987). On the adaptive control of robot manipulators. International Journal of Robotics Research, 6(3), 49–59.

9

Slotine, J.-J. E., & Li, W. (1989). Composite adaptive control of robot manipulators. Automatica, 25(4), 509–519. Slotine, J.-J. E., & Li, W. (1991). Applied nonlinear control. Englewood Cliffs, NJ: Prentice-Hall. Son, H. I., Franchi, A., Chuang, L. L., Kim, J., Bülthoff, H. H., & Giordano, P. R. (2013). Human-centered design and evaluation of haptic cueing for teleoperation of multiple mobile robots. IEEE Transactions on Cybernetics, 43(2), 597–609. Sontag, E. D. (1998). Comments on integral variants of ISS. Systems & Control Letters, 34(1–2), 93–100. Spong, M. W., Hutchinson, S., & Vidyasagar, M. (2006). Robot modeling and control. Hoboken, NJ: Wiley. Vidyasagar, M. (1993). Nonlinear systems analysis (2nd ed.). Englewood Cliffs, NJ: Prentice-Hall. Wang, H. (2014). Consensus of networked mechanical systems with communication delays: A unified framework. IEEE Transactions on Automatic Control, 59(6), 1571–1576. Wang, H. (2017). Adaptive control of robot manipulators with uncertain kinematics and dynamics. IEEE Transactions on Automatic Control, 62(2), 948–954. Wang, H. (2018). Towards manipulability of bilateral teleoperators with time-varying delay. In Chinese automation congress (pp. 946–951). Xi’an, China. Wang, H. (2020). Differential-cascade framework for consensus of networked Lagrangian systems. Automatica, 112, 108620. Wang, H., & Xie, Y. (2018). Task-space consensus of networked robotic systems: Separation and manipulability. arXiv preprint arXiv:1702.06265. Weiss, L. E., Sanderson, A. C., & Neuman, C. P. (1987). Dynamic sensorbased control of robots with visual feedback. IEEE Journal of Robotics and Automation, RA-3(5), 404–417. Yokokohji, Y., Imaida, T., & Yoshikawa, T. (1999). Bilateral teleoperation under time-varying communication delay. In Proceedings of the IEEE/RSJ international conference on intelligent robots and systems (pp. 1854–1859). Kyongju, South Korea. Yokokohji, Y., & Yoshikawa, T. (1994). Bilateral control of master–slave manipulators for ideal kinesthetic coupling-formulation and experiment. IEEE Transactions on Robotics and Automation, 10(5), 605–620.

Hanlei Wang received the B.S. degree in Mechanical Engineering and Automation from Shijiazhuang Railway Institute, Shijiazhuang, China, in 2004, the M.S. degree in Mechatronic Engineering from Harbin Institute of Technology, Harbin, China, in 2006, and the Ph.D. degree in Control Theory and Control Engineering from Beijing Institute of Control Engineering, China Academy of Space Technology, Beijing, China, in 2009. He joined Beijing Institute of Control Engineering as an Engineer in 2009. Since 2011, he has been a Senior Engineer in Beijing Institute of Control Engineering. From 2015 to 2016, he was a visiting scholar with the Department of Electrical and Computer Engineering, University of California, Riverside, CA. His research interests include robotics, spacecraft, networked systems, teleoperation, and nonlinear control.