This paper appears in IEEE Transactions on Control Systems Technology, 2014. http://dx.doi.org/10.1109/TCST.2014.2301840 Absolute Stability of Multi-DOF Multi-lateral Haptic Systems Jian Li, Student Member, IEEE, Mahdi Tavakoli, Member, IEEE, and Qi Huang, Senior Member, IEEE Abstract—Multi-degree-of-freedom (DOF) multi-lateral haptic systems involve teleoperation of several robots in physical environments by several human operators or collaborative interaction of several human operators in a virtual environment. An m-DOF n-lateral haptic system can be modeled as an n-port network where each port (terminal) connects to a termination defined by m inputs and m outputs. The stability analysis of such systems is not trivial due to dynamic coupling across the different DOFs of the robots, the human operators, and the physical/virtual environments, and unknown dynamics of the human operators and the environments exacerbate the problem. Llewellyn’s criterion only allows for absolute stability analysis of 1-DOF bilateral haptic systems (m = 1, n = 2), which can be modeled as two-port networks. The absolute stability of a general m-DOF bilateral haptic system where m > 1 cannot be obtained from m applications of Llewellyn’s criterion to each DOF of the bilateral system. Also, if we were to use Llewellyn’s criterion for absolute stability analysis of a general 1-DOF nlateral haptic system where n > 2, we would need to couple n − 2 terminations of the n-port network to (an infinite number of) known impedances in order to reduce it to an equivalent two-port network; this is a cumbersome process that involves an infinite number of applications of Llewellyn’s criterion. In this paper, we present a straightforward and convenient criterion for absolute stability analysis of a class of m-DOF n-lateral haptic systems for any m ≥ 1 and n ≥ 2. As case studies, a 1-DOF trilateral and a 2-DOF bilateral haptic system are studied for absolute stability with simulations and experiments confirming the theoretical stability conditions. Index Terms—Multi-port network, multi-lateral haptic system, absolute stability. I. I NTRODUCTION Multi-lateral haptic systems have recently found applications in tele-medicine , , cooperative robotics for humanrobot lunar exploration , and multi-robot systems , . An m-DOF n-lateral haptic system can be modeled as an n-port network where each port (terminal) connects to an m-DOF termination. In the special case of m = 1 and n = 2, this is a bilateral teleoperation system modeled as a two-port network. Multi-port networks are widely used in other applications such as radio-frequency and microwave circuits to analyze their absolute stability (sometimes called unconditional stability) , . For a teleoperation system consisting of a teleoperator comprised of master(s), slave(s) and controllers coupled to terminations consisting of human operator(s) and environment(s), This research was supported by the Natural Sciences and Engineering Research Council (NSERC) of Canada, by the Natural Science Foundation of China (NFSC, Grant No. 51277022), and by the China Scholarship Council (CSC) under grant 3005. Jian Li is with the School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731 China. Jian Li and Mahdi Tavakoli are with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, Alberta, T6G 2V4 Canada. Qi Huang is with the School of Energy Science and Engineering, University of Electronic Science and Technology of China, Chengdu, Sichuan, 611731 China. E-mail: [email protected], [email protected], [email protected] closed-loop stability is critical for safe and effective teleoperation. Investigation of teleoperation system stability using common closed-loop stability analysis tools in the control systems literature is not possible because the models of the human operator(s) and the environment(s) are usually unknown, uncertain, and/or time-varying. However, research has shown that it is still possible to draw stability conditions for a haptic teleoperation system under unknown “terminations” as long as they are passive. These stability conditions can be categorized as passivity and absolute stability criteria. For stability analysis of 1-DOF n-lateral haptic systems, passivity is used in ,  for n = 2, in ,  for n = 3, and in  for any n ≥ 2. Specifically, in , Raisbeck’s method is useful as a passivity criterion for 1DOF bilateral teleoperation systems based on the immittance matrix of the teleoperator. Shahbazi et al. in  performed stability analysis for a dual-user (trilateral) teleoperation system based on the passivity definition for a three-port network. In , Panzirsch et al. proposed a time-domain passivity observer/passivity controller approach for a dual-user (trilateral) teleoperation system. In , Mendez and Tavakoli presented a criterion (necessary and sufficient) for passivity of general n-port networks, which can model 1-DOF n-lateral haptic systems. Passivity of a multi-port network is a conservative condition for its coupled stability. A less conservative condition, absolute stability is discussed in ,  for n = 2, in ,  for n = 3, and in  for any n ≥ 2. Specifically, Llewellyn in  proposed an absolute stability criterion for two-port networks, which model 1-DOF bilateral teleoperation systems, based on the immittance matrix of network (the teleoperator). In , Khademian et al. analyzed absolute stability of a dualuser (trilateral) teleoperation system by reducing the three-port network to an equivalent two-port network, paving the way for the applications of Llewellyn’s criterion. Li et al. in  presented an absolute stability criterion for a class of trilateral haptic systems. In , Ku studied n-port network stability if the impedance matrix of the n-port network conforms to the tri-diagonal Jacobian form. The above research only addresses the absolute stability analysis of 1-DOF haptic teleoperators. In past research, for stability analysis of multi-DOF bilateral  and tri-lateral haptic systems , the multi-DOF systems are decoupled to 1-DOF systems. Then, various stability criteria for 1-DOF n-lateral haptic systems are used. This poses difficulties in terms of decoupling a coupled haptic system especially because the human operator(s) and the environment(s) terminations are also coupled themselves. In this paper, we present a criterion to analyze the absolute stability of multi-DOF multi-lateral haptic systems directly and without a need for decoupling. As a case study, we consider a 2-DOF bilateral haptic system and use the proposed absolute stability criterion to design stabilizing controllers for the system. The rest of the paper is organized as follows: The next 2 section gives mathematical definitions and lemmas for analysis of absolute stability. Section III introduces simple motivating examples to show that the absolute stability criteria for 1-DOF bilateral haptic systems fail to analyze the absolute stability of m-DOF bilateral haptic systems. Next, in Section IV, the proposed absolute stability criterion for m-DOF n-lateral networks is derived. Then, as a case study to show how the resulting absolute stability criterion can be utilized, in Section V, a 1-DOF trilateral haptic system is considered, the absolute stability conditions in terms of system parameters including controller gains are found, and simulations to verify the validity of the calculated absolute stability conditions are conducted. In another case study, in Section VI, a 2-DOF bilateral teleoperation system with position-position control is considered, the absolute stability conditions in terms of system parameters including controller gains are found, and experiments to verify the validity of the calculated absolute stability conditions are presented. Section VII contains concluding remarks. II. M ATHEMATICAL P RELIMINARIES Notation 1. a is a scalar, A is a vector, A is a matrix, and A is a block matrix (i.e., with matrix elements). Definition 1.  A multi-port network is passive if the total energy delivered to the network at its ports is non-negative. Property 1.  A gyration operator, which transforms one immittance matrix to another, preserves the passivity property. Definition 2.  A n × n proper rational transfer matrix G(s) is positive real if i) Poles of all elements of G(s) are in Re[s] ≤ 0, ii) Any pure imaginary pole jω of any element of G(s) is a simple pole and the residue matrix lims→jω (s − jω)G(s) is positive semidefinite Hermitian, iii) For all real ω for which jω is not a pole of any element of G(s), the matrix G(jω)+GT (−jω) is positive semidefinite. Property 2.  The Hermitian part of a symmetric matrix is a real matrix. A real matrix is positive semidefinite if its principal minors are all nonnegative. Lemma 1.  A linear time-invariant minimal realization model with transfer matrix G(s) is passive if G(s) is positive real. Definition 3.  A multi-port network is absolutely stable if the coupled system remains bounded-input bounded-output stable under all possible passive terminations. Otherwise, it is potentially unstable. Lemma 2.  Let Z = ZT be the impedance matrix of a reciprocal n-port network. Then, the network is passive if and only if it is absolutely stable. Lemma 3.  Let Z1 and Z2 be the impedance matrices of two n-port networks. Then, if Z1 and Z2 possess identical principal minors of all orders, then Z1 is absolutely stable if and only if Z2 is absolutely stable. III. M OTIVATION Llewellyn’s criterion has been used to analyze the absolute stability of 1-DOF bilateral teleoperation systems. In the following, using two examples, we show why it cannot be used for coupled m-DOF (m > 1) bilateral teleoperation systems. For absolute stability, both terminations of the twoport network need to be passive. For a coupled 2-DOF bilateral teleoperation system, consider the following termination for its first port: 5 8 − s+1 s+3 T1 = (1) 5 1 − s+1 s+3 According to Definition 2 and Property 2, we find that al1 8 and s+3 along each of the first though the terminations s+3 two DOFs are passive, the coupled 2-DOF termination T1 is non-passive. Therefore, viewing the termination impedances along each of the DOFs separately can result in misleading results in terms of absolute stability. As another case, consider a coupled 2-DOF bilateral teleoperator modeled as Fh Vh =Z (2) Fe Ve where Fh = [fhx , fhy ]T , Fe = [fex , fey ]T , Ve = [vex , vey ]T , and Z11 Z12 Z = Z21 Z22 5 1 9 − s+1 s+3 − s+5 − 5 9 1 − s+3 s+5 s+3 = 1 1 1 − s+1 − s+3 s+3 1 9 1 − s+1 − s+1 − s+2 Vh = [vhx , vhy ]T , 1 − s+2 1 − s+1 9 − s+1 1 (3) s+3 Assume the terminations of this teleoperator are always passive. For using Llewellyn’s criterion once along the x direction and once along the y direction, we have to consider the following two subsystems of (2): vhy vhx fhy fhx (4) , = Zy = Zx vey vex fey fex where Zx = Zy = 9 s+3 1 − s+1 1 − s+1 1 s+3 (5) While the subsystems involving Zx and Zy always satisfy Llewellyn’s criterion (see Appendix for Llewellyn’s criterion), as shown next, the coupled 2-DOF teleoperator (2) is not absolutely stable. In general, for checking the absolute stability of a two-port network such as a bilateral teleoperator, the port #2 (environment port) can be connected to passive terminations while the input energy at the port #1 (operator port) is measured. The bilateral teleoperator is absolutely stable if and only if, at all times t > 0, we have : Z t Es (t) = FhT (τ )Vh (τ ) dτ ≥ 0. (6) 0 Similarly, the subsystems involving Zx is absolutely stable if and only if, at all times t > 0, we have Z t fhx (τ )vhx (τ ) dτ ≥ 0. (7) Es (t) = 0 As shown in Figure 1, which plots Es (t) for the teleoperator (2) (solid) and each of the two subsystems (4) (dash-dot), 3 Energy 10 where Zij , i, j = 1, 2, · · · , n, are m × m matrices given in (12). On the other hand, the n pairs of m-dimensional terminations are represented by 0 −10 −20 T = diag[T1 , T2 , · · · , Tn ] Absolutely stable Potentially unstable 0 5 10 Time (s) 15 20 Figure 1. Simulation results for analysis of absolute stability of the 2DOF bilateral teleoperator (3). While the subsystems in (4) always satisfy Llewellyn’s criterion as evidenced by the nonnegative energy plot (dashdot), the coupled 2-DOF teleoperator in (3) is actually potentially unstable as evidenced by the negative energy plot (solid). T1 V1 Vn + + F1 Fn - … … Z Vi+1 Vi + Ti Tn - + Fi Ti+1 Fi+1 - - Figure 2. An n-port network where each port (terminal) connects to an mDOF termination. simulations confirmed that each of the two 1-DOF subsystems are absolutely stable while the 2-DOF teleoperator is not absolutely stable (i.e., is potentially unstable). From the above two examples, it is clear that for a 2-DOF haptic system, using Llewellyn’s criterion twice in each DOF is not useful as it ignores the coupling that may exist in the terminations (i.e., human operators and environments) and the teleoperator. While we showed that 1-DOF n-lateral teleoperator absolute stability analysis methods do not work for m-DOF n-lateral teleoperators for the special case of n = 2, the same holds for any n > 2. To the best of our knowledge, no work has been done on direct absolute stability analysis of m-DOF (m ≥ 2) n-lateral coupled teleoperators. Motivated by these facts, we propose a new absolute stability criterion of m-DOF n-lateral haptic systems. IV. M AIN R ESULT: A N A BSOLUTE S TABILITY C RITERION FOR M ULTI -DOF M ULTI - LATERAL H APTIC S YSTEMS An m-DOF n-lateral teleoperation system can be modeled as an n-port network where each port (terminal) connects to an m-DOF termination as shown in Figure 2. The network impedance model will be F = ZV (8) where F = [ F1 , F2 , · · · , Fn ] V = [ V1 , V2 , · · · , Vn ] T (9) T (10) and Fi and Vi , i = 1, 2, · · · , n, represent the m × 1 vectors of force and velocity at the ith port of the network, respectively. The impedance matrix of the network will be Z11 Z12 · · · Z1n Z21 Z22 · · · Z2n Z= (11) .. .. ... . ··· . Zn1 Zn2 ··· Znn n×n (14) where Ti , i = 1, 2, · · · , n, represents the m × m impedance matrix of the ith m-dimensional termination. Theorem 1. An m-DOF n-lateral haptic system with impedance matrix Z in (11) satisfying the symmetrization conditions A) zi,j zj,k zk,i = zj,i zk,j zi,k , where i, j, k = 1, 2, · · · , m × n, i 6= j 6= k, and i 6= k. B) Zℓℓ is symmetric, where ℓ = 1, 2, · · · , n. is absolutely stable if and only if C) The elements of Z matrix in (12) have no poles in the right-half plane (RHP). D) Any poles of the elements of the Z ′ matrix in (13) on the imaginary axis are simple, and the principal minors of the residues matrix of the Z ′ matrix at these poles are greater than or equal to zero. E) For all real values of frequencies ω, the principal minors of the real part of the Z ′ matrix in (13) are greater than or equal to zero, or equivalently Re(zi,i ) ≥ 0, i = 1, 2, · · · n × m |z1,2 z2,1 | + Re(z1,2 z2,1 ) ≥0 Re(z1,1 )Re(z2,2 ) − 2 .. . (15b) det(Re(Z ′ )) ≥ 0 (15c) (15a) Proof. Consider a linear time-invariant system with impulse response h(t). The system’s transfer function is the Laplace transform of h(t) defined as Z ∞ H(s) = h(t)e−st dt (16) 0 where s = σ + jω. H(s) is stable if every bounded input produces a bounded output and this happens if the poles of H(s) have negative real parts. This stability definition is equivalent to the absolute convergence (defined below) of H(s) in the region Re(s) ≥ 0. If h is locally integrable, Rthen H(s) is r said to converge if the limit H(s) = limr→∞ 0 h(t)e−st dt Also, H(s) is said to converge absolutely if the integral Rexists. ∞ −st |h(t)e |dt exists. The set of values of s for which H(s) 0 converges is known as the region of convergence (ROC) and is of the form Re(s) ≥ a, where a is a real constant. Importantly, if H(s) converges at s = s0 , then it automatically converges for all s with Re(s) > Re(s0 ). The above means that for stability analysis it suffices to focus on the convergence of H(s) when Re(s) = 0, i.e., on the jω axis. This is sometimes referred to as real-frequency stability. Thus, as a linear timeinvariant system, the stability of an m-DOF n-lateral haptic system coupled to an m-DOF termination at each of its ports needs to only be analyzed for s = jω. An n-port network is stable if the port currents I1 , I2 , · · · , In are zero under all passive terminations z1 , z2 , · · · , zn for ports . In other words, an n-port network with an impedance matrix Zn×n is stable if and only if the equation (Z + Z0 )I = 0, where I = [I1 , I2 , · · · , In ]T and Z0 = diag[z1 , z2 , · · · , zn ] has only the trivial solution I = 0 4 z(i−1)m+1,(j−1)m+1 z(i−1)m+2,(j−1)m+1 Zij = .. . zim,(j−1)m+1 z(i−1)m+1,(j−1)m+2 z(i−1)m+2,(j−1)m+2 .. . zim,(j−1)m+2 √ γ1 z1,2 z2,1 z2,2 .. . √ γ2m×n−3 z2,m×n zm×n,2 (m × n)(m × n − 1) i = 1, 2, · · · , 2 z √ 1,1 γ1 z1,2 z2,1 Z′ = .. . √ γm×n−1 z1,m×n zm×n,1 where, γi = ±1, for every passive choice of Z0 ; this happens if and only if det(Z + Z0 ) 6= 0. On the other hand, according to , if two n × n matrices Z1 and Z2 have identical principal minors of all orders, then det(Z1 + Z0 ) = det(Z2 + Z0 ) (17) for any Z0 = diag[z1 , z2 , · · · , zn ]. This implies that the stability of two n-port networks with impedance matrices Z1 and Z2 will happen at the same time (Lemma 3). Now, if there exists a reciprocal n-port network with impedance matrix Z ′ that has the same stability characteristics as the original nonreciprocal n-port network with impedance matrix Z, then det(Z ′ + T ) = det(Z + T ) (18) for any passive T in (14). The above is to hold for any passive T . By induction, it is easy to show that calculating the two determinants and equating the coefficients of T1 , T2 , · · · , Tn gives the matrix Z ′ in (13) as well as the symmetrization conditions A and B. On the other hand, according to Lemma 2, the reciprocal n-port network with impedance matrix Z ′ is absolutely stable if and only if it is passive. In turn, according to Lemma 1, Z ′ is passive if and only if it is positive real, which can be verified through Definition 2. From the above, we conclude that the original nonreciprocal n-port network with impedance matrix Z is absolutely stable if and only if the equivalent reciprocal n-port network’s impedance matrix Z ′ is positive real. Obviously, the Hermitian part of Z’ is a real symmetric matrix. In this context, it is straightforward to show that Conditions C and D in Theorem 1 are the same as Conditions i) and ii) in Definition 2. Also, according to Condition iii) of Definition 2, the Hermitian part T Z ′ (jω) + Z ′ (−jω) = 2Re(Z ′ (jω)) (19) needs to be positive semidefinite for the n-port network with impedance matrix Z to be absolutely stable. Using Property 2 , and simplifying the conditions by r |zi,j zj,i | + Re(zi,j zj,i ) √ (20) (Re( zi,j zj,i )) = 2 where i, j = 1, 2, · · · , m × n, we arrive at conditions (15a)(15c). This concludes the proof. Remark 1. Theorem 1 holds not only for the impedance matrix of a general network but also for its other immittance ··· ··· ··· ··· ··· ··· ··· ··· z(i−1)m+1,jm z(i−1)m+2,jm .. . zim,jm m×m √ γm×n−1 z1,m×n zm×n,1 √ γ2m×n−3 z2,m×n zm×n,2 .. . zm×n,m×n (12) (13) (admittance, hybrid, and transmission) matrices. The reason for this is that according to Property 1, a gyration operators transform one immittance matrix to another and preserves passivity. Remark 2. When m = 1 and n = 2, Theorem 1 is the same as Llewellyn’s absolute stability criterion. Also, for the case m = 1 and n ≥ 2, the matrix Zij will reduce to a scalar, which is always symmetric, meaning that Condition B of Theorem 1 is always satisfied. V. C ASE S TUDY 1: A BSOLUTE S TABILITY OF A 1-DOF T RILATERAL H APTIC S YSTEM In this section, the aim is to apply the proposed absolute stability criterion for a 1-DOF trilateral haptic system. In the following, we begin by reviewing a dual-user teleoperation system in position-position control structure. A. A 1-DOF dual-user teleoperation system In a linear time-invariant (LTI) 1-DOF dual-user teleoperation control system, the goal is that two users collaboratively control a robot to perform a desired task on a remote environment. Such a system consists of two master robots for the two users and one slave robot that is in contact with the environment. As elaborated by , a Masters Correspondence with Environment Transfer (MCET) dual-user teleoperation system creates for training purposes a correspondence between the masters positions while transferring the environment dynamics to the two users. As seen in Figure 3, this is done via a weight parameter α ∈ [0, 1] that specifies the relative authority of each operator over the slave robot’s reference position. In such a system, the dynamics of the two masters and the slave in contact with the two users and the environment are zmi vhi = fhi + fcmi zs ve = fe + fcs (21a) (21b) where zmi (i = 1, 2) and zs are the impedances of the two masters and the slave, respectively. Also, fhi denotes the interaction force between each user and its corresponding master and fe denotes the interaction force between the slave and the environment. Lastly, vhi , and ve are the users and the environment velocities and fcmi and fcs are the control signals for the two masters and the slave, respectively. Since in (21) the impedances relate force to velocity (and not position), modeling each robot by a mass-spring-damper 2 2 s s+ks . For results in zmi = mmi s +bsmi s+kmi and zs = ms s +b s 5 Table I T HE CONTROLLERS GAINS OF THE POSITION - POSITION DUAL - USER TELEOPERATION SYSTEM USED IN SIMULATIONS . Environment fe 2 1(fh2, vh2) (fh1, vh1) (A) fe 2 vh1 User 1 User 2 Figure 3. Masters Correspondence with Environment Transfer (MCET) architecture dual-user haptic teleoperation. this dual-user teleoperation system, the four-channel control laws can be written as : fcmi = −cmi vhi − c4mi vhid + c6mi fhi − c2mi fhid fcs = −cs ve + c1 ved − c5 fe + c3 fed (22a) (22b) where cmi and cs are local position controllers, c6mi and c5 are local force controllers, and c1 , c2mi , c3 , and c4mi are feedforward and feedback compensators. Also, vhid and ved are reference velocities and fhid and fed are references forces for the two masters and the slave selected according to vh2d = vh1 ved = αvh1 + (1 − α)vh2 , fh2d = 1 fe , 2 fh1d = 1 fe 2 fed = αfh1 + (1 − α)fh2 (23) For simplicity, let us consider the position-position control laws as a special case of the above four-channel control. We will have c1 = cs , c4m1 = −cm1 , and c4m2 = −cm2 . Also, c3 , c5 , c2mi , and c6mi are zero. Also, let us model each robot by a mass only. Again, since in (21) the impedances relate force to velocity, normally PD position controllers show up as PI velocity controllers: kpm1 + kvm1 s , s kps + kvs s cs = s cm1 = cm2 = kpm2 + kvm2 s , s fh1 fh2 fe # = " cm1 + zm1 c4m2 −αc1 Slave kps 150 kvs 85 c4m1 cm2 + zm2 −(1 − α)c1 k11 = kpm1 ≥ 0 k22 = kpm2 ≥ 0 k33 = kps ≥ 0 k11 k22 − k12 k21 = 0 k11 k33 − k13 k31 = 0 k22 k33 − k23 k32 = kpm1 kps ≥ 0 k11 k22 k33 − k11 k23 k32 − k22 k13 k33 − k33 k12 k21 + k13 k21 k32 + k12 k23 k31 = 0 (26) (27) (28) (29) (30) (31) (32) Conditions (26)-(32) are satisfied if we choose the proportional control gains to be nonnegative, thus, Conditions C and D of Theorem 1 are readily fulfilled. With s = jω, it is possible to see that the conditions (15a)-(15c) become kvm1 ≥ 0 kvm2 ≥ 0 kvs ≥ 0 − (kvm1 kpm2 − kpm1 kvm2 )2 ≥ 0 kvm1 kvs ≥ 0 kvm2 kvs ≥ 0 − kvs (kvm1 kpm2 − kpm1 kvm2 )2 ≥ 0 (33) (34) (35) (36) (37) (38) (39) Clearly, condition (36) and (39) will be fulfilled for all frequencies ω if we choose the derivative control gains to be nonnegative and kpm1 kpm2 = kvm1 kvm2 (40) So, a necessary and sufficient, frequency-independent, and compact condition for absolute stability of the above-described position-position dual-user teleoperation systems is given by (40), where all control gains are nonnegative. Note that the ratios in (40) are merely artifacts of our presentation of the absolute stability conditions meaning that division by zero can be avoided. (24) To get the impedance matrix of position-position control dualuser teleoperation system, first substitute (23) in (22) and then substitute the result in (21) to get " Master #2 kpm2 120 kvm2 48 Analysis of the residues leads to vh2 vh1d = vh2 , Master #1 kpm1 160 kvm1 64 or 80 0 0 cs + z s #" # vh1 vh2 ve (25) Now, let us perform the stability analysis via Theorem 1, where m = 1 and n = 3. Evidently, for the impedance matrix (25), the symmetrization condition A and B of Theorem 1 holds for any α because z13 z21 z32 − z12 z23 z31 is identical to zero. All poles from elements of (25) are equal to zero. B. Simulations The position-position dual-user teleoperation system has been simulated in MATLAB/Simulink. There is no time delay in the communication channel between the masters and the slave. Three 1-DOF robots as the two masters and the slave are modeled by masses mm1 = 0.7, mm2 = 0.5, and ms = 1.5, respectively. According to (40), the stability of the positionposition dual-user teleoperation system should depend on the controllers gains. In the simulations, the controllers gains kpm1 , kvm1 , kpm2 , kvm2 , kps , and kvs were chosen according to Table I. Evidently, (40) is satisfied for kvm1 = 64 and violated for kvm1 = 80. Also, α ∈ [0, 1]. In simulations, to check the absolute stability of the threeport network, the master #2 and the slave ports are connected 1 , which are to LTI terminations with transfer functions s+1 6 2 2 Absolutely stable Potentially unstable Energy 0.2 0.1 0 −0.1 −0.2 0 20 40 60 80 100 Time (s) Angular position (rad) Figure 4. Simulation results for the dual-user teleoperation system. Input energy at the master #1’s port is shown. Simulation parameters are listed in Table I for the absolute stability case with kvm1 = 64, and for the potentially unstable case with kvm1 = 80. 0.2 0.15 0.1 where i = m, s correspond to the master and the slave, respectively. The subscripts are chosen to correspond to the y − z plane while any other plane can be chosen. Also, Fh = [fhy , fhz ]T denotes the interaction force vector between the user and the master and Fe = [fey , fez ]T denotes the interaction force vector between the slave and the environment. Lastly, Vh = [vhy , vhz ]T and Ve = [vey , vez ]T are the user and the environment velocities. Similar to the previous case study, let us consider positionposition control laws : Masters Slave 0.05 0 s s+Ks , where Zm = Mm s +Bs m s+Km , and Zs = Ms s +B s are 2 × 2 impedance matrices of the master and the slave, respectively. Assume miyy miyz biyy biyz Mi = , Bi = , miyz mizz biyz bizz kiyy kiyz Ki = kiyz kizz 0 2 4 6 8 10 Time (s) Figure 5. Simulation results for the dual-user teleoperation system. The desired and actual positions for the slave are shown. A sinusoidal force was applied to the master #1 while the master #2 and the slave were connected to passive terminations. Simulation parameters are listed in Table I for the absolutely stable case of kvm1 = 64. 1 passive because, for s = jω, we have Re( s+1 ) = ω21+1 > 0. Port 1 is open and a sine-wave input fh1 is applied to the master #1. The input energy Es (t) in (6) is plotted in Figure 4 for kvm1 = 64 and kvm1 = 80 while the rest of the control gains are listed in Table I. As it can be seen, if the control gains are selected according to (40), e.g., as listed in Table I with kvm1 = 64, then the input energy at port 1 is non-negative at all times, indicating the absolute stability of the trilateral haptic system. However, when we change kvm1 to 80, which violates (40), the input energy Es (t) will become negative at least for a period of time, indicating potential instability of the trilateral haptic system. For the case of kvm1 = 64, Figure 5 depicts the linear combination of the two masters positions based on authority factor α (i.e., the desired position for the slave) versus the slave position. The above show that there is agreement between the theoretical absolute stability condition (40) and the simulations. VI. C ASE S TUDY 2: A BSOLUTE S TABILITY B ILATERAL H APTIC S YSTEM OF A 2-DOF In this section, the aim is to apply the proposed absolute stability criterion to a coupled 2-DOF bilateral haptic teleoperation system. Then, experiments will be conducted for verifying the theoretical absolute stability conditions. (42a) (42b) where the normally PD position controllers show up as PI velocity controllers: # " kpmyy +kvmyy s s kpmzy +kvmzy s s Cm = Cs = " kpsyy +kvsyy s s kpszy +kvszy s s kpmyz +kvmyz s s kpmzz +kvmzz s s (43a) # (43b) kpsyz +kvsyz s s kpszz +kvszz s s By substituting (42) in (41), the impedance matrix representation of the 2-DOF teleoperator is found as Fh Cm + Zm −Cm Vh = (44) Fe −Cs Cs + Zs Ve Now, let us investigate the absolute stability of the teleoperator via Theorem 1 for the case of m = 2 and n = 2. With s = jω, the symmetrization conditions of A and B boils down to the following four conditions involving the control gains and the frequency ω: j(kpmzy − kpmyz ) =0 ω j(kpszy − kpsyz ) kvsyz − kvszy + =0 ω ω 2 (kvmyz kvsyy − kvsyz kvmyy ) + jω(kvsyz kpmyy + kpsyz kvmyy − kvmyz kpsyy − kpmyz kvsyy ) + kpsyz kpmyy − kpmyz kpsyy = 0 ω 2 (kvmzz kvsyz − kvszz kvmyz ) + jω(kvszz kpmyz + kpszz kvmyz − kvmzz kpsyz − kpmzz kvsyz ) + kpszz kpmyz − kpmzz kpsyz = 0 kvmyz − kvmzy + (45a) (45b) (45c) (45d) Conditions (45) will be fulfilled for all frequencies ω if the gains of the PD controllers (43) satisfy A. A 2-DOF bilateral teleoperation system In a 2-DOF LTI bilateral teleoperation system, the dynamics of the master and the slave in contact with the user and the environment, respectively, are Zm Vh = Fh + Fcm Zs Ve = Fe + Fcs Fcm = −Cm Vh + Cm Ve Fcs = −Cs Ve + Cs Vh (41a) (41b) kpmyz = kpmzy , kvmyz = kvmzy (46a) kpsyz = kpszy , kvsyz = kvszy (46b) kvmyz kpmyy kpmyz kvmzz kpmzz kvmyy = = = = = kvsyy kvsyz kpsyy kpsyz kvszz kpszz (46c) 7 It is easy to see that, under (46), all the elements of the impedance matrix (44) have only a simple pole on the imaginary axis, thus satisfying Condition C. Define kij , i, j = 1, 2, 3 as the elements of residues matrix K, analysis of the residues according to Condition D leads to the following constraints: kmyy + kpmyy ≥ 0 kmzz + kpmzz ≥ 0 ksyy + kpsyy ≥ 0 kszz + kpszz ≥ 0 Qm + Qpm + Wmpm ≥ 0 Qm (ksyy + kpsyy ) + Qpm (ksyy + kmyy ) + ksyy Wmpm ≥ 0 Qm (kszz + kpszz ) + Qpm (kszz + kmzz ) + kszz Wmpm ≥ 0 Qs (Qm + Qpm + Wmpm ) + Qm (Qps + Wsps ) kpmyy Qps Wms ≥ 0 + kpsyy (47a) (47b) (47c) (47d) (47e) i = m, s, pm, ps (47g) (47h) (48) Now, let us deal with Condition E of Theorem 1. Condition (15a) turns out to state bmyy + kvmyy ≥ 0 bmzz + kvmzz ≥ 0 bsyy + kvsyy ≥ 0 bszz + kvszz ≥ 0 (49a) (49b) (49c) (49d) Under (46) and (48), the second principal minor condition, i.e., (15b), gives Um + Qvm + bmyy kvmzz + bmzz kvmyy ≥ 0 Us + Qvs + bsyy kvszz + bszz kvsyy ≥ 0 (50) (51) where Um = bmyy bmzz − b2myz , Us = bsyy bszz − b2syz , 2 2 Qvs = kvsyy kvszz − kvsyz , and Qvm = kvmyy kvmzz − kvmyz . Similarly, the third principal minor condition requires bmyy kvmyy Qvm + bsyy Wmvm ≥ 0 kvsyy (52) bsyy kvsyy Qvs + bmyy ) + + bmyy Wsvs ≥ 0 (53) kvmyy Um (kvsyy + bsyy ) + Us (kvmyy where Wmvm = kmyy kvmzz + kmzz kvmyy − 2kmyz kvmyz and Wsvs = ksyy kvszz + kszz kvsyy − 2ksyz kvsyz . Finally, the fourth principal minor condition, i.e., (15c), mandates Us (Um + Qs 2 kvmyy kvmyy )≥0 ) + Qvs (Um + 2 kvsyy kvsyy (54) All in all, one can see that conditions (49)-(54) will be fulfilled for all frequencies ω if bmyy bmzz ≥ b2myz , bsyy bszz ≥ b2syz , 2 kvmyy kvmzz ≥ kvmyz kvsyy kvszz ≥ 2 kvsyz Master Slave (47f) where Wmpm = kmyy kpmzz + kmzz kpmyy − 2kmyz kpmyz , Wsps = ksyy kpszz + kszz kpsyy − 2ksyz kpsyz , Wms = kmyy kszz + kmzz ksyy − 2kmyz ksyz . Also, Qi = kiyy kizz − 2 kiyz , where i = m, s, pm, ps. It is easy to see that, the condition set (47) holds if 2 kiyy kizz ≥ kiyz , Table II T HE CONTROLLERS GAINS OF THE 2-DOF BILATERAL TELEOPERATION SYSTEM USED IN EXPERIMENTS . (A1) AND (A2) A BSOLUTELY STABLE , (B) P OTENTIALLY UNSTABLE . (55a) (55b) kpmyy kvmyy kpmzz kvmzz kpmyz kvmyz kpsyy kvsyy kpszz kvszz kpsyz kvsyz A1 500 300 500 300 450 250 500 300 500 300 450 250 A2 1000 800 800 600 50 25 1000 800 800 600 50 25 B 500 300 500 250 550 300 500 300 500 250 550 300 So, a sufficient, frequency-independent, and compact condition set for absolute stability of the above-described 2-DOF bilateral teleoperator is bmyy bmzz ≥ b2myz , 2 kiyz , bsyy bszz ≥ b2syz (56a) kiyy kizz ≥ i = m, s, pm, ps, vm, vs (56b) kpmyz = kpmzy , kvmyz = kvmzy (56c) kpsyz = kpszy , kvsyz = kvszy (56d) kvmyy kvmyz kpmyy kpmyz kvmzz kpmzz = = = = = kvsyy kvsyz kpsyy kpsyz kvszz kpszz (56e) where all control gains are nonnegative. Again, the ratios in (56) are merely artifacts of our presentation of the absolute stability conditions meaning that division by zero can be avoided. An alternative to the above stability analysis is to use the extended Z-W criterion . However, in that approach, the stability conditions often need to be evaluated numerically rather than being in closed form. B. Experiments We use a 2-DOF bilateral teleoperation system comprising two 3-joint Phantom Premium 1.5A robots (Sensable Technologies/Geomagic, Wilmington, MA) as the master and as the slave. Out of the three joints of each robot, the second (y) and the third (z) joints, which form a vertical plane, are considered. The first joint (x), which corresponds to rotation of the vertical plane about an axis, is locked using highgain control. The experimental setup is shown in Figure 6, where a human user interacts with the master while the slave is physically connected via a 2D passive spring array to a stiff wall. The stiffness of the springs were chosen such that sufficient displacements result as the robot end-effector applies forces. Given the limited maximum continuous output force of 1.4N of the Phantom Premium the stiffness was selected to be 45 N/m. Even though we will only implement positionposition teleoperation control, the master is equipped with a JR3 6-DOF force/torque sensor (product model: 50M31, JR3, Inc., Woodland, CA) for measuring the applied forces such that Es (t) in (6) can be quantified. With knowledge of Phantom 1.5A robot dynamics from , gravity compensation for each robot arm is performed by calculating the gravity vector at each point within the workspace and feeding it forward. All data logging and robot control actions occurred with a 1 kHz sampling frequency. According to the condition set (56), the absolute stability of the 2-DOF bilateral teleoperator should depend on the control 8 Position (mm) 50 Joint 2 (y direction) 0 −25 −50 0 10 Position (mm) 50 0 10 20 30 Time (s) Figure 7. Experimental results for the 2-DOF bilateral teleoperation system. Input energy at the master’s port is shown while the slave is physically connected via a 2D passive spring array to a stiff wall. The control gains are listed in Table II(A1) and (A2) for the absolutely stable case and in Table II(B) for the potentially unstable case. gains. In the experiments, the control gains were chosen according to either case A1, A2 or case B listed in Table II. For these cases, the input energy Es (t) in (6) is plotted in Figure 7. As it can be seen, if the control gains are selected according to (56), i.e., as listed in Table II(A1) and (A2), then the input energy Es (t) in (6) at the master’s port are non-negative at all times, indicating the absolute stability of the bilateral teleoperator. However, when the control gains are selected as Table II(B), which violates (56), the input energy Es (t) in (6) will become negative at least for a period of time, indicating the potential instability of the bilateral teleoperator. Figure 8 and 9 depicts the master position versus the slave position for each of the two joints for the parameters listed in Table II(A1) and (A2). The above show that there is agreement between the theoretical absolute stability condition (56) and the experiments. Note that as long as the control gains satisfy (56), the system will be stable. However, the exact values of such stabilizing control gains influence position tracking performance shown in Figure 8 and 9. Also, the force tracking performance is shown in Figure 10, where the master-side forces are measured by a JR3 force sensor, while the slaveside forces are calculated through multiplying the stiffness of the springs by the slave robot’s displacement. Joint 3 (z direction) 30 Master Slave 0 0 10 20 30 Figure 8. Experimental results for the 2-DOF bilateral teleoperation system. The master and slave positions in terms of their second and the third joints are shown when using the control gains listed in Table II(A1), which amount to absolute stability of the bilateral teleoperator. 20 Position (mm) −50 20 Time (s) Joint 2 (y direction) Master Slave 10 0 −10 −20 0 10 Time (s) 20 30 20 30 20 Position (mm) Energy 0 Time (s) 25 −25 Figure 6. Experimental setup where the master is controlled by a human user and the slave is physically connected via a 2D passive spring array to a stiff wall. 150 Absolutely stable (A1) 100 Absolutely stable (A2) Potentially unstable (B) 50 Master Slave 25 Master Slave Joint 3 (z direction) 10 0 −10 0 10 Time (s) Figure 9. Experimental results for the 2-DOF bilateral teleoperation system. The master and slave positions in terms of their second and the third joints are shown when using the control gains listed in Table II(A2), which amount to absolute stability of the bilateral teleoperator. VII. C ONCLUSIONS A ND F UTURE W ORKS haptic system by terminating some of its terminals such that Llewellyn’s absolute stability criterion can be used is cumbersome. This paper presented a closed-form and easyto-use absolute stability criterion for multi-DOF multi-lateral haptic systems. Through two case studies, we elaborated on its application in absolute stability analysis of a 1-DOF trilateral and a 2-DOF bilateral haptic system. Through simulations and experiments, the proposed absolute stability criterion was validated. In the future, a possible investigation is the stability-transparency trade-offs for haptic systems in light of the proposed stability criterion. Also, investigating the stability of multi-dof multi-lateral haptic systems in which multiple master robots control a higher-DOF slave robot for performing a dexterous task through collaboration of several human operators is an interesting direction for future research. In the beginning of this paper, it was shown via an example that applying Llewellyn’s absolute stability criterion once in each DOF of a multi-DOF bilateral haptic system cannot guarantee the absolute stability as this method ignores the coupling between DOFs that may exist in the system. Also, reducing a multi-lateral haptic system to a bilateral VIII. A PPENDIX Llewellyn’s criterion: If pmn = rmn + jxmn , m, n = 1, 2, represents any of the four immittance parameters (z, y, h, and g) of a two-port network, for all real values of frequencies ω, the network is absolutely stable if and only if 9 Force (N) 2 Joint 2 (y direction) 1 0 Master Slave −1 −2 0 10 20 30 Time (s) Force (N) 2 Joint 3 (z direction) 0 −2 Master Slave −4 0 10 20 30 Time (s) Figure 10. Experimental results for the 2-DOF bilateral teleoperation system. 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