An Adaptive Full-State Feedback Controller for Bilateral Telerobotic Systems

Transcription

1 21 American Control Conference Marriott Waterfront Baltimore MD USA June 3-July 2 21 FrB16.3 An Adaptive Full-State Feedback Controller for Bilateral Telerobotic Systems Ufuk Ozbay Erkan Zergeroglu and Beytullah Okur Abstract In this paper we propose a new adaptive controller scheme for the bilateral telerobotic/teleoperation system. The proposed controller achieves asymptotic tracking despite the parametric uncertainties of both the master and slave robots while ensuring the passivity of the closed loop system. Extensive simulation studies are presented to illustrate the feasibility and efficiency of the proposed adaptive controller. I. INTRODUCTION A telerobotic system is basically a dual robot system in which a remote slave robot tracks the motion of a master robot that is commented by a human operator. A typical telerobotic system is composed of a human operator maneuvering a master robot and a slave robot working on a remote and/or hard to reach environment. Motivated by the large variety of applications ranging from space explorations to mining medical applications to nuclear waste contamination telerobotics have been studied extensively. For a complete overview of the past research on various aspects of telerobotics readers are referred to 1 and 2. Safe interaction and coupling stability of the teleoperation can be established by the passivity of the closed loop system Therefore ensuring energetic passivity (passivity with the mechanical power as the supply rate) of the closed loop telerobotic system as in 6 and 7 would be a way to ensure safe operation for both the human user and the environment. Another issue for telerobotics is the nonlinear dynamics of the system. Though there has been some successful control schemes for the linear telerobotic systems8 9 in order for a telerobotic system operate effectively it is imperative that the nonlinear dynamics of (both master and slave) robots into the controller design 1. Owing these a controller for a telerobotic system should nonly compensate the nonlinearities of the overall system but also have to satisfy energetic passivity conditions of the closed-loop system in order to ensure safety. Recently there have been some attention on the nonlinear controller design for a passive telerobotic system In this study we propose a new adaptive controller scheme for the nonlinear telerobotic system. The proposed controller achieves asymptotic trajectory tracking despite the presence of parametric uncertainties in both the master and the slave arm. Lyapunov based techniques are used to ensure the stability and passivity of the closed loop system. Introducing This work was mainly supported by the grands provided from The Scientific and Technological Research Council of Turkey - TÜBİTAK Project No: 14E61 U.Ozbay E.Zergeroglu and B.Okur are with Departmenf Computer Engineering Gebze Institute of Technology Gebze Kocaeli 414 Turkiye ezerger@bilmuh.gyte.edu.tr novel integral inequalities in conjunction with the Lyapunov type analysis we were able to satisfy all of the controller objectives (stability and passivity) at the same time via a simple controller structure. The resf the paper is organized as follows. Section 2 presents the overall system model and problem statement including the main and secondary controller objectives and the open loop error dynamics. The controller design with the desired end effector profile generation are given in Section 3. Simulation results are presented in Section 4 and Section 5. contains the concluding remarks. II. SYSTEM MODEL AND PROBLEM STATEMENT The corresponding dynamics for a telerobotic system composed of an n-dof master and an m-dof slave robotic mechanisms can be represented as follows 13 γ {M m (q m ) q m +C m (q m q m ) q m +F m ( q m )+G m (q m )} = γτ m + γf m (1) M s (q s ) q s +C s (q s q s ) q s + F s ( q s )+G s (q s )=τ s + F s where q m q m q m R n and q s q s q s R m are the position velocity and acceleration vectors of the master and slave robot respectively γ > is a force scaling factor through the master (human operator) to the slave work environment. M i (q i ) are the symmetric positive definite inertia matrices C i (q i q i ) are the matrix containing the centripetal and Coriolis effects F i ( q i ) are used for the friction vector G i (q i ) represent the vector of gravitational forces τ i are the control torque input vectors and F i are the vectors representing the joint space projection of the forces applied to the tip of the robots from the environment and/or the human operator where {( ) i i = m for the master robot i = s for the slave robot}. Assuming that both the forces applied by the human operator to the master robot and environment to the slave robot are measurable the forces applied to the end effectors of the master and slave robot can be calculated as F H = γ (J m ) T F m F E =(J s ) T (2) F s where F H is the force applied to the end effector of the master by the operator and F E is the force applied to the slave robot by the surrounding environment J m J s are the corresponding manipulator Jacobian matrices of the master and the slave robot respectively. It is also well known that using the manipulator Jacobian the end effector velocity can be calculated using ẋ m = J m q m ẋ s = J s q s (3) /1/$ AACC 636

2 formulation where x m (t) and x s (t) denote the end effector position and orientations of the master and the slave robots respectively. To ease the presentation of the controller design and the subsequent stability analysis the joint space representation of the dynamics given in (1) can be reformulated to have the following advantageous form (q) q + C (q q) q + F ( q)+ḡ(q)=τ + F (4) M where the newly defined combined matrices matrices M (q) C (q q) R (n+m) (n+m) F ( q)ḡ(q)fτ R n+m are explicitly defined as M (q) = F ( q) = F = m n γfm M s γmm n m F s γf1 γcm C (q q)= n m G s γgm Ḡ(q)= γτ1 F 2 and τ = τ 2 m n C s (5) and the newly defined combined joint space vector q R n+m is defined to have the following form q = (q m ) T (q s ) T T. (6) It is straight forward to show that the newly formed combined dynamics (5) does have the same useful properties as the standard robot dynamics (as. boundendness of the inertia matrix and the skew symmetric relationship between the time derivative of the inertia matrix and the matrix representing the centripetal and Coriolis forces). Also to ease the presentation of the stability analysis we will define a special vector function Tanh( ) R n such that Tanh(ξ ) tanh(ξ 1 )... tanh(ξ n ) T (7) for the vector ξ =ξ 1... ξ n T R n. Using the above definition and the properties of the hyperbolic functions it is straight forward to show that the following properties holds n n ξ i ξ T Tanh(ξ ) ξ T Tanh(ξ )dτ Ω <. ξ i dτ t where Ω R + is a positive bounding constant. A. Control Objective The main control objective for the telerobotic system of (1) is to ensure that the end effector of the master arm follows a desired trajectory while ensuring that the end effector of the slave manipulator follows the trajectory formed by the master manipulator at the same time. To quantify this control objective we defined the tracking error signal e(t) R 2p where p is the dimension of the operation space (containing both position and orientation information of the end effector) as follows xd x e = m (9) x s x m where x d (t) R p is the yet to be formulated the desired end effector trajectory that the human operator has to follow (8) and the terms x m (t) and x s (t) were defined in (3). The error signal of (9) can also be represented to have the following advantageous form e = xd n xd = n }{{} X d xm x m x s Ip p p I p I p }{{} S xm x s }{{} } {{ } X x (1) where the auxiliary signals X d x and X are explicitly defines as follows X d = (x d ) T ( p ) T T x = (x m ) T (x s ) T T (11) X = Sx also notice that the auxiliary transformation S R p p defined in (1) is always invertible due to its definition. Our secondary control objective is to ensure that the closed-loop dynamics of the teleoperator system remains passive with respect to the user and the physical/virtual environment forces which in operation space can be represented as 6 ( ) ẋ T s F E + ẋ T m F H dτ c 2 (12) where c R is a bounding constant. Similarly from (3) and the fact that F H = γjm T F 1 F E = Js T F 2 equation (12) can also be represented in the following form ( ) q T s F s(τ)+γ q T m F m(τ) dτ c 2. (13) B. Error Dynamics Taking the time derivative of (1) the open loop dynamics of the error signal can be obtain to have the following form ẋm ė = Ẋ d Ẋ = Ẋ d (14) ẋ m ẋ s Inserting for the (3) we can rearrange the error dynamics as ė = Ẋ d J q (15) where the generalized composite Jacobian matrix is defined as follows Jm J = (16) J m J s and q(t) is the time derivative of the joint space position vector given in (6). Adding and subtracting αe term to the right hand side of (15) where α is a positive definite diagonal gain matrix with proper dimension we can further rearrange the open loop error dynamics as follows ė = αe + Jr. (17) In (17) the auxiliary signal r (t) R n+m similar to thaf 15 is explicitly defined as r = J + (Ẋ d + αe) q (18) 637

3 where J + is the pseudo inverse of the composite Jacobian matrix defined in (16). Based on the structure of (17) we are motivated to regulate r (t) in order to regulate e(t); hence we need to calculate the open-loop dynamics for r (t). To this end we take the time derivative of (18) premultiply by the combined inertia matrix M (q) of (5) and substitute (4) to yield the following advantageous form for the open-loop dynamics M (q) ṙ = C (q q)r +Y (q qẍ d eė)θ u (19) where the regression matrix parameter vector formulation Y θ is defined as follows Y θ = (q) M dt d { J + (Ẋ d + αe) } + C (q q) { J + (Ẋ d + αe) } +F ( q)+ḡ(q) (2) where Y (q qẍ d eė) R (n+m) q denoted the regression matrix and θ R q denotes the constant but uncertain system parameters. u(t) R n+m in (19) is the the yet to be designed auxiliary control input signal formed as follows u = τ + F. (21) Before going into the control design and analysis we also would like to define an auxiliary function denoting the combined user and physical/virtual environment forces as F = S T FH F E = FH + F E F E (22) where the transformation S was previously defined in (1). III. CONTROL DESIGN AND ANALYSIS A. Desired Operation Space Trajectory Similar to the design of 12 the desired end effector trajectory for the master robot x d (t) in (1) is generated using the following dynamics M T ẍ d + B T ẋ d + K T x d = F H + F E (23) where M T B T K T are positive definite constant diagonal matrices. We will also utilize the standard assumption that the forces applied by the user on the master robot side and by the environmenn the slave robot side are bounded functions of time (i.e. F H F E L ). From this assumption and (23) it is straight forward to prove that ẍ d ẋ d and x d are bounded functions of time (i.e. ẍ d ẋ d x d L ). Now we are ready to propose the following Lemma for the passivity of the telerobot system: Lemma 1: For the combined system dynamics of (4) when the desired user trajectory signal x d on the master robot is designed according to (23) and the time derivative of the error signal defined in (1) satisfies ė(t) L 1 then the the telerobotic system is passive with respect to the forces applied by user on the master robot and physical and/or virtual environment as given in (12). Proof: To prove the above Theorem we introduce a nonnegative bounded scalar function V p (t) R + as follows V p = 1 2ẋT d M T ẋ d xt d K T x d (24) where the variables M T and K T were previously defined. Taking the time derivative of (24) and inserting for M T ẍ d from (23) we can upper bound the time derivative of V p (t) as V p ẋ T d (F H + F E ) (25) where the fact that the term ẋ T d B T ẋ d is always positive is utilized. Integrating both sides of (25) the following integral inequality can be obtained c 1 V p (t) V p ( ) ẋ T d (F H + F E )dτ (26) where c 1 R is a positive scaler bounding constant. Using the definitions of (1) and (22) we can rearrange the given integral inequality as c 1 Ẋd T Fdτ (27) Since from its definition it is obvious that X T Fdτ = Ẋd T Fdτ ė T Fdτ (28) From the above definition and the assumption that the user and the environment forces are bounded we can conclude that whenever the integral of the error signal is bounded i.e. ė(t) L 1 the following integral inequality holds for some positive bounding constant c 2 X T Fdτ c 2 (29) which after same mathematical manipulations can be written in the following form x T S T S T FH dτ c F 2 (3) E When the definitions (11) and S T S T = I 2p are utilized (12) is achieved which proves the Theorem. B. Controller Design As a consequence of Lemma 1 we have an additional control objective that is to ensure that ė(t) L 1 for this purpose we propose the following adaptive controller u = Y ˆθ + Kr+ β 1 Tanh(r)+J T (e + Tanh(e)) (31) where Kβ 1 are positive controller gain matrices with proper dimensions ˆθ (t) denotes the dynamic parameter estimates of θ computed on-line according to the following update rule ˆθ = ΓY T r (32) with Γ being the positive definite diagonal gain matrix. In addition the difference between the actual and estimated system parameters are defined as follows θ = θ ˆθ (t). (33) 638

4 Substituting the adaptive controller of (31) into the open-loop error dynamics of (19) the closed-loop error system for r (t) can now be obtained as M (q) ṙ = C (q q)r+y θ Kr β 1 Tanh(r) J T (e + Tanh(e)). (34) We are now ready to propose the following Theorem: Theorem 1: Given the telerobotic system dynamics of (4); the adaptive controller of (31) with the parameter estimates (32) ensures that the tracking error signal (1) asymptotically stable in the sense that lim e(t)= t while ensuring that the time derivative of (1) satisfies ė(t) L 1. Proof: To prove Theorem 1 a nonnegative scalar function V (t) is defined as follows V = 1 2 rt M (q)r+ 1 2 et e+ 1 2 θ T Γ 1 θ + 2p ln(cosh(e i )) (35) Taking the time derivative of V (t) inserting for the closedloop dynamics of (34) and the dynamics of the parameter updates (32) the following expression can be obtained V = r T Kr e T αe β 1 r T Tanh(r) e T αtanh(e) (36) where the property d dt { 2p } ln(cosh(e i )) = ė T Tanh(e) has been utilized. Notice that using the fact that ξ T Tanh(ξ ) ξ R n (36) can be upper bounded to have the following form V r T Kr e T αe. (37) From (35) and (37) we can conclude that r (t)e(t) L 2 L and from the definition of (17) ė(t) L 2 L can be obtained and finally from direct application of Barbalat s Lemma 14 we can conclude that lim t e =. Standard signal chasing arguments can be applied to show that all signals in the closed-loop dynamics are bounded (assuming that both robots operate inside a singularity free operational region). Additionally integrating both sides of equation (36) we obtain V ( ) V (t) β 1 r T Tanh(r)dτ + e T αtanh(e)dτ (38) and using the fact that V (t) is bounded we can conclude that the integrals of the terms r T Tanh(r) and e T Tanh(e) are bounded. From direct application of the integral inequalities given in (8) we can conclude that 2n t o r i dτ Ω 1 2n t o e i dτ Ω 2 (39) where Ω 1 Ω 2 R + are positive bounding constants. Therefore we can conclude that the auxiliary tracking signal and 639 the tracking error signal are integrable i.e. r (t)e(t) L 1. We can use (17) to prove that ė(t) L 1. To sum it up the controller proposed ensures that the tracking error signal asymptotically converges to zero (lim t e = ) while also ensuring the passivity of the overall system of (4) when the desired end-effector profile for the master robot is generated through the dynamics given by (23) which concludes the proof. IV. SIMULATION RESULTS To illustrate the performance and the effectiveness of the proposed adaptive controller we have simulated a telerobotic system composed of a 2-link planar master robot and a 3-link planar slave robot (the dynamics of the simulated robots are taken from 16 and 15 respectively). The best performance was obtained when the controller gains are selected as α = diag { } K = diag { } β 1 = diag { } with the adaptation gains were set to { Γ = diag with the initial value of the system parameters set to zero. The desired end-effector trajectory of the master robot x d was calculated according to (23) with the parameters set to M T = I B T = K T = while the forces applied by the human operator force and the actual/virtual environment were assumed to be measurable.the control frequency used in the simulation studies were 1KHz and a constant delay of 6ms between the master and the slave robots were imposed to the overall simulation to simulate the effects of time delay. Figure 1 illustrates the outcome of the desired end effector trajectory for the master robot. Figure 2 and Figure 3 illustrates the end-effector tracking error signals for the master and the slave robot respectively while some of the adaptation of the master and the slave robot system parameters (only the dynamic friction parameters) are presented at Figure 4 and Figure 5. The resf the parameter updates for the mechanical parameters were similar to those of the friction values anr are not presented here due to space considerations. V. CONCLUSION In this paper we have presented a new adaptive controller scheme for the bilateral telerobotic system composed of an n DOF master and m DOF slave robot. The proposed method guarantees asymptotic end effector tracking despite the uncertainties in the dynamics of both the master and the slave robots provided that user and environment input forces are measurable. Lyapunov-based techniques were used to prove the stability of the closed loop system. Simulation results were presented to illustrate the effectiveness of the proposed method. Future work will concentrate on removing the need for the input force measurements. }

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6 8 6 F d F d Time sec Fig. 4. Parameter Estimates of the Master Robot 4 F d F d F d Time sec Fig. 5. Parameter Estimates of the Slave Robot 641