Journal of Advanced Research in Dynamical and Control Systems Vol. 1, Issue. 2, 2009, pp. 46-61 Online ISSN: 1943-023X Adaptive control of robot manipulator having friction and uncertainties Jyoti Ohri∗ and Lillie Dewan National Institute of Technology, Kurukshetra, India. Abstract. A novel approach for adaptive control of robot manipulators having friction and other uncertainties using exponential estimation laws has been proposed. Proposed estimation law is based on time varying parameters and depends upon the system dynamics and tracking error. Friction is an important aspect for control design of mechanical systems, including robotics, because it can lead to tracking errors, limit cycles, and undesired stick-slip motion. The developed error derived adaptive compensator ensures global position tracking when applied to an n degree of freedom manipulator perturbed by friction forces, random external disturbances, measurement noise, and all the system parameters (robot and friction model) unknown. Simulation results show an acceptable performance of our compensator compared with another adaptive friction compensator previously reported. Keywords: Friction; Adaptive controller; Dynamic; Exponential law; Tracking control; Non-linear system; Nonlinear uncertainties. AMS subject classifications: 93C40, 93C10, 74M10, 68T40, 68T35. 1 Introduction Trajectory tracking control of robot manipulators is of practical significance and is the simplest but most fundamental task in robot control. Practically, parameters of the system such as gravitational load vary from a task to another, and, may not be precisely known in advance. The system may also be subjected to uncertain nonlinearities such as external disturbances and joint friction. On the whole, a good control strategy should take into account both parametric uncertainties and uncertain nonlinearities such as friction and external disturbances. ∗ Correspondence to: Jyoti Ohri, Department of Electrical Engineering, National Institute of Technology, Kurukshetra, India. Email: ohrijyoti@rediffmail.com † Received: 19 February 2009, revised: 23 May 2009, accepted: 15 July 2009. http://www.i-asr.org/jardcs.html 46 c °2009 Institute of Adnanced Scientific Research Jyoti Ohri and Lillie Dewan 47 In motion control systems unmodeled friction is usually the cause of performance degradation especially in low velocities when the friction terms dominate the mechanical dynamics. However, modeling nonlinear friction effects is not straightforward since the friction is caused as a result of complex interactions between the surface and nearsurface regions of the two interacting materials as well as other substances present such as lubricants and may also change with time. Although there are some efforts for creating a dynamic model for friction [1, 2], common consensus is that friction is a static nonlinear function of velocity [3] and can be represented mathematically as a sum of Coulomb friction- a constant opposing torque for nonzThe coefficients of the various friction-related effects are usually very difficult to measure. A number of methods for friction estimation and compensation have been proposed: adaptive control, joint torque control, learning control, variable structure control, and soft techniques and so on (see [4] and the references there in). Friedland et al. [5] proposed a reduced-order nonlinear observer to estimate the velocity dependent coefficient of the classical nonlinear friction model with velocity measurements. Tomei [6] considered the tracking problem for robot manipulators with unknown parameters and dynamic friction model using fullstates information. Adaptive friction compensation is investigated by Ge et al. [7] using both model- based and neural network (non-model-based) parameterization techniques. An adaptive neural-network based tracking control with guaranteed H 8 performances was proposed for robotic systems with plant uncertainties and external disturbances by Chang and Cheng [8].Canudas de Wit et al. [9] designed an observer based adaptive friction compensation schemes for systems with generalized position /velocity static characteristics based on the full state measurement. The proposed controller guarantees the global asymptotic stability of the tracking error while preserving boundedness of all the internal signals .The nonlinear friction is approximated either by a linear span of continuous known function, or by a neural network of bounded basis function. Putra et al. [10] proposed an observer-based friction compensation for a class of the kinetic friction models with known system parameters based on the strictly positive real condition. In the case of partial state measurements, an adaptive estimator that does not require the strictly positive realness of the plant is needed. Ray et al. [11] presented a non-model- based friction estimation method using extended Kalman –Bucy filtering. The filter is used to estimate a friction force with the full state measurements. Marton et al. [12] solved the control of the robot with unknown payload mass and friction parameters, sliding mode control algorithm was proposed combined with robust parameter adaptation techniques. Xu et al. [13] proposed an adaptive robust control (ARC) scheme based friction compensation strategy. In contrast to existing deterministic robust control (DRC) and robust adaptive control (AC) schemes, the proposed ARC scheme utilizes both the structural information of the dynamic friction model and the a priori information of the system, such as the bounds on the parameters and unmeasured internal friction state. Mitsunaga et al [14] proposed an adaptive differential filter to estimate the velocity and dynamic friction force is compensated by a fuzzy adaptive controller with position measurements. A new Coulomb friction compensator is proposed for servo control systems in [15]. The novelty of the new approach lies in its capability of assigning 48 Adaptive control of robot manipulator having friction and uncertainties the eigen values of the resulting closed loop system while attacking the problem. The friction compensation scheme proposed in [15] improves and extends the work of [16], guarantees exponential convergence of the closed loop error system. In this paper, we propose new friction compensation strategy without the requirement of large control effort and exact knowledge of the friction parameters. A new adaptive control law is derived for n-link robot manipulators based on the Lyapunov based theory. Parameter estimation laws are updated using exponential functions of manipulator kinematics, inertia parameters, tracking errors and friction. Stability of the system is established by the Lyapunov function, and a control law that guarantees the system stability is derived as a result of analytical solution. The advantage of this approach is that, parameter adaptation laws are dynamic and take system dynamics and tracking error into account. Superior performance of the proposed approach is shown by extensive simulation. The rest of the paper is organized as follows: Robot dynamics with friction model is given in section 2. Section 3 presents the proposed adaptive estimation law both for manipulator parameters and friction. Section 4 presents a case study of tracking control of two degree of freedom manipulator having friction and other uncertainties with the proposed control law. Finally concluding remarks are given in last section. ero velocities and viscous friction - the frictional force opposing the motion and is proportional to the velocity. 2 Preliminaries The dynamics of revolute joint type of robot can be described by following nonlinear differential equation [17, 18] τ = M(q)q̈ + C(q, q̇)q̇ + G(q) + F(q̇) + τd (2.1) Rn where q , q̇ , q̈ ∈ denotes the link position, velocity and acceleration vectors respectively, τ is vector of input torques, M (q) is the inertia matrix which is symmetric and positive definite,C(q,q̇)is the coriolis and centripetal coefficient matrix, G(q) includes the gravitational torque vectors, F(q̇) represents friction forces acting independently in each joints and τ d is vector of any generalized input due to disturbances or unmodeled dynamics. The following standard properties of the robot dynamic model are assumed [17, 18]: (i) The mass matrix is bounded from above and below, h i (ii) Ṁ(q) − 2C(q,q̇) is a skew-symmetric matrix, and σ T Ṁ(q) − 2C(q,q̇) σ = 0 (iii) In the absence of friction or other disturbances, the dynamics of the robot can be linearly parameterized as M(q)q̈ + C(q,q̇)q̇ + G(q) = Y(q,q̇,q̈)π, (2.2) where π is the vector of unknown manipulator parameters and Y is a known regressor matrix, which is a function of joint positions, velocities and accelerations. Jyoti Ohri and Lillie Dewan 49 (iii) The disturbances are bounded. The classical model for friction involves incorporating coulomb and viscous frictions. The following model of friction is considered in this paper [1] F(q̇) = Fv q̇+Fc sgn(q̇) (2.3) where Fv is a diagonal matrix with diagonal elements as static friction constants, and similarly Fc is a diagonal matrix representing coulomb friction constants for each link of the robot system.. Consider model (2.1) and define the tracking errors asq̃ = qd −q where qd is a desired bounded trajectory for q, with bounded first and second derivatives. The following filtered terms are defined as usual σ = q̇r − q̇ = q̃˙ + Λq̃, q̇r = q̇d + Λq̃ (2.4) where Λ represents a diagonal positive definite gain matrix. Consider the following adaptive control law with K as diagonal positive- definite matrix τ = M̂ (q)q̈r + Ĉ(q, q̇)q̇ r + Ĝ(q) + F̂v q̇r + F̂c sgn(q̇r ) + Kσ = Y (q, q̇, q̇r , q̈r )π̂ + Yf (q̇r )β̂f (2.5) The term Kσ is equivalent to a PD action on the error. (ˆ) denote the estimated terms in the dynamic model. βf is the vector of unknown friction parameters and Yf is a known regressor matrix, which is a function of filtered joint velocities as given below. Yf (q̇r )=[Y1 , Y2 ]and Y 1 and Y 2 are chosen as below Y1 = diag(q̇r1 , ........, q̇rn ), Y2 = diag(sgn(q̇r1 ), .........sgn(q̇rn )) (2.6) Since Fv and Fc are diagonal, the elements of the parameter vector βf are the diagonal elements of Fv, followed by the diagonal elements of Fc. 3 Proposed adaptive law for manipulator and friction parameters with time varying function A new adaptive control strategy is proposed below for the purpose of trajectory control of robot manipulator based on the use of exponential function [20] for estimation. These parameter estimation laws depend on the manipulator kinematics, dynamic parameters, friction parameters, and tracking errors and are dynamic in nature. The adaptive estimates of manipulator parameter i.e. π̂and friction parameter i.e.β̂f to be used in control law (2.5) are proposed as new control input as following 1 π̂ = 2(e− 2 R T Y σdt − e− R T Y σdt )+π (3.1) 50 Adaptive control of robot manipulator having friction and uncertainties 1 β̂f = 2(e− 2 R T Yf σdt − e− R T Yf σdt ) + βf (3.2) where π̂(0) = π andβ̂f (0) = βf , as the initial conditions are assumed to be known or assumed as zero. Theorem 3.1. For the purpose of trajectory control of robot manipulator, if the adaptation functionsπ̂and β̂f in control law (2.5) is chosen as (3.1) and (3.2) respectively, a ˙ new control strategy is obtained, by, such that the tracking errors q̃, q̃and the estimate of parameter π̂ and the estimate of friction parameter β̂f are bounded. Proof. In order to derive the parameter estimation laws, defined in (3.1) and (3.2) that satisfy the stability of the closed system and ensures limited tracking errors, the following Lyapunov function candidate is proposed. 1 1 1 V(σ, q̃, π̃, β̃f ) = σ T M(q)σ + q̃T B q̃ + π̃ T Ω (t)π̃ + β̃fT λ(t)β̃f > 0∀σ, q, π̃, β̃f 6= 0 (3.3) 2 2 2 In this proposed algorithm Ω (t) and λ(t)are positive definite time varying diagonal matrices, whereas these are always chosen as constants in previous classical standard approach of adaptive estimation laws e.g. proposed by Slotine and Li [19]. The time derivative of (3.3) is written as 1 T 1 T 1 T T ˙ (3.4) V̇ = σ T M(q)σ̇ + σ T Ṁ(q)σ + q̃T Bq̃˙ + π̃ Ω̇π̃ + π̃ Ωπ̃˙ β̃f λ̇β̃f + β̃f λβ̃f 2 2 2 h i Taking B =2ΛK, and using the propertyσ T Ṁ(q) − 2C(q, q̇) σ = 0, (3.4) can be rewritten as below ´ ³ V̇ = −q̃˙ T Kq̃ − q̃T ΛKΛq̃ + π̃ T Ωπ̃˙ + 12 Ω̇π̃ − YT (q, q̇, q̇r , q̈r )σ ³ ´ (3.5) ˙ + β̃fT λβ̃f + 12 λ̇β̃f − YfT (q,q̇,q̇r ,q̈r )σ Since K> 0 and Λ> 0, the following expression holds true: −q̃˙ T Kq̃˙ − q̃T ΛKΛq̃ ≤0 (3.6) The stability of the equilibrium can be guaranteed if V̇ < 0 and so the last two terms in (3.5) must be separately equal to zero i.e. 1 Ωπ̃˙ + Ω̇π̃ − YT σ = 0 2 (3.7) 1 ˙ λβ̃f + λ̇β̃f − YfT σ = 0 2 (3.8) and Jyoti Ohri and Lillie Dewan 51 Now, it is very important to choose the functions Ω and λ in order to solve the (3.7) and (3.8). As there is no certain rule for selection of Ω and λ, for this system, we use system state and friction parameters for appropriate function of Ω and λ as a solution of the first order differential in equations (3.7) and (3.8). Hence Ω and λ and their derivatives are chosen as exponential functions as given below respectively ³ R T ´ ³ ´ R T Ω = e Y σdt I; Ω̇ = YT σ e Y σ dt I (3.9) and µ R ¶ µ ¶ R T T T Yf σdt Yf σdt λ= e I; λ̇ = Yf σ e I (3.10) where I is a suitable dimension identity matrix. Substitution of (3.9) into (3.7) and (3.10) into (3.8) yields e R R T 1 T R T T π̂ + Y σe Y σdt (π̂ − π) = Y σ 2 Y σdt ˙ (3.11) 1 T R T T ˙ β̂f + Yf σe Yf σdt (β̂f − βf ) = Yf σ (3.12) 2 R T ˙ ˙ ˙ As π̃˙ = π̂(πis constant) and alsoβ̃f = β̂f ( βf is constant). Dividing (3.11) by e Y σdt e R and (3.12) by e T Yf σdt T Yf σdt yields R T 1 T 1 T T π̂˙ + Y σ π̂ = e− Y σdt Y σ + Y σ π 2 2 (3.13) R T 1 T 1 T T ˙ β̂f + Yf σ β̂f = e− Yf σdt Yf σ + Yf σ βf 2 2 (3.14) 1 R T Multiplying (3.13) by the factor e 2 Y following (3.15) and (3.16) respectively 1 R 1 R e2 e2 T σdt 1 and (3.14) by the factore 2 R T Yf σdt result into R T R T R T 1 1 1 T 1R T 1 T T π̂ + Y σe 2 Y σdt π̂ = e 2 Y σdt e− Y σdt Y σ + Y σ e 2 Y σdt π (3.15) 2 2 Y σdt ˙ T Yf σdt R T R T R T 1 1 1 T 1R T T ˙ 1 T β̂f + Yf σ e 2 Yf σdt β̂f = e 2 Yf σdt e− Yf σdt Yf σ + Yf σe 2 Yf σdt βf (3.16) 2 2 (3.15) can be rearranged as (3.17) µ R ¶ R T R T T 1 1 1 d 1 T T e 2 Y σdt π̂ = e− 2 Y σdt Y σ + Y σ e 2 Y σdt π dt 2 (3.16) can be rearranged as (3.18) (3.17) 52 Adaptive control of robot manipulator having friction and uncertainties d dt ¶ µ R R T R T T 1 1 1 1 T T e 2 Yf σdt β̂f = e− 2 Yf σdt Yf σ + Yf σ e 2 Yf σdt βf 2 (3.18) Integrating both sides of (3.17) and (3.18) yields respectively following 1 e2 R T Y σdt 1 R 1 R π̂ = −2e− 2 T Y σdt 1 + π e2 R T Y σdt + C1 (3.19) and 1 e2 R T Yf σdt β̂f = −2 e− 2 1 Dividing both sides of (3.19) by e 2 are obtained: π̂ = −2e− β̂f = −2e− R R R Y T σdt T Y σdt T Yf σdt T Yf σdt R 1 + βf e 2 YfT σdt R 1 , and (3.20) by e 2 1 + π + C 1 e− 2 R T Yf σdt (3.20) the following results T Y σdt R 1 + βf + C2 e− 2 + C2 (3.21) T Yf σdt (3.22) If the conditions of π̂(0) = π andβ̂f (0) = βf are taken as known initial conditions, the constants C 1 and C 2 are equivalent to 2. Hence, the parameter and friction adaptation laws respectively are derived as following π̂ = −2(e− β̂f = −2(e− R R T Y σdt T Yf σdt 1 e− 2 R 1 − e− 2 T Y σdt R )+π T Yf σdt (3.23) ) + βf (3.24) ˙ In order to show that this result satisfies the stability condition, π̃ and π̃are drawn from (3.23), respectively as µ ¶ µ ¶ R T R T R T R T T − 12 Y σdt − Y σdt − Y σdt − 12 Y σdt ˙ ˙ π̃ = π̂−π = 2 e −e (31)π̃ = π̂ = Y σ 2e −e (3.25) ˙ Similarly β̃f and β̃f can be obtained from (3.24), respectively as 1 β̃f = β̂f − βf = 2(e − 2 R T Yf σdt − e− R T Yf σdt ) (3.26) R T R T 1 T ˙ ˙ β̃f = β̂f = Y σ(2 e− Yf σdt − e− 2 Yf σdt ) (3.27) Substituting the results obtained above into (3.5), the following is obtained Jyoti Ohri and Lillie Dewan 53 · ¸ R T R T T T − 21 Y σdt − Y σdt ˙ ˙ −e V̇ = −q̃ Kq̃ − q̃ ΛKΛq̃ + 2 e µ ¶ ¸ · R µ ¶ ³ T ´ R T R T R T R T R T T 1 1 T T e Y σdt (Y σ) 2e− Y σdt − e− 2 Y σdt + 12 Y σ e Y σdt 2 e− 2 Y σdt − e− Y σdt − Y σ · ¸ R T R T − 12 Yf σdt − Y σdt f +2 e −e µ µ ¶ · R ¶ ¸ ³ T ´ R T ³ ´ R T R T R T R T T 1 1 T T 2e− Yf σdt − e− 2 Yf σdt + 12 Yf σ e Yf σdt 2 e− 2 Yf σdt − e− Yf σdt − Yf σ e Yf σdt Yf σ (3.28) After some arrangements, (3.28) is rewritten in the following form: · ¸h i R T R T 1 T T − Y σdt − Y σdt V̇ = −q̃˙ Kq̃˙ − q̃ ΛKΛq̃ + 2 e 2 −e 2Y σ − 2Y σ + · ¸h i R T R T T T − 21 Yf σdt − Y σdt f 2 e −e 2Yf σ − 2Yf σ T T (3.29) As seen from (3.29), if the last two terms on the right sides is equal to zero and then the time derivative of the Lyapunov function can be written as T T V̇ = −q̃˙ Kq̃˙ − q̃ ΛKΛq̃ < 0 since Λ>0 and K>0, the closed system is stable at the equilibrium point q̃˙ ≡ 0, q̃ ≡ 0. Eq. (3.12) shows that V is a positive continuous function and V (0) = 0, that is lower bounded by zero whenπ̃ = π̂ − π = 0 andβ̃f = β̂f − βf = 0and at equilibrium points q̃˙ ≡ 0, q̃ ≡ 0. V tends to a constant as t → 0 and therefore V remains bounded. Hence ˙ all the internal signals q̃, σ, Ω, λ, π̂, and β̂f are bounded. As q̃and q̃are bonded and R converge to zero, this implies that σis bounded and converges to zero. Thus Y T σdt is bounded and this implies that Ω, λ, π̂, and β̂f are bounded. 4 Simulation study This section discusses the implementation of the proposed exponential adaptive controller on a two-link planar manipulator arm having revolute joints [17, 18]. The robot link parameters are: 2 2 π1 = m1 lc1 + m2 l12 + I1 , π2 = m2 lc2 + I2 , π3 = m2 l1 lc2 , π4 = m1 lc1 , π5 = m2 l1 , π6 = m2 lc2 With this parameterization, the components Yi j of Y (q, q̇, q̈) in (2.2) are given as Y11 = q̈1 ; Y12 = q̈1 + q̈2 ; Y13 = cos(q2 )(2q̈1 + q̈2 ) − sin(q2 )(2q̇1 q̇2 + q̇22 − 2); Y14 = gc cos(q1 ); 54 Adaptive control of robot manipulator having friction and uncertainties Y15 = gc cos(q1 ); Y16 = gc cos(q1 +q2 ); Y21 = 0; Y22 = q̈1 + q̈2 ; Y23 = cos(q2 )q̈1 +sin(q2 )(q̇12 ); Y24 = 0; Y25 = 0; V26 = gc cos(q1 + q2 ). With this parameterization, the components Y (q, q̇, q̇r , q̈r ) are given as Y11 = q̈r1 ; Y12 = q̈r1 + q̈r2 ; Y13 = cos(q2 )(2q̈r1 + q̈r2 ) − sin(q2 )(2q̇1 q̇r2 + q̇2 q̇r2 ); Y14 = gc cos(q1 ); Y15 = gc cos(q1 ); Y16 = gc cos(q1 + q2 ); Y21 = 0; Y22 = q̈1 + q̈2 ; Y23 = cos(q2 )q̈1 + sin(q2 )(q̇12 ); Y24 = 0; Y25 = 0; Y26 = gc cos(q1 + q2 ) The initial estimation of vector π̂is computed on the basis of the nominal parameters given in Table 1. m1 10 m2 5 l1 1 l2 1 l c1 0.5 lc2 0.5 I1 0.833 I2 0.416 Table 1: Nominal parameters of the two link planar robot. π1 8.33 π2 1.67 π 3 2.5 π 4 5 π5 5 π 6 2.5 Table 2: Initial estimation of parameters π̂for the two link planar model. Fv 1 11 Fv 2 1.00 Fc1 5 Fc 2 1.00 Table 3: Nominal parameters of the Friction for two link planar robot. The friction constants matrix βf is chosen as = [F v(1, 1), F v(2, 2), F c(1, 1), F c(2, 2)]T Manipulator is commanded to trace the following trajectory: A straight line trajectory with parabolic blends at end given by cubic polynomial with initial joint angle configuration as q1(0) = 0 radians and q2 (0) = 0 radians and final joint angle configuration q1(Tf) = 2.5 radians and q2 (Tf) = 2.5 radians in Tf =3 second is the desired trajectory[17]. It is slow speed trajectory, having static friction as predominant friction effect. The sampling time is chosen as ∆t =0.001sec. The disturbance torque τ d = [5 Sin (200t); 5 Sin (200t)] is applied. In addition to this deterministic noise, simulations with uniformly distributed random noise are also performed. And pay load of 5 Kg. is applied at t = 1 seconds, i.e. ∆ m2 = 5 Kg; For the controller tuning, gain matrix parameters are tuned as K=diag (105, 105) and Λ= diag (110, 110). Jyoti Ohri and Lillie Dewan βf 1 0 βf 2 0 βf 3 0 55 βf 4 0 Table 4: Initial estimation of parameters β̂f for the two link planar model. Simulations are performed with these gain matrices for following cases: Case1: Adaptive controller of (2.5) is applied for the tracking control of manipulator dynamics of (2.1), having friction, disturbance and pay load variations, with the assumption that all the parameters are unknown. Parameters of model are estimated on line using the exponential estimation algorithm of (3.23) and (3.24). Tracking of the desired trajectory is shown in the Figs.1 and 2, which show good performance. Tracking of joint angle1 vs time 3 2.5 joint angle1 in radians 2 desired actual 1.5 1 0.5 0 -0.5 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 1: Tracking of Desired Trajectory of Joint angle 1 in case1 Tracking of joint angle2 vs time 2.5 jo in ta n g le 2inra d ia n s 2 1.5 desired actual 1 0.5 0 -0.5 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 2: Tracking of Desired Trajectory of Joint angle 2 in case1 Case2: Integral type adaptive controller as proposed by Slotting and Li [14] is applied for the tracking control of manipulator, with the similar conditions as above for the comparison purpose. Tracking of the desired trajectory is shown in the Figs.3 and 4 56 Adaptive control of robot manipulator having friction and uncertainties for this case. tracking of joint angle1 3 desired actual 2.5 joint angle1 in radians 2 1.5 1 0.5 0 -0.5 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 3: Tracking of Desired Trajectory of Joint angle 1 in case2 Tracking of joint angle2 vs time 2.5 jo in ta n g le 2inra d ia n s 2 1.5 desired actual 1 0.5 0 -0.5 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 4: Tracking of Desired Trajectory of Joint angle 2 in case2 Tracking error trajectory of joint angles 1 and 2, for both the above cases are shown in Figs 5 and 6 respectively. The comparisons of these results are shown in Table 5 in terms of various performance measures. As commonly used performance measures such as rising time, damping and steady state error, are not adequate for nonlinear systems like robots .Hence the RT scalar valued L2 norm given by L2 [q̃(t)] = ( T1f 0 f kq̃(t)k2 dt)1/2 is used as an objective numerical measure of tracking performance for an entire error curvesq̃(t), which are shown in Figs. 7 and 8. However it is an average measure and peaks of errors during the initial transient stage can not be predicted. Thus the maximum absolute value of tracking error of each joint is also used an index of measure of transient performance, in R Tf T which q̃im = maxt∈[0,Tf ] {|q̃i (t)|}.Norm of the control effort is also found as 0 τ τ dt and shown in Fig 9, which shows the lesser and smoother control effort in case of proposed controller. Actual control efforts as τ 1, τ 2 for this case is shown in Fig.10. Convergence of some of the few friction model parameters are shown in Figs. 11 and 12. Jyoti Ohri and Lillie Dewan 57 Tracking error in joint angle 1 0.02 error in joint angle 1 0 -0.02 case 1 case 2 -0.04 -0.06 -0.08 -0.1 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 5: Tracking error in joint angle 1 for case 1 and 2 Tracking error in joint angle 2 0.005 0 -0.005 error in joint angle 2 case1 case2 -0.01 -0.015 -0.02 -0.025 -0.03 -0.035 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 6: Tracking error in joint angle 2 for case 1 and 2 Hence from these comparisons it is clear that on introducing dynamic exponential estimation law in conventional adaptive control, there is a significant improvement in the tracking performance as compared to standard adaptive controllers with integral estimation laws. The aim of the control law is to decrease σ as minimum as possible. The other problem is to increase the parameter convergence rate while Y T σdecreases .When tracking error converges to zero,Y T σ also converges to zero, whereas the exponential term R T Y σdt varies very slowly. Hence the parameter estimation speed and the convergence rate can not be decreased as much as the integral algorithm of [19] and the proposed algorithm has faster parameter estimation speed and the convergence rate comparing to the algorithm of [19]. 58 Adaptive control of robot manipulator having friction and uncertainties Tracking error for above two cases Max(Absolute value of Tracking error) 2 norm of Tracking error 2 norm of Control effort Joint angle 1 Joint angle 2 Joint angle 1 Joint angle 2 Joint angle 1 Joint angle 2 Case1 0.0202 0.0057 0.5717 0.2156 1.2053e+4 3.7087e+3 Case2 0.0976 0.0346 4.1794 1.4124 3.6566e+4 1.2609e+4 Table 5: Performance Measures of Tracking of joint angle 1 and 2. 2 norm of the tracking error in joint angle 1 4.5 4 3.5 3 2.5 2 1.5 1 0.5 0 case1 case2 Figure 7: 2norm of the tracking error in joint angle 1 2 norm of thracking error in joint angle 2 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0 case1 case2 Figure 8: 2norm of the tracking error in joint angle 2 5 Conclusion Friction dynamics is a major cause of performance degradation in precise motion control tasks. A new adaptive algorithm compensating the effect of uncertainties and friction in Jyoti Ohri and Lillie Dewan 8 norm of control input 15 59 norm of control input x 10 10 case1 case2 5 0 0 0.5 Figure 9: 1 R Tf 0 1.5 time in sec 2 2.5 3 τ T τ dt of control effort in case 1 and 2 control input vs time in case 2 500 tau 1 tau 2 contorl input in joint angles 400 300 200 100 0 -100 -200 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 10: Control efforts τ 1 and τ 2 in case 1 dynamics using exponential estimation laws is proposed. This algorithm is derived on the basis of the Lyapunov stability criterion hence the stability is guaranteed and friction is also well compensated. The developed, error derived estimation rules act as dynamic compensator, that is, estimate the most appropriate values of parameters, so as to solve the direct adaptive control problem for finding a control law that ensures limiting errors, and not to determine the actual parameters of the system (as in indirect adaptive control law).The proposed scheme was compared with the popular integral type adaptive control of Slotting and Li. Results show the superior performance of the proposed adaptation laws in terms of tracking error, and lesser and smoother control effort than the integral type adaptation laws. The proposed algorithm also has faster parameter speed and faster convergence rate. It is also confirmed that friction compensation is essential for obtaining low trajectory tracking errors. 60 Adaptive control of robot manipulator having friction and uncertainties friction parameter Fc1 vs. time in sec. 6 frictionparameter Fc1 5 4 3 2 1 0 0 0.5 1 1.5 time in sec. 2 2.5 3 Figure 11: Convergence of the friction parameter Fc1 friction parameter Fv2 vs time in sec. 1.4 frictionparameter Fv2 1.2 1 0.8 0.6 0.4 0.2 0 0 0.5 1 1.5 time in sec 2 2.5 3 Figure 12: Convergence of the friction parameter Fv 2 Acknowledgments The author thanks the reviewers for their valuable comments and suggestions. References [1] B. Armstrong-Velour, P. DuPont, C. Canudas. A survey of models, analysis tool and compensation methods for the control of machines with friction. Automatica, 1994, 30: 10831138. [2] C. Canudas de Wit, H. Olsson, K. J. Astor, P. Lichens. A new model for control of systems with friction. IEEE Trans. on Automatic Control, 1995, 40: 419-425. [3] M.K. Ciliz. 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