Advanced Strategies For Robot Manipulators Part 8 doc

Collision Detection and Control of Parallel-Structured Flexible Manipulators Based on Unscented Kalman Filter 201 The Hamilton’s principle is described by the following equation: {} 2 1 () () () () () =0, t nc t Tt Vt W t stδψ tdt δδδ −+ + ∫ (9) where t 1 and t 2 are arbitrary times; s(t) represents the Lagrange multiplier which is equivalent to the external force generated by the collision with the obstacle; and δ W nc (t) denotes the virtual work due to the nonconservative forces, e.g. internal damping forces of the beams, the control torque and external disturbances. ψ(t) describes the geometric constrained condition between the unlooked-for obstacle and the flexible beam, i.e. 0 ()= (, ) tan{ ()} 0. cc ψ tutx x t ϕ θ − −≡ (10) Let us assume that | ϕ 0 – θ (t)| is sufficiently small. We can regard tan{ ϕ 0 – θ (t)} ≅ ϕ 0 – θ (t). Then, (10) can be rewritten into 0 0 ()= (, ) ( ) { ()} 0. cc ψ tutxxxdxx t δϕθ − −−≡ ∫ A (11) δ W nc (t) is expressed by ( ) = ( ) ( ) ( ) nc Wt t t g t θθ δ μθ δθ τ δθ γ δθ −++  3 2 00 (, ) (, ) , D f i utx cudxgtxudx xt δγδ ⎛⎞ ∂ ′′ −+ ⎜⎟ ∂∂ ⎝⎠ ∫∫ AA (12) where the prime denotes the differentiation with respect to x; μ denotes the damping coefficient corresponding to the damping force acting at the shaft of the servomotor; τ (t) the control torque; γ θ (t) the random disturbance acting at the rotation of the arm; γ (t, x) the distributed random disturbance along the beam due to the aerodynamic resistance; and g θ and g f are constants. As the generalized coordinates, we consider the following variables: θ (t), ( )t θ  , u(t, x), (, )utx  , u’(t, x), u’(t, x), u’’(t, x), (, ),utx ′ ′  u(t, ℓ), (,)ut  A . Substituting (7), (8) and (11) into (10); in addition, performing a large amount of calculations, we have the following nonlinear differential equations as the mathematical model of the approximated dynamical model corresponding to the parallel-structured single-link flexible manipulator: 254 2 1 24 4 (, ) (, ) (, ) =()()(,)(,) D utx utx utx ScIEI SxetSetutxgtx txtx ρρργ ∂∂∂ ++ −+ + ∂∂∂∂   2 2 2 (, ) ()( ) ( )() ()(, ) ( ), c utx st x x m het m me tutx x t δδ ⎧⎫ ∂ + −− + + − − ⎨⎬ ∂ ⎩⎭   AA (13) where γ (t, x) the distributed random disturbance modeled by the white Gaussian noise; g 1 and h are constants; δ (·) denotes the Dirac’s delta function; and s(t) the magnitude of collision input. Assuming that the collision occurs momentarily, the magnitude of collision is assumed to be expressed by s(t) := s 0 δ (t – t c ), where s 0 and t c are all unknown. The initial and boundary conditions of (13) are Advanced Strategies for Robot Manipulators 202 (0, ) : (0,)= =0 ux IC u x t ∂ ∂ (14) 3 3 (,0) (,) (,) :(,0)= = = =0. ut ut ut BC u t xxx ∂∂∂ ∂∂∂ AA (15) The initial condition of the error of rotation e(t) is given by e(0) = θ 0 – θ d , where θ 0 is initial angle position of the arm. The dynamics of rotation is given by the following nonlinear differential equation: 22 01 0 (,) ( ) (, ) ()J J mut mh h Sutxdxet ρ ⎡⎤ ++ + ++ ⎢⎥ ⎣⎦ ∫ A  AA 0 2 (,)(,) 2(, )(, ) () (,)mu t u t S u t x u t x dx e t mhu t θ μρ ⎡⎤ ++ + + ⎢⎥ ⎣⎦ ∫ A  AA A 54 2 44 00 (, ) (, ) (, ) (,) () D utx utx S u t x dx m u t e t x c I EI dx xt x ρ ⎡ ⎤ ∂∂ ⎡⎤ −− − + ⎢ ⎥ ⎢⎥ ⎣⎦ ∂∂ ∂ ⎣ ⎦ ∫∫ AA  AA 1 0 ( , ) ( ) ( ) = 0. gx txdx t g t θθ γτγ +−− ∫ A (16) The observation data is obtained by means of P strain sensors pasted at x = ξ j , (j = 1, … ,P) and a potentiometer installed at the shaft of the hub, i.e. 00 00 ()= () () y tcete t β + (17) 2 2 (, ) ()= (), b js jj jj j utx y tc dxe t x ξ ξ β + ∂ + ∂ ∫ (18) where c j and e j are constants; and β j (t), (j =0, 1, … ,P) represents the observation noise which is modeled by the white Gaussian noise. In order to use the finite-dimensional controller and state estimator, the dynamics of the arm described by (13) and (16) are converted into the stochastic finite-dimensional state space model via the modal expansion technique, =1 (, )= () ( ), N kk k utx u t x φ ∑ (19) where {u k (t)} k=1,… ,N denote the modal displacements; N the large positive number; φ k (x) the eigenfunction (mode function) of the following eigenvalue problem with respect to the operator A = {(EI)/( ρ S)}(d 4 /dx 4 ): ()= (). kkk xx φ λφ A (20) Introducing the state vector defined by v(t) = [u 1 (t),… ,u N (t), 1 u  (t), … , N u  (t), e(t), e  (t)] T , the state space model of the approximated flexible arm can be described by the following stochastic differential equation: ()= ( )() ( )() ( )() ( ; )() cc vt Avvt bv t Gv t g vx st τ γ + ++  (21) Collision Detection and Control of Parallel-Structured Flexible Manipulators Based on Unscented Kalman Filter 203 ( ) = ( ) ( ), yt Cvt E t β + (22) where γ (t) = [ γ 1 (t), … , γ N (t), γ θ (t)] T ; γ k (t) = ∫ 0 A γ (t, x) φ k (x)dx; β (t) := [ β 0 (t), β 1 (t), … , β P (t)] T ; E{ γ (t) γ T ( τ )g} = W δ (t – τ ); E{ β (t) β T ( τ )} = V δ (t – τ ) (E{·}: mathematical expectation). 3. Nonlinear state estimation using UKF The state space model described by (21) and (22) is a stochastic nonlinear system with the collision input. In order to control the tip position and to reduce the random vibration of the whole flexible manipulator, the information of the state v(t) is required. However, the collision input affects as an unknown disturbance. For achieving these purposes, the unscented Kalman filter (UKF) for the following collision-free system is employed: ()= ( ) () ( ) () ( )(), ffff f vt Avvt Bv f tGv t γ + +  (23) where v f (t) denotes the state vector of collision-free system. The UKF is constructed for the discrete-time nonlinear stochastic system given by equation (23) that is the continuous-time system. Equation (23) and (22) can be converted into the discretized version of the system. By using the Runge-Kutta method, (22) and (23) are rewritten into the following nonlinear discrete-time system: ( 1) = ( ( ), ( ), ( )) ff vk Fvk k k τ γ + (24) ()= () (), y kCvkEk β + (25) where k denotes the time-step; Δt the time interval; and F(·) the nonlinear function defined by 12 ( ( ), ( ), ( )) = ( ) { ( ( ), ( ), ( )) 2 ( ( ), ( ), ( )) 6 fff f t Fvkkkvk Hvkkk Hvkkk τγ τγ τγ Δ ++ 34 2 ( (),(),()) ( (),(),())} f Hvk k k Hvk k k τ τ γτγ + + (26) 1 ( (),(),())= ( ()) () ( ())() ( ())() fffff Hvk k k Avkvk Bvk k Gvk k τ γτγ + + 21 ( (),(),())= ( ()){ () () /2} ( ())() ( ())() fff ff Hvk k k Avk vk H t Bvk k Gvk k τ γτγ + ⋅Δ + + 32 ( ( ), ( ), ( )) = ( ( )){ ( ) ( ) /2} ( ( )) ( ) ( ( )) ( ) fff ff Hvk k k Avk vk H t Bvk k Gvk k τ γτγ + ⋅Δ + + 43 ( ( ), ( ), ( )) = ( ( )){ ( ) ( ) } ( ( )) ( ) ( ( )) ( ). fffff Hvk k k Avk vk H t Bvk k Gvk k τ γτγ + ⋅Δ + + The algorithm of UKF is summarized the three steps as follows: Step 1. The (2N + 2)-dimensional random variable v f (k) is approximated by 2(2N + 2) + 1 sigma points X i with weight coefficients W i . 0 ˆ = ( | ) f vkkX (27) 0 = 22 W N κ κ ++ (28) Advanced Strategies for Robot Manipulators 204 ˆ = (|) { (2 2 )(|)} i f i vkk N Pkk κ +++X (29) 1 = 2(2 2 ) i W N κ ++ (30) 22 ˆ = (|) {(2 2 )(|)} iN f i vkk N Pkk κ ++ −++X (31) 22 1 = , ( = 1, ,2 2) 2(2 2 ) iN WiN N κ ++ + ++ " (32) where κ denotes the integer scaling parameter; W i the weight coefficient that is associated with the i-th point and {(2 2 )(|)} i NPkk κ ++ represents the i-th column of the matrix U satisfying M = UU T if M := (2N + 2 + κ )P(k|k). In this paper, the matrix U is calculated via the incomplete Cholesky decomposition (Saad, 1996). Step 2. Transform each point through the nonlinear function F(·), and the predicted mean, covariance and observation, the innovation covariance P yy (k + 1|k) and the cross correlation matrix P xy (k + 1|k) are computed as follows: (1|)=((|),(),0) ii kkFkkk τ + XX (33) 2(2 2) =0 ˆ (1|)= (1|) N fii i vk k W k k + ++ ∑ X (34) 2(2 2) =0 ˆ (1|)= {(1|) (1|)} N ii f i Pk k W k k v k k + ++−+ ∑ X TT ˆ {( 1|) ( 1|)} if kkvkkGWG×+−+ +X (35) ( 1| ) = ( 1| ) ii kkCkk + +YX (36) 2(2 2) =0 ˆ (1|)= (1|) N iii i y kk Wkk + ++ ∑ Y (37) 2(2 2) =0 ˆ (1|)= {(1|)(1|)} N yy i i i Pk k W k k y kk + ++−+ ∑ Y TT ˆ { ( 1| ) ( 1| )} i kkykkEVE×+−+ +Y (38) 2(2 2) =0 ˆ ( 1|)= { ( 1|) ( 1|)} N vy i i f i Pk k W k k vk k + ++−+ ∑ X T ˆ {( 1|) ( 1|)}. i kkykk×+−+Y (39) Step 3. The state estimate and covariance are given through updating the prediction by the linear update rule which is specified by the weights chosen to minimize the mean squared error of the estimate. The update rule is Collision Detection and Control of Parallel-Structured Flexible Manipulators Based on Unscented Kalman Filter 205 ˆˆ ˆ (1|1)=(1|)(1){(1)(1|)} ff vk k vk k Kk y k y kk + +++++−+ (40) T (1|1)=(1|)(1)(1|)(1|), yy Pkk PkkKkPkkKkk++ + −+ + + (41) where K(k + 1) is the Kalman filter gain given by 1 (1)=(1|)(1|). vy yy Kk P k kP k k − ++ + (42) 4. Collision detection algorithm It is an undesirable accident that the flexible manipulator collides with an unknown obstacle, because the collision input s(t) affects the state of flexible manipulators as the disturbance. The problem of the collision detection is considered as a detection problem of the abrupt change from the collision-free system to the system with the collision. The change of the systems can be detected using the observation data measured by the piezoelectric sensors pasted at the root of the flexible arm. In other words, the collision is detected by making a decision whether the observation data y(t) is provided from the collision included model or the collision-free model. For this purpose, the intensity of the innovation process is used. The innovation process of the UKF, μ (k), is defined by the difference between the actual observation data and the estimated observation data measuring collision-free system, i.e., ˆ ()= () (|), f k y kCvkk μ − (43) where ˆ (|) f vkk is the estimate of v f (k) which is calculated by the UKF mentioned in the previous Section. Substituting (25) into (43), we have ()= () (),kCzkEk μ β + (44) where z(k) is the estimation error defined by z(k) := v(k) – ˆ (|) f vkk. If the collision does not occurr, the state vector v(k) is equal to v f (k). However, if the collision occurrs, v(k) is equal to the state vector of collision model. In this case, z(k) becomes large because of the collision input. In order to detect the collision, the following scalar function (collision detection function) is introduced: T ()= ()().rk k k μμ (45) If the collision detection function r(k) exceeds a threshold ε, then the collision has occurred. In fact, it is assumed that the estimation error when the collision has occurred is separated as ()=() (),zk zk k α +  (46) where ()zk  is the estimation error of the UKF based on the collision-free system; and α (k) the estimation error caused by the collision input. Substituting (46) into (44), the innovation process μ (k) is rewritten into ()= () () ().kCzkCkEk μ αβ + +  (47) Advanced Strategies for Robot Manipulators 206 From equations (45) and (47), the mathematical expectation of the collision detection function r(k) is evaluated by TTTT {()}=tr{ () } tr{ } tr{ ()() }, z rk CP kC EVE C k kC αα ++E (48) where T () { ()()} z Pk z kzk:=  E . The first two terms in the right-hand side of (48) is the bias depending on the observation and the system noises. The third term is caused by the collision input which is the deterministic process. If the third term becomes sufficiently large, then r(k) becomes also large at the time when the collision occurs. 5. Position and suspend control The purpose of controller is to generate the control torque for the servomotor so that the tip position of the flexible manipulator follows the reference trajectory. In this work, the sliding mode controller is employed (Utkin, 1992). The flexible manipulator is controlled so that its state converges to the equilibrium sate. The sliding mode controller is constructed for the following deterministic collision-free system: ()= ( ()) () ( ())() ffff vt Avtvt Bvt t τ +  (49) which has no system noise term. If the flexible manipulator is well controlled, the error state vector v f (t) is assumed to sufficiently be small, i.e. &v f (t)& 1. In this work, we consider that the matrices A(v f (t)) and B(v f (t)) can be approximated as A(v f (t)) ≅ A(0) and B(v f (t)) ≅ B(0). Using these approximations, the error system is rewritten into the following equation: ()= () (), fefe vt Avt B t τ +  (50) where A e and B e are constant matrices defined by (0), (0). ee AA BB:= := (51) The objectives of the sliding mode controller are to control the tip position, to suppress the random vibration of the whole manipulator, and to suspend the motion of the manipulator when a collision is detected. The control torque τ (t) is generated by the sliding mode controller. The output of the controller can be separated into two parts, i.e.: ()= () (), eq nl t f t f t τ + (52) where f eq (t) represents the term of linear control input called the equivalent control input in the sliding mode; and f nl (t) the term of nonlinear control input in the reaching mode. The switching function σ (t) is given by ()= (), f tSvt σ (53) where S represents the gradient of the switching plane. In the sliding mode, the switching function σ (t) holds the following conditions: ()=0t σ (54) Collision Detection and Control of Parallel-Structured Flexible Manipulators Based on Unscented Kalman Filter 207 ()=0.t σ  (55) The equivalent control input is obtained using (50), (53) and (55) as: 1 ()= ( ) (). eq e e f f tSBSAvt − − (56) Substituting (56) into (50), the equivalent control system can be described by 1 ()={ ( ) } () (), feeeefef vt A BSB SAvt Avt − −≡   (57) where 1 () eeee e A ABSBSA − := −  . In order to find the switching plane, we consider the cost functional defined by 22 12 00 = { [{(,)} {(,)} t J q usx q usx+ ∫∫ A  2222 34 56 { ( , )} { ( , )} ] ( ) ( )} , q usx q usx dx q es q esds ′′ ′′ ++ ++  (58) where q i (i = 1, … 6) are positive constants. Substituting the solution of the system described by the equation (13) given by (19) into this, it is rewritten into the following expression: T 0 =()(), T ff JvtQvtdt ∫ (59) where Q := diag{ Θ 1 , Θ 2 , q 5 , q 6 }; Θ 1 = q 1 I N + q 3 Ψ; Θ 2 = q 2 I N + q 4 Ψ; (I N : N-dimensional unit matrix); and 11 1 00 1 00 () () () () =. () () () () N i NNN x x dx x x dx x x dx x x dx φφ φφ φφ φφ ⎡ ⎤ ′′ ′′ ′′ ′′ ⎢ ⎥ ⎢ ⎥ Ψ ⎢ ⎥ ⎢ ⎥ ′′ ′′ ′′ ′′ ⎣ ⎦ ∫∫ ∫∫ AA AA ## (60) The gradient of the switching plane S must be decided so that the eigenvalues of e A  becomes stable. There are a method to choose S as a feedback gain of the optimal control. Namely, S is determined as follows (Y S. Chen & Wakui (1989)): T = e SBP (61) TT =0. ee ee PA A P PB B P Q+− + (62) The nonlinear control input in reaching mode is considered. The sliding mode control is regarded as variable structure control as a required condition. So, using the switching function σ (t), the term of nonlinear control input f nl (t) is defined by ()= s g n( ( )), nl f tF t σ − (63) where F is the nonlinear controller gain; and sgn(·) the signum function. Therefore, the control input f (t) is given by Advanced Strategies for Robot Manipulators 208 1 ( ) = ( ) ( ) sgn( ( )). eef tSBSAvtF t τσ − −− (64) In order to achieve σ (t)→0 (t→∞), the Lyapunov function for the switching function is chosen as T 1 ()= () (). 2 Vt t t σσ (65) The time derivative of the the Lyapunov function defined by (65) is described with (50) and (64) by T ()= () s g n( ( )). e Vt tSBF t σσ −  (66) If ( )Vt  < 0, the switching function converges to zero. Hence, the nonlinear control gain F must satisfy the following condition: >0 : if >0 <0 : if <0. e e SB F SB ⎧ ⎨ ⎩ (67) At the neighborhood of the switching plane, the signum function raises the chattering. So, the signum function is approximated as follows: () () s g n( ( )) = , () () tt t tt σσ σ σ σδ ≈ + &&&& (68) where δ is a positive constant. As a result, (64) is rewritten into 1 () ()= ( ) () . () eef t tSBSAvtF t σ τ σ δ − −− + && (69) Because of using the unscented Kalman filter, it is necessary that the switching function σ (t) and the controller input τ (t) given by the sliding mode controller with the UKF are converted into the discretized version given by ˆ ()= () f kSvk σ (70) 1 () ˆ ()= ( ) () . () eef k kSBSAvkF k σ τ σ δ − −− + && (71) When the collision occurs, it is necessary that the flexible manipulator is suspended because of absorbing the impact of collision. The proposed flexible manipulator is controlled by tracking the reference trajectory using the sliding mode controller. For suspending the motion of the manipulator, we consider that the reference trajectory (position control) is changed into a steady position when the collision occurs. The angle position of flexible manipulator at the time t c when the collision occurs is given by ()= , cc t θ θ (72) Collision Detection and Control of Parallel-Structured Flexible Manipulators Based on Unscented Kalman Filter 209 After the collision occurred, the reference trajectory is changed into the trajectory given by the following equation: () . dc t θ θ ≡ (73) The desired position becomes the position that the flexible manipulator collides with the obstacle. Then, the flexible manipulator can be suspended at this position. 6. Simulation studies In this section, several numerical results are presented. The flexible beam is assumed to be made with the phosphor bronze. The physical parameters and the coefficients of the flexible manipulator are listed in Table 1. The observation data is supposed to be measured by piezoelectric sensors with their length b s = 3×10 –2 [m] and width 1.2×10 –2 [m] pasted at ξ 1 = 3×10 –3 [m] and potentiometers installed at each hub. The parameters of the observation system were set as c 0 = 10, c 1 = 1, e 0 = e 1 = 1. The parameter for the UKF κ was set as κ = 1. The covariance matrices for the system and observation noises were given by W = 10 –5 × I 2N+2 , V = 10 –8 × I 2 . The number of modes of the system was set as N = 2. The time partition in numerical simulation was set as Δt = 1 × 10 –3 [s]. The initial condition of state vector were set as u(0, x)=0[m], u  (0, x)=0[m/s θ  ], ˙ θ (0)=0[rad/s] and θ (0) = 0[rad]. The initial condition of the state vector of the control error system was also set as zero. The weight coefficients of cost functional for the hyperplane S were set as the values; q 1 = 100, q 2 = 100, q 3 = 100, q 4 = 5, q 5 = 4500, q 6 = 200. The nonlinear controller gain was F = 4, δ = 10 and the simulation study was carried out for 5 seconds. Parameters Values ℓ 1 E S ρ c D J 0 J 1 m h g 1 g 2 0.3[m] 1.1×10 5 [MPa] 2×10 –5 [m 2 ] 8.8×10 3 [kg/m 2 ] 4.84×10 7 [N·s/m 2 ] 5 [kg·m 2 ] 0.08[kg·m 2 ] 0.61[kg] 0.026[m] 0.05 0.4 Table 1. Physical parameters of the flexible manipulator. 6.1 Position control The simulation results of the position control in the collision-free case are shown in Figs.3-6. Fig.3 depicts the angle θ (t) and its estimate computed by the UKF ˆ ()t θ . The estimation error with respect to θ (t) sufficiently small. The controlled angle has converged at the desired position using the UKF based sliding mode control. Figs. 4 and 5 presents the observation Advanced Strategies for Robot Manipulators 210 data of the strain measured by the piezoelectric sensor y 1 (t) and the displacement of the tip mass u(t,ℓ), respectively. Furthermore, Fig.5 also depicts the estimate of the tip-mass displacement ˆ (,)utA which is calculated by =1 ˆˆ (,)= () () N ii i ut u t φ ∑ AA. The estimation error of the tip-displacement based on the noisy observation data (see Figure 4) is adequately small for suppressing the vibration of the tip-mass. Fig.6 shows the response of the control torque τ (t). 0 1 2 3 4 5 -0.2 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 Angle True value Estimate Time t[s] θ(t)[rad/s] Fig. 3. Behavior of the rotational angle θ (t) and its estimate ˆ ()t θ obtained using the UKF in the collision-free case. The solid line and the dashed line depict the true state of the angle θ (t) and its estimate ˆ ()t θ , respectively 0 1 2 3 4 5 -2 -1 0 1 2 3 4 x 10 -4 Observation (Strain) Time t [ s ] y 1 ( t ) Fig. 4. Observation data of the strain measured by the piezoelectric sensor, y 1 (t) in the collision-free case. [...]... Conference 20 08, Tokyo pp 3250– 3255 Kaneko, M.; Kanayama, N & Tsuji, T (19 98) Active antenna for contact sensing, IEEE Transactions on Robotics and Automation Vol 14(No 2): 2 78 291 Moorehead, S & Wang, D (1996) Active antenna for contact sensing, Proceedings of IEEE International Conference on Robotics and Automation pp 80 4 80 9 Julier, S.; Uhlmann, J & Durrant-Whyte, H F (2000) A new method for the nonlinear... IEEE/ASME International Conference on Advanced Intelligent Mechanics pp 177– 182 Utkin, V I (1992) Sliding modes in optimization and control problems, Springer, New York 216 Advanced Strategies for Robot Manipulators Chen, Y.-S.; Ikeda, H.; Mita, T & Wakui, S (1 989 ) Trajectory control of robot arm using sliding mode control and experimented results, Journal of the Robotics Society of Japan Vol 7(No 6):... regulation problem of robot manipulators with bounded inputs have been presented by Zavala & Santibañez (2006), Zavala & Santibañez (2007), Dixon (2007), Alvarez-Ramirez et al., (2003), and Alvarez–Ramirez et al., (20 08) An adaptive 2 18 Advanced Strategies for Robot Manipulators approach involving task–space coordinates, and considering the uncertainities of the kinematic model of the robot manipulator... PID Controllers for Industrial Robots: The PA10 Robot Arm as Case of Study 219 Fig 1 Practical nonlinear PID controller with bounded torques for robot manipulators COMPUTER DRIVER Fig 2 Variant of the practical PID controller with bounded torques for robot manipulators loop system This result guarantees that exponential stability of the classical PID linear regulator in industrial robots is preserved... prallel-structured single-link flexible manipulator and the unlooked -for obstacle occurs at tc = 1.24[s] The value of r(t) before the collision occurs is very small When the collision occurs at t = tc, the value of r(t) abruptly increases As seen in Fig 8, the 212 Advanced Strategies for Robot Manipulators 1 x 10 -11 Collision Detection Function 0.9 0 .8 0.7 0.6 r ( t) 0.5 0.4 0.3 0.2 0.1 0 0 1 2 3 4 5 Time t[s]... λmax{A(x)}, for all x ∈ Rn, that is, λmax{A} = supx∈Rn λmax{A(x)} The norm of vector x is defined as x = x T x and that of matrix A(x) is defined as the corresponding induced norm A( x ) = λmax { A( x )T A( x )} 2 Preliminaries 2.1 Robot dynamics The dynamics of a serial n–link rigid robot, without the effect of friction, can be written as (Spong & Vidyasagar, 1 989 ): 220 Advanced Strategies for Robot Manipulators. .. Sat[Sat(K p q + x + g ( q d ))] − g( q d − q ) ≡ 0 ⇒ q ≡ h 1( x ) 2 28 Advanced Strategies for Robot Manipulators Therefore, from LaSalle’s Invariance Principle we conclude that the equilibrium point ⎡q T ⎣ T T q T ⎤ = ⎡ h 1( x )T ⎦ ⎣ 0T ⎤ ∈ R 2 n of (35) is globally asymptotically stable ⎦ 5.1.2 Local exponential stability analysis Before proceeding with the stability analysis of this section, we recall... Collision estimation for a flexible cantilevered-beam subject to random disturbance, Proceedins of 34th ISCIE Int Symp on Stochastic Systems Theory and Its Application pp 183 – 188 Sawada, Y (2002c) Detection of collisions for a flexible beam subject to random disturbance, Proceedins of 41st SICE Annual Conference pp 2 68 273 Sawada, Y (2004 (in Japanese)) On-line collision detection for flexible cantilevered... Experimental testing of a gauge based collision detection mechanism for a new three-degree-of-freedom flexible robot, Journal of Robotics Systems Vol 20(No 6): 271– 284 Payo, I.; Feiu V & Cortazar, O D (2009) Force control of very lightweight single-link flexible arm based on coupling torque feedback, Mechatronics Vol 19: 334–337 Kondo, J & Sawada, Y (20 08) Collision detection and suspend control of parallel-structured... bounded for all q ∈ Rn This means that there exist finite constants γ i ≥ 0 such that (Craig, 19 98) : sup gi ( q ) ≤ γ i q ∈R n i = 1, 2, , n, (3) where gi(q) stands for the i-th element of g(q) Equivalently, there exists a constant k′ such that g(q) ≤ k′ , for all q ∈ Rn Furthermore, there exists a positive constant kg such that ∂ g( q ) ≤ kg , ∂q for all q ∈ Rn, and g(x) − g(y) ≤ kg x − y , for all . al., (20 08) . An adaptive Advanced Strategies for Robot Manipulators 2 18 approach involving task–space coordinates, and considering the uncertainities of the kinematic model of the robot manipulator. Preliminaries 2.1 Robot dynamics The dynamics of a serial n–link rigid robot, without the effect of friction, can be written as (Spong & Vidyasagar, 1 989 ): Advanced Strategies for Robot Manipulators. Conference on Advanced Intelligent Mechanics pp. 177– 182 . Utkin, V. I. (1992). Sliding modes in optimization and control problems, Springer, New York. Advanced Strategies for Robot Manipulators