Using the finite - Time disturbance observer (fto) for robotic manipulator almega 16

Nội dung text: Using the finite - Time disturbance observer (fto) for robotic manipulator almega 16

1. KHOA H ỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 USING THE FINITE - TIME DISTURBANCE OBSERVER (FTO) FOR ROBOTIC MANIPULATOR ALMEGA 16 DÙNG BỘ QUAN SÁT NHIỄU VỚI THỜI GIAN HỮU HẠN CHO TAY MÁY ROBOT ALMEGA 16 Vo Thu Ha machine not to stick exactly to the given trajectory.To ABSTRACT know the exact external noise components, it is necessary This paper presents build a finite time observator (FTO) and applies it to the to incorporate a noise observation device (DOB) to Almega16 robot motion system. The main content of the article is to design a estimate these disturbances. When applying the DOB noise FTO so that the observation of the external noise of the Almega16 robot motion monitor in the mechanical hand movement system, control system will converge to the desired true value over a period of time. finite, is can be based on the noise monitor [1-3], estimate and done by estimating the external noise quantities and then feeding them into the compensate the friction component [4,5], control force or available Robot controller. The advantage when applying the FTO disturbance tissue. non-sensor torque [6-8], error diagnosis and monitor is that it is possible to eliminate the inverse inertia matrix component in isolation (FDI) [9-11]. The DOB turbulence monitor has been the dynamic equation. The results achieved showed that the Almega16 robot widely used in hand machine motion control for a variety of movement system ensures that the errors of the rotating joints quickly reach purposes. The basic idea of DOB is to use the motion state zero with a small transition time, making the closed system stable according to variables of the robot and the torque of the joints as input Lyapunov standards. values and then estimate all the unknown internal and Keywords: Robot Almega 16, Finite - time observer, Lyapunov standards. external torque. In [5], the Nonlinear Noise Observer (NDOB) was established to estimate the friction component TÓM TẮT so that accurate real friction component values can be Bài báo trình bày xây dựng bộ quan sát nhiễu với thời gian hữu hạn (FTO) và known with fast time. The NDOB is done by choosing a ứng dụng cho hệ chuyển động Robot Almega16. Nội dung chính bài báo là thiết certain nonlinear function. But the downside of the NDOB kế bộ quan sát nhiễu với thời gian hữu hạn (FTO) sao cho việc quan sát các nhiễu is that choosing such a nonlinear function is not ngoại của hệ thống chuyển động Robot Almega16 sẽ hội tụ về giá trị thực mong straightforward. In [9], the use of the generalized muốn với một khoảng thời gian hữu hạn, được thực hiện bằng cách là ước lượng momentum observer (GMO) has the advantage of not only các đại lượng nhiễu ngoại sau đó đưa vào bộ điều khiển Robot có sẵn. Ưu điểm avoiding acceleration calculations to reduce the effect of khi ứng dụng bộ quan sát nhiễu FTO là có thể loại bỏ thành phần ma trận quán noise in site measurements, but also creating disturbance tính nghịch đảo trong phương trình động lực học. Kết quả đạt được, cho thấy hệ observations at superlative form. GMOs are able to realize chuyển động Robot Almega16 đảm bảo sai số của các khớp quay nhanh chóng FDI such as predicting random effects as well as saturation đạt tới không với thời gian quá độ nhỏ, làm cho hệ thống kín ổn định theo tiêu actuator error. The GMO Observer is easy to implement and chuẩn Lyapunov. has reliable results and the GMO has become a popular and Từ khóa: Robot Almega 16, bộ quan sát nhiễu với thời gian hữu hạn, tiêu widely used method in many hand-operated applications. chuẩn Lyapunov. However, the downside of GMOs is that the failure to return diagnostic results and slow response isolation (FDI) results University of Economics - Technology for Industries in reduced sensitivity and response speed when the GMO is Email: vtha@uneti.edu.vn used in the case of collision detection. In [10] there was a Received: 16/4/2021 solution for the GMO set, by treating the collision detection Revised: 20/5/2021 case as an extrinsic perturbation. Although many DOB Accepted: 25/6/2021 observers have been developed and used for mechanical hand movement systems [5,9,10,13,14]. However, this DOB observer shows that the asymptotic convergence rate and 1. INTRODUCTION the estimated bias of the perturbations will not converge In the kinetic equation of industrial manipulator [26], quickly to zero. So for the conventional DOB convergence there are always external noise components and internal rate is is best exponentially while the FTO can achieve a noise. Especially the external noise components inside are faster convergence rate with convergence in finite time. unknown or not exactly known and these are the Given their finite time characteristics, a number of FTOs components that cause the movement of the hand have been designed and applied to different systems with 44 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 57 - Số 3 (6/2021) Website:
2. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY versatile applications [15-20]. In this paper, a new DOB is planetary gear and Small air gap. It is influenced by friction shown. form is based on the requirement of fast and such as static friction, friction, viscous friction and so on. accurate estimation of robot disturbances. Based on the Therefore, the first three joints are the basic chain that robot's dynamic model, the finite time control concept is ensures movement in 3D (X, Y, Z) space. The basis for the used in the observer design. The resulting FTO can allow for study of the next steps in robot manipulator motion estimation of the disturbance for a finite time, this ensures systems. The problem with the controller is that: should that the estimated error can disappear after a certain time. design the quality control ensures precise orbit grip that The proposed FTO also eliminates the need for acceleration does not depend on the parameters of the model calculation. Its finite-time convergence feature makes uncertainty and the impact on channel mix between observing perturbations more accurate and fast. The match-axis error between joint angles and the angle joints structure of the paper is presented as follows: Part 1, actually put a small (< 0.1%). problematic, Part 2 is the introduction of the Almega robot 3. PRELIMINARIES control object 16. Part 3 is detailed about robot dynamics 3.1. Robot Dynamic Model and stability in time. finite time. Part 4 is an introduction to the FTO Observer. Part 5 is the application of the FTO The dynamic of an n-link rigid manipular, [1-4, 6] can be controller to the Almega 16. Robot motion system. Part 6 is written as    the conclusion. τ + τd = M(q)q + C(q, q)q+ G(q)+ τ f (1) 2. OBJECT CONTROL Where q is the n x 1 joint variable vector, τ is a n x 1 The Almega 16 robot is shown in Figure 1, as follows generalized torque vector M(q) is the nxn inertia matrix, [25]. This is a vertical welding robot with fast, rhythmic and H(q,q) is the n x 1 Coriolis/centripetal vector, G(q) is the n n precise movement characteristics, including six–link axes, x1 gravity vector. τd R denotes the lumped friction effect each one link axes is equipped with a permanent magnet from both the motor and link sides and are always synchronous servo motor and closed loop control. In the described with the following Coulomb-viscous model, article using only three-link axes as the research object, namely specifically the main specifications of the three joints as τ = F sgn(q) +F q  (2) follows. f c v with Fc diag F c1, , F cn , F v diag F v1 , , F vn , F ci , F vi ( 1 i n) are the Coulomb and viscous friction coefficients for the ith joint. Such a friction model could capture most dynamic property of the friction in a rigid joint. The equivalent motor torque at the link side through a reduced n n amplification is denoted as τ RR, τd is the internal/external disturbances which could be an external force, unmodeled or uncertain robot dynamics. The exact meaning of τd decides on the specific application. The observed disturbance for a manipulator can be further utilized in FDI and disturbance rejection control. For example, τ is deemed as the physical impact with the d environment for collision detection scenario and, thus, τ Figure 1. Six-link Almega 16 arm d can indicate the occurrence of the collision. The robot 0 First joint: Rotation angle: 135 . Center tops from top dynamics model in Equation (1) has the following property. to bottom: 28cm. Center line of axis I to the center of the In which 1: The matrix M(q) - 2C(q,q) is skew-symmetry cylinder: 35cm. Second joint: Rotation angle: 1350. The [21], and it follows that length between the center of the axis I and II is 65cm. Third T joint: Angle of rotation: 900 and -450. The length between M(q) = C(q, q) + C (q,q) (3) the two centers of axis I and II is 47cm. The total volume of 3.2. Disturbance Observer GMO 3 the Almega16 Robot: V = 0.12035m . Total weight of the In order to estimate the external beeb-type noise robot: 250kg. The mass of joints is as follows: m0 = 100kg, components for hand-operated systems, there are many m1 = 67kg, m2 = 52kg, m3 = 16kg, m4 = 10kg, m5 = 4kg, m6 = 1kg. different monitors. One of the most commonly used The motion system Almega16 Robot is a nonlinear observers is the observed (GMO) observed in Reference [9]. system that has constant model parameters and is Combined with the generalized momentum p, in Equation interfering with the channel between the component (1) could be rewritten to motion axes. According to the literature as follows [26], the T   p τ +C (q,q)q- G(q)+ τd (4) first three joints have fully integrated the dynamics of the freedom arm. The motor connected to the joint is usually a The GMO component p is estimated as follows: Website: Vol. 57 - No. 3 (June 2021) ● Journal of SCIENCE & TECHNOLOGY 45
3. KHOA H ỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 pˆ = τ +CT (q,q)q-   G(q)+ τˆ (5)  -1 α2 d zb2 = z b3 + M (q)τ a +K 2 e b (14) τˆ = K (p - p)ˆ (6) α3 d 0  zb3 = K 3 e b (15) where ( . )denotes the estimated value and ˆ ˆ -1 Where zb1 = q,ˆ z b2 = q, z b3 = M (q)τ d and K0 diag k 0i 0 . So the estimate of the external ˆ nxn disturbance is given is: e=q-qK,K,Kb, 1 2 3 R are diagonal gain matrices. ˆ T   ˆ (7) Moreover, the corresponding powers are selected as τd = K 0 (τ + C (q,q)q- G(q)+ τ d )dt 2 From equations (5) and (6), are determined: α1 =α,α 2 =2α-1,α 3 =3α-1 and 0 (16) K 0 Consequently, the disturbance observation τˆ d is τˆ d = τ d (9) s +K0 computed as According to reference [9] shows that the component τˆ d = M(q)z b3 (17) τˆ is a first order inertial function τ . So the external d d From (17) shows, the proposed FTO is a ternary system perturbation estimation τˆ d component of the GMO will that can simultaneously estimate the joint velocity and the converge exponentially and depend on the observation external perturbation component. It shows that the joint matrix Ko. Therefore, the GMO observer always has an velocity can be obtained instantly from the robot control estimated bias in the outer perturbations. system. From the formula (13) - (15), it is possible to 3.3. Consider steady state in finite time downgrade the observational equation for external disturbance state variables leading to a reduction in the Consider the following nonlinear system computational heavy process. Therefore, the downgrade x = f(x), x Rn ,f(0) = 0 (10) FTO is determined as follows: where f satisfies the locally Lipschitz continuous  -1 α1 zr1 = z r2 + M (q)τ a +K 1 e r (18) condition. Some basic knowledge about finite time  α2 homogeneity and stability (FTS) in the document [22,23]. zr2 = K 2 e r (19) 4. FINITE-TIME OBSERVER OF ROBOTIC DISTURBANCE -1 Where zb1 = q,zˆ b2 = M (q)τ d and The main content of this paper is to design a finite time 1 observer so that the observation of the noise td can e = q- q,ˆ α = α, α = 2α -1, < α < 1. From formula (19), b 1 2 2 converge to its true value in a finite time. In this section 4 we determine the formula to calculate the estimate of the will be presented on the content of constructing the FTO external disturbance is determined as follows: observer to estimate the external perturbations. After the ˆ estimation is complete, the state variables estimate the τd M(q)z r2 (20) external perturbations to the existing control system such The decremented FOT observer will estimate the as the PID controller, When the FTO Observer is perturbed state variables faster than the original connected, the calculation and elimination will be reduced. unremarked design. And from formula (12) shows that still remove the inverse inertial matrix in the kinetic equation. exists the inverse matrix component of M(q). To remove 4.1. Finite-Time Observer Design the inverse inertial matrix component of M(q), it is From Equation (1), the acceleration q can be written as necessary to rearrange the original system from equation (12) into a transformed equation with different state  1 1   q= M (q)τd M (q)(τ C(q,q)q- G(q) - τ f ) (11) variables. Multiplying both sides of Equation (12) by M(q)   yields the following: Put: τa = τ - C(q, q)q- G(q) - τ f , then (11) is rewritten as M(q)q = τ + τ (21) follows as: a d q = M-1 (q)τ +M -1 (q)τ (12) Additionally, the left side of Equation (21) could be d a altered using the generalized momentum p, namely -1 where M (q)τd is treated as the system disturbances   p - M(q)q = τa + τ d (22) with M-1 (q)τ the system input. According to reference a Reorganizing Equation (22) and employing Property 1, [20], the FTO monitor for manipulators is specifically the derivative of the generalized momentum p is rewritten designed as follows: as  α1  zb1 = z b2 +K 1 e b (13) p = τd + τ p (23) 46 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 57 - Số 3 (6/2021) Website:
4. P-ISSN 1859-3585 E-ISSN 2615-9619 SCIENCE - TECHNOLOGY Where τ = M(q)q+  τ = τ +φ(q,q)  = CT (q,q)q-   G(q) - τ . V p a f   V =T e The system should observe that the externally e n α2 implemented perturbation variables have been altered and α α 1T 2 have different state variables. A set of FTOs reduced from = k2i e pi e di - K 1 e pi - e d K 2 e pi i=1 (29) tertiary to quadratic is replaced as follows: n TTα2 α 1 +α 2 α 2 α1 = e K e - k k e - e K e z = z + τ + K e (24) d 2 pi  1i 2i pi d 2 pi m1 m2 p 1 m i=1 n α α +α z = K e 2 (25) 1 2 m2 2 m = - k1i k 2i e pi 0 i=1 ˆ ˆ where zm1 = q, z m2 = τˆ d and em = q- q . Where k1i is a positive defined diagonal element of K1 The control structure diagram with FTO is shown in and such that ep = ed = 0. Apply to LaShalle's theorem, to figure 2. ensure that the asymptotic convergence of the deviation e to 0 is guaranteed. The next content of the paper will present about demonstrating the observer's finite convergence of time. According to Definition 2 [27], Equations (28) and (29) are orderly homogeneous with respect to weight. Hence, consider equations (28) and (29) that have a negative identity. From Theorem 1, [27], the error system is the global FTO. In other words, the estimated deviations of the turbulent state variables will disappear for a finite time. From that it can be concluded, the proposed downgrade FTO is stable and with convergent efficiency in finite time 5. SIMULATION RESULTS Afer building up the algorithms and control programs, we will proceed to run the simulation program to test computer program. The FTO was Simulink with Table 1. Figure 2. The control structure diagram with FTO of robotic disturbance Table 1. The Parameter of FTO finite-time observer The Parameter value of the Symbol The parameter The obtained FTO given in Equations (24) and (25) is joint axis structurally similar to the GMO defined in Equations (5) and q Desired joint position q = q = q = sint (6) as both observer shares the same system states and d d1 d2 d3 input. The obtained FTO observer is represented in K1 Scalar K1 = [200, 200, 200] Equations (26) and (27) and has an estimation structure of K2 Constant K2 = [10000, 10000, 10000] external perturbation variables similar to that identified in α Power coefficient α = 1 Equations (5) and (6), since both of these observers use system state variables and as input signals. τd Disturbance sint 4.2. Consider the stability and convergence of the FTO After simulation we have results position and position tracking error is depicted Figure 3÷ 7. The observation errors are given as: α * Desired joint position is sin(t)  1 ep = e d -K 1 e p (26) α  2 ed = -K 2 e p (27) Where ep = p -p,eˆ d = τ d -τˆ d ,e=[e p ,e] d . Using Lyapunov standard to prove the stability of the FTO is proposed as follows; Given a positive defined function, (28): nTT n epi α e ek e e V = k τ 2 dτ +d d =2i eα2 +1 + d d (28) 2i 0  pi i=12 i=1 α2 +1 2 where k2i is the ith diagonal element of K2. Then its derivative is Joint 1 Website: Vol. 57 - No. 3 (June 2021) ● Journal of SCIENCE & TECHNOLOGY 47
5. KHOA H ỌC CÔNG NGHỆ P-ISSN 1859-3585 E-ISSN 2615-9619 0.004 0.003 0.002 0.001 0 -0.001 -0.002 e = q q 3e2 = d3q d2 3 q 2 -0.003 -0.004 0 1 2 3 4 5 6 7 8 9 10 Joint 2 Time(s) Figure 4. Express the response between the set angles q and real q 3000 2500 2000 1500 1000 500 0 -500 -1000 -1500 -2000 Joint 3 20 40 60 80 100 120 140 160 180 200 q q d1 Figure 5. Express the control moments of the controller joints Figure 3. Performing deviation between the angles q set and q real 1.2 0.004 1 0.003 0.8 0.002 0.6 0.001 0.4 Joint Angle (Rad) (Rad) Angle Joint 0 0.2 -0.001 0 -0.002 e1 = q d1 q 1 -0.2 0 2 4 6 8 10 12 14 16 18 20 -0.003 Time (s) -0.004 Figure 6. Performing the response of the actual joint position and the actual 0 1 2 3 4 5 6 7 8 9 10 joint position Time(s) 0.04 0.004 0.035 0.003 0.03 0.002 0.025 0.001 0 0.02 0.015 -0.001 -0.002 0.01 e2 = q d2 q 2 -0.003 0.005 -0.004 0 0 1 2 3 4 5 6 7 8 9 10 0 5 10 15 20 Time(s) Time(s) Figure 7. The deviation response controls the actual joint position and estimated joint position 48 Tạp chí KHOA HỌC VÀ CÔNG NGHỆ ● Tập 57 - Số 3 (6/2021) Website:
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