Design Artificial Intelligence-Based Switching PD plus

I.J. Engineering and Manufacturing, 2013, 1, 38-57 Published Online June 2013 in M ECS (http://www.mecs-press.net) DOI: 10.5815/ ijem.2013.01.04 Available online at http://www.mecs-press.net/ijem

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System Farzin Piltana, Mahdi Jafarib, Mehdi Eramb, Omid Mahmoudib, Omid Reza Sadrniab a

Senior Researcher at Research and Development Department of SanatkadeheSabze Pasargad Company (S.S.P. Co), Shiraz, Iran b Research and Development Unit, SanatkadeheSabze Pasargad Company (S.S.P. Co), Shiraz, Iran

Abstract Refer to this research, an intelligent fuzzy parallel switching Proportional-Derivative (PD) p lus gravity controller is proposed for highly nonlinear continuum robot manipulator. Design a nonlinear controller for second order nonlinear uncertain dynamical systems is one of the most impo rtant challenging works. In order to provide high performance in nonlinear systems, switching partly slid ing mode plus gravity controller is selected. Pure switching partly slid ing mode p lus gravity controller can be used to control of partly known nonlinear dynamic parameters of continuum robot manipulator. Conversely, this method is used in many applications; it must to solve chattering phenomenon which it can cause some problems such as saturation and heat the mechanical parts of continuum robot man ipulators or drivers. In order to solve the chattering phenomenon, implement easily and avoid mathemat ical model base controller, Mamdani’s performance/erro rbased fuzzy logic methodology with two inputs and one output and 49 rules is parallel applied to pure switching partly sliding mode plus gravity controller. The results demonstrate that this method is a model-free controllers which works well in certain and partly uncertain system. Index Terms: Fu zzy inference system, Mamdani fu zzy inference engine, slid ing mode control, continuum robot, PD p lus gravity. © 2013 Published by MECS Publisher. Selection and/or peer review under responsibility of the Research Association of Modern Education and Computer Science.

1. Introduction The international organization defines the robot as “an automatically controlled, reprogrammable, mu ltipurpose manipulator with three or mo re axes.” Robot manipulator is a collection of lin ks that connect to each other by joints, these joints can be revolute and prismat ic that revolute joint has rotary mot ion around an axis and prismat ic joint has linear mot ion around an axis. Each jo int provides one or more degrees of freedo m * Corresponding author. E-mail address: *[email protected]; *2 [email protected]

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

39

(DOF). Fro m the mechanical point of view, robot man ipulator is div ided into two main groups, which called; serial robot links and parallel robot links. In serial robot man ipulator, links and jo ints is serially connected between base and final frame (end-effector). Most of continuum robots are serial links, which in 𝑛𝑛 degrees of freedom serial lin k robot manipulator the axis of the first three jo ints has a known as major axis, these axes show the position of end-effector, the axis number four to six are the minor axes that use to calculate the orientation of end-effector and the axis nu mber seven to 𝑛𝑛 use to reach the avoid the difficu lt conditions (e.g., surgical robot and space robot manipulator). Dynamic modeling of continuu m robot manipulators is used to describe the behavior of continuum robot manipulator such as linear or nonlinear dynamic behavior, design of model based controller such as pure sliding mode controller and pure computed torque controller which design these controller are based on nonlinear dynamic equations, and for simulat ion. The dynamic modeling describes the relationship between joint motion, velocity, and accelerations to force/torque or current/voltage and also it can be used to describe the particular dynamic effects (e.g., inertia, corio lios, centrifugal, and the other parameters) to behavior of system[1]. Continuum robots represent a class of robots that have a biologically inspired form characterized by flexib le backbones and high degrees-of-freedo m structures [1]. The idea of creating “trunk and tentacle” robots, (in recent years termed continuu m robots [1]), is not new [2]. Inspired by the bodies of animals such as snakes [3], the arms of octopi [4], and the trunks of elephants [5], [6], researchers have been building prototypes for many years. A key motivation in this research has been to reproduce in robots some of the special qualities of the bio logical counterparts. This includes the ability to “slither” into tight and congested spaces and (of part icular interest in this wo rk) the ab ility to grasp and man ipulate a wide range of objects, via the use of “whole arm man ipulation” i.e. wrapping their bodies around objects, conforming to their shape profiles. Hence, these robots have potential applicat ions in whole arm grasping and manipulat ion in unstructured environments such as rescue operations. Theoretically, the compliant nature of a continuum robot provides infinite degrees of freedom to these devices. However, there is a limitation set by the practical inability to incorporate infinite actuators in the device. Most of these robots are consequently underactuated (in terms of nu mbers of independent actuators) with respect to their anticipated tasks. In other wo rds they must achieve a wide range of configurations with relat ively few control inputs. This is partly due to the desire to keep the body structures (which, unlike in conventional rigid-link man ipulators or fingers, are required to directly contact the environment) “clean and soft”, but also to exploit the ext ra control authority available due to the continuum contact conditions with a min imu m number of actuators. For examp le, the Octarm VI continuum manipulator, discussed frequently in this paper, has nine independent actuated degrees-of-freedom with only three sections. Continuum man ipulators differ fundamentally fro m rigid-link and hyper-redundant robots by having an unconventional structure that lacks lin ks and joints. Hence, standard techniques like the Denavit-Hartenberg (D-H) algorith m cannot be directly applied for developing continuum arm kinematics. Moreover, the design of each continuum arm varies with respect to the flexib le backbone present in the system, the positioning, type and number of actuators. The constraints imposed by these factors make the set of reachable configurations and nature of movements unique to every continuum robot. This makes it difficult to formulate generalized kinematic or dynamic models for continuum robot hardware. Chirikjian and Burd ick were the first to introduce a method for modeling the kinemat ics of a continuum structure by representing the curve-shaping function using modal functions [6]. Mochiyama used the Serret- Frenet formulae to develop kinematics of hyper-degrees of freedom continuum manipulators [5]. For details on the previously developed and more manipulator-specific kinemat ics of the Rice/Clemson “Elephant trunk” man ipulator, see [1], [2], [5]. For the Air Octor and Octarm continuum robots, more general forward and inverse kinematics have been developed by incorporating the transformations of each section of the manipulator (using D-H parameters of an equivalent virtual rigid link robot) and expressing those in terms of the continuum manipulator section parameters [4]. The net result of the work in [6], [3]-[5] is the establishment of a general set of kinematic algorith ms fo r continuum robots. Thus, the kinemat ics (i.e. geo metry based modeling) of a quite general set of prototypes of continuum man ipulators has been developed and basic control strategies now exist based on these. The development of

40

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

analytical models to analyze continuum arm dynamics (i.e. physicsbased models involving forces in addition to geometry) is an active, ongoing research topic in this field. Fro m a p ractical perspective, the modeling approaches currently available in the literature prove to be very comp licated and a dynamic model which could be conveniently imp lemented in an actual device’s real-time controller has not been developed yet. The absence of a computationally tractable dynamic model for these robots also prevents the study of interaction of external forces and the impact of collisions on these continuum structures. This impedes the study and ultimate usage of continuum robots in various practical applicat ions like grasping and manipu lation, where impulsive dynamics [1], [4] are important factors. Although continuum robotics is an interesting subclass of robotics with promising applicat ions for the future, fro m the current state of the literature, this field is still in its stages of inception. Controller is a device which can sense information fro m linear or nonlinear system (e.g., continuum robot man ipulator) to improve the systems performance [3]. The main targets in designing control systems are stability, good disturbance rejection, and small t racking error[5]. Several continuum robot manipulators are controlled by linear methodologies (e.g., Proportional-Derivative (PD) controller, Proportional- Integral (PI) controller or Proportional- Integral-Derivative (PID) controller), but when continuum robot manipulator works with various payloads and have uncertainty in dynamic models this technique has limitations. In some applications continuum robot manipu lators are used in an unknown and unstructured environment, therefore strong mathemat ical tools used in new control methodologies to design nonlinear robust controller with an acceptable performance (e.g., minimu m error, good trajectory, disturbance reject ion). Slid ing mode controller is an influential nonlinear controller to certain and uncertain systems which it is based on system’s dynamic model. Sliding mode controller is used to control of highly nonlinear systems especially fo r continuum robot man ipulators. The important problem of the pure sliding mode controller with switching function was chattering phenomenon in certain and uncertain systems. In th is research the chattering phenomenon problem can be reduced in certain system by using parallel fuzzy log ic theory. To eliminate the continuum robot man ipulator’s dynamic of system and chattering, 49 rules Mamdani inference system is design and applied to PD partly sliding mode plus gravity methodology with switching function. This methodology is worked based on applied fu zzy logic parallel with gravity nonlinear dynamic part to eliminate unknown dynamic parameters. This paper is organized as fo llows; section 2, is served as an introduction to the PID controller, feedback linearization co mpensator, Mamdani fuzzy inference engine to estimate the FLC and its application to control of continuum robot and dynamic of continuum robot. Part 3, introduces and describes the methodology. Section 4 presents the simulation results and discussion of this algorithm applied to a continuum robot and the final section is describing the conclusion. 2. Theory 1.1. Dynamic Formulation of Continuum Robot The Continuu m section analytical model developed here consists of three modules stacked together in series. In general, the model will be a more precise replication of the behavior of a continuum arm with a greater of modules included in series. However, we will show that three modules effectively represent the dynamic behavior of the hardware, so more co mplex models are not motivated. The kinetic energy (T) of the system comprises the sum of linear kinetic energy terms (constructed using the above velocities) and rotational kinetic energy terms due to rotation of the rig id rod connecting the two actuators, and is given below as

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

2

41

2

T = (0.5) m1 ṡ 1 2 + (0.5)m2 ��ṡ 2 sinθ1 + s 2 cosθ1 θ̇1 � + �ṡ 1 + ṡ 2 cosθ1 − s 2 sinθ1 θ̇1 � �

+ (0.5) m3 ��ṡ 2 sinθ1 + s 2 cosθ1 θ̇1 + ṡ 3 sin(θ1 + θ2 ) + s 3 cos (θ1 + θ2 ) θ̇1 + s 3 cos (θ1 + θ2 )θ̇ 2 � 2

2

+ �ṡ 1 + ṡ 2 cosθ1 − s 2 sinθ1 θ̇1 + ṡ 3 cos (θ1 + θ2 ) − s 3 sin(θ1 + θ2 ) θ̇1 − s 3 sin(θ1 + θ2 )θ̇ 2 � � 2 2 2 + (0.5) I1 θ̇1 + (0.5) I2 � θ̇1 + θ̇2 �

2 2 + (0.5) I3 � θ̇1 + θ̇ 2 2 + θ̇3 �

(1)

The potential energy (P) of the system co mprises the sum of the grav itational potential energy and the spring potential energy. A small angle assumption is made throughout the derivation. This allows us to directly express the displacement of springs and the velocities associated with dampers in terms of system generalized coordinates. 2 1 P = −m1 gs 1 − m2 g( s 1 + s 2 cosθ1 ) − m3 g �s 1 + s 2 cosθ1 + s 3 cos (θ1 + θ1 )� + (0.5) k 11 �s 1 + � � θ1 − s 01 � 2 + (0.5) k 21 (s 1 + ( 1⁄2 ) θ1 − s 01 )2 + (0.5)k 12 ( s 2 + ( 1⁄2 )θ2 − s 02 )2 +(0.5)k 22 ( s 2 + ( 1⁄2 ) θ2 − s 02 )2 + (0.5) k 13 (s 3 + ( 1⁄2 ) θ3 − s 03 )2 + (0.5)k 23 ( s 3 + ( 1⁄2 ) θ3 − s 03 ) 2 (2) where,S01 , S02 , S03 are the in itial values ofS1 , S2 , S3 respectively. Due to viscous damping in the system, Rayliegh’s dissipation function [6] is used to give damping energy 2

2

2

D = (0.5) c11 �ṡ 1 + (1/2) θ̇ 1 � + (0.5) c21 �ṡ 1 + (1/2)θ̇ 1 � + (0.5) c12 �ṡ 2 + (1/2) θ̇2 � + (0.5)c22 �ṡ 2 + (1/2)θ̇ 2 � 2 + (0.5)c13 �ṡ 3 + (1/2) θ̇ 3 � 2

+ (0.5)c23 �ṡ 3 + (1/2) θ̇3 � .

(3)

𝑄𝑄𝑠𝑠1 = 𝐹𝐹11 + 𝐹𝐹21 + ( 𝐹𝐹12 + 𝐹𝐹22 )𝑐𝑐𝑐𝑐𝑐𝑐𝜃𝜃1 + ( 𝐹𝐹13 + 𝐹𝐹23 )𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 + 𝜃𝜃2 )

(4)

The generalized forces in the system corresponding to the generalized co-o rdinates are exp ressed as appropriately weighted combinations of the input forces.

𝑄𝑄𝑠𝑠2 = 𝐹𝐹12 + 𝐹𝐹22 + ( 𝐹𝐹13 + 𝐹𝐹23 )𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 )

(5)

𝑄𝑄𝜃𝜃1 = ( 1⁄2 )( 𝐹𝐹11 − 𝐹𝐹21 ) + (1⁄2 )( 𝐹𝐹12 − 𝐹𝐹22 ) + (1⁄2 )( 𝐹𝐹13 − 𝐹𝐹23 ) + 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠𝜃𝜃2 (𝐹𝐹13 + 𝐹𝐹23 )

(7)

𝑄𝑄𝑠𝑠3 = 𝐹𝐹13 + 𝐹𝐹23

𝑄𝑄𝜃𝜃1 = ( 1⁄2 )( 𝐹𝐹12 − 𝐹𝐹22 ) + (1⁄2 )( 𝐹𝐹13 − 𝐹𝐹23 ) 𝑄𝑄𝜃𝜃1 = ( 1⁄2 )( 𝐹𝐹13 − 𝐹𝐹23 )

(6) (8) (9)

2

42

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

It can be ev inced fro m the force exp ressions that the total input forces acting on each module can be resolved into an additive co mponent along the direction of extension and a subtractive component that results in a torque. For the first module, there is an additional torque produced by forces in the third module. The model resulting fro m the application of Lagrange’s equations of motion obtained for this system can be 𝐹𝐹𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐𝑐 𝜏𝜏 = 𝐷𝐷 �𝑞𝑞 � 𝑞𝑞̈ + 𝐶𝐶 �𝑞𝑞 � 𝑞𝑞̇ + 𝐺𝐺 � 𝑞𝑞�

(10)

represented in the form

where τ is a vector of input forces and q is a vector of generalized co-ordinates. The force coefficient matrix Fcoeff transforms the input forces to the generalized forces and torques in the system. The inertia matrix, D is composed of four block matrices. The block matrices that correspond to pure linear accelerations and pure angular accelerations in the system (on the top left and on the bottom right) are symmetric. The matrix C contains coefficients of the first order derivatives of the generalized co-ordinates. Since the system is nonlinear, many elements of C contain first order derivatives of the generalized co-o rdinates. The remaining terms in the dynamic equations resulting from gravitational potential energies and spring energies are collected in the matrix G. The coefficient mat rices of the dynamic equations are given below, 1 ⎡ ⎢ 0 0 𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹𝐹 = ⎢ ⁄ 1 ⎢ 2 ⎢ 0 ⎣ 0 𝐷𝐷 �𝑞𝑞 � =

1 0 0 −1⁄2 0 0

⎡ 𝑚𝑚1 + 𝑚𝑚2 ⎢ +𝑚𝑚3 ⎢ ⎢ 𝑚𝑚2 𝑐𝑐𝑐𝑐𝑐𝑐 ( 𝜃𝜃1 ) ⎢ ⎢ +𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐 ( 𝜃𝜃1 ) ⎢ ⎢ 𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 + 𝜃𝜃2 ) ⎢ ⎢ ⎢ −𝑚𝑚2 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃1 ) ⎢ −𝑚𝑚3 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃1 ) ⎢ −𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 + 𝜃𝜃2 ) ⎢ ⎢ ⎢ −𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 + 𝜃𝜃2 ) ⎢ ⎢ ⎣ 0

𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 ) 1 0 1⁄2 1⁄2 0

𝑐𝑐𝑐𝑐𝑐𝑐( 𝜃𝜃1 ) 𝑐𝑐𝑐𝑐𝑐𝑐 ( 𝜃𝜃1 + 𝜃𝜃2 ) 1 𝑐𝑐𝑐𝑐𝑐𝑐( 𝜃𝜃2 ) 0 1 −1⁄2 1⁄2 + 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 ) −1⁄2 1⁄2 0 1⁄2

𝑚𝑚2 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 ) +𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 )

𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐 ( 𝜃𝜃1 + 𝜃𝜃2 )

𝑚𝑚2 + 𝑚𝑚3

𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐( 𝜃𝜃2 )

𝑚𝑚3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 )

𝑚𝑚3

−𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )

𝑚𝑚3 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )

−𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )

0

0

0

𝑐𝑐𝑐𝑐𝑐𝑐( 𝜃𝜃1 + 𝜃𝜃2 ) ⎤ 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 ) ⎥ 1 ⎥ −1⁄2 + 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃2 ) ⎥ ⎥ −1⁄2 ⎦ −1⁄2

(11)

−𝑚𝑚2 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 ) −𝑚𝑚3 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 ) −𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃1 + 𝜃𝜃2 )

−𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃1 + 𝜃𝜃2 )

𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃2 )

0

−𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 ( 𝜃𝜃2 )

−𝑚𝑚3 𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )

𝑚𝑚2 𝑠𝑠22 + 𝐼𝐼1 + 𝐼𝐼2 +𝐼𝐼3 + 𝑚𝑚3 𝑠𝑠22 + 𝑚𝑚3 𝑠𝑠32 +2𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐(𝜃𝜃2 ) 𝑠𝑠2

𝐼𝐼2 + + 𝐼𝐼3 +𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐( 𝜃𝜃2 )𝑠𝑠2

𝐼𝐼3

𝐼𝐼3

𝐼𝐼2 + 𝑚𝑚3 𝑠𝑠32 + 𝐼𝐼3 +𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 ) 𝑠𝑠2 𝐼𝐼

𝑚𝑚3 𝑠𝑠32

𝐼𝐼2 + 𝑚𝑚3 𝑠𝑠32 + 𝐼𝐼3

⎤ 0⎥

⎥ ⎥ ⎥ 0⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ 𝐼𝐼3 ⎥ ⎥ ⎥ ⎥ 𝐼𝐼3 ⎥ ⎥ ⎥ 𝐼𝐼3 ⎦

(12)

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System 𝐶𝐶 �𝑞𝑞�

⎡ ⎢ ⎢ −2𝑚𝑚2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1)𝜃𝜃1̇ ⎢ 𝑐𝑐 11 + 𝑐𝑐 21 −2𝑚𝑚3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1)𝜃𝜃1̇ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ 0 𝑐𝑐 12 + 𝑐𝑐 22 ⎢ ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ 0 2𝑚𝑚3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2)�𝜃𝜃1̇ � ⎢ ⎢ ⎢ 2𝑚𝑚3 𝑠𝑠3𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 )�𝜃𝜃̇1 � ⎢ ⎢ (1 ⁄ 2 ) −2𝑚𝑚3𝑠𝑠2 �𝜃𝜃1̇ � ⎢(𝑐𝑐 11 + 𝑐𝑐 21) +2𝑚𝑚2𝑠𝑠2 �𝜃𝜃1̇ � ⎢ ⎢ ⎢ (1 ⁄2)(𝑐𝑐12 + 𝑐𝑐 22) + ⎢ 0 2𝑚𝑚3 𝑠𝑠3𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 )�𝜃𝜃̇1 � ⎢ ⎢ ⎢ 0 0 ⎣

−2𝑚𝑚3𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 + 𝜃𝜃2 ) �𝜃𝜃1̇ + 𝜃𝜃̇2 �

−2𝑚𝑚3𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 ) �𝜃𝜃1̇ + 𝜃𝜃̇2 �

𝑐𝑐 13 + 𝑐𝑐 23 2𝑚𝑚3𝑠𝑠3 �𝜃𝜃1̇ + 𝜃𝜃2̇ � −2𝑚𝑚3𝑠𝑠2 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2 ) �𝜃𝜃1̇ + 𝜃𝜃2̇ � 2𝑚𝑚3 𝑠𝑠3 �𝜃𝜃1̇ + 𝜃𝜃2̇ �

(1⁄2)(𝑐𝑐13 − 𝑐𝑐 23)

−𝑚𝑚2 𝑠𝑠2 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1)�𝜃𝜃1̇ � +(1 ⁄2)(𝑐𝑐11 + 𝑐𝑐 21) −𝑚𝑚3 𝑠𝑠2 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1)�𝜃𝜃1̇ � −𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃1 + 𝜃𝜃2 )�𝜃𝜃̇1� −𝑚𝑚3𝑠𝑠3 �𝜃𝜃̇1� + (1 ⁄ 2 ) (𝑐𝑐 12 + 𝑐𝑐 22) −𝑚𝑚3𝑠𝑠2 �𝜃𝜃̇1� −𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2)�𝜃𝜃1̇ � −𝑚𝑚3𝑠𝑠3 𝑠𝑠2 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2)�𝜃𝜃1̇ � −𝑚𝑚3𝑠𝑠3 �𝜃𝜃̇1� 2𝑚𝑚3𝑠𝑠3 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )�𝜃𝜃2̇ � + ( 1 2⁄ 4 ) (𝑐𝑐 11 + 𝑐𝑐 21) 𝑚𝑚3 𝑠𝑠3 𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2 )�𝜃𝜃1̇ � 0

−𝑚𝑚3𝑠𝑠3 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃1 + 𝜃𝜃2)

−2𝑚𝑚3𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2)�𝜃𝜃̇1� −𝑚𝑚3 𝑠𝑠3 𝑐𝑐𝑐𝑐𝑐𝑐 (𝜃𝜃2)�𝜃𝜃̇2� −2𝑚𝑚3 𝑠𝑠3�𝜃𝜃̇1� −𝑚𝑚3𝑠𝑠3 �𝜃𝜃2̇ � 𝑚𝑚3 𝑠𝑠3𝑠𝑠2 𝑠𝑠𝑠𝑠𝑠𝑠 (𝜃𝜃2)�𝜃𝜃2̇ � ( 1 2⁄ 4 ) (𝑐𝑐 12 + 𝑐𝑐 22) 0

43

⎤ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ ⎥ (13) (1 ⁄ 2 ) ⎥ ⎥ (𝑐𝑐 13 + 𝑐𝑐 23) ⎥ ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ⎥ 0 ⎥ ⎥ ⎥ ( 1 2⁄ 4 ) ⎥ (𝑐𝑐 13 + 𝑐𝑐 23) ⎦

−m1 g − m2 g + k 11 (s 1 + ( 1⁄2 )θ1 − s 01 ) + k 21 (s 1 − ( 1⁄2 ) θ1 − s 01 ) − m3 g ⎡ ⎤ ⎢ ⎥ ⎢ −m2 gcos (θ1 ) + k 12 (s 2 + ( 1⁄2 )θ2 − s 02 ) + k 22 (s 2 − ( 1⁄2 )θ2 − s 02 ) − m3 gcos ( θ1 ) ⎥ ⎢ ⎥ −m3 gcos ( θ1 + θ2 ) + k 13 (s 3 + (1⁄2 )θ3 − s 03 ) + k 23 (s 3 − (1⁄2 ) θ3 − s 03 ) ⎢ ⎥ ⎢ ⎥ G � q� = ⎢ m2 s 2 gsin( θ1 ) + m3 s 3 gsin( θ1 + θ2 ) + m3 s 2 gsin( θ1 ) + k 11 (s 1 + (1⁄2 ) θ1 − s 01 )( 1⁄2 ) ⎥ ⎢ ⎥ +k 21 (s 1 − ( 1⁄2 )θ1 − s 01 )(−1⁄2 ) ⎢ ⎥ ⎢ ⎥ ⎢ m3 s 3 gsin(θ1 + θ2 ) + k 12 (s 2 + (1⁄2 ) θ2 − s 02 )(1⁄2 ) + k 22 (s 2 − (1⁄2 ) θ2 − s 02 )(−1⁄2 ) ⎥ ⎢ ⎥ ⎣ ⎦ k 13 (s 3 + ( 1⁄2 )θ3 − s 03 )( 1⁄2 ) + k 23 ( s 3 − ( 1⁄2 )θ3 − s 03 )( −1⁄2 )

(14)

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

44

1.2.

Design Switching PD-plus Gravity Sliding Mode Controller

A time-vary ing sliding surface 𝒔𝒔(𝒙𝒙, 𝒕𝒕) in the state space 𝑹𝑹𝒏𝒏 is given by [6]:

s (x, t) = (

d

dt

+ λ)n −1 x� = 0

(15)

where λ is the positive constant. To further penalize tracking erro r, integral part can be used in sliding surface part as follows [6]: 𝑡𝑡 𝑑𝑑 𝑠𝑠(𝑥𝑥, 𝑡𝑡) = ( + 𝜆𝜆)𝑛𝑛 −1 �� 𝑥𝑥� 𝑑𝑑𝑑𝑑� = 0 (16) 𝑑𝑑𝑑𝑑 0

The main target in this methodology is kept the sliding surface slope 𝒔𝒔 (𝒙𝒙, 𝒕𝒕) near to the zero. Therefore, one of the common strategies is to find input 𝑼𝑼 outside of 𝒔𝒔(𝒙𝒙, 𝒕𝒕) [6]. 1 𝑑𝑑

2 𝑑𝑑𝑑𝑑

𝑠𝑠 2 (𝑥𝑥, 𝑡𝑡) ≤ −𝜁𝜁 |𝑠𝑠(𝑥𝑥, 𝑡𝑡) |

where ζ is positive constant. If S(0)>0→

𝑑𝑑

𝑑𝑑𝑑𝑑

𝑆𝑆(𝑡𝑡 ) ≤ −𝜁𝜁

if St reach = S(0) → error (x − xd ) = 0

(17) (18) (19)

suppose S is defined as 𝑑𝑑 𝑠𝑠(𝑥𝑥, 𝑡𝑡) = ( + 𝜆𝜆) 𝑥𝑥� = ( 𝑥𝑥̇ − 𝑥𝑥̇ 𝑑𝑑 ) + 𝜆𝜆 (𝑥𝑥 − 𝑥𝑥 𝑑𝑑 ) 𝑑𝑑𝑑𝑑

The derivation of S, namely, 𝑆𝑆̇ can be calculated as the following;

(20)

𝑆𝑆̇ = (𝑥𝑥̈ − 𝑥𝑥̈ 𝑑𝑑 ) + 𝜆𝜆(𝑥𝑥̇ − 𝑥𝑥̇ 𝑑𝑑 )

(21)

ẍ = f + u → Ṡ = f + U − ẍ d + λ (ẋ − ẋ d )

(22)

� = −f̂ + ẍ d − λ( ẋ − ẋ d ) U

(23)

suppose the second order system is defined as;

� is Where 𝒇𝒇 is the dynamic uncertain, and also since 𝑆𝑆 = 0 𝑎𝑎𝑎𝑎𝑎𝑎 𝑆𝑆̇ = 0, to have the best appro ximation ,𝑼𝑼 defined as

Design Artificial Intelligence-Based Switching PD plus Gravity for Highly Nonlinear Second Order System

45

A simple solution to get the slid ing condition when the dynamic parameters have uncertainty is the switching control law [52-53]: � − K(x�⃗, t) ∙ sgn(s ) Udis = U

where the switching function 𝐬𝐬𝐬𝐬𝐬𝐬(𝐒𝐒) is defined as [1, 6] 1 sgn (s ) = � −1 0

s>0 s