İki uzuvlu bir robot kolunun hesaplanmış moment metoduyla kontrolu

Abstract

ÖZET Anahtar kelimeler: Robot, Robot dinamiği, Hesaplanmış moment algoritması. Yakın zamana kadar robot kontrol metodlarının bir çok tipi üzerinde durulmuş fakat son yıllarda hesaplanmış moment kontrolörlerinin kullanımı ağırlık kazanmıştır. Hesaplanmış moment aynı zamanda nonlineer sistemlerin geri beslemeli lineerleştirme işleminin özel bir uygulamasıdır. Bu metodda eklemlere uygulanacak T momenti, her eklem için ayrı olarak hesaplanır. Hesaplanmış moment metodu kumanda sinyali olarak eklem ivmesini alır. T = M(qd -u) + N. Bu kanundan yola çıkılarak robot kolu kontrolünün simülasyonu yapılmıştır. Robot kolundan değişik yörüngeler takip etmesi istenerek değişik çalışma periyotları seçilmiş, böylelikle eklem açılarının, torklarının, hızlarının ve izleme hatasının öngörülen ve gerçekleşen değerleri eğriler yardımıyla mukayese edilmiştir. Sonuç olarak bu kontrol metodunun hassasiyeti sınanmıştır.

SUMMARY Keywords: Robot, Robot dynamics, Computed torque control algoritm. The robot as a programmable.multi-functionaJ manipülatör designed to move material, parts, tools, ör specialized devices trough variable programmed motions for the performance of a variety of tasks. This is a widely accepted definition of an indusrial robot. The emphasis is on the programmable facilıties by which the robot can perform simple tasks without human assistance. Robot arm consists of several rigid links /vhich are connected to each other in seriesby revaluate ör prismatic joints. Öne end of the chain is attached to a supporting base, while the other end is free and equipped with a tool to perform various tasks. The motion of the robot arm is accomplished by driving the joint with actuators. Robot classification may be considered on the following basics: 1l)Structural configuration and robot motion (2)Trajectories based on motion control Classification based on structural configuration and robot motion -Revaluate (jointed arm) robot -Polar (spherical) robot -Cartesian (rectangular) robot -Cylindirical robot Classification based on path control -Point to point control -Continuous path control. Robot Arm Kinematics Robot arm kinematics deals with the analytical study of the geometry of motion of a robot arm with respect to a fixed reference coordinate system without regard to forces/moments that cause the motion. Thus, kinematic deals wıth the analytical description of the spatıal displacement of the robot as a funcn'on of time, in particular the relations between the joint vanablespace and the pozitıon and orientation of the end-effector of a robot arm. There are fimdamental problems in robot arm kinematics. The first problem is usually referred to as the direct (ör fonvard) kinematics problem,while the second problem is the invers kinematik problem, while the second problem is the invers kinematics viproblem. The direct kinematics problem is to find the position and orientation of the end effector of a manipülatör vvith respect to a reference coordinate system, given the Joint-angle vector q=(qı,q:,q3,....,q6)7of the robot arm. The inverse kinematics problem (ör arm solution) is to calculate the joint-angle vector q given the position and orientation of the end effector with respect to the reference coordinate system. Since the independent variables in a robot arm are the joint angles, and a task is generally stated in terms of the base ör world coordinate system, the inverse kmematics solution ıs used more frquently in control of robot arm. The homogenıus matrix T0ı, which specifies the position and orientation of the end point of link i with respect to the base coordinate system, is the chain product of succesive coordinate transformation matrices of A,1.,, expressed as Tâ = AXA;, = f]Arlj ; fort= 1,2n )-> -IX y, z- P.I.FR; pJ ~Lo o o ıj-[o ıj(1) where; [x,.y,'zJ= Orientation matrix of the i th coordinate system established at link i with respect to the base coordinate system. P, = Position vector which points from the origin of the base coordinate system. Specifically for i=6, we obtain the T matnx, T= A*, which specifies the position an orientation of the end point of a manipülatör vvith respect to the base coordinate system. This T matrix is used so frequentiy in robot arm kinematics that it is called the `arm matrix`. Consider the T matrix as T=[X> y< Z< P<l = [n S H P](2) [o o o ıJ [o o o ıJ^} where; n = The normal vector of the hand, s = The sliding vector of the hand, a = The approach vector of the hand, p = The position vector of the hand. Robot Arm Dynamics The dynamic behavior of robot arm is described in terms of the time rate of the arm configuration in relation to the joint torques exerted by the actuators. This relations can be expressed by a set of differential equation that gövem the dynamic response of the arm linkage to input joint torques. viiTwo dynamics models that are widely used to describe the dynamics behavior of a manipulator are the Lagrange-Euler (LE) formulation and the Newton-Euler (NE) formulation. The LE formulation generates a set of closed from differential equations to describe the motion. However, the LE formulation is computationally very inefficient with an order 0 ( n4 ) relation ship to the number of degrees of freedom of a manipulator. For a PUMA robot arm, where n=6, this corresponds to approximately 78000 addition and 102000 multiplications, in general, the Lagrange-Euler equations of motion can be written in a compact vector matrix notation as t(t) = D(q)q(t) + h(q,q) + G(q) (3) where i(t) is an nxl applied torque vector for joint actuators; q(t) is an nxl joint velocity vector; q(t) is an nx 1 joint acceleration vector; G(q) is an nx 1 gravitational force vector; h(q,q) is an nxl coriolis and centrifugal force vector; and D(q) is an nxn accerelation related matrix. Studies have shown that, with current hardware, implementation of this algoritm would be prohibitively expensive for real-time control. The NE formulauon yields a computationally efficient set of forward and backward recursive equations of motion.The forward equations propagate linear velocity, linear acceleration, angular velocity, angular acceleration, and total link forces and moments at each link's center of mass from the base coordinate frame to the end effector coordinate frame of the manipulator. The backward equations propagate the forces and moments exerted on each link from the end effector coordinate frame to the' base coordinate frame of the manipulator. To compute the joint torques, the NE equations of motion have an order 0(n) relation ship to the number of degrees of freedom of the manipulator. Solving for all torques using the NE formulation requires 666 additions and 804 multiplications for n=6. Robot Arm Control A basic problem in controlling robots is to make the manipulator follow a preplanned desired trajectory. Before the robot can do any useful work, we must position it in the right place at the right instances. In this research discussed computed-torque control, which yields a family of easy-to-understand control schemes that often work well in practice. These schemes involve the decomposition of the controls design problem into an inner-loop design and an outer-loop design. We assume here the robot is moving in free space, having no contact with its environment. Contact results in the generation of forces. We will also assume in this research that the robot is a well known rigid system, thus designing controllers based on a fairly well-known model. Control in the presence of uncertainties or unknown model. Control in the presence of uncertainties or unknown parameters (e.g., friction, payload mass) requires refined approaches. This problem is dealt using robust control and adaptive control. An actual robot manipulator may have flexibility in its links, or compliance in its gearing (joint flexibility). VIIIBefore we can control a robot arm, it is necessary to know the desired path for performing a task. There are many issues associated with the path planning problem, such as avoiding obstacles and making sure that the planned path does not require exceeding the voltage and torque limitations of the actuators. To reduce the control problem to its basic components, in this research we assume that the ultimate control objective is to move the robot along a prescribed desired trajectory. We do not concern ourselves with the actual trajectory-planning problem; we do, however, show how to reconstruct a continuous desired path from a given table of desired points the end effector should pass t rough. In most practical situations robot controllers are implemented micro processors, particularly in view of the complex nature of modern control schemes. Throughout, we demonstrate how to simulate robot controllers on a computer. This should be done to verify the effectiveness of any proposed control scheme prior to actual implementation on a real robot manipulator Through the years there have been proposed many sorts of robot control schemes. As it happens, most of them can be considered as special cases of the class of computed torque controllers. Computed torque, at the same time, is a special application of feed back linearization of nonlinear systems, which has gained popularity in modern system theory [Hunt et al. 1983, Gilbert and Ha 1984]. In fact, one way to classify robot control schemes is to divide them as `computed-torque-like`. Computed-torque-like controls appear in robust control, adaptive control, learning control, and so on. IX