KINEMATIC ANALYSIS AND SIMULATION OF 3-PRR PLANAR PARALLEL

MANIPULATORS WITH NON-PLANAR LINKS

ABSTRACT

In conventional 3-RRR planar parallel manipulator, the links are planar. This makes them vulnerable to cantilever action during vertical loading. To overcome this difficulty, a new design has been proposed with non-planar links [19]. This design is found to have better stiffness [20] and optimized workspace at lesser mass than the conventional manipulator and also overcomes the cantilever action. The components of the manipulator are designed in Pro-E software and the working model of the manipulator has been manufactured using aluminium-6061 and assembled where the links are inclined at an optimized angle of 45o. High torque dc motors are used as active joints and ball bearings are used as passive joints. The manipulator is serially controlled with 8051 micro-controller board and MATLAB software. A Graphical User Interface (GUI) has been developed using MATLAB software where the motors can be controlled individually and the code is dispatched to the manipulator to follow the given command using serial programming [18].

The advantage of non-planar links in 3-PRR manipulator has also been explored. In 3-PRR planar parallel manipulator with non-planar links, inverse kinematics has been performed and the inverse kinematic equations have been generalized. The workspace of the manipulator has been plotted using MATLAB software. The inverse kinematic equations and workspace of the manipulator have been verified using geometric method. A MATLAB GUI for simulation of the manipulator is made and it has been programmed to trace a sinusoidal path

KEY WORDS: Cantilever action; Non-planar links; Serial programming; Inverse Kinematics; Workspace ; MATLAB GUI.

INTRODUCTION:

A mechanical manipulator is a kinematic mechanism which is made up of several links connected by joints. One link is fixed to the ground, while another is designated as output link. Some of the joints in the manipulator are actuated; the others are passive. A robot is said to be a serial manipulator or an open-loop manipulator if its kinematic structure takes the form of an open-loop chain, a parallel manipulator is made up of closed loop chain and a hybrid manipulator has both closed loop and open loop chains. Another classification of robot is on the basis of their motion characteristics. A manipulator

is called a planar manipulator if its end effector motion is a planar mechanism. They are useful for manipulating objects in a plane. A manipulator is called a spatial manipulator if atleast one of the links in the mechanism possesses a general spatial motion. Parallel robots have been under intensive study for over more than one decade. Among the advantages are greater load carrying capacities as total load can be shared by number of parallel links connected to fixed base, low inertia, higher structural stiffness, reduced sensitivity to certain errors (high accuracy), easy controlling and built-in redundancy but

smaller and less dexterous workspace due to link interference.

1.1 COMPARISON TO SERIAL MANIPULATORS:

Parallel manipulators are widely popular recently even though conventional serial manipulators possess large workspace and dexterous maneuverability. The basic problems with serial one are their cantilever structure makes them susceptible to bending at high load and vibration at high speed leading to lack of precision and many other problems .Most robot applications require rigidity. Serial robots may achieve this by using high-quality rotary joints that permit movement in one axis but are rigid against movement outside this. Any joint permitting movement must also have this movement under deliberate control by an actuator. A movement requiring several axes thus requires a number of such joints. Unwanted flexibility or sloppiness in one joint causes a similar sloppiness in the arm: there is no opportunity to brace one joint's movement against another. Their inevitable hysteresis and off-axis flexibility accumulates along the arm's kinematic chain; a precision arm is a compromise between precision, complexity and cost of these joints

1.2 APPLICATIONS OF PARALLEL MANIPULATORS:

Parallel manipulators have a lot of practical applications like worktable, a camera orienting device, a wrist or a motion simulator, to perform operations in medical field, as flight simulators, satellite trackers etc.Parallel Robots can offer many advantages for high-speed laser operations due to their structural stiffness and limited moving masses with less power consumption.

1.3 PLANAR PARALLEL MANIPULATORS:

A planar parallel manipulator is one in which the end-effector motion is a planar motion. The end-effector moves along a plane. The base platform and end-effector are connected by a parallel connection of links and joints. In conventional planar parallel manipulators the links are also planar. In the proposed design, the links are non-planar, but it is still a planar parallel manipulator only since the end-effector motion is planar.

1.3.1 3-RRR planar parallel manipulator:

The moving platform of a planar 3-DOF 3-RRR parallel manipulator is connected to its legs by three revolute joints Pi (i = 1,2,3) (Fig. 1) [25]. Each leg comprises two links connected by revolute joints Ai (i = 1,2,3) and they are mounted on the frame by revolute joints Oi (i = 1, 2, 3). The input parameters of such a manipulator are defined by the joint angles Hi (i = 1, 2,3) of each leg and the output parameters by the pose of the moving platform, i.e. its orientation and position of one point of the moving platform, by example, the centre of mass of the moving platform (Xo, Yo).The 3-RRR planar manipulator is a preferred practical manipulator owing to its simplicity in design. Here, the actuators being fixed at the base reduces the inertia of the mobile body. One reason for the workspace limitation for such parallel manipulators is interference of the legs with each other. The arrangement of three legs on a single plane inevitably restricts the mobility and accessibility of the platform

1.3.2 3-PRR planar parallel manipulator:

A 3-PRR planar parallel manipulator has 3legs in parallel between the top platform and base platform. Each leg has 2 links connected by 1 active prismatic joint and 2 passive rotary joints. The moving platform of a planar 3-DOF 3-PRR parallel manipulator is connected to its legs by three passive revolute joints. Each leg comprises two links connected by passive revolute joints and they are mounted on the frame by prismatic joints.The 3-PRR and 3-RRR planar parallel manipulators find its application for fast positioning or assembly operations. Such applications make use of the high speed capability, large stiffness and low inertia of parallel manipulator with minimum positioning error structure.

1.4 MANIPULATOR WITH PLANARLINKS:

Planar parallel manipulators with planar links have their links oriented in one plane. All the links are oriented in one plane only and have their motion in that plane. But these type of manipulators with planar links are vulnerable to cantilever action during vertical loading. When the load is applied in a direction normal to the plane of motion, these manipulators are susceptible to failure due to cantilever action. Also there is a risk of collision between the links since all the links are in the same plane.

1.5 MANIPULATOR WITH NON-PLANAR LINKS: To overcome the above drawbacks due to planar links, a new design idea is proposed where the links are non-planar. The links are given an angle of inclination ‘Ɵ’ to make them non-planar. This allows the links to carry more load than the conventional manipulator with planar links. Also the proposed manipulator with non-planar links overcomes the cantilever nature of links during vertical loading since the links are inclined. Planar parallel manipulators with non-planar links offer

significant advantages over the conventional manipulator in that they have better stiffness along all the axes, since additional stiffness is introduced as the links are close to vertical plane. The modified manipulator has less mass, more workspace and higher stiffness than the conventional manipulator. Also, since the links are not in a single plane there is no collision of links.

REVIEW OF LITERATURE

This Project involves several research areas, such as parallel manipulators, kinematic analysis of manipulators and mechanisms with link flexibility, and Serial control of planar parallel manipulators using MATLAB GUI and Simulation in MATLAB. This section will review the research literature relative to these research topics.

2.1 PARALLEL ROBOT MANIPULATORS:

In the applications where high load carrying capacity, high-speed, and precise positioning are of paramount importance, it is desirable to have an alternative to conventional serial manipulators. In general, it is expected that manipulators have the end-effector connected to the ground via several chains having actuation in parallel, and therefore have greater rigidity and superior positioning capability. This makes the parallel manipulators attractive for certain applications and the last two decades have witnessed considerable research interest in this direction. The research efforts mainly involved inverse position kinematics, direct position kinematics, singularities, workspace and dexterity, and dynamics and control, etc. (Dasgupta and Mruthyunjaya [10]). A workspace analysis method of the Stewart platform is developed by Yang and Lee [12]

to determine some particular sections of the positional workspace with constant orientation for very specialized structures of the manipulator. Luh et al. [13] presented a general formulation for the workspace and dexterity analysis of parallel manipulators in terms of rank-deficiency of the Jacobian of the constraints by incorporating inequality constraints of slack variables. The work reported by Masory and Wang [14] addressed the problem of determining workspace sections including the constraints of joint angle limits and leg interface. Compared to the vast literature on the kinematics of the parallel manipulators, research reports on the dynamics and control of parallel manipulators are relatively few. Do and Yang [15]presented the inverse dynamics of the Stewart platform using the Newton-Euler approach. Liu etal. [16] developed Lagrangian equations of motion with simplified assumptions regarding the Chapter 1 Introduction geometry and inertia distribution of the parallel manipulator. A dynamic control strategy was proposed in work (Hatip and Ozgoren [17]) for the control for the Stewart platform assumed to be mounted on a ship and used as a motion stabilizer. To summarize, the research on dynamics and control of parallel manipulators has accomplished much, with significant issues yet to be resolved. With the consideration of link flexibility, the dynamic formulation of parallel manipulators is more complicated and challenging.

2.2 KINEMATIC MODELING OF PLANAR PARALLEL MANIPULATORS:

Due to the great number of advantages offered by planar parallel manipulators like greater load carrying capacities as total load can be shared by number of parallel links connected to fixed base, low inertia, higher structural stiffness, reduced

sensitivity to certain errors (high accuracy), easy controlling and built-in redundancy but the planar parallel manipulators are vulnerable to cantilever action under vertical loading. Therefore, the research interest in the kinematics and control of planar parallel manipulators and mechanisms has increased significantly in recent decades in order to fully exploit the potential offered by parallel manipulators.Raza UR-REHMAN, St´ephane CARO, Damien CHABLAT, Philippe WENGER presented their work on multi-objective design optimization of 3-PRR planar parallel manipulator in which they determined the optimum structural and geometric parameters of parallel manipulators. A workspace analysis method of the Stewart platform is developed by Yang and Lee [12]to determine some particular sections of the positional workspace with constant orientation for very specialized structures of the manipulator.

3.3 KINEMATIC ANALYSIS AND WORKSPACE PLOTTING OF 3-PRR PLANAR PARALLEL MANIPULATOR:

Inverse kinematics was performed for the 3-PRR planar parallel manipulator and the inverse kinematic equations obtained were generalized in terms of prismatic actuation and angle of rotation of intermediate links by manually solving the equations. The inverse kinematic equations were solved and generalized in MATLAB for verification purpose. The Workspace of the manipulator was plotted using a MATLAB code written, which checks for valid reachable points by the centroid of the end-effector by scanning the X and Y co-ordinates and by applying the constraints specified for maximum actuation and angle of rotation of each joint and plotting those points.

4.9 INVERSE KINEMATICS OF 3-PRR PLANAR PARALLEL MANIPULATOR:

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