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**INVERSE GEOMETRY AND WORKSPACE OF ROBOT MECHANISMS**

T. Bajd and M. Mihelj

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Triangle Triangle plays an important role in Euclidean geometry.

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Triangle Triangle plays an important role in geometry of robot mechanisms.

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**Two-segment planar robot**

When solving the inverse geometry of robot, we calculate the joint angles 𝜗 1 and 𝜗 2 from the known position of the robot end-point 𝑥, 𝑦.

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**Two-segment planar robot**

The angle in the second joint of the two-segment robot is calculated by the use of the law of cosines.

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**Two-segment planar robot**

The angle in the first joint is calculated as the difference of the angles 𝜗 1 and 𝜗 2 .

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**Two-segment planar robot**

When calculating the joint angles we have two configurations, „elbow-up“ and „elbow-down“.

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**Three-segment planar robot**

When solving the inverse geometry of robot, we calculate the internal coordinates 𝑞 1 , 𝑞 2 , 𝑞 3 from the known position 𝑝 1 , 𝑝 2 and orientation 𝑝 3 of the end-effector.

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**Three-segment planar robot**

While defining 𝑙 2 = 𝑝 𝑥 2 + 𝑝 𝑦 2 the two solutions for the second joint angle 𝑞 2 are obtained by the law of cosines.

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**Three-segment planar robot**

The solutions for the angle in the first joint are obtained by law of cosines. They depend on the selected solution for the second joint angle 𝑞 2 .

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**Three-segment planar robot**

Usually there exist two configurations. When the second joint is extended ( 𝑞 2 =0), only single solution exists. When 𝑙 1 = 𝑙 2 and 𝑞 2 = ±𝜋 , there is infinite number of configurations.

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**Two-dimensional robot workspace**

The workspace of a robot mechanism is the spatial volume which is reachable by its end-point.

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**Two-dimensional robot workspace**

The workspace of a robot mechanism depends on the number of degrees of freedom, their arrangement, the lengths of the segments and constraints in the motion of particular joint coordinates.

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**Two-dimensional robot workspace**

The reachable workspace of a planar mechanism with two rotational joints (2R) is determined with arc ℎ 2 which is expanded around the first rotational axis along the arc ℎ 1 .

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**Workspace of 2R robot mechanism**

The work space can be described by a mesh of two types of circles. The circles depending on the angle 𝜗 1 have their radii of equal length while their centers travel around the origin of the coordinate frame. The circles depending on 𝜗 2 angle have all their centers in the origin of the frame, while their radii depend on the lengths of both segments and the angle between them.

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**Workspace of 2R robot mechanism**

The shape of workspace is presented for 𝑙 1 = 𝑙 2 0°≤ 𝜗 1 ≤180° 0°≤ 𝜗 2 ≤180° and 0°≤ 𝜗 1 ≤60° 60°≤ 𝜗 2 ≤120° The area of the workspace can be replaced by the area of a corresponding sector of a ring.

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**Workspace of 2R robot mechanism**

Different values of the working areas are obtained for equal ranges of the angle 𝜗 2 , 0°≤ 𝜗 1 ≤30°, and for 𝑙 1 = 𝑙 2 =1.

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**Workspace of 2R robot mechanism**

The largest working area of the 2R mechanism occurs for equal lengths of both segments.

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**Workspace of 3R planar robot mechanism**

The reachable robot workspace represents all the points that can be reached by the robot end-point. The dexterous workspace comprises all the points that can be reached with an arbitrary orientation of the robot end-effector.

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**Three-dimensional robot workspace**

When adding translation to 2T mechanism, the Cartesian mechanism is obtained. When adding rotation to 2T mechanism, the cylindrical mechanism is obtained.

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**Three-dimensional robot workspace**

When adding translation to RT mechanism, the cylindrical mechanism is obtained. When adding rotation to RT mechanism, the spherical mechanism is obtained.

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**Three-dimensional robot workspace**

When adding translation to RR mechanism, the so called SCARA mechanism is obtained. When adding rotation to RR mechanism, the anthropomorphic mechanism is obtained.

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Robot workspace The robot manufacturer is required to clearly show the maximal reachable workspace of an industrial robot in at least two planes.

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Robot workspace Robot workspace plays an important role when selecting an industrial robot for an anticipated task.

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T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 ROBOT ASSEMBLY T. Bajd and M. Mihelj.

T. Bajd, M. Mihelj, J. Lenarčič, A. Stanovnik, M. Munih, Robotics, Springer, 2010 ROBOT ASSEMBLY T. Bajd and M. Mihelj.

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