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Robots: 1. Velocity 2. Jacobian Matrix Concept 3. Singularity Concept

Robots: 1. Velocity 2. Jacobian Matrix Concept 3. Singularity Concept 4. Static forces in manipulators. Velocity. They are a vector and represented in frame {i}. Velocity of the origin of frame {i}. Angular velocity of link i. The expression of is: ???. Velocity.

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Robots: 1. Velocity 2. Jacobian Matrix Concept 3. Singularity Concept

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  1. Robots: 1. Velocity 2. Jacobian Matrix Concept 3. Singularity Concept 4. Static forces in manipulators Jabobian Matrix, Handout 6

  2. Velocity They are a vector and represented in frame {i}. Velocity of the origin of frame {i} Angular velocity of link i Jabobian Matrix, Handout 6

  3. The expression of is: ??? Velocity Viewpoint: angular velocity link i+1 can be viewed as experiencing two consecutive rotations: and Jabobian Matrix, Handout 6

  4. Velocity Likewise, the velocity of the origin of frame {i+1} can be viewed as superposition of the velocity of the origin of frame {i} and angular velocity of link i, namely we have: Jabobian Matrix, Handout 6

  5. Velocity When the joint i+1 is prismatic one, then we have: Jabobian Matrix, Handout 6

  6. Singularity Observation: the relation between the joint velocity and the end-effector’s velocity is linear. This means that the relation can be written as a matrix. V = J , where J: Jacobian matrix. A reasonable question: Is this J-matrix invertible? or Non-singularity? Singularity: give V, at certain configurations, joint velocity may not be exist. Jabobian Matrix, Handout 6

  7. At the workspace boundary, there exist singularities. • Singularities may take place in the interior of workspace. • When a manipulator is in a singular configuration, it has lost one or more degrees of freedom of motion, viewed from the Cartesian space. • Loss of degree of freedom along a particular motion direction implies that whatever joint velocities are, it is impossible to move the end-effector along that direction in the Cartesian space. Jabobian Matrix, Handout 6

  8. Example: Suppose that a Jacobian is shown in the following: To find the singular points of a manipulator. Solution: To let DET of Jacobina is equal to zero. We can get the following Jabobian Matrix, Handout 6

  9. The physical meaning of the singularities in this example: Singularity exists when is 0 or 180 =0: The arm is stretched straight out. In this configuration, motion of the end-effector is possible only along one Cartesian direction (the one perpendicular to the arm. Jabobian Matrix, Handout 6

  10. The arm is folded completely back on itself. There is only one direction of motion in this case as well. =180: Static forces in manipulators We consider how forces and moments “propagate” from one link to the next. Typically, the robot is pushing on something in the environment with the chain’s free end (the end-effector) or is perhaps supporting a load at the hand. We wish to solve for joint torques needed to support a static load acting at the end-effector. Jabobian Matrix, Handout 6

  11. Figure below should define notations: Jabobian Matrix, Handout 6

  12. Summing the forces and setting them equal to zero: Summing torques about the origin of frame {i} and setting them equal to zero: If we start with a description of the force and moment applied by the hand, we can calculate the force and moment applied by each link working from the last link down to the base, link 0. Please fill it from the text book of Craig Please fill it from the text book of Craig Jabobian Matrix, Handout 6

  13. We write the two equations above as such: In order to write these equations in terms of forces and moments defined within their own link frames, we transform with the rotation matrix describing frame {I+1} relative to frame {I}. This leads to the following equations: Please fill it from the text book of Craig Jabobian Matrix, Handout 6

  14. Suppose we know the load at the end-effector. What torques are needed at the joints in order to balance the reaction forces and moments acting on the links. The joint torque can be calculated by the following equation: Please fill it from the text book of Craig Please fill it from the text book of Craig Jabobian Matrix, Handout 6

  15. Jacobians in the force domain: According to virtual work principle where Please fill it from the text book of Craig Jabobian Matrix, Handout 6

  16. It is noted that the Jacobian matrix is frame-dependent and in particular, it depends on the frame, with respect to which, the vectors at both sides of the equation are written. Please fill it from the text book of Craig Jabobian Matrix, Handout 6

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