The verification of these relationships ensures the correct orientation and movement of various components of the robotic cell injection system, that is, injection manipulator, working plat, microscope, camera, etc., and is vital considering the safety-critical nature of the underlying system. The proof process of Theorem 5.3 is mainly based on Theorems 5.1 and 5.2 along with some properties of the vectors and matrices. Finally, the conclusion captures the relationship between the image and stage coordinate frames. Similarly, Assumption A6 provides the camera-stage coordinate frame interrelationship. Assumption A5 models the image-camera coordinates interrelationship. Assumptions A1–A4 present the design constraints for the image-stage coordinate frame interrelationship. Where ⁎ ⁎ represents the operator for the multiplication of a matrix with a vector and vice versa. : 0 < dx ∧ : 0 < dy ∧ : 0 < fx ∧ : 0 < fy ∧ : two_dim_coordin u v t = display_resol_matrix fx fy ⁎⁎ two_dim_coordin xc yc t ∧ : two_dim_coordin xc yc t = rotat_matrix α ⁎⁎ two_dim_coordin x y t + displace:vector dx dy ⇒ two_dim_coordin u v t = transform_matrix fx fy α ⁎⁎ two_dim_coordin x y t + fx ⁎ dx fy ⁎ dy Theorem 5.3 Image-stage coordinates interrelationship
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