![]() ![]() d v d t real_differentiable atreal t) ∧ : (∀t. d u d t real_differentiable atreal t) ∧ : (∀t. : 0 < dx ∧ : 0 < dy ∧ : 0 < fx ∧ : 0 < fy ∧ : invertible (transform_matrix fx fy α) ∧ : (∀t. ⊢ thm ∀xc yc u v x y fx fy dx dy mx my mp taux tauy fexd feyd α. Theorem 5.5 Image-stage coordinates interrelationship For this purpose, we need to model the positioning table and inertia matrices: We verify an alternate form of the image-stage coordinate frame interrelationship, which depends on the dynamical behavior of the motion stage ( Definition 5.5) and is characterized as a vital property for the analyzing the robotic cell injection systems. #MOTRIX ROTATIONS VERIFICATION#The verification of Theorem 5.4 involves the properties of derivatives of the real-valued functions, transcendental functions, vectors, and matrices along with some arithmetic reasoning. Finally, the conclusion provides the dynamical behavior of the 2-DOF motion stage of the underlying system. Assumptions A10–A11 capture the values of the xy stage coordinates at any time t. Assumptions A8–A9 provide the condition that the torque and the force vectors are zero. ![]() ![]() Assumptions A4–A7 model the initial conditions, that is, the values of the stage coordinates x and y and their first-order derivatives d x d t and d y d t at t = 0. y(t) = (y0 + yd0 ⁎ (my + mp)) - yd0 ⁎ (my + mp) ⁎ e − 1 m y + m p t ) ⇒ dynamics_2dof_motion_stage mx my mp x y t taux tauy fexd feydĪssumptions A1–A3 present the nonnegativity of the masses mx, my, and mp, respectively. ⊢ thm ∀x y mx my mp taux tauy fexd feyd alpha x0 y0 xd0 yd0. ![]() Theorem 5.4 Verification of solution of dynamics of motion stage ( 2), is formalized in HOL Light as follows: The dynamics of the cell injection system, that is, Eq. For the sake of simplicity, we consider 2-DOF motion stage of the system, which considers the process of cell injection in the xy plane only. Next, we model the dynamics of the robotic cell injection systems, which are generally modeled as a set of differential equations and formally verify the solution of these differential equations. 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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