Abstract: Control systems and methods for an earthmoving machine and control architecture to generate a surface associated with a curved surface of an earthmoving implement, project the surface onto a two-dimensional (2D) plane associated with the earthmoving implement, determine a continuous curve based on the surface projected on to the 2D plane, determine a derivative of the continuous curve, project a design plane normal n* of a ground surface for smoothing onto the 2D plane associated with the earthmoving implement, determine a point of perpendicular intersection between the derivative of the continuous curve and the design plane normal n* of the ground surface projected onto the 2D plane, and operate the earthmoving machine using one or more linkage assembly actuators and the point of perpendicular intersection.

Abstract: Grade control systems and methods for an earthmoving machine and control architecture to generate a continuous differential surface associated with a rear curved surface of an earthmoving implement, project the continuous differential surface onto a two-dimensional (2D) plane associated with the earthmoving implement, determine a piecewise-derivative continuous curve based on the continuous differential surface projected on to the 2D plane, determine a derivative of the piecewise-derivative continuous curve, project a design plane normal n* of a ground surface for smoothing onto the 2D plane associated with the earthmoving implement, determine a point of perpendicular intersection between the derivative of the piecewise-derivative continuous curve and the design plane normal n* of the ground surface projected onto the 2D plane, and operate the earthmoving machine using one or more linkage assembly actuators and the point of perpendicular intersection to smooth the ground surface.

Abstract: Grade control systems and methods for an earthmoving machine and control architecture to generate a continuous differential surface associated with a rear curved surface of an earthmoving implement, project the continuous differential surface onto a two-dimensional (2D) plane associated with the earthmoving implement, determine a piecewise-derivative continuous curve based on the continuous differential surface projected on to the 2D plane, determine a derivative of the piecewise-derivative continuous curve, project a design plane normal n* of a ground surface for smoothing onto the 2D plane associated with the earthmoving implement, determine a point of perpendicular intersection between the derivative of the piecewise-derivative continuous curve and the design plane normal n* of the ground surface projected onto the 2D plane, and operate the earthmoving machine using one or more linkage assembly actuators and the point of perpendicular intersection to smooth the ground surface.