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Using the curvature theory for the ruled surfaces a technique for robot trajectory planning is presented. This technique ensures the calculation of robot’s next path. The positional variation of the Tool Center Point (TCP), linear velocity, angular velocity are required in the work area of the robot. In some circumstances, it may not be physically achievable and a re-computation of the robot trajectory might be necessary. This technique is suitable for re-computation of the robot trajectory. We obtain different robot trajectories which change depending on the darboux angle function and define trajectory ruled surface family with a common trajectory curve with the rotation trihedron. Also, the motion of robot end effector is illustrated with examples.

Let [Formula: see text] be a parameter and an asymptotic curve on a surface [Formula: see text] We obtain conditions for offsets [Formula: see text] of [Formula: see text] such that the image [Formula: see text] of the curve [Formula: see text] is a common asymptotic on each offset. We illustrate the method with an example.

There is a unique curve called striction curve that is not present on a surface except a ruled surface. This curve is defined as the shortest distance with the help of a common perpendicular line between two adjacent rulings. In this work, we present Lorentz forces and magnetic striction curves produced by the geodesic Frenet frame {}ē,0.1emtruet¯,0.1emtrueg¯$$ \left\{\bar{e},\overline{t},\overline{g}\right\} $$ on the striction curve of the ruled surface defined by the spherical indicatrix curve in a magnetic field. We calculate magnetic vector fields of magnetic striction curves for truee‾,0.1emtruet¯$$ \overline{e},\overline{t} $$, and trueg¯$$ \overline{g} $$. Furthermore, we define magnetic flux surfaces constructed by magnetic vector fields along magnetic striction curves. We obtain developability conditions for these surfaces according to their curvature functions. Finally, we give some examples about magnetic flux surfaces.

Some situations that change the parameters of the kinematic structure may cause the robot end effector to deviate [Merlet [2005] Parallel Robots, Vol. 128 (Springer Science & Business Media, Germany)] from the desired trajectory. This effect is called the robustness of the robot by Merlet. One of the ways to correct the robustness is by updating the robot trajectory. The jerk vector of the robot end effector is the third-order positional variation of the TCP and defined as thus the time derivative of the acceleration vector. If there is a high curvature on the transition curve trajectory of robot, then there is a tangential jerk along the trajectory. In this study, the geometrically offset trajectory of the robot end effector from the current trajectory was obtained by using the curvature theory. The angular velocity and angular acceleration of the offset trajectory were calculated. An example of the main trajectory of robot end effector and its offset is given. Also, the jerk of the robot end effector of the offset trajectory was calculated according to the curvature of the trajectory surface in case of a jerk problem caused by a high curvature in the transition curve along the offset trajectory curve.

In this paper, we investigate the ruled surfaces generated by a straight line according to rotation minimizing frame (RMF). Using this frame of a straight line, we obtained the necessary and sufficient conditions when the ruled surface is developable. Also, we give some new results and theorems related to be the asymptotic curve, the geodesic curve and the line of curvature of the base curve on the ruled surface.

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Developable surfaces are defined to be locally isometric to a plane. These surfaces can be formed by bending thin flat sheets of material, which makes them an active research topic in computer graphics, computer aided design, computational origami and manufacturing architecture. We obtain condition for developable and minimal ruled surfaces using rotation frame. Also, the validity of the theorems is illustrated with examples.