Trajectory Planning Using the Ferguson Curve Model and Curvature Theory of a Ruled Surface

1990 ◽  
Vol 112 (3) ◽  
pp. 377-383 ◽  
Author(s):  
B. S. Ryuh ◽  
G. R. Pennock
Keyword(s):  
Trajectory Planning ◽  
Geometric Modeling ◽  
Ruled Surface ◽  
End Effector ◽  
Robot Trajectory ◽  
Curve Model ◽  
Curvature Continuity ◽  
Curvature Theory ◽  
Pass Through ◽  
Differential Properties

The trajectory of a robot end-effector is completely described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The robot can then be programmed so that the end-effector will follow the prescribed trajectory with high accuracy. In the case where the ruled surface cannot be described by an explicit analytic function, the ruled surface may be represented by a geometric modeling technique. Since a ruled surface can be expressed in terms of a single parameter, a curve generating technique is used to represent the ruled surface. The technique presented in this paper is the Ferguson curve model which guarantees that the trajectory will pass through all of the set points and will have curvature continuity at each set point. To illustrate the proposed method of robot trajectory planning, a practical example is included in the paper.

Trajectory Planning Using the Ferguson Curve Model and Curvature Theory of Ruled Surface

1989 ◽  
Author(s):  
B. S. Ryuh ◽  
G. R. Pennock
Keyword(s):  
Trajectory Planning ◽  
Geometric Modeling ◽  
Ruled Surface ◽  
End Effector ◽  
Robot Trajectory ◽  
Curve Model ◽  
Curvature Continuity ◽  
Curvature Theory ◽  
Pass Through ◽  
Differential Properties

Abstract The trajectory of a robot end-effector is completely described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The robot can then be programmed so that the end-effector will follow the prescribed trajectory with high accuracy. In the case where the ruled surface can not be described by an explicit analytic function, the ruled surface may be represented by a geometric modeling technique. Since a ruled surface can be expressed in terms of a single parameter, a curve generating technique is used to represent the ruled surface. The technique presented in this paper is the Ferguson curve model which guarantees that the trajectory will pass through all of the set points and will have curvature continuity at each set point. To illustrate the proposed method of robot trajectory planning, a practical example is included in the paper.

scholarly journals A study on motion of a robot end-effector using the curvature theory of dual unit hyperbolic spherical curves

Filomat ◽  
2016 ◽  
Vol 30 (3) ◽  
pp. 791-802
Author(s):  
Burak Sahiner ◽  
Mustafa Kazaz ◽  
Hasan Ugurlu
Keyword(s):  
Trajectory Planning ◽  
Ruled Surface ◽  
End Effector ◽  
Robot Trajectory ◽  
Spherical Curve ◽  
Timelike Ruled Surface ◽  
Curvature Theory ◽  
Differential Properties ◽  
Robot Trajectory Planning

In this paper we study the motion of a robot end-effector by using the curvature theory of a dual unit hyperbolic spherical curve which corresponds to a timelike ruled surface with timelike ruling generated by a line fixed in the end-effector. In this way, the linear and angular differential properties of the motion of a robot end-effector such as velocities and accelerations which are important information in robot trajectory planning are determined. Moreover, the motion of a robot end-effector which moves on the surface of a right circular hyperboloid of one sheet is examined as a practical example.

Accurate Motion of a Robot End-Effector Using the Curvature Theory of Ruled Surfaces

1988 ◽  
Vol 110 (4) ◽  
pp. 383-388 ◽  
Author(s):  
B. S. Ryuh ◽  
G. R. Pennock
Keyword(s):  
Trajectory Planning ◽  
Interpolation Method ◽  
Ruled Surface ◽  
Degree Of Freedom ◽  
End Effector ◽  
Straight Path ◽  
Curvature Theory ◽  
Position And Orientation ◽  
Six Degree Of Freedom ◽  
Differential Properties

In robotics, there are two methods of trajectory planning: the joint interpolation method which is appropriate for fast transition of the robot end-effector; and the cartesian interpolation method which is appropriate for slower motion of the end-effector along straight path segments. Neither method, however, is sufficient to allow a smooth, differentiable, transition of position and orientation of the end-effector. In this paper, we propose a method of trajectory planning that will permit more accurate motion of a robot end-effector. The method is based on the curvature theory of a ruled surface generated by a line fixed in the end-effector, referred to as the tool line. The orientation of the end-effector about the tool line is included in the analysis to completely describe the six degree-of-freedom motion of the end-effector. The linear and angular properties of motion of the end-effector, determined from the differential properties of the ruled surface, are utilized in the trajectory planning.

scholarly journals On Motion of Robot End-Effector Using the Curvature Theory of Timelike Ruled Surfaces with Timelike Rulings

2008 ◽  
Vol 2008 ◽  
pp. 1-19 ◽  
Author(s):  
Cumali Ekici ◽  
Yasin Ünlütürk ◽  
Mustafa Dede ◽  
B. S. Ryuh
Keyword(s):  
Ruled Surface ◽  
Ruled Surfaces ◽  
End Effector ◽  
Timelike Ruled Surface ◽  
Curvature Theory ◽  
Differential Properties

The trajectory of a robot end-effector is described by a ruled surface and a spin angle about the ruling of the ruled surface. In this way, the differential properties of motion of the end-effector are obtained from the well-known curvature theory of a ruled surface. The curvature theory of a ruled surface generated by a line fixed in the end-effector referred to as the tool line is used for more accurate motion of a robot end-effector. In the present paper, we first defined tool trihedron in which tool line is contained for timelike ruled surface with timelike ruling, and transition relations among surface trihedron: tool trihedron, generator trihedron, natural trihedron, and Darboux vectors for each trihedron, were found. Then differential properties of robot end-effector's motion were obtained by using the curvature theory of timelike ruled surfaces with timelike ruling.

scholarly journals The adjoint trajectory of robot end effector using the curvature theory of ruled surface

Filomat ◽  
2020 ◽  
Vol 34 (12) ◽  
pp. 4061-4069
Author(s):  
Fatma Güler
Keyword(s):  
Fixed Point ◽  
Angular Velocity ◽  
Angular Acceleration ◽  
Ruled Surface ◽  
End Effector ◽  
Fixed Frame ◽  
Robot Trajectory ◽  
Curvature Theory

The ruled surface is formed by the movement of a director based on a curve. The point P not on the director vector at fixed frame o-ijk draws a curve. However, each position of this point on the curve always corresponds to position of director on the ruled surface, or this point is adjoint to director vector. Thus, the curve is adjoint to the ruled surface. In this study, we expressed the adjoint trajectory of robot end effector. We can change the trajectory of the robot movement by defining the adjoint trajectory when it may not be physically achievable and not re-computation of the robot trajectory. We investigated the angular acceleration and angular velocity of adjoint trajectory of the robot end effector. Also, we obtained the condition that moving point is a fixed point.

A path planning method for robot end effector motion using the curvature theory of the ruled surfaces

2018 ◽  
Vol 15 (03) ◽  
pp. 1850048 ◽  
Author(s):  
Fatma Güler ◽  
Emin Kasap
Keyword(s):  
Angular Velocity ◽  
Path Planning ◽  
Trajectory Planning ◽  
Ruled Surfaces ◽  
End Effector ◽  
Robot Trajectory ◽  
Positional Variation ◽  
Angle Function ◽  
Work Area ◽  
Curvature Theory

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.

Offset Trajectory Planning of Robot End Effector and its Jerk with Curvature Theory

2021 ◽  
pp. 2150050
Author(s):  
Fatma Güler
Keyword(s):  
Time Derivative ◽  
Angular Acceleration ◽  
Parallel Robots ◽  
Transition Curve ◽  
Third Order ◽  
End Effector ◽  
Robot Trajectory ◽  
Positional Variation ◽  
Curvature Theory ◽  
Business Media

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.

scholarly journals Industrial robot trajectory planning based on improved pso algorithm

2021 ◽  
Vol 1820 (1) ◽  
pp. 012185
Author(s):  
Shunjie Han ◽  
Xinchao Shan ◽  
Jinxin Fu ◽  
Weijin Xu ◽  
Hongyan Mi
Keyword(s):  
Trajectory Planning ◽  
Industrial Robot ◽  
Pso Algorithm ◽  
Robot Trajectory ◽  
Improved Pso ◽  
Robot Trajectory Planning

The Curvature Theory of Line Trajectories in Spatial Kinematics

1981 ◽  
Vol 103 (4) ◽  
pp. 718-724 ◽  
Author(s):  
J. M. McCarthy ◽  
B. Roth
Keyword(s):  
Planar Motion ◽  
Spatial Motion ◽  
Third Order ◽  
Local Shape ◽  
The Third ◽  
Moving Body ◽  
Physical Representation ◽  
Curvature Theory ◽  
Spatial Kinematics ◽  
Differential Properties

This paper develops the differential properties of ruled surfaces in a form which is applicable to spatial kinematics. Derivations are presented for the three curvature parameters which define the local shape of a ruled surface. Related parameters are also developed which allow a physical representation of this shape as generated by a cylindric-cylindric crank. These curvature parameters are then used to define all the lines in the moving body which instantaneously generate speciality shaped trajectories. Such lines may be used in the synthesis of spatial motions in the same way that the points on the inflection circle and cubic of stationary curvature are used to synthesize planar motion. As an example of this application several special sets of lines are defined: the locus of all lines which for a general spatial motion instantaneously generate helicoids to the second order and the locus of lines generating right hyperboloids to the third order.

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