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HOME > No.16, Feb 2019 > Smooth Jerk-limited Trajectory Generation and Nonlinear Friction Compensation for Feed Drive Systems

Smooth Jerk-limited Trajectory Generation and Nonlinear Friction Compensation for Feed Drive Systems

The International Federation of Automatic Control (IFAC) 2017, Application Paper Prize


The systems engineering laboratory in the department of mechanical engineering at Toyohashi University of Technology in collaboration with the institute for system dynamics at the University of Stuttgart, has developed methods for generating smooth motion trajectories for feed drive systems for efficient manufacturing and energy saving. The method is based on quintic Bézier curves and the altered bang-bang approach to ensure smooth velocity transitions within the motion trajectory. In addition, a contouring controller with a feedforward friction compensator was applied in order to smoothly track the designed trajectory by canceling out the effect of friction forces.

The result of this study was presented during the 20th World Congress, The International Federation of Automatic Control (IFAC), Toulouse, France, July, 2017 and published by ELSEVIER. Consequently, the researchers of this study were awarded with the IFAC Application Paper Prize (Finalist). The prize is given for outstanding technical contributions at an IFAC Congress in the area of control applications. The IFAC congress is held once in every three years and during the 20th IFAC congress, 2,800 out of 4,267 submissions were accepted for presentation. From the 2,800 accepted submissions, five papers including our paper were selected and awarded as the application paper prize finalists.

Details of the research

Computer numerical control (CNC) machines are widely used to manufacture complex components that are required to be very precise to comply with the rising demand. In order to achieve the desired manufacturing quality, motion trajectories must be traversable without violating systems’ constraints such as a permissible acceleration and jerk. Generally, the trajectories have to be at least second order differentiable in order to achieve a continuous velocity and acceleration. Linearly interpolated trajectories have been used for decades due to their simplicity in design and implementation. However, these linear trajectories cause undesired performance, such as stops between trajectories’ segments, unnecessary time and energy consumptions and wear on the system parts.

For improvement of motion trajectories, many methods have been proposed in the literature and most of them focus on smoothing the linear interpolated tool-path points using curve fitting techniques. For densely tool-path points, curve fitting techniques exhibit oscillations in the trajectory because high-order spline curves are numerically unstable. On the other hand, cubic parametric spline curves are used to smoothly interpolate the linear tool-path points. They are infinitely differentiable when used as a single curve, however, their differentiability is limited to first-order when two or more curves are connected.

In this study, a method for generating smooth motion trajectories using quintic Bézier curves and smooth velocity profiles based on the altered bang-bang approach was proposed. The altered bang-bang approach allows for smooth velocity transitions between the spline curves, such that low-velocity values can be used during the motion start and end, and in all areas with high curvatures. Since it is well known that mechanical systems cannot instantly accelerate to high velocities and that the velocity is inversely proportional to the curvature, smooth reference velocity ensures that the motion acceleration and jerk are within the permissible range.

Fig.1 represents a method to generate smooth trajectories using Bézier curves. From the left side, the first figure represents the 3D surface to be machined, the middle figure is the 3D curve extracted from the 3d surface. The last figure is the zoomed portion of the 3D curve (point Gi), where the dashed and solid lines referrer to the original and modified curves, respectively. Points Gi, i = 1, 2,… are the G-code points used to define the original trajectory, γ and µi are the tolerance in the contour error owing to the induced geometrical error and the tangential unit vectors corresponding to the motion direction, respectively. Points P0i to P5i and B0i to B5i, are Bézier control points for each linear trajectory-segment and the inserted curve at the corner to create the modified curve (solid line). αi and βi are the fractions of the length li, and they are used to design the curvature of the inserted curve. The modified curve ensures that the induced geometrical errors do not exceed the allowed tolerance γ.

In addition to the velocity profile matter, mechanical systems experience friction forces which vary nonlinearly with velocities. Therefore, on implementing a smooth velocity profile, a contouring controller with a feed forward friction compensator was applied in order to smoothly track the designed trajectory by canceling out the effect of friction forces. Experimental verification was conducted and results showed that the proposed method is effective for improving the performance of CNC systems.

Behind-the-scenes story

We had previously successfully developed a method to automatically generate smooth trajectories for mobile robots based on the assigned via-points. In addition, other members of our laboratory studied and developed a nonlinear motion controller with "a" friction compensator for feed drive systems. The performance of the designed controller was found dependent on the nature of the applied trajectory. Tracking performance was much better if the trajectory is smooth enough to be traversed by the system. On the other hand, poor performance was observed when traversing trajectories with high curvatures. It was in this light that we studied the method previously used for mobile robots and modified it for application to feed drive systems.

Future Research

The research team of this study believe that the proposed method will be adopted for industrial applications to enhance manufacturing performance. In addition, this study can be applied to other motion systems, such as autonomous mobile robots and driverless cars.

This study is partly sponsored by the Japan Society for the Promotion of Science (JSPS).


Kenneth Renny Simba, Gunter Heppeler, Ba Dinh Bui, Yogi Muldani Hendrawan, Oliver Sawodny, Naoki Uchiyama, Bézier Curve Based Trajectory Generation and Nonlinear Friction Compensation for Feed Drive Contouring Control, IFAC-PapersOnLine, Volume 50, Issue 1, (2017), Pages 1944-1951.


International Federation of Automatic Control (IFAC) 2017, Application Paper Prize

By シンバ ケニス レニー


本成果は、20th World Congress, The International Federation of Automatic Control (IFAC), Toulouse, France, July, 2017に報告し、Elsevier社の論文に採択されました。さらに、制御応用分野の論文を対象としたIFAC Application Paper Prize (Finalist)として表彰されました。The IFAC World congressは3年に一度開催される制御分野の世界的な会議であり、20回目を迎える今回は4,267の投稿論文のうち2,800件が採択され、本論文を含む全部で5編の論文が同賞を受賞しました。










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Student Profile

Affiliation Department of Mechanical Engineering Current: Musashi Seimitsu Industry Co., Ltd.
Title Doctor course student (when he got the prize)
Fields of Research precision control, motion planning, feed drives