## 7 Conclusions  
For the industrial robots' PTP joint motion planning with velocity constraint, cubic polynomial planning has the problem of angular acceleration discontinuity, which will cause vibrations and impacts. As for quintic polynomial planning, angular accelerations at target points need to be given in advance and because it is hard to give appropriate angular accelerations in advance, it will lead to large velocity fluctuations which are not conducive to velocity control.  
In order to solve these problems in joint planning, a kind of methodology based on modified Hermite interpolation is proposed. On the basis of Hermite interpolation, new target knots are reconstructed from the initial target knots. Then, newly-built knots instead of old initial knots are used to carry out joint motion planning. Using the proposed methodology, not only does the motion planning meet the requirements for angular displacement and angular velocity at the initial knots, but also the angular acceleration of the whole planning is continuous without broken points. Moreover, the maximum order of the planning functions is three, which avoids the velocity fluctuation caused by the higher-order functions such as quintic polynomial. Therefore, the problems existing in cubic polynomial planning and quintic polynomial planning can be solved well.  
A study case is given to verify the rationality and effectiveness of the proposed method. The case study results show that, compared with the other two planning methods, the proposed methodology is applied well to PTP joint motion planning with velocity constraints, and is conducive to the working performance and service life of industrial robots.  
In the future, efforts will be made on optimization investigation as continued research in the area. The basic motion planning of industrial robots, including improved motion planning, focuses on the continuity and smoothness of the trajectory. However, efficiency, energy, impact and other problems in the actual operation requirements are not considered. Thus, optimization of single or multiple objectives among efficiency, energy, the impact  
Figure 15: Angular acceleration comparison: (**a**) Proposed methodology planning; (**b**) Cubic polynomial planning; (**c**) Quintic polynomial planning.  
will be the further potential studies on the basis of the motion planning. Moreover, motion planning needs to be realized through motion control. Due to strong nonlinear coupling between each joint and end-effector, there will be disturbances, errors and other uncertain factors, so optimal control will also be considered as a further investigation to guarantee motion planning realized precisely.  
Conceptualization, Y.S.; Investigation, X.L. and P.Z.; Funding acquisition, Y.S. and P.Z.; Writing--original draft, Y.P.; Validation, W.Z. and Y.P.; Methodology, Y.P. and W.Z.; Writing--review and editing, Y.P., Y.S. and X.L. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the \"National Science and Technology Major Project of China, grant number 2017-VII-0002-0095\" and the \"Aeronautical Science Foundation of China, grant number 2020Z045053001\".  
Not applicable.  
The authors declare no conflict of interest.