·660. 智能系统学报 第14卷 划。与直线部分类似，S型速度规划可保证加速度 参考文献： 连续。 [1]LIU Huashan,LAI Xiaobo,WU Wenxiang.Time-optimal 加速度为600mm/s2 加速度为2800mm/s and jerk-continuous trajectory planning for robot manipu- 450 lators with kinematic constraints[J].Robotics and com- puter-integrated manufacturing,2013,29(2):309-317. 50 [2]GIBERTI H,SBAGLIA L.URGO M.A path planning al- 5420 gorithm for industrial processes under velocity constraints 410 with an application to additive manufacturing[J].Journal of 550 manufacturing systems,2017,43:160-167. Y/mm -100 450500 [3]ABU-DAKKA F J,ASSAD I F,ALKHDOUR R M,et al. -200350 400 X/mm Statistical evaluation of an evolutionary algorithm for min- 图7圆弧插补运动的位移曲线 imum time trajectory planning problem for industrial ro- Fig.7 The displacement curve of arc interpolation motion bots[J].The international journal of advanced manufactur- ing technology,2017,891/2/3/4):389-406. [4]VULLIEZ M,LAVERNHE S,BRUNEAU O.Dynamic 加速度为600mm/s2 50 -加速度为2800mm/s approach of the feedrate interpolation for trajectory plan- ning process in multi-axis machining[J].The international 0 0.51.01.52.02.53.03.54.04.5 的 journal of advanced manufacturing technology,2016, 88(5/6/7/8):2085-2096. 1000 500 一加速度为600mm/s 一加速度为2800mm/s2 [5]ZHAO Huan,ZHU Limin,DING Han.A real-time look- 0 ahead interpolation methodology with curvature-continu- -500 -100 ous B-spline transition scheme for CNC machining of short 00.51.01.52.02.53.03.54.04.5 line segments[J].The international journal of machine s tools and manufacture.2013.65:88-98. 图8圆弧插补运动的速度与加速度曲线 [6]许健，梅江平，段晓斌，等.一种工业机器人连续轨迹规 Fig.8 Curve of velocity and acceleration of are interpola- 划过渡算法.工程设计学报，2016.23(6)：537-543 tion motion XU Jian,MEI Jiangping,DUAN Xiaobin,et al.An al- gorithm for segment transition in continuous trajectory planning of industrial robot[J].Chinese journal of engin- eering design,.2016,23(6):537-543. [7]李黎，尚俊云，冯艳丽，等.关节型工业机器人轨迹规划 研究综述.计算机工程与应用，2018,54(5)：36-50 LI Li,SHANG Junyun,FENG Yanli,et al.Research of tra- jectory planning for articulated industrial robot:a 图9圆弧插补运动喷绘实验效果 Fig.9 Experimental effect diagram of arc interpolation review[J].Computer engineering and applications,2018, motion spray painting 545):36-50 [8]HU Jun,XIAO Liangjian,WANG Yuhan,et al.An optim- 4结论 al feedrate model and solution algorithm for a high-speed machine of small line blocks with look-ahead[J].The inter- 1)采用S型速度规划，基于四元数的直线与 national journal of advanced manufacturing technology. 圆弧运动的位姿模型，可实现笛卡尔空间规划。 2006.28(9/10:930-935. 2)根据路径长度约束条件，能将速度规划类 [9]史步海，孙会会.基于新S型速度规划的B样条曲线算 型分为4种情况，可实现速度和加速度的自动调整。 法研究).机床与液压，2016,2016,44(15)：72-79. 3)本文设计的S型速度规划算法可保证加速 SHI Buhai,SUN Huihui.B-spline curve algorithm re- 度连续，减小机械冲击。 search based on new S velocity planning[J].Machine tool 在下一步工作中，将增加误差等约束条件，提 and hydraulics,.2016.2016,44(15):72-79 高控制精度；进一步优化机器人控制系统软硬件 [10]JAHANPOUR J,ALIZADEH M R.A novel acc-jerk-lim- 平台，提高系统实时性。 ited NURBS interpolation enhanced with an optimized S.划。与直线部分类似，S 型速度规划可保证加速度 连续。 400100 410 420 550 Z/mm430 0 500 440 450 −100 450 400 −200 350 加速度为 600 mm/s2 加速度为 2 800 mm/s2 Y/mm X/mm 图 7 圆弧插补运动的位移曲线 Fig. 7 The displacement curve of arc interpolation motion 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 t/s 0 50 100 150 速度/(mm·s−1 ) 加速度为 600 mm/s2 加速度为 2 800 mm/s2 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 t/s −1 000 −500 0 500 1 000 加速度/(mm·s −2 ) 加速度为 600 mm/s2 加速度为 2 800 mm/s2 图 8 圆弧插补运动的速度与加速度曲线 Fig. 8 Curve of velocity and acceleration of arc interpola￾tion motion 图 9 圆弧插补运动喷绘实验效果 Fig. 9 Experimental effect diagram of arc interpolation motion spray painting 4 结论 1) 采用 S 型速度规划，基于四元数的直线与 圆弧运动的位姿模型，可实现笛卡尔空间规划。 2) 根据路径长度约束条件，能将速度规划类 型分为 4 种情况，可实现速度和加速度的自动调整。 3) 本文设计的 S 型速度规划算法可保证加速 度连续，减小机械冲击。 在下一步工作中，将增加误差等约束条件，提 高控制精度；进一步优化机器人控制系统软硬件 平台，提高系统实时性。 参考文献： LIU Huashan, LAI Xiaobo, WU Wenxiang. Time-optimal and jerk-continuous trajectory planning for robot manipu￾lators with kinematic constraints[J]. Robotics and com￾puter-integrated manufacturing, 2013, 29(2): 309–317. [1] GIBERTI H, SBAGLIA L, URGO M. A path planning al￾gorithm for industrial processes under velocity constraints with an application to additive manufacturing[J]. Journal of manufacturing systems, 2017, 43: 160–167. [2] ABU-DAKKA F J, ASSAD I F, ALKHDOUR R M, et al. Statistical evaluation of an evolutionary algorithm for min￾imum time trajectory planning problem for industrial ro￾bots[J]. The international journal of advanced manufactur￾ing technology, 2017, 89(1/2/3/4): 389–406. [3] VULLIEZ M, LAVERNHE S, BRUNEAU O. Dynamic approach of the feedrate interpolation for trajectory plan￾ning process in multi-axis machining[J]. The international journal of advanced manufacturing technology, 2016, 88(5/6/7/8): 2085–2096. [4] ZHAO Huan, ZHU Limin, DING Han. A real-time look￾ahead interpolation methodology with curvature-continu￾ous B-spline transition scheme for CNC machining of short line segments[J]. The international journal of machine tools and manufacture, 2013, 65: 88–98. [5] 许健, 梅江平, 段晓斌, 等. 一种工业机器人连续轨迹规 划过渡算法 [J]. 工程设计学报, 2016, 23(6): 537–543. XU Jian, MEI Jiangping, DUAN Xiaobin, et al. An al￾gorithm for segment transition in continuous trajectory planning of industrial robot[J]. Chinese journal of engin￾eering design, 2016, 23(6): 537–543. [6] 李黎, 尚俊云, 冯艳丽, 等. 关节型工业机器人轨迹规划 研究综述 [J]. 计算机工程与应用, 2018, 54(5): 36–50. LI Li, SHANG Junyun, FENG Yanli, et al. Research of tra￾jectory planning for articulated industrial robot: a review[J]. Computer engineering and applications, 2018, 54(5): 36–50. [7] HU Jun, XIAO Liangjian, WANG Yuhan, et al. An optim￾al feedrate model and solution algorithm for a high-speed machine of small line blocks with look-ahead[J]. The inter￾national journal of advanced manufacturing technology, 2006, 28(9/10): 930–935. [8] 史步海, 孙会会. 基于新 S 型速度规划的 B 样条曲线算 法研究 [J]. 机床与液压, 2016, 2016, 44(15): 72–79. SHI Buhai, SUN Huihui. B-spline curve algorithm re￾search based on new S velocity planning[J]. Machine tool and hydraulics, 2016, 2016, 44(15): 72–79. [9] JAHANPOUR J, ALIZADEH M R. A novel acc-jerk-lim￾ited NURBS interpolation enhanced with an optimized S- [10] ·660· 智 能 系 统 学 报 第 14 卷