1、會計學1機器人學基礎機器人軌跡機器人學基礎機器人軌跡(guj)規劃蔡自規劃蔡自興興第一頁，共54頁。2Ch.7 Trajectory Planning of Robots7.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5Summary第1頁/共53頁第二頁

2、，共54頁。37.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第2頁/共53頁第三頁，共54頁。47.1 General considerations第3頁/共53頁第四頁

3、，共54頁。57.1 General considerations第4頁/共53頁第五頁，共54頁。6General Considerations - Solution Space7.1 General considerations第5頁/共53頁第六頁，共54頁。7General Considerations - Solution Space7.1 General considerationsInitial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.第6頁/共53頁第七頁，共5

4、4頁。87.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第7頁/共53頁第八頁，共54頁。97.2 Interpolated Calculation of Joint Tr

5、ajectories 關節關節(gunji)軌跡的插值計軌跡的插值計算算7.2 JointSpace Schemes第8頁/共53頁第九頁，共54頁。10Joint-Space Schemes 7.2 JointSpace SchemesSeveral possible path shapes for a single joint.第9頁/共53頁第十頁，共54頁。blends for a path with via points11Joint-Space Schemes 7.2 JointSpace Schemes第10頁/共53頁第十一頁，共54頁。127.2.1 Cubic Polyno

6、mials 三次三次(sn c)多項式插值多項式插值(7.1)(7.2)fft)()0(00)(0)0(ft7.2 JointSpace Schemes t第11頁/共53頁第十二頁，共54頁。137.2.1 Cubic Polynomials(7.5)7.2 JointSpace Schemes23211332210003200fffffftataaatatataaa00120230303()2()ffffaaatat (7.6)第12頁/共53頁第十三頁，共54頁。147.2.1 Cubic Polynomials332210)(tatataatThe joint velocity and

7、 acceleration along this path are:taattataat32232162)(32)( (7.4)(7.3)7.2 JointSpace Schemes第13頁/共53頁第十四頁，共54頁。157.2.1 Cubic Polynomials7.2 JointSpace Schemes第14頁/共53頁第十五頁，共54頁。167.2.1 Cubic Polynomials0012023031503()202()4.44ffffaaatat 232301232212323( )15204.44( )234013.33( )264026.66taa ta ta tttt

8、aa ta ttttaa tt7.2 JointSpace Schemes第15頁/共53頁第十六頁，共54頁。177.2.1 Cubic Polynomials7.2 JointSpace Schemes23230123( )15204.44taa ta ta tttStarts at 15 degrees and ends at 75 degrees!第16頁/共53頁第十七頁，共54頁。187.2.1 Cubic Polynomials7.2 JointSpace Schemes22123( )234013.33taa ta tttStarts and ends at rest!第17頁

9、/共53頁第十八頁，共54頁。197.2.1 Cubic Polynomials7.2 JointSpace Schemes23( )264026.66taa ttAcceleration profile is linear!第18頁/共53頁第十九頁，共54頁。207.2.2 Cubic polynomials with via points 過路徑過路徑(ljng)點的三次多項式插值點的三次多項式插值7.2 JointSpace Schemesfft)()0(00010200230032321()21()()fffffffffaaatttatt Solutions:232110332210

10、0032ffffffftataaatatataaa第19頁/共53頁第二十頁，共54頁。(average of 2 sides etc.).nThe system automatically chooses the velocities at the via points in such a way as to cause the acceleration at the via points to be continuous.217.2 JointSpace Schemes7.2.2 Cubic polynomials with via points第20頁/共53頁第二十一頁，共54頁。22

11、7.2.3 Higher-order polynomials高階多項式插值高階多項式插值5544332210)(tatatatataat(7.10)7.2 JointSpace Schemes第21頁/共53頁第二十二頁，共54頁。237.2.3 Higher-order polynomials(7.11)35243220453423211055443322100020126225432ffffffffffffffftatataaatatatataaatatatatataaa 7.2 JointSpace Schemes第22頁/共53頁第二十三頁，共54頁。247.2.3 Higher-or

12、der polynomials(7.12)00100220003320004420005522020(812)(3)23030(1416)(32)21212(66)()2ffffffffffffffffffaaattatttatttat7.2 JointSpace Schemes第23頁/共53頁第二十四頁，共54頁。257.2.4 Linear function with parabolic blends 用拋物線過渡用拋物線過渡(gud)的線性插值的線性插值7.2 JointSpace SchemesDiscontinuous velocity - can not be controlle

13、d!第24頁/共53頁第二十五頁，共54頁。267.2.4 Linear function with parabolic blends7.2 JointSpace Schemes第25頁/共53頁第二十六頁，共54頁。277.2.4 Linear function with parabolic blends7.2 JointSpace Schemes第26頁/共53頁第二十七頁，共54頁。287.2.4 Linear function with parabolic blendsbhbbthbttt(7.13)7.2 JointSpace Schemes第27頁/共53頁第二十八頁，共54頁。2

14、97.2.4 Linear function with parabolic blends0)(02fbbttt (7.15) 2)(42022fbttt(7.16)20)(4tf (7.17)The acceleration chosen must be sufficiently high, to ensure the existence of a solution:7.2 JointSpace Schemes第28頁/共53頁第二十九頁，共54頁。307.2.5 Linear function with parabolic blendsfor a path with via points過路

15、徑過路徑(ljng)點的用拋物線過渡的線性插值點的用拋物線過渡的線性插值7.2 JointSpace SchemesMulti-segment linear path with blends.第29頁/共53頁第三十頁，共54頁。317.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points第30頁/共53頁第三十一頁，共54頁。327.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a

16、 path with via pointskjdjkjkkjkklkkjkklkdjkjkjktttttt2121)sgn( 第31頁/共53頁第三十二頁，共54頁。337.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points211212112121211221212111212121)(2)sgn(tttttttttdddd 第32頁/共53頁第三十三頁，共54頁。347.2 JointSpace Schemes7.2.5 Linear function with p

17、arabolic blendsfor a path with via points1)1()1()1(1)1(112)1()1(12121)(2)sgn(nnnndnnnnndnnnnnnnnndnndnnnnnttttttttt 第33頁/共53頁第三十四頁，共54頁。357.2 JointSpace Schemes7.2.5 Linear function with parabolic blendsfor a path with via points第34頁/共53頁第三十五頁，共54頁。367.1General Considerations in Robot Trajectory Pla

18、nning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第35頁/共53頁第三十六頁，共54頁。7.3 Cartesian-Space Schemes377.3 Cartesian-Space Schemes第36頁/共53頁第三十七頁，共54頁。7.3 Cartesian-S

19、pace Schemes387.3 Cartesian-Space Schemes第37頁/共53頁第三十八頁，共54頁。7.3 Cartesian-Space Schemes397.3 Cartesian-Space Schemes10001000iziziziziyiyiyiyixixixixiiiiiBpaonpaonpaonPpaon1000100011111111111111111ziziziziyiyiyiyixixixixiiiiiiBpaonpaonpaonPpaon第38頁/共53頁第三十九頁，共54頁。7.3 Cartesian-Space Schemes407.3 Car

20、tesian-Space Schemes第39頁/共53頁第四十頁，共54頁。41Initial (A) and Goal (B) Points are reachable, but intermediate points (C) unreachable.7.3 Cartesian-Space Schemes7.3 Cartesian-Space Schemes第40頁/共53頁第四十一頁，共54頁。42Approaching singularities some joint velocities go to causing deviation from the path.7.3 Cartes

21、ian-Space Schemes7.3 Cartesian-Space SchemesCartesian planning difficulties (2/3):第41頁/共53頁第四十二頁，共54頁。43Start point (A) and goal point (B) are reachable in different joint space solutions (The middle points are reachable from below.)7.3 Cartesian-Space Schemes7.3 Cartesian-Space SchemesCartesian planning difficulties (3/3):第42頁/共53頁第四十三頁，共54頁。447.1General Considerations in Robot Trajectory Planning7.2Interpolated Calculation of Joint Trajectories 7.3Planning of Cartesian Path Trajectories7.4Real Time Generation of Planning Trajectories7.5SummaryCh.7 Trajectory Planning of Robots第43