1 Introduction
2 Analytical solution based on joint angle parameterization
2.1 Kinematics modeling of the EMM
Tab.1 D–H parameters of the experimental module manipulator |
Link | /m | /( ) | /m | /( ) |
---|---|---|---|---|
1 | 0 | 90 | 0 | |
2 | 0 | 90 | 0 | |
3 | 0 | –90 | -90 | |
4 | 0 | 0 | ||
5 | 0 | 90 | ||
6 | 0 | 90 | 0 | |
7 | 0 | –90 | 0 | |
8 | 0 | 90 | 0 | 90 |
Note: is the length of link, which is defined as the distance from to measured along positive direction; is the torsion angle of link, which is defined as the angle from to measured about positive direction; is the offset distance of link, which is defined as the distance from to measured along positive direction; and is the rotation angle of link, which is defined as the angle from to measured about positive direction. Link 1 is relative to coordinate system , and link 8 is relative to coordinate system . |
2.2 Analytical solution of each joint angle
3 Approximate analytical solution based on arm angle parameterization
3.1 Definition of arm angle and arm plane
3.2 Analytical solution of the SRS manipulator based on arm angle parameterization
3.2.1 Analytical solution of
3.2.2 Analytical solution of
3.2.3 Analytical solution of , , and
3.2.4 Analytical solution of , , and
3.3 Approximate analytical solution of the EMM based on arm angle parameterization
3.3.1 Relationship of joint angles between the EMM and the SRS manipulator
Tab.2 Relationship of joint angles between the experimental module manipulator (EMM) and the spherical–roll–spherical (SRS) manipulator |
Joint angle | EMM | SRS manipulator | Relationship |
---|---|---|---|
, | , | ||
The EMM and and the SRS manipulator . Therefore, , , and has no relationship with and ; hence, | |||
Note: , ; , . |
3.3.2 Approximate analytical solution of each joint angle
3.3.3 Case analysis
Tab.3 Eight approximate solutions calculated by nominal arm angle and their actual arm angle in Case 1 |
Inverse solution | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | Actual arm angle /(° ) |
---|---|---|---|---|---|---|---|---|
1 | –75.0144 | 77.9015 | –35.0088 | –67.7947 | –106.9448 | 108.4550 | 78.4236 | 129.7819 |
2 | –75.0144 | 77.9015 | –32.9623 | –93.9476 | 97.1616 | –108.4550 | –101.5764 | 130.1939 |
3 | 104.9856 | –77.9015 | 145.4975 | –44.7304 | –130.5154 | 108.4550 | 78.4236 | 142.4985 |
4 | 104.9856 | –77.9015 | 157.5737 | –83.3832 | 76.0612 | –108.4550 | –101.5764 | 142.5272 |
5 | 65.0858 | 129.3636 | –170.4417 | 76.7806 | –83.3659 | 87.0906 | –58.9323 | 126.6470 |
6 | 65.0858 | 129.3636 | –160.6816 | 39.4032 | 124.2514 | –87.0906 | 121.0677 | 125.3823 |
7 | –114.9142 | –129.3636 | 12.8840 | 96.8902 | –106.8012 | 87.0906 | –58.9323 | 140.8476 |
8 | –114.9142 | –129.3636 | 10.8205 | 75.3257 | 96.8267 | –87.0906 | 121.0677 | 140.3597 |
4 Precise semi-analytical solution based on arm angle parameterization
4.1 Solution process
4.2 Case correction and analysis
Tab.4 Eight precise solutions corrected to nominal arm angle and their actual arm angle in Case 1 |
Inverse solution | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | Actual arm angle /(° ) |
---|---|---|---|---|---|---|---|---|
1 | –79.6594 | 80.0057 | –31.7645 | –68.5655 | –107.4127 | 112.4957 | 81.6077 | 134.9996 |
2 | –79.2564 | 79.8118 | –29.8156 | –94.8141 | 96.7095 | –112.1390 | –98.6737 | 135.0003 |
3 | 111.1286 | –75.5521 | 143.0353 | –46.8091 | –128.5043 | 103.3635 | 74.3511 | 135.0000 |
4 | 111.4466 | –75.4437 | 154.0291 | –84.2827 | 77.8483 | –103.1072 | –105.8565 | 134.9999 |
5 | 57.8508 | 124.0487 | –170.7387 | 75.5782 | –86.7634 | 81.7248 | –53.1516 | 135.0002 |
6 | 56.9938 | 123.3800 | –162.8315 | 40.3714 | 119.9489 | –81.0841 | 127.5950 | 134.9995 |
7 | –109.5662 | –132.8739 | 14.2388 | 97.0154 | –104.6965 | 90.9200 | –62.6529 | 134.9997 |
8 | –109.8352 | –132.7062 | 12.9465 | 74.4857 | 98.9454 | –90.7312 | 117.5244 | 135.0001 |
5 Superiority of the arm angle parameterization method in parameter selection
Tab.5 Eight approximate solutions calculated by nominal arm angle and their actual arm angle in Case 2 |
Inverse solution | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | Actual arm angle /(° ) |
---|---|---|---|---|---|---|---|---|
1 | –74.5529 | 41.1411 | 59.0447 | –147.0799 | 39.7875 | 96.3458 | –25.6682 | 135.8676 |
2 | –74.5529 | 41.1411 | 33.3418 | –123.9961 | –137.5933 | –96.3458 | 154.3318 | 122.7169 |
3 | 105.4471 | –41.1411 | –127.4815 | –121.8101 | 21.0440 | 96.3458 | –25.6682 | 146.4159 |
4 | 105.4471 | –41.1411 | –151.1755 | –97.6415 | –159.4307 | –96.3458 | 154.3318 | 136.0138 |
5 | 25.7588 | 113.7663 | 103.5857 | 97.3836 | 139.3401 | 121.8922 | 98.1554 | 138.8542 |
6 | 25.7588 | 113.7663 | 85.7979 | 125.7015 | –51.1899 | –121.8922 | –81.8446 | 131.6935 |
7 | –154.2412 | –113.7663 | –99.9945 | 121.7516 | 138.5523 | 121.8922 | 98.1554 | 139.3173 |
8 | –154.2412 | –113.7663 | –124.9401 | 145.1176 | –39.8680 | –121.8922 | –81.8446 | 125.0305 |
Tab.6 Eight precise solutions corrected to nominal arm angle and their actual arm angle in Case 2 |
Inverse solution | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | /(° ) | Actual arm angle /(° ) |
---|---|---|---|---|---|---|---|---|
1 | –74.2549 | 40.5634 | 58.1106 | –147.0877 | 40.4691 | 96.8771 | –25.9693 | 135.0011 |
2 | -77.9949 | 51.0466 | 43.4482 | –125.4135 | –143.7159 | –87.7922 | 159.8987 | 134.9980 |
3 | 113.0101 | –31.4347 | –137.9357 | –123.3949 | 26.0090 | 105.8662 | –30.5921 | 135.0012 |
4 | 105.9051 | –40.2642 | –151.8906 | –97.6617 | –159.0967 | –97.1529 | 153.8738 | 135.0007 |
5 | 23.3078 | 116.7033 | 100.6682 | 97.9066 | 140.0811 | 118.3647 | 96.9436 | 135.0007 |
6 | 27.5818 | 111.2742 | 88.3354 | 125.4589 | –52.2753 | –124.7980 | –80.9160 | 134.9985 |
7 | –156.4982 | –116.4861 | –103.1285 | 121.4000 | 140.5163 | 118.6301 | 97.0378 | 135.0014 |
8 | –150.2092 | –107.8211 | –117.2833 | 146.4267 | –46.1905 | –128.7216 | –79.7563 | 135.0004 |
Tab.7 Changes in actual arm angle in the interval |
Combination of joint angles | Type of interval | Interval of actual arm angle /(° ) |
---|---|---|
, , | Non-solution | |
, , | Repetition-solution | |
, , | Non-solution | |
, , | Repetition-solution | |
, , | Non-solution | |
, , | Repetition-solution | |
, , | Non-solution | |
, , | Repetition-solution |