Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities

Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities Stéphane CARO Méthodes de subdivisions pour les systèmes singuliers December 15, 2014

Context Figure: A 6-dof parallel manipulator Figure: A lower-mobility parallel manipulator S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 2 / 39

Lower-Mobility Parallel Manipulators S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 3 / 39

Lower-Mobility Parallel Manipulators Great number of architectures and motion patterns S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 3 / 39

Lower-Mobility Parallel Manipulators Great number of architectures and motion patterns Motion mode changing S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 3 / 39

Lower-Mobility Parallel Manipulators Great number of architectures and motion patterns Motion mode changing Constraint singularities S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 3 / 39

Lower-Mobility Parallel Manipulators Great number of architectures and motion patterns Motion mode changing Constraint singularities tools to analyze and compare manipulators at the conceptual design stage S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 3 / 39

Scope Geometrical analysis Analyze the constraints applied on the moving platform S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 4 / 39

Scope Geometrical analysis Analyze the constraints applied on the moving platform Parallel singularities Loss of control of the moving platform They affect the safety and the performance of the robot S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 4 / 39

Scope Geometrical analysis Analyze the constraints applied on the moving platform Parallel singularities Loss of control of the moving platform They affect the safety and the performance of the robot Conceptual design To consider the parallel singularities at the conceptual design stage S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 4 / 39

Outline 1 Singularity Analysis using Grassmann-Cayley Algebra 2 Superbracket Decomposition 3 Case Study : the 4-RUU Parallel Manipulator 4 Grassmann-Cayley Algebra and Grassmann Geometry 5 Conclusions and Future Works S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 5 / 39

Singularity Analysis using GCA Objectives Determine the parallel singularity conditions Describe the motion of the moving platform in these configurations S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 7 / 39

Singularity Analysis using GCA Objectives Determine the parallel singularity conditions Describe the motion of the moving platform in these configurations Tools The superbracket of Grassmann-Cayley algebra (GCA) S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 7 / 39

Parallel Singularities (n<6)-dof parallel manipulator Actuators apply a n-actuation wrench system W a Limbs apply a (6 n)-constraint wrench system W c J T E = [ˆ$ 1 a... ˆ$n }{{} a ˆ$ c... ˆ$6 n c }{{} a basis of W a 1 a basis of W c ] S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 8 / 39

Parallel Singularities (n<6)-dof parallel manipulator Actuators apply a n-actuation wrench system W a Limbs apply a (6 n)-constraint wrench system W c J T E = [ˆ$ 1 a... ˆ$n }{{} a ˆ$ c... ˆ$6 n c }{{} a basis of W a 1 a basis of W c ] Parallel singularities rank(j E ) < 6 det(j E ) = 0 Constraint singularities W c degenerates gain of some extra dof Actuation singularities (W a + W c ) degenerates while W c does not actuators cannot control the motion of the moving platform S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 8 / 39

Determination of Parallel Singularities Classical methods det(j E ) is a large expression cannot provide insight into geometric conditions for parallel singularities S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 9 / 39

Determination of Parallel Singularities Classical methods det(j E ) is a large expression cannot provide insight into geometric conditions for parallel singularities Alternative methods Grassmann-Cayley algebra (GCA) Grassmann geometry (GG) S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 9 / 39

Grassmann-Cayley Algebra The GCA was developed by H. Grassmann (1809-1877) as a calculus for linear varieties operating on extensors with the join and meet operators. GCA makes it possible to work at the symbolic level, to produce coordinate-free algebraic expressions for the singularity conditions of spatial PMs. The 4-dimensional vector space V associated with P 3 Extensors of step 2 Plücker lines (screws of zero/infinite-pitch) S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 10 / 39

Superbracket Decomposition J T E = [ˆ$ 1 a... ˆ$n }{{} a ˆ$ c... ˆ$6 n c }{{} a basis of W a 1 a basis of W c ] Superbracket S = [ab cd ef gh ij kl] = 24 i=1 y i S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 12 / 39

Superbracket Decomposition J T E = [ˆ$ 1 a... ˆ$n }{{} a ˆ$ c... ˆ$6 n c }{{} a basis of W a 1 a basis of W c ] Superbracket S = [ab cd ef gh ij kl] = 24 i=1 y i y 1 = [abcd][efgi][hjkl] y 23 = [abch][defj][gikl]. y 2 = [abcd][efhi][gjkl] y 24 = [abdh][cefj][gikl] S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 12 / 39

Superbracket Decomposition J T E = [ˆ$ 1 a... ˆ$n }{{} a ˆ$ c... ˆ$6 n c }{{} a basis of W a 1 a basis of W c ] Superbracket S = [ab cd ef gh ij kl] = 24 i=1 y i y 1 = [abcd][efgi][hjkl] y 23 = [abch][defj][gikl]. y 2 = [abcd][efhi][gjkl] y 24 = [abdh][cefj][gikl] A bracket, [a b c d] vanishes whenever : 1 two points are repeated 2 three points are collinear 3 the four points are coplanar S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 12 / 39

Formulation of the Superbracket S = [ab ad ef gh ij kl] = 24 i=1 y i y 1 = [abcd][efgi][hjkl] y 23 = [abch][defj][gikl]. y 2 = [abcd][efhi][gjkl] y 24 = [abdh][cefj][gikl] Simplifications Reduce the number of distinct points Use collinear points Use coplanar points, points at infinity P f j l m k Difficulties Several possible formulations Effect of the permutation of lines Long hand calculations R 3 c g a e S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 13 / 39

Graphical User Interface S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 14 / 39

Applications 3-RPS Exechon Orthoglide 3-UPU wrist H4 5-RPUR S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 15 / 39

Case Study z U U R Figure: The 4-RUU parallel manipulator S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 17 / 39

Constraint Analysis Limb wrench systems z A pure constraint moment M i = (0; m i z) A pure actuation force F i = (f i ; r Ci f i ) C i m i Platform wrench systems F i r Ci z F 1, F 2, F 3, F }{{} 4, M 1, M 2, M 3, M }{{} 4 4-system 2-system the PM is over-constrained z x z O y Superbracket Superbracket [F 1, F 2, F 3, F 4, M I, M II ] six possible superbrackets B i m i Figure: A RUU limb A i S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 18 / 39

Superbracket Points S 1 = [ab, ef, cd, gh, ij, kj] S 2 = [ab, ef, cd, gh, ij, mj] P R 3 h f j d b i l m k S 3 = [ab, ef, cd, gh, ij, lj] z S 4 = [ab, ef, cd, gh, lj, mj] S 5 = [ab, ef, cd, gh, lj, kj] F 2 F 4 S 6 = [ab, ef, cd, gh, mj, kj] m 1 F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 19 / 39

Superbracket Points S 1 = [ab, ef, cd, gh, ij, kj] P R 3 h f j d b k i z F 2 F 4 m 1 F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 19 / 39

Superbracket Points S 1 = [ab, ef, cd, gh, ij, kj] P h f j d b k i R 3 F 1 z z P 1 F 2 F 4 m 1 F 1 F 3 Figure: The 4-RUU PM Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 19 / 39

Superbracket Points S 1 = [ab, ef, cd, gh, ij, kj] z P h f j d b k i R 3 F 1 F 2 z z P 1 P2 F 2 F 4 m 1 F 1 F 3 Figure: The 4-RUU PM Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 19 / 39

Superbracket Points S 1 = [ab, ef, cd, gh, ij, kj] z P h f j d b k i F 1 F 2 R 3 z T 12 T 34 c g a e z P 1 P2 F 2 F 4 m 1 F 1 F 3 Figure: The 4-RUU PM Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 19 / 39

Wrench Graph S 1 = [ab, ef, cd, gh, ij, kj] P h f j d b k i Points at infinity {b, d, f, h, i, j, k} R 3 z T 12 T 34 c g a e [a, c, j, x] = 0 F 2 F 4 [e, g, j, x] = 0 m 1 F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 20 / 39

Superbracket Decomposition ( ) S j = A j [a d f h][e c b j] [a b f h][e c d j] }{{} B ; j = 1,..., 6 A 1 = [g i k j], A 2 = [g i l j], A 3 = [g i m j], A 4 = [g k l j], A 5 = [g k m j], A 6 = [g l m j] S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 21 / 39

Superbracket Decomposition ( ) S j = A j [a d f h][e c b j] [a b f h][e c d j] }{{} B ; j = 1,..., 6 A 1 = [g i k j], A 2 = [g i l j], A 3 = [g i m j], A 4 = [g k l j], A 5 = [g k m j], A 6 = [g l m j] Two parallel singularity cases 1 All terms A j vanish simultaneously Constraint singularity S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 21 / 39

Superbracket Decomposition ( ) S j = A j [a d f h][e c b j] [a b f h][e c d j] }{{} B ; j = 1,..., 6 A 1 = [g i k j], A 2 = [g i l j], A 3 = [g i m j], A 4 = [g k l j], A 5 = [g k m j], A 6 = [g l m j] Two parallel singularity cases 1 All terms A j vanish simultaneously Constraint singularity 2 Term B vanishes Actuation singularity S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 21 / 39

Constraint Singularities m 1 m 2 m 3 m 4 z m i z 1 z Figure: A constraint singular configuration Figure: Coupled motions S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 22 / 39

Actuation Singularities ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 P h f j d b i l m k R 3 z T 12 T 34 c g a u e F 2 F 4 m 1 F 1 F 3 S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 23 / 39

Actuation Singularities (Cont d) ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 F 2 F 4 F 3 F 2 F 1 F 4 F 1 C 1 B 1 F 3 Figure: Two actuation forces F 1 and F 2 are parallel. Figure: All actuation forces are coplanar. S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 24 / 39

Correspondence between GCA and GG Objectives Use Grassmann Geometry (GG) with points and lines at infinity Highlight the correspondence between GCA and GG S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 26 / 39

Correspondence between GCA and GG Objectives Use Grassmann Geometry (GG) with points and lines at infinity Highlight the correspondence between GCA and GG Tools The classification of linear varieties S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 26 / 39

Use of Grassmann Geometry M c P F 1 F 2 F 3 F 4 F 5 S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 27 / 39

Use of Grassmann Geometry M c P F 1 F 2 F 3 F 4 F 5 Rank Class Linear line variety 3 planes (3a) a regulus (3b) the union of two flat pencils having a line in common but lying in distinct planes and with distinct centers (3c) all lines through a point (3d) all lines in a plane 4 congruences 5 complexes (5a) non singular complex ; generated by five skew lines (5b) singular complex ; all the lines meeting one given line S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 27 / 39

Rank Deficiency of the Jacobian Matrix J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 f 1 f 2 R 3 z T 12 T 34 c g a u e P j L M II M I b d F 2 F 4 T 12 F 2 c R 3 m 1 F 1 a F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 28 / 39

Rank Deficiency of the Jacobian Matrix J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 f 1 f 2 R 3 z T 12 T 34 c g a u e P j L M II M I b d F 2 F 4 T 12 F 2 c R 3 m 1 F 1 a F 1 Figure: Wrench graph of the 4-RUU PM F 3 ( Condition (3b) of GG ) : dim span(f 1, F 2, M I, M II ) 3 S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 28 / 39

Unconrolled Motions J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 (f 1 f 2 ) (f 3 f 4 ) j i F 2 R 3 z T 12 T 34 c g a u e m 1 F 4 P R 3 F 2 F 1 d b c a T 12 h k f T 34 e g F3 F 4 F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 29 / 39

Unconrolled Motions J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 (f 1 f 2 ) (f 3 f 4 ) j i F 2 R 3 z T 12 T 34 c g a u e m 1 F 4 P R 3 F 2 F 1 d b c a T 12 h k f T 34 e g F3 F 4 F 1 Figure: Wrench graph of the 4-RUU PM F 3 Condition (5b) of GG : A line at infinity intersects all the lines of J T E S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 29 / 39

Unconrolled Motions (Cont d) J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 j u z = 0 i F 2 R 3 z T 12 T 34 c g a u e F 4 P F 2 R 3 F 1 e k T 12 T 34 c a F 3 m 1 g F 4 F 1 F 3 Figure: Wrench graph of the 4-RUU PM S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 30 / 39

Unconrolled Motions (Cont d) J T E = [F 1, F 2, F 3, F 4, M I, M II ] P j i h f d b k ( ) (f 1 f 2 ) (f 3 f 4 ) (u z) = 0 j u z = 0 i F 2 R 3 z T 12 T 34 c g a u e F 4 P F 2 R 3 F 1 e k T 12 T 34 c a F 3 m 1 g F 4 F 1 Figure: Wrench graph of the 4-RUU PM F 3 Condition (5b) of GG : A finite line intersects all the lines of J T E S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 30 / 39

Correspondence between GCA and GG Case Singularity condition Corresponding case GCA of GG (a) f i f j condition (3b) (b) u kl ij z condition (5b) (c) (f i f j) (u kl ij z) condition (5b) (d) (f i f j) (f k f l ) conditions (5b), (3d) (e) (f ( i f j) (u kl ij z) ) (f k f l ) condition (5b) (f) (f i f j) (f k f l ) (u kl ij z) condition (5a) (g) m i m j m k m l conditions 1, (5b) S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 31 / 39

Correspondence between GCA and GG The use of GCA and GG as complementary approaches to better understand the singularities Amine, S., Tale Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., 2011 Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures, ASME Journal of Mechanisms and Robotics, DOI : 10.1115/1.4005336 Amine, S., Tale Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., Singularity Analysis of 3T2R Parallel Mechanisms using Grassmann-Cayley Algebra and Grassmann Line Geometry, Mechanism and Machine Theory, 10.1016/j.mechmachtheory.2011.11.015 S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 32 / 39

Correspondence between GCA and GG The use of GCA and GG as complementary approaches to better understand the singularities How to consider the parallel singularities at the conceptual design stage? Amine, S., Tale Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., 2011 Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures, ASME Journal of Mechanisms and Robotics, DOI : 10.1115/1.4005336 Amine, S., Tale Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., Singularity Analysis of 3T2R Parallel Mechanisms using Grassmann-Cayley Algebra and Grassmann Line Geometry, Mechanism and Machine Theory, 10.1016/j.mechmachtheory.2011.11.015 S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 32 / 39

Contributions 1 Wrench graph : a framework to visualize wrenches in the projective space S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 34 / 39

Contributions 1 Wrench graph : a framework to visualize wrenches in the projective space 2 An efficient singularity analysis method using Grassmann-Cayley algebra enumerate the parallel singularity conditions describe the behavior of the moving platform S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 34 / 39

Contributions 1 Wrench graph : a framework to visualize wrenches in the projective space 2 An efficient singularity analysis method using Grassmann-Cayley algebra enumerate the parallel singularity conditions describe the behavior of the moving platform 3 A graphical user interface for the parallel singularity analysis S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 34 / 39

Future Works Identify the limits of Grassmann-Cayley algebra by studying other motion patterns such as 3R2T, 3R1T and 2R2T Singularity analysis of PMs with finite-pitch wrenches Analysis of the motion mode changing for lower-mobility PMs S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 35 / 39

References Amine, S., Tale-Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., 2012, Singularity Analysis of 3T2R Parallel Mechanisms using Grassmann-Cayley Algebra and Grassmann Line Geometry, Mechanism and Machine Theory, Vol. 52, pp. 326 340. Amine, S., Caro, S., Wenger, P. and Kanaan, D., 2012, Singularity Analysis of the H4 Robot using Grassmann-Cayley Algebra, Robotica, doi :10.1017/S0263574711001330. hal-00642230 Amine, S., Tale-Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., 2012, Singularity Conditions of 3T1R Parallel Manipulators with Identical Limb Structures, ASME Journal of Mechanisms and Robotics, Vol. 4(1), pp. 011011-1 011011-11, doi :10.1115/1.4005336. hal-00642238 Kanaan, D., Wenger, P., Caro, S. and Chablat, D., 2009, Singularity Analysis of Lower-Mobility Parallel Manipulators Using Grassmann-Cayley Algebra, IEEE Transactions on Robotics, Vol.25, N o.5, pp. 995 1004. Amine, S., Tale-Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., Singularity Analysis of the 4-RUU Parallel Manipulator based on Grassmann-Cayley Algebra and Grassmann Geometry, Proceedings of the ASME 2011 International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, August 29-31, 2011, Washington, DC, USA. Amine, S., Tale-Masouleh, M., Caro, S., Wenger, P., and Gosselin, C., Singularity Analysis of 5-DOF Parallel Mechanisms 3T2R using Grassmann-Cayley Algebra, 13th World Congress in Mechanism and Machine Science, Guanajuato, México, June 19-25, 2011. S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 36 / 39

References (Cont d) Amine, S., Kanaan, D., Caro, S. and Wenger, P., Constraint and Singularity Analysis of Lower-mobility Parallel Manipulators with Parallelogram Joints, Proceedings of ASME Design Engineering Technical Conferences, August 15-18, 2010, Montreal, QC., Canada. Amine, S., Kanaan, D., Caro, S. and Wenger, P. Singularity Analysis of Lower-Mobility Parallel Robots with an Articulated Nacelle, Advances in Robot Kinematics, Piran-Portoroẑ, Slovenia, June 27 - July 1, 2010. S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 37 / 39

Conclusions & Future Works Thank you! stephane.caro@irccyn.ec-nantes.fr S. CARO Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities 15/12/2014 38 / 39

Lower-Mobility Parallel Manipulators: Geometrical Analysis and Singularities Stéphane CARO Méthodes de subdivisions pour les systèmes singuliers December 15, 2014