Diapositive 1

UMR CNRS 6597

Institut de Recherche en Communications et Cybernétique de Nantes

Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot D. CHABLAT, R. JHA, F. ROUILLIER, G. MOROZ Institut de Recherche en Communications et en Cybernétique de Nantes, France INRIA Paris-Rocquencourt Institut de Mathématiques de Jussieu , France INRIA Nancy-Grand Est, France 1

Problem statement  There are parallel robots with several operation modes [Zlatanov]  Can we extend the definitions of the aspects for this kind of robots?  Application: 3-RPS [Husty]  Can we easily see if, in the workspace, a trajectory is made between two assembly modes in the same aspect?  What is the image of this trajectory in the joint space? 7/1/2014

D. Chablat – ARK 2014 - June 30 - July 3

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Outline of the presentation  Mechanism under study  Inverse and direct kinematic problem / Operation mode  Singularities and aspects / Operation mode  Characteristic surfaces / Operation mode  Non-singular assembly mode changing trajectory  Conclusions

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The 3-RPS parallel robot  Actuators 𝜌𝜌1, 𝜌𝜌2, 𝜌𝜌3

 End-effector 𝑥𝑥, 𝑦𝑦, 𝑧𝑧, 𝑞𝑞1, 𝑞𝑞2, 𝑞𝑞3, 𝑞𝑞4  Parameters 𝑔𝑔 = ℎ = 1  With

|| Ai  Bi || i

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with i  1,2, 3


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Kinematics  Coordinates of the joints in local and global reference frame  g / 2   g / 2  g            A1   0 A2  g 3 / 2 A 3  g 3 / 2      0 0 0            h / 2   h / 2  h            b1   0 b2  h 3 / 2 b3  h 3 / 2      0 0 0          u  v w  x x  x Bi  P  Rbi for i  1,2, 3 and R  uy vy wy  u v w   x z z  7/1/2014


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Kinematics  Six constraint equations || Ai  Bi || i uy h  y  0

with i  1,2, 3

y  uy h / 2  3vy h / 2  3x  3ux h / 2  3vx h / 2  0 y  uy h / 2  3vy h / 2  3x  3ux h / 2  3vx h / 2  0  Simplification from 6 to 4 parameters y  huy x h

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

3ux  3vy  3uy  3vx



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Operation modes  Definition of the orientation matrix by the unit quaternions  2q 2  2q 2  1 2q q  2q q   q q q q 2 2 2 1 4 2 3 1 3 2 4   1 R   2q1q 4  2q2q 3 2q12  2q 32  1 2q1q2  2q 3q 4    2 2      q q q q q q q q q q 2 2 2 2 2 2 1  1 3 2 4 1 2 3 4 1 4  with q12  q22  q 32  q 42  1  Two operation modes with q1  0

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or

q4  0


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Operation modes  In the workspace W, for each operation mode, the 𝑊𝑊𝑂𝑂𝑂𝑂𝑗𝑗 is defined such that WOM  W j

X  WOM ,OM is constant j

 Inverse and direct kinematic problem g j (X)  q

g j 1(q)  X  (X, q)  OM j  In our case: 7/1/2014

OM 1 : q1  0

or OM 2 : q 4  0


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Algebraic tools  SIROPA library (ANR SIROPA, J-P. Merlet, P. Wenger)  Groëbner basis elimination – Computation of the singularities and characteristic surfaces

 Cylindrical Algebraic Decomposition (CAD) – Representation of the aspects and basic regions

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Singularities  Differentiating with respect to time the constraint equations leads . . to the velocity model: A t B q  0 SOM : q 4 (8q2q 32q 46  2q2q 48  64zq 36q 4  96zq 34q 43  36zq 32q 45  6zq 47 1

24z 2q2q 32q 42  6z 2q2q 44  32q2q 32q 44  10q2q 46  2z 3q 43  96zq 34q 4 72zq 32q 43  23zq 45  16z 2q 2q 32  10z 2q2q 42  8q2q 44  z 3q 4  36zq 32q 4 21zq 43  4z 2q2  4zq 4 )  0 SOM : q12 (6q17q 3  8q15q 33  2zq16  36zq14q 32  96zq12q 34  64zq 36 2

18z 2q13q 3  24z 2q1q 33  18q15q 3  16q13q 33  2z 3q12  3zq14  72zq12q 32 96zq 34  18z 2q1q 3  12q13q 3  z 3  3zq12  36zq 32  4z )  0 7/1/2014


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Singularities  For 𝑧𝑧 = 3 and 𝑞𝑞4 > 0 𝑜𝑜𝑜𝑜 𝑞𝑞1 > 0

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Aspect for an operation mode  Definition for « classical » parallel robot, an aspect WAi is a maximal singularity free set defined such that WAi  W WAi is connected X  WAi , det(A)  0 and det(B)  0  For a given operation mode j, an aspect WAij is a maximal singularity free set defined such that WAij  WOM

j

WAij is connected X  WAij , det(A)  0 and det(B)  0 7/1/2014


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Aspects for an operation mode  Four aspects, two for each operation mode

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Direct kinematics  Up to 16 real solutions

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Characteristic surfaces for an operation mode  Let 𝑊𝑊𝐴𝐴𝑖𝑖𝑖𝑖 be one aspect for the operation mode j.

 The characteristic surfaces, denoted by 𝑆𝑆𝐶𝐶 (𝑊𝑊𝐴𝐴𝑖𝑖𝑖𝑖 � SC (WAij )  g j 1 g(WAij ) WAij



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


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Non-singular assembly mode changing trajectories  Can I connect three configurations?  When the robot is changing assembly mode?  Example:

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Non-singular assembly mode changing trajectories  Two trajectories P1P2P3 and P5P6P7 for z=3

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Non-singular assembly mode changing trajectories  Images in the joint space

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Conclusions  A study of the joint space and workspace of the 3-RPS parallel robot  Two non-singular assembly mode changing trajectories are made.  The two operation modes are divided into two aspects  This mechanism admits a maximum of 16 real solutions to the direct kinematic problem, eight for each operation mode.  The basic regions for each operation mode are defined.  A path is defined through several basic regions which are images of the same basic component with 8 solutions for the DKP.  The proof is made by the analysis of the determinant of Jacobian.

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Thank you for your attention [email protected] www.irccyn.ec-nantes.fr/~chablat

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