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
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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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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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