CCToMM 2017 Symposium on Mechanisms, Machines, and Mechatronics (2015 CCToMM M3 Symposium) Venue: Montreal, Canada Abstract: This paper presents the variations in the workspace, singularities and joint space with respect to the design parameter $k$ of the 3-R\underline{P}S parallel manipulator. Also, the influence on the parasitic motions due to the design parameters is studied, which plays an important role in the selection of the manipulator for the desired task. The cylindrical algebraic decomposition method and Gr\"{o}bner based computations are used to model the workspace and joint space with the parallel singularities in 2R1T and 3T projection spaces, where the orientation of the mobile platform is represented by using quaternions. These computations are useful to select the optimum value for the design parameter such that the parasitic motions can be limited to specific values. Depending on the design parameter $k$, three different configurations of the 3-R\underline{P}S parallel robot are analyzed.

1. Ranjan Jha1, Damien Chablat2, Luc Baron1
1Department of Mechanical Engineering, École Polytechnique de Montréal, H3C A37 Québec,
Canada
2Laboratoire des Sciences du Numérique de Nantes, UMR CNRS 6004, Nantes, France.
Canadian Committee for the Theory of Machines
and Mechanisms
2017 CCToMM M3 Symposium
MAY 25-26, 2017 MONTREAL,
CANADA
Influence of Design Parameters on the Singularities And
Workspace of a 3-RPS Parallel Robot

2. Problem Statements
Parasitic Motions: 3-RPS
Influence of Design Parameter on Singularities
Influence of Design Parameter on Workspace
Influence of Design Parameter on Joint Space
25 May 2017Ranjan JHA - CCToMM 2017
2

3. Structure
Mechanism Under Study: 3-RPS
Kinematics: IKP & DKP
Influence of Design Parameter on Parallel Singularities
Joint Space Analysis Of The 3-RPS Parallel Robot
Workspace Analysis Of The 3-RPS Parallel Robot
Conclusions
25 May 2017Ranjan JHA - CCToMM 2017
3

4. 25 May 2017
Ranjan JHA - CCToMM 2017
4
Mechanism Under Study
3-RPS Parallel Robot
3- dof robot with revolute, prismatic and spherical Joints

5.  3-RPS parallel robot with 3 DOF
 The transformation from the moving frame to the fixed frame can be
described by a position vector p = OP and a 3 × 3 rotation matrix R
 Coordinates of the joints in local and global reference frame
𝐴1 =
𝑔
0
0
𝐴2 =
−
𝑔
2
𝑔 3
2
0
𝐴3 =
−
𝑔
2
−𝑔 3
2
0
𝑏1 =
ℎ
0
0
𝑏2 =
−
ℎ
2
ℎ 3
2
0
𝑏3 =
−
ℎ
2
−ℎ 3
2
0
𝑅 =
𝑢 𝑥 𝑣 𝑥 𝑤 𝑥
𝑢 𝑦 𝑣 𝑦 𝑤 𝑦
𝑢 𝑧 𝑣𝑧 𝑤𝑧
3RPS Parallel Mechanism
𝐵𝑖 = 𝑃 + 𝑅𝑏𝑖 𝑤ℎ𝑒𝑟𝑒, 𝑖 = 1, 2, 3 𝑎𝑛𝑑
Moving Platform
Fixed Platform
 Coordinates of the joints of moving platform in global reference frame
25 May 2017Ranjan JHA - CCToMM 2017
5

6. 𝐴𝑖 − 𝐵𝑖 = 𝑟𝑖 ∀ 𝑖 = 1, 2, 3
𝑢 𝑦ℎ + 𝑦 = 0
𝑦 −
𝑢 𝑦ℎ
2
+
3𝑣 𝑦ℎ
2
+ 3𝑥 −
3𝑢 𝑥ℎ
2
+
3𝑣 𝑥ℎ
2
= 0
𝑦 −
𝑢 𝑦ℎ
2
−
3𝑣 𝑦ℎ
2
− 3𝑥 +
3𝑢 𝑥ℎ
2
+
3𝑣 𝑥ℎ
2
= 0
𝑅 =
2𝑞1
2
+ 2𝑞2
2
− 1 −2𝑞1 𝑞4 + 2𝑞2 𝑞3 2𝑞1 𝑞3 + 2𝑞2 𝑞4
2𝑞1 𝑞4 + 2𝑞2 𝑞3 2𝑞1
2
+ 2𝑞3
2
− 1 −2𝑞1 𝑞2 + 2𝑞3 𝑞4
−2𝑞1 𝑞3 + 2𝑞2 𝑞4 2𝑞1 𝑞2 + 2𝑞3 𝑞4 2𝑞1
2
+ 2𝑞4
2
− 1
𝑞1
2
+ 𝑞2
2
+ 𝑞3
2
+ 𝑞4
2
= 1
Kinematics
 Six constraint equations
 Definition of the orientation matrix by the unit quaternions
3RPS Parallel Mechanism
 Workspace : [𝒑 𝒙, 𝒑 𝒚, 𝒑 𝒛, 𝒒 𝟏, 𝒒 𝟐, 𝒒 𝟑, 𝒒 𝟒]
 Joint space : [𝝆 𝟏, 𝝆 𝟐, 𝝆 𝟑]
25 May 2017Ranjan JHA - CCToMM 2017
6

7. Assembly modes
(No. of Solns.: 16)
• Study of all the geometrical and time-based properties of the motion
Working modes
(No. of Solns.: 4)
Kinematics : IKP & DKP
Given Pose Variables
Compute Pose Variables
Compute Joint Variables
Workspace
Given Joint Variables
Jointspace
Direct kinematics Problem
Inverse kinematics Problem
1
2
𝜌 = γ(X)
X = β(𝝆)
𝐹(𝑋, 𝜌)
25 May 2017Ranjan JHA - CCToMM 2017
7
[𝒑 𝒙, 𝒑 𝒚, 𝒑 𝒛, 𝒒 𝟏, 𝒒 𝟐, 𝒒 𝟑, 𝒒 𝟒] [𝝆 𝟏, 𝝆 𝟐, 𝝆 𝟑]

8. 25 May 2017Ranjan JHA - CCToMM 2017
8
 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
Algebraic Tools

9. Operation Modes
 In the workspace W, for each operation mode, the 𝑊𝑂𝑀 𝑗
is
defined such that
• 𝑊 𝑂𝑀 𝑗 ⊂ 𝑊
• ∀𝑋 ∈ 𝑊 𝑂𝑀 𝑗, 𝑂𝑀 𝑖𝑠 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
𝑔𝑗 𝑋 = 𝑞
𝑔𝑗
−1
𝑞 = (𝑋/(𝑋, 𝑞) ∈ 𝑂𝑀 𝑗
)
 Inverse and direct kinematic problem
 The two Operation modes
• 𝑂𝑀1 ∶↦ 𝑞1 = 0
• 𝑂𝑀2
∶↦ 𝑞4 = 0
 Analysis : 3𝑻 ↦ 𝒑 𝒙, 𝒑 𝒚 , 𝒑 𝒛
2𝑹1𝑻 ↦ [𝒒 𝟐, 𝒒 𝟑, 𝒑 𝒛 ]
25 May 2017Ranjan JHA - CCToMM 2017
9

10. k =1k =0.5 k =1.5
Design Parameter h/g=k
g = 1
25 May 2017Ranjan JHA - CCToMM 2017
10

11. Uniqueness domains for an operation mode
Uniqueness domain 𝑊𝑢 𝑘
𝑗
𝑊𝑢 𝑘
𝑗
= (∪𝑖∈𝐼, 𝑊𝐴𝑏𝑖
𝑗
) ∪ 𝑆 𝐶
𝑗
(𝐼′)
Ranjan JHA - Ph.D. Thesis Defense
11
25 May 2017
Basic regions Basic components
∪ 𝑊𝑎𝑏𝑖
𝑗
= 𝑊𝐴𝑖
𝑗
− 𝑆 𝐶
𝑗
𝑄𝑎𝑏𝑖
𝑗
= 𝑔𝑗(𝑊𝐴𝑏𝑖
𝑗
)

12. Separation of singularity surface based on +ve & -ve valus of q4
25 May 2017Ranjan JHA - CCToMM 2017
12

13. k =1k =0.5 k =3k =2
q1 = 0 Operation Mode 1
Parallel Singularities : 3T [px, py, pz] Projection space [TOP VIEW]
25 May 2017Ranjan JHA - CCToMM 2017
13

14. k =1k =0.5 k =3k =2
q1 = 0 Operation Mode 1
Parallel Singularities : 3T [px, py, pz] Projection space [FULL VIEW]
pz [1, 3]
py [-0.5, 0.5]
px [-0.5,0.5]
pz [1, 3]
py [-1, 1]
px [-1, 1]
pz [1, 3]
py [-2, 2]
px [-2, 2]
pz [1, 3]
py [-3, 3]
px [-3, 3]
25 May 2017Ranjan JHA - CCToMM 2017
14

15. k =1k =0.5 k =3k =2
q4 = 0 Operation Mode 2
Parallel Singularities : 3T [px, py, pz] Projection space [TOP VIEW]
25 May 2017Ranjan JHA - CCToMM 2017
15

16. k =1k =0.5 k =3k =2
q4 = 0 Operation Mode 2
Parallel Singularities : 3T [px, py, pz] Projection space [FULL VIEW]
25 May 2017Ranjan JHA - CCToMM 2017
16pz [1, 3]
py [-0.5, 0.5]
px [-0.5,0.5]
pz [1, 3]
py [-1, 1]
px [-1, 1]
pz [1, 3]
py [-2, 2]
px [-2, 2]
pz [1, 3]
py [-3, 3]
px [-3, 3]

17. Direct kinematics
 Up to 16 real solutions
 Slice of jointspace for 𝜌1 = 3
Ranjan JHA - Ph.D. Thesis Defense
17
25 May 2017

18. k =1k =0.5 k =1.5
Jointspace ρ1 = 3
25 May 2017Ranjan JHA - CCToMM 2017
18

19. k =1k =0.5 k =3k =2
q1 = 0 Operation Mode 1Singularity curves with workspace slice [pz=2]
+ve 4 solutions IKP
-ve 4 solutions IKP
25 May 2017Ranjan JHA - CCToMM 2017
19

20.  A study of the joint space and workspace of the 3-RPS parallel robot
 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 uniqueness domains for each operation mode are defined.
 Able to separate (differentiate) the singularity surfaces for the +ve and – ve values of q4 and q1 for OM1 and
OM2 respectively.
 Comparison of the singularities, workspace and Joint space for the different design parameters . (Helps
engineer to select the specific design parameters for specific task).
Conclusions
25 May 2017Ranjan JHA - CCToMM 2017
20

21. Ranjan.Jha@polymtl.ca
www.ranjan-jha.com
25 May 2017Ranjan JHA - CCToMM 2017
21