This document summarizes the workspace and singularity analysis of a delta-like family robot. It describes the kinematics and singularities of different delta robot architectures including Orthoglide, Hybridglide, Triaglide, and UraneSX. It presents the equations for parallel and serial singularities for each architecture. It also compares the complexity of singularities, computation of workspace boundaries, and joint space analysis between the different architectures. The document concludes with suggestions for future work on computing singularity-free robot paths.

1. Ranjan Jha1, Damien Chablat1, Fabrice Rouillier2, Guillaume Moroz3
Workspace and Singularity Analysis of
a Delta Like Family Robot
1Institut de Recherche en Communications et Cybernetique de Nantes, UMR CNRS 6597, France
2INRIA Paris-Rocquencourt, Institut de Mathematiques de Jussieu, UMR CNRS 7586, France
3INRIA Nancy-Grand Est, France
Futuroscope, Poitiers – June 2015

2. Presentation Outline
• Mechanism Architecture
• Singularities – Delta-Like Robot
• Complexity Analysis – Parallel & Serial Singularities
• Workspace Computation & Analysis
• Workspace Comparison
2

3. Assembly modes
(No. of Solns.)
• Study of all the geometrical and time-based properties of the motion
Working modes[1]
(No. of Solns.)
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)
3

4. A1
B1
A2
B2
A3
B3
Manipulator Architecture
• Orthoglide A1B1 A2B2 A3B3
• Hybridglide A1B1 A2B2 A3B3
• Triaglide A1B1 A2B2 A3B3
• UraneSX A1B1 A2B2 A3B3
• In different Planes
4

5. Orthoglide Architecture & Kinematics
(𝑥 − 𝜌1)2
+ 𝑦2
+ 𝑧2
= 𝐿2
𝑥2 + (𝑦 − 𝜌2)2+ 𝑧2 = 𝐿2
𝑥2 + 𝑦2 + (𝑧 − 𝜌3)2= 𝐿2
A3
A1
B1
A2
B2
B3
P
• The kinematics constraints imposed by the limbs will lead to a series of three
constraint equations[10]:
=
𝑥
𝑦
𝑧
𝐴𝑖 𝐵𝑖 = 𝜌𝑖
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝜌) : 0 < 𝜌i ≤ 2𝐿 Joint Limits
5

6. Hybridglide Architecture & Kinematics
(𝑥 − 1)2
+ (𝑦 − 𝜌1)2
+ 𝑧2
= 𝐿2
(𝑥 + 1)2 + (𝑦 − 𝜌2)2 + 𝑧2 = 𝐿2
𝑥2 + 𝑦2 + (𝑧 − 𝜌3)2= 𝐿2
• The kinematics constraints imposed by the limbs will lead to a series of three
constraint equations:
A3
A1
B1
A2
B2
B3
P =
𝑥
𝑦
𝑧
𝐴𝑖 𝐵𝑖 = 𝜌𝑖
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝜌) : 0 < 𝜌i ≤ 2𝐿 Joint Limits
6

7. Triaglide Architecture & Kinematics
• The kinematics constraints imposed by the limbs will lead to a series of three
constraint equations:
A3
A1
B1A2
B2
B3
P =
𝑥
𝑦
𝑧
(𝑥 − 1)2
+ (𝑦 − 𝜌1)2
+ 𝑧2
= 𝐿2
(𝑥 + 1)2 + (𝑦 − 𝜌2)2 + 𝑧2 = 𝐿2
𝑥2 + (𝑦 − 𝜌3)2 + (𝑧)2= 𝐿2
𝐴𝑖 𝐵𝑖 = 𝜌𝑖
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝜌) : 0 < 𝜌i ≤ 2𝐿 Joint Limits
7

8. UraneSX Architecture & Kinematics
• The kinematics constraints imposed by the limbs will lead to a series of three
constraint equations:
A3
A1
B1
A2
B2
B3
P =
𝑥
𝑦
𝑧
(𝑥 − 1)2
+ (𝑦)2
+ (𝑧 − 𝜌1)2
= 𝐿2
(𝑥 + 1/2)2
+ (𝑦 − 3/2)2
+ (𝑧 − 𝜌2)2
= 𝐿2
(𝑥 + 1/2)2 + (𝑦 + 3/2)2 + (𝑧 − 𝜌3)2= 𝐿2
𝐴𝑖 𝐵𝑖 = 𝜌𝑖
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝜌) : 0 < 𝜌i ≤ 2𝐿 Joint Limits
8

9. A 𝑿 + B 𝝆 = 0 where 𝐀 =
𝜕Ϝ
𝜕𝑋
and 𝐁 =
𝜕Ϝ
𝜕𝜌
 Point at which a given mathematical object is not defined, or where it fails to be well-behaved in
some particular way
 MAPLE -- SIROPA library -- Groëbner based elimination[13]
 Differentiating kinematics constraint equation with respect to time leads to the velocity model:
• det(A) = 0 : Type2 Singularity or
Parallel Singularities
• Mapping tool velocity to joint velocity
(ill-conditioned)
• Loses stifness, motion exists when
actuators are locked[10]
• det(B) = 0 : Type1 Singularity or Serial
Singularities
• Mapping joint velocity to tool velocity
(ill-conditioned)
• Loses one degree of freedom
Singularities
9

10. A 𝑿 + B 𝝆 = 0 where 𝐀 =
𝜕Ϝ
𝜕𝑋
and 𝐁 =
𝜕Ϝ
𝜕𝜌
Singularities
𝜕𝜑1
𝜕𝑥
𝜕𝜑1
𝜕𝑦
𝜕𝜑1
𝜕𝑧
𝜕𝜑2
𝜕𝑥
𝜕𝜑2
𝜕𝑦
𝜕𝜑2
𝜕𝑧
𝜕𝜑3
𝜕𝑥
𝜕𝜑3
𝜕𝑦
𝜕𝜑3
𝜕𝑧
𝜌1 = 𝜑1(x, y, z)
𝜌2 = 𝜑2(x, y, z)
𝜌3 = 𝜑3(x, y, z)
𝐀 =
• Easy to compute in this form for family of
Deltla-like robots
10

11. Workspace JointspaceOrthoglide Singularities
det 𝐀 = -8 𝝆1 𝝆2 𝝆3 + 8 𝝆1 𝝆2 𝒙 + 8 𝝆1 𝝆3y + 8 𝝆2 𝝆3 x
det 𝐁 = 8 (𝝆1 - x)(𝝆2 – y)(𝝆3 - z)
det 𝐀 → 𝜉(𝝆) det(𝐀) → µ(𝑿)
Projection in Joint space Projection in Workspace
det(𝐁) → ƒ(𝝆) det(𝐁) → ß(𝑿)
Projection in Joint space Projection in Workspace
11

12. Workspace JointspaceHybridglide Singularities
det 𝐀 = -8𝝆1 𝝆3x + 8𝝆2 𝝆3 𝒙 - 8𝝆1 𝝆3 + 8𝜌1z - 8𝝆2 𝝆3 + 8𝜌2z +16𝝆3 y
det 𝐁 = 8 (𝝆1 - y)(𝝆2 – y)(𝝆3 - z)
det 𝐀 → 𝜉(𝝆) det(𝐀) → µ(𝑿)
Projection in Joint space Projection in Workspace
det(𝐁) → ƒ(𝝆) det(𝐁) → ß(𝑿)
Projection in Joint space Projection in Workspace
12

13. Workspace JointspaceTriaglide Singularities
det 𝐀 = 8 𝝆1z + 8 𝝆2 𝒛 - 16 𝝆3z
det 𝐁 = 8 (𝝆1 - y)(𝝆2 – y)(𝝆3 -y)
det 𝐀 → 𝜉(𝝆) det(𝐀) → µ(𝑿)
Projection in Joint space Projection in Workspace
det(𝐁) → ƒ(𝝆) det(𝐁) → ß(𝑿)
Projection in Joint space Projection in Workspace
13

14. Workspace JointspaceUraneSX Singularities
det 𝐀 = 4 𝟑(3z-𝝆1-𝝆2-𝝆3 + 𝜌2x+𝜌3x-2𝝆1x)+12𝜌3y-12𝜌2y
det 𝐁 = 8 (𝝆1 - z)(𝝆2 – z)(𝝆3 - z)
det 𝐀 → 𝜉(𝝆) det(𝐀) → µ(𝑿)
Projection in Joint space Projection in Workspace
det(𝐁) → ƒ(𝝆) det(𝐁) → ß(𝑿)
Projection in Joint space Projection in Workspace
14

15. Manipulators Types of
Singularities
Plotting Time
(s)
Degrees No. of Terms Binary SIze No. of Cells
Orthoglide Parallel 0037.827 18[10,10,10] 097 015 [02382,0272]
Serial 0005.133 18[12,12,12] 062 012 [00044,0004]
Hybridglide Parallel 5698.601 20[16,08,12] 119 017 [28012,1208]
Serial 0007.007 18[12,12,12] 281 017 [00158,0027]
Triaglide Parallel 0010.625 03[00,00,03] 002 002 [00138,0004]
Serial 0005.079 06[06,06,06] 042 007 [00077,0017]
UraneSX Parallel 0022.625 06[06,04,00] 015 040 [02795,0070]
Serial 0018.391 12[12,12,12] 252 151 [00392,0142]
Complexity of Singularities
Parallel : Projection in Workspace
Serial : Projection in Joint space
Highest power [Degrees in diff Variables]
Cylindrical Algebraic Decomposition(CAD)Cylindrical Algebraic Decomposition(CAD)
Discriminant variety, Groebner Bases
Computation Total No. Of terms in Function
log2 𝑀𝑎𝑥(𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡 𝑖𝑛 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛)
15

16. Workspace Computation
Sample Points
Cells
 Cylindrical Algebraic Decomposition (CAD)
 Cell Formulation
 Different Aspects can be extracted based on
the cell properties
 Different regions with different number of
solutions of IKP
16

17. 17

18. 18

19. Workspace Analysis
• Orthoglide • Hybridglide
• Triaglide • UraneSX 19

20. Joint space Analysis
• 2 DKP
• Orthoglide • Hybridglide
• Triaglide • UraneSX 20

21. Conclusions
• Singularity Computations using algebraic tools
• Singularity surfaces with the projections of joint limits
• Accurate computation of workspace Boundaries
• Complexity analysis of singularities
• Workspace computations and comparison
21

22. Future Work
P1
P2
• Singularity Surface
• Singularity Free path
• To compute the path, which do not intersect the
singularity surface anywhere.
22

23. References
1] Chablat D., Wenger Ph., Working Modes and Aspects in Fully-Parallel Manipulator, Proceeding IEEE International Conference on Robotics and
Automation, pp. 1964-1969, May 1998.
[2] Chablat D., Jha R., Rouillier F., Moroz G., Workspace and joint space analysis of the 3-RPS parallel robot, In: Proceedings of ASME 2014
International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Buffalo, United States, pp. 1–10,
2014
[3] Chablat D., Jha R., Rouillier F., Moroz G., Non-singular assembly mode changing trajectories in the workspace for the 3-RPS parallel robot, 14th
International Symposium on Advances in Robot Kinematics, Ljubljana, Slovenia, 2014.
[4] Merlet J-P., Parallel robots, Springer, Vol. 74, 2001.
[5] Siciliano, B. and Khatib, O., Springer handbook of robotics, Springer, 2008.
[6] Chablat D., Wenger P., Majou F. et Merlet J.P., An Interval Analysis Based Study for the Design and the Comparison of 3-DOF Parallel Kinematic
Machines, International Journal of Robotics Research, pp. 615-624, Vol 23(6), juin 2004.
[7] Ilian A. Bonev, Jeha Ryu, A new approach to orientation workspace analysis of 6-DOF parallel manipulators, Mechanism and Machine Theory, V
ol 36/1, pp. 15–28,2001.
[8] Sefrioui J., Gosselin C., Singularity analysis and representation of planar parallel manipulators, Robots and autonomous Systems 10, pp. 209-224,
1992
[9] Chablat D., Moroz G., Wenger P., Uniqueness domains and non singular assembly mode changing trajectories, Proc. IEEE Int. Conf. Rob. and
Automation, May 2011.
[10] Pashkevich A., Chablat D., Wenger P., Kinematics and workspace analysis of a three-axis parallel manipulator the Orthoglide, Robotica, Vol.
24(01), pp. 39–49, 2006.
[11] Wenger Ph., Chablat D., Uniqueness Domains in the Workspace of Parallel Manipulators, IFAC-SYROCO, Vol. 2, pp 431-436, 3-5 Sept.,
[12] Innocenti C., Parenti-Castelli V., Singularity-free evolution from one configuration to another in serial and fully-parallel manipulators, Robotics,
Spatial Mechanisms and Mechanical Systems, ASME 1992.
[13] Cox D.A., Little J., O’Shea D.,Using Algebraic Geometry, 2nd edition, Graduate Texts in Mathematics, Springer,
2004.
[14] McAree P.R., Daniel R.W., An explanation of never-special assembly changing motions for 3-3 parallel manipulators, International Journal of
Robotic Research, 18 (6), pp. 556-574., 1999.
[15] Zein M., Wenger P. Chablat D., Non-Singular Assembly-mode Changing Motions for 3-RPR Parallel Manipulators, Mechanism and Machine
Theory, Vol 43/4, pp. 480–490, 2008.
[16] Wenger P., Chablat D., Workspace and assembly modes in fully parallel manipulators: a descriptive study, In: ARK, Strobol, pp. 117-126, 1998.
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24. Thank you very much…
For your attention 
Ranjan.jha@irccyn.ec-nantes.fr
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