Ph.D. Thesis : Ranjan JHA : Contributions to the Performance Analysis of Parallel Robots

Contributions To The Performance Analysis of Parallel
Robots
M. Jean-Pierre MERLET, Directeur de Recherche, INRIA / Rapporteur
M. Grigore GOGU, Professeur, IFMA / Rapporteur
M. Said ZEGHLOUL, Professeur, Université de Poitiers / Examinateur
M. Damien CHABLAT, Directeur de Recherche, CNRS / Directeur de Thèse
M. Fabrice ROUILLIER, Directeur de Recherche, INRIA / Co-directeur de Thèse
M. Guillaume MOROZ, Chargé de Recherche, INRIA / Co-encadrant de Thèse
Ranjan Jha
by
JURY
7th July 2016

15 May 2017 Ranjan JHA - Ph.D. Thesis Defense 2
Contributions To The Performance Analysis of Parallel
Robots

 Parallel Robot = Parallel mechanism applied to
machining ( milling, drilling, …. )
- Flight Simulator (Stewart, 1964)
- Machine Tools (Giddings & Lewis,1994)
 First parallel mechanisms constructions :
- Tyre test machine (Gough, 1962)
Parallel Robots

Problem Statements
Modeling of the Workspace & Joint space
Computations of the singularities & their Projections
Singularity- Free Path Planning
Accuracy Analysis of the Parallel Manipulators
Ranjan JHA - Ph.D. Thesis Defense 415 May 2017

Structure
Mathematical Tools & Robotic Definitions
Analysis of Delta-Like Family Robot
Analysis of 3RPS Parallel Robot
Singularity-free Path Planning
Prospective work: Accuracy Analysis
Conclusions & Future Work

Mathematical Tools & Definitions

 Variable’s elimination , projection of solutions
 Rational univariate representation
 Representation of the aspects and basic regions
Algebraic Tools
 Computation of the singularities and characteristic surfaces
 Partition of space w.r.t sign condition
Gröbner basis
Discriminant Variety
Cylindrical Algebraic Decomposition
Certified Computations (Root Isolation)

Gröbner Bases
Why the Name so ?
Wolfgang Gröbner
Bruno Buchberger
 Given a set 𝐹 of polynomials in 𝐾[𝑥1, 𝑥2 … , 𝑥 𝑛]
 Transform 𝐹 into another set 𝐺 of polynomials  Gröbner
basis
 𝐺 generates the same ideal as 𝐹 (𝐺 has the same zeros as 𝐹)
𝐺 is minimal w.r.t monomial ordering
 Canonical function for “reducing” a polynomial modulo the
ideal
𝑃 ∈ 𝐼 ⇔ 𝑃 → 𝐺
∗
0
 Monomial Ordering : Univariate Polynomial (degree)
Multivariate (Lexicographic Ordering) Example 𝒙 𝟓
𝒚 𝟐
𝒙 𝟑
𝒚 𝟒
For Lex Ordr. x > y -- [5,2] > [3,4]
For Lex Ordr. y > x -- [3,4] > [5,2]

G is the GB for the lexicographical ordering x1<…xk….<xn
𝐺 𝑘 = 𝐺 ∩ 𝑄 𝑥1 … . 𝑥 𝑘
The projection of the solutions of F on x1….xk, are dense subset
of the solutions of G_k
Properties and Applications Of Gröbner Bases
Elimination
Dimension of the solutions
Parameterization
Dimension of F : 0 implies finite set of points
1 - curve
2 - surfaces
If dimension of F is 0

 Set of equations 𝐹 𝑋, 𝜌 = 0 , 𝜌 are the parameters and X are the unknowns
such that for any 𝜌 , system has finite number of solutions (has dimension zero)
 Computing polynomial in 𝜌
 Discriminant variety is defined by polynomial equations in 𝜌, divides the
parameter space into open, full-dimensional cells such that the number of
solutions of the system is constant for parameter values chosen from the same
open cell
 To describe the connected cells of the complement of the discriminant variety ,
we use CAD, connected cell is the union of cells with constant non zero sign
Discriminant Variety

Cylindrical Algebraic Decomposition
𝑃1, 𝑃2, 𝑃3 cylindrical algebraic decomposition
adapted to p1,p2,p3
Partition of space in Cells of constant sign

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)
Aspect
Aspects
Singularities
Singularities
𝐹(𝑋, 𝜌)

Delta-Like Family Robot

A1
B1
A2
B2
A3
B3
Manipulator Architecture of Delta-Like Robot[Majou:2004]
• Orthoglide A1B1 A2B2 A3B3
• Hybridglide A1B1 A2B2 A3B3
• Triaglide A1B1 A2B2 A3B3
• UraneSX A1B1 A2B2 A3B3
• In different Planes

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:
=
𝑥
𝑦
𝑧
𝐴𝑖 𝐵𝑖 = 𝜌𝑖
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝜌) : 0 < 𝜌i ≤ 2𝐿 Joint Limits

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
A3
A1
B1
A2
B2
B3
P =
𝑥
𝑦
𝑧
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :

Triaglide Architecture & Kinematics
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
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :

UraneSX Architecture & Kinematics
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
𝐵𝑖 𝑃 = L i = 1, 2, 3
Ϝ(X, 𝜌) :

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

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)

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)

Workspace JointspaceTriaglide Singularities
det 𝐀 = 8 𝝆1z + 8 𝝆2 𝒛 - 16 𝝆3z
det 𝐁 = 8 (𝝆1 - y)(𝝆2 – y)(𝝆3 -y)

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)

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)
Ranjan JHA - Ph.D. Thesis Defense
24

Workspace Plot
Sample Points
Cells
 Cylindrical Algebraic Decomposition (CAD)
 [Constraint Eqn ] [Pose variables]
 GB  Discriminant Variety  CAD
 Different Aspects can be extracted based on
the cell properties
 Different regions with different number of
solutions of IKP
Orthoglide Workspace

Workspace Computation Orthoglide Workspace

Workspace Analysis
• Orthoglide • Hybridglide
• Triaglide • UraneSX 27

Joint space analysis
• 2 DKP
• Orthoglide • Hybridglide
• Triaglide • UraneSX 28
 Regular joint space shape
 Complex joint space shape

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 comparisons

3RPS Parallel Robot

 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

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]

k =1k =0.5 k =1.5
Design Parameter h/g=k
g = 1

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 ]

Singularities of 3RPS
• Differentiating with respect to time the constraint equations leads to the velocity model:
1
2 6 8 6 4 3 2 5 7
4 2 3 4 2 4 3 4 3 4 3 4 4
2 2 2 2 4 2 4 6 3 3 4
2 3 4 2 4 2 3 4 2 4 4 3 4
2 3 5 2 2 2 2 4 3 2
3 4 4 2 3 2 4 2 4 4 3 4
3 2
4 2 4
: (8 2 64 96 36 6
24 6 32 10 2 96
72 23 16 10 8 36
21 4 4 ) 0
OM
S q q q q q q zq q zq q zq q zq
z q q q z q q q q q q q z q zq q
zq q zq z q q z q q q q z q zq q
zq z q zq
+ - - - -
- - - - + +
+ + + + + - -
- - + =
2
2 7 5 3 6 4 2 2 4 6
1 1 3 1 3 1 1 3 1 3 3
2 3 2 3 5 3 3 3 2 4 2 2
1 3 1 3 1 3 1 3 1 1 1 3
4 2 3 3 2 2
3 1 3 1 3 1 3
: (6 8 2 36 96 64
18 24 18 16 2 3 72
96 18 12 3 36 4 ) 0
OM
S q q q q q zq zq q zq q zq
z q q z q q q q q q z q zq zq q
zq z q q q q z zq zq z
+ - + + +
- - - - + + -
- + + - + + - =
• For 𝑧 = 3 and 𝑞4 > 0 𝑜𝑟 𝑞1 > 0
𝐴 𝑡 + 𝐵 𝑞 = 0
𝑂𝑀1 𝑂𝑀2

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
• 𝑊𝐴𝑖
𝑗
⊂ 𝑊 𝑂𝑀 𝑗
• 𝑊𝐴𝑖
𝑗
𝑖𝑠 𝐶𝑜𝑛𝑛𝑒𝑐𝑡𝑒𝑑
• ∀𝑋 ∈ 𝑊𝐴𝑖
𝑗
, det 𝑨 ≠ 0 and det(𝑩) ≠ 0
 For a given operation mode 𝑗, an aspect 𝑊𝐴𝑖
𝑗
is a maximal
singularity free set defined such that
Aspects for OM1
Aspects for OM2
det(A) < 0 det(A) > 0
det(A) < 0 det(A) > 0

Direct kinematics
 Up to 16 real solutions
 Slice of jointspace for 𝜌1 = 3

Characteristic surfaces for an operation mode
 Let 𝑊𝐴𝑖
𝑗
be one aspect for the operation mode 𝑗.
 The characteristic surfaces, denoted by 𝑆𝑐
𝑗
(𝑊𝐴𝑖
𝑗
)
𝑆𝑐
𝑗
𝑊𝐴𝑖
𝑗
= 𝑔𝑗
−1
(𝑔𝑗 𝑊𝐴𝑖
𝑗
) ∩ 𝑊𝐴𝑖
𝑗
𝑂𝑀1 𝑂𝑀2

Uniqueness domains for an operation mode
Uniqueness domain 𝑊𝑢 𝑘
𝑗
𝑊𝑢 𝑘
𝑗
= (∪𝑖∈𝐼, 𝑊𝐴𝑏𝑖
𝑗
) ∪ 𝑆 𝐶
𝑗
(𝐼′
)
Basic regions Basic components
∪ 𝑊𝑎𝑏𝑖
𝑗
= 𝑊𝐴𝑖
𝑗
− 𝑆 𝐶
𝑗
𝑄𝑎𝑏𝑖
𝑗
= 𝑔𝑗(𝑊𝐴𝑏𝑖
𝑗
)

Uniqueness domains
 Slice of the workspace for 𝑧=3
1
OM
2
OM

Uniqueness domains for 𝑶𝑴 𝟏
det(A)>0
1 1 1 1 1
1 1 2 3 4
1 1 1 1 1
2 1 2 3 5
1 1 1 1 1
3 1 2 3 6
W u W A W A W A W A
W u W A W A W A W A
W u W A W A W A W A
= È È È
= È È È
= È È È

Uniqueness domains for 〖𝑂𝑀〗^1 det("A" )<0
1 1 1 1 1
4 7 11 12 12
1 1 1 1 1
5 8 11 12 12
1 1 1 1 1
6 9 11 12 12
1 1 1 1 1
7 10 11 12 12
W u W A W A W A W A
W u W A W A W A W A
W u W A W A W A W A
W u W A W A W A W A
= È È È
= È È È
= È È È
= È È È

Non-singular assembly mode changing trajectories
 Can I connect three configurations?
 When the robot is changing assembly mode?
 Two trajectories P1P2P3 and P5P6P7 for z=3

• Images in the joint space

Separation of singularity surface based on +ve & -ve valus of q4

Parallel Singularities : 2R1T [q1 ,q2 ,pz] Projection space Operation Mode 1 q1=0 [TOP VIEW]
k =1k =0.5 k =1.5
q4<0 q4<0 q4<0q4>0 q4>0 q4>0

k =1k =0.5 k =1.5
q1<0 q1<0 q1<0q1>0 q1>0 q1>0
Parallel Singularities : 2R1T [q1 ,q2 ,pz] Projection space Operation Mode 2 q4=0

k =1k =0.5 k =3k =2
q1 = 0 Operation Mode 1
Parallel Singularities : 3T [px, py, pz] Projection space [TOP VIEW]

k =1k =0.5 k =3k =2
Parallel Singularities : 3T [px, py, pz] Projection space [FULL VIEW]
px [1, 3]
py [-0.5, 0.5]
pz [-0.5,0.5]
px [1, 3]
py [-1, 1]
pz [-1, 1]
px [1, 3]
py [-2, 2]
pz [-2, 2]
px [1, 3]
py [-3, 3]
pz [-3, 3]

k =1k =0.5 k =3k =2
Parallel Singularities : 3T [px, py, pz] Projection space [TOP VIEW]

k =1k =0.5 k =3k =2
Parallel Singularities : 3T [px, py, pz] Projection space [FULL VIEW]
px [1, 3]
py [-0.5, 0.5]
pz [-0.5,0.5]
px [1, 3]
py [-1, 1]
pz [-1, 1]
px [1, 3]
py [-2, 2]
pz [-2, 2]
px [1, 3]
py [-3, 3]
pz [-3, 3]

k =1k =0.5 k =1.5
Jointspace ρ1 = 3

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

 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.
 The uniqueness domains 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.
 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

An Algebraic Method to Check the Singularity-Free Paths
for Parallel Robots

𝟊(𝜌, 𝑋) = 0
A 𝑿 + B 𝝆 = 0 where 𝐀 =
𝜕Ϝ
𝜕𝑋
and 𝐁 =
𝜕Ϝ
𝜕𝜌
det(𝑨) ↦ ξ(𝑿)
det(𝑨) ↦ ε(𝝆)
χ 𝝆 ↦ μ 𝑿 ∀𝜌 ∈]𝜌 𝑚𝑖𝑛, 𝜌 𝑚𝑎𝑥]
𝑋 ↦ φ(𝑡)
• Set of (algebraic) equations that connects input values 𝝆 to output values X
METHODOLOGY
𝝆 → set of all actuated joint variables
𝑿 → set of position and orientation variables
• Differentiating kinematic constraint equations with respect to time leads to the velocity model :
Projection of parallel singularities in workspace
• Projection of joint limits in Workspace :
• Trajectory definition in workspace
Equations in parametric form
Groëbner based elimination
Projection of parallel singularities in joint space

METHODOLOGY
Ψ 𝑿, 𝝆, 𝒕 = [φ 𝒕 − X, 𝟊 𝝆, 𝑿 ]
Ψ 𝑿, 𝝆, 𝒕 ↦ Υ(𝝆, 𝒕)
• Projection of trajectories in Jointspace
System Of equations
Trajectory equations
Constraint equations
𝝆 ← Υ(𝝆, 𝒕)
• Singularity Analysis
ξ(φ 𝒕 ) → Vanishing values of Jacobian as a function of time
 Solutions contain the singular points on trajectory φ 𝒕 and
also few spurious points due to projection of det(A) in Cartesian
space
 Spurious singular points can be differentiated from real singular
points by substituting φ 𝒕 and Υ 𝝆, 𝒕 in det(A)
Projections in the joint space, are the set of Algebraic equations

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
=
𝑥
𝑦
𝑧
𝐵𝑖 𝑃 = L = 2 i = 1, 2, 3
Ϝ(X, 𝜌) :
χ(𝝆) : 0 < 𝜌i ≤ 2𝐿 Joint Limits

Parallel Singularities
det 𝐀 = -8 𝛒1 𝛒2 𝛒3 + 8 𝛒1 𝛒2 𝐱 + 8 𝛒1 𝛒3y + 8 𝛒2 𝛒3 x
det(𝑨) ↦ ξ(𝑿)
det(𝑨) ↦ ε(𝝆)
Projection of parallel singularities in the workspace
Projection of parallel singularities in the joint space
χ 𝝆 ↦ μ 𝑿 ∀𝜌 ∈ (𝜌 𝑚𝑖𝑛, 𝜌 𝑚𝑎𝑥]
• Projection of joint limits in Workspace :
χ 𝝆 = [𝝆1, 𝝆2, 𝝆3, −4 + 𝝆1, −4 + 𝝆2, −4 + 𝝆3]
μ 𝑿 ∶ 𝜌𝑖 ↦ 𝑥2 + 𝑦2 + 𝑧2 − 4 𝑖 = 1,2,3
−4 + 𝝆1 ↦ 𝑥2 + 𝑦2 + 𝑧2 − 8𝑥 + 12
−4 + 𝝆2 ↦ 𝑥2 + 𝑦2 + 𝑧2 − 8𝑦 + 12
−4 + 𝝆3 ↦ 𝑥2 + 𝑦2 + 𝑧2 − 8𝑧 + 12

ξ(𝑿)ξ 𝑿 + μ 𝑿
Parallel singularity surface
with joint limits
Parallel singularity
surface

Workspace & Jointspace : ORTHOGLIDE
Workspace Jointspace
1 IKP
(Yellow Region)
2 DKP
(Red Region)
8 IKP
(Green Region)
 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

Trajectory Definitions in the Workspace
φ1(t) : x =
8
7
𝒔3(t)
y =
13
14
𝒄 t −
5
14
𝒄 2t −
1
10
𝒄 3t −
1
14
𝒄(4t)
z = 1
φ2(t) : x =
4
5
𝒔3(t)
y =
13
20
𝒄 t −
1
4
𝒄 2t −
1
20
𝒄 3t −
1
20
𝒄(4t)
z = 1
φ3(t) : x = 𝒔(t)
y = 𝒄 t
z =
1
20
t
Trajectory 1
Trajectory 2
Trajectory 3
𝑺 𝒕 → 𝑺𝒊𝒏 𝒕
𝑪(𝒕) → 𝑪𝒐𝒔(𝒕)

Projection In Jointspace
Projection
Corresponds
to 8 IKP
1
2
3
4
5
6
7
8
φ(𝑡) ↦ Υ(𝝆, 𝒕)
Feasible trajectory
inside joint space
Ψ2 𝑿, 𝝆, 𝒕 = [φ2 𝒕 − X, 𝟊 𝝆, 𝑿 ]
Ψ1 𝑿, 𝝆, 𝒕 = [φ1 𝒕 − X, 𝟊 𝝆, 𝑿 ]
Ψ1 𝑿, 𝝆, 𝒕 ↦ Υ1 𝝆, 𝒕 ∀𝑡 ∈ [−𝜋, 𝜋]
Ψ2 𝑿, 𝝆, 𝒕 ↦ Υ2 𝝆, 𝒕 ∀𝑡 ∈ [−𝜋, 𝜋]

Singularity Analysis
S1S2
S3
S4
Spurious Singular Points
μ1 𝒕 = ξ 𝝋 𝟏 𝒕 ∀𝑡 ∈ [−𝜋, 𝜋]
μ2 𝒕 = ξ 𝝋 𝟐 𝒕 ∀𝑡 ∈ [−𝜋, 𝜋]
μ3 𝒕 = ξ 𝝋 𝟑 𝒕 ∀𝑡 ∈ [ 0, 20]
𝑡 = [−1.51, −0.97,0.97,1.51] ← μ1 𝒕 = 0 ∀𝑡 ∈ [−𝜋, 𝜋]
𝑡 = [0.97,1.51] ← 𝑑𝑒𝑡 𝑨 = 0 ∀𝑡 ∈ [−𝜋, 𝜋]
𝑡 = [ ] ← μ2 𝒕 = 0 ∀𝑡 ∈ [−𝜋, 𝜋]
𝑡 = ← μ3 𝒕 = 0 ∀𝑡 ∈ [ 0, 20]
 Single analysis of the singularities in a projection
space can introduce spurious singularities
 Singularity analysis should be done in the cross
product of the workspace and the joint space
 Determinant of the Jacobian for the
trajectories
 Singularity free: Trajectory 2 & 3

Singularity Analysis
S x y z ρ1 ρ2 ρ3
S1 -1.13 0.35 1.00 0.55 1.66 2.60
S2 -0.65 0.80 1.00 0.88 2.40 2.71
S3 0.65 0.80 1.00 2.18 2.40 2.71
S4 1.13 0.35 1.00 2.83 1.66 2.60

Conclusions
• The proposed method enables us to project the joint constraint in the
workspace of manipulator
• Helps to analyse the projection of the singularities in the workspace with
joint constraints
• Significant change in the shape of workspace and singularity surfaces due
to the joint constraints
• Ensures the existence of a singular configuration between two poses of
the end-effector unlike other numerical or discretization methods
• Single analysis of the singularities in a projection space can introduce
spurious singularities
• Singularity analysis should be done in the cross product of the workspace
and joint space for parallel robots with several assembly and working
modes

Evaluation Of the Accuracy
Prospective Work

𝑵 = − 𝜌1
6
𝜌2
4
− 𝜌1
6
𝜌2
2
𝜌3
2
− 𝜌1
4
𝜌2
6
− 2𝜌1
4
𝜌2
4
𝜌3
2
− 𝜌1
4
𝜌2
2
𝜌3
4
−
𝜌1
2
𝜌2
6
𝜌3
2
− 𝜌1
2
𝜌2
4
𝜌3
4
+ 384400(𝜌1
4
𝜌2
4
+ 𝜌1
4
𝜌2
2
𝜌3
2
+ 𝜌1
2
𝜌2
4
𝜌3
2
Kinematics
𝑥 =
1
2𝜌1
𝜌1
4
𝜌2
2
+ 𝜌1
4
𝜌3
2
− 𝑵𝜌3
𝜌1
2
𝜌2
2
+ 𝜌1
2
𝜌3
2
+ 𝜌2
2
𝜌3
2
𝑦 =
1
2𝜌2
𝜌1
4
𝜌2
2
+ 𝜌2
4
𝜌3
2
− 𝑵𝜌3
𝜌1
2
𝜌2
2
+ 𝜌1
2
𝜌3
2
+ 𝜌2
2
𝜌3
2
𝑧 =
1
2𝜌3
𝜌1
4
𝜌3
2
+ 𝜌1
4
𝜌3
2
− 𝑁𝜌3
𝜌1
2
𝜌2
2
+ 𝜌1
2
𝜌3
2
+ 𝜌2
2
𝜌3
2
𝛽 𝝆 :
A3
A1
B1
A2
B2
B3
P =
𝑥
𝑦
𝑧
(𝑥 − 𝜌1)2
+ 𝑦2
+ 𝑧2
= 𝐿2
𝑥2
+ (𝑦 − 𝜌2)2
+ 𝑧2
= 𝐿2
𝑥2
+ 𝑦2
+ (𝑧 − 𝜌3)2
= 𝐿2
𝐹 𝑋, 𝜌 :
𝜌1 = 𝑥 ± −𝑦2 − 𝑧2 + 96100
𝜌2 = 𝑦 ± −𝑥2 − 𝑧2 + 96100
𝜌3 = 𝑧 ± −𝑥2 − 𝑦2 + 96100
𝛾 𝑿 :
Direct Kimematics Model
Inverse Kimematics Model
Constraint Equations

Trajectory Planning
Γ𝑖 = 𝑀( 𝜌𝑖 + 𝐾 𝑃 𝜌𝑖
𝑑
− 𝜌𝑖 + 𝐾𝐼
𝑡0
𝑡
𝜌𝑖
𝑑
− 𝜌𝑖 + 𝐾 𝐷( 𝜌𝑖
𝑑
− 𝜌𝑖)
𝐾 𝑃 = 19200 𝑟𝑎𝑑2
/𝑠2
𝐾𝐼 = 512000 𝑟𝑎𝑑3
/𝑠3
𝐾 𝐷 = 240 𝑟𝑎𝑑/𝑠 𝑀 = 91.6278 𝐾𝑔
𝜙𝑖 𝑡 ∶ 𝜏 𝑥𝑖 = 𝑐 𝑥𝑖 + 40 sin 2𝑡3
6𝑡2
− 15𝑡 + 10 𝜋 − 𝑥
𝜏 𝑦𝑖 = 𝑐 𝑦𝑖 + 40 cos 2𝑡3
6𝑡2
− 15𝑡 + 10 𝜋 − 𝑦
𝜏 𝑧𝑖 = 𝑐 𝑧𝑖 − 𝑧 ∀𝑡 ∈ [0, 1]
Parallel Singularities
Location of the trajectories φi (t ) along with the parallel singularity surfaces
ξ(X). CAD algorithm is used to plot these surfaces for a single working mode
 Control Loop in Joint space
 Torque Control, sampling rate : 1.5 KHz
 Trajectories φi (t ): [τxi τyi, τzi] are defined in
parametric form in the Cartesian space using
fifth order polynomial equation with different
center locations

𝑥 𝑚𝑎𝑥 = 𝑦 𝑚𝑎𝑥 = 5 𝑚/𝑠2
𝑥 𝑚𝑎𝑥 = 𝑦 𝑚𝑎𝑥 = 0.45 𝑚/𝑠
Trajectory Planning
Cartesian velocities and accelerations along the
trajectories φi (t )
𝜏 𝑥𝑖 = 2400𝑡2
𝜋 𝑡2
− 2𝑡 + 1 cos(2𝑡3
6𝑡2
− 15𝑡 + 10 𝜋)
𝜏 𝑦𝑖 = −2400𝑡2
𝜋 𝑡2
− 2𝑡 + 1 sin(2𝑡3
6𝑡2
− 15𝑡 + 10 𝜋)
𝜏 𝑧𝑖 = 0 ∀𝑡 ∈ [0, 1]
Ψ𝑖 𝑿, 𝝆, 𝒕 = [[𝜏 𝑥𝑖, 𝜏 𝑦𝑖, 𝜏 𝑧𝑖], 𝟊 𝝆, 𝑿 ]
Ψ𝑖 𝑿, 𝝆, 𝒕 ↦ Υ1(𝝆, 𝒕)
𝝆 ← Υ(𝝆, 𝒕)

72
Parallel Singularity
Orthoglide
Trajectory 1
Workspace

73
Parallel Singularity
Orthoglide
Trajectory 5
Workspace

Error Analysis

Δ𝜌𝑖
𝑚
= 𝑘1𝑖 + 𝑘2𝑖 𝜌𝑖 + 𝑘3𝑖 𝜌𝑖
𝑘11 =
1
250000
𝑘21 =
1
2000000
𝑘31 =
1
8000000
𝑘12 =
1
250000
𝑘22 =
1
2000000
𝑘32 =
33
250000000
𝑘13 =
1
250000
𝑘23 =
1
2000000
𝑘33 =
1
500000000
Error Model
 Based on experimental data and the dynamics, i.e, joint velocities and
accelerations of the Orthoglide 5-axis
 Constant values represents the static error, the friction and the dynamic effect

Projection of Joint Errors in Workspace
Δ𝑥2
=
𝛿𝑥
𝛿𝜌1
2
𝑒1
2
+
𝛿𝑥
𝛿𝜌2
2
𝑒2
2
+
𝛿𝑥
𝛿𝜌3
2
𝑒3
2
Δ𝑦2
=
𝛿𝑦
𝛿𝜌1
2
𝑒1
2
+
𝛿𝑦
𝛿𝜌2
2
𝑒2
2
+
𝛿𝑦
𝛿𝜌3
2
𝑒3
2
Δ𝑧2
=
𝛿𝑧
𝛿𝜌1
2
𝑒1
2
+
𝛿𝑧
𝛿𝜌2
2
𝑒2
2
+
𝛿𝑧
𝛿𝜌3
2
𝑒3
2

Influence of Starting positions on the joints errors

Trajectory 1 Trajectory 2 Trajectory 3
Trajectory 4
Trajectory 5
Different Starting
Positions

Conclusions
 Proposed error model based on the dynamic properties helps in estimating the error in the
Cartesian workspace.
 Trajectory closer to the singularity surface (by taking dynamic properties in account) admitted
the maximum error and approx. twice than the trajectory closest to the isotropic posture.
 Significant change in the accuracy by changing the starting position of the manipulator.
 Among the five trajectories, which are defined for the analysis, trajectory T5 has approximately
five times pose error than the trajectory T3, which is closest to the isotropic posture.
 Optimal Starting point and location for the trajectory so that error in the pose of end effector can
be minimized.

Conclusion and Future Work

General Conclusion
 Workspace and joint space analysis of the 3-RPS parallel robot
 The uniqueness domains for parallel robot with several operation modes and
assembly modes
 Non-singular assembly mode changing trajectories in the workspace for the 3-RPS
parallel robot
 An Algebraic Method to Check the Singularity-Free Paths for Parallel Robots
 Workspace and Singularity analysis of a Delta like family robot
 Influence of the trajectory planning on the accuracy of the Orthoglide 5-axis

Publications
1. Ranjan Jha, Damien Chablat, Fabrice Rouillier, Guillaume Moroz. Influence of the trajectory planning on the accuracy of the orthoglide 5-axis. ASME
International Design Engineering Technical Conference and the Computer and Information in Engineering Conference (IDETC/CIE), Aug 2016, Charlotte,
NC, United States. Proceedings of the ASME 2016 International Design Engineering Technical Conference and the Computer and Information in
Engineering Conference (IDETC/CIE), 2016. <hal-01309190>
2. Damien Chablat, Ranjan Jha, Stephane Caro. A Framework for the Control of a Parallel Manipulator with Several Actuation Modes. 14th International
Conference on Industrial Informatics , Jul 2016, Poitiers, France. IEEE, 2016. <hal-01313376>
3. Ranjan Jha, Damien Chablat, Fabrice Rouillier, Guillaume Moroz. Workspace and Singularity analysis of a Delta like family robot. 4th IFTOMM
International Symposium on Robotics and Mechatronics, Jun 2015, Poitiers, France. 2015. <hal-
01142465>
4. Ranjan Jha, Damien Chablat, Fabrice Rouillier, Guillaume Moroz. An algebraic method to check the singularity-free paths for paral lel robots.
International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Aug 2015, Boston, United States. 2015.
<hal-01142989>
5. Damien Chablat, Ranjan Jha, Fabrice Rouillier, Guillaume Moroz. Workspace and joint space analysis of the 3-RPS paral lel robot. ASME 2013
International Design Engineering Technical Conferences & Computers and Information in Engineering Conference, Aug 2014, Buffalo, United States.
Volume 5A, pp.1-10, 2014. <hal- 01006614>
6. Damien Chablat, Ranjan Jha, Fabrice Rouillier, Guillaume Moroz. Non-singular assembly mode changing trajectories in the workspace for the 3-RPS
parallel robot. 14th International Symposium on Advances in Robot Kinematics, Jun
2014, Ljubljana, Slovenia. pp.149 - 159, 2014. <hal-00956325>
7. Ranjan Jha, Singularity-Free Trajectory and Workspace Analysis for Parallel Robots Using Algebraic Tool. 16th Annual Day of ED STIM. [won 2nd prize
for presentation].

Future Work
A3
A1
B1
A2
B2
B3
P =
𝑥
𝑦
C3
C1
C2
• Direct Kinematics - Method I
α
β
• 3 RRR Mechanism – NaVaRo robot
• Ci in terms of moving platform angle and pose variables
• Bi in terms of actuated angle
• Distance constraint BiCi = L
• Direct Kinematics - Method II
• Ci in terms of angle α and β
• Bi in terms of angle α
• Distance constraint C1 C2 = C2 C3 = C3 C1 = L
• P as centroid, P = (C1+ C2+ C3 )/3
• Angle θ as the angle between moving and base platform
θ
• Pose Variables  [ x, y, θ ] • Articular Variables  [α1, α2, α3] • Passive Variables  [ β1, β2, β3 ]
15-05-2017 84

• Error in angles α1, α2, α3, β1, β2, β3
• Error in pose variables x, y, θ
Deviation from the original path due to the error
in the positions of Bi and Ci
15-05-2017 85

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

Ph.D. Thesis : Ranjan JHA : Contributions to the Performance Analysis of Parallel Robots

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