1. TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
ABSTRACT
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
Petar Pavešić, University of Ljubljana
In a mechanical device, like a robot arm, the forward kinematic map relates the configuration space of the joints to the working space of the device. A typical kinematic map is smooth but with certain singularities, and it admits only partial inverses. We are going to describe a general setting for the study of the motion planning problem in this context, and discuss the complexity of some simple joint configurations.
Lisboa 1

2. TOPOLOGICAL COMPLEXITY
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
TOPOLOGICAL COMPLEXITY
Motion planning in configuration spaces:
𝒞 - configuration space of a mechanical device
Find robust rules that product paths connecting any two given configurations in 𝒞.
𝑐 𝑖𝑛𝑖𝑡 , 𝑐 𝑓𝑖𝑛 ∈𝒞×𝒞 query
𝜌:𝒞×𝒞→𝑃𝒞, 𝜌 𝑐 𝑖𝑛𝑖𝑡 , 𝑐 𝑓𝑖𝑛 0 = 𝑐 𝑖𝑛𝑖𝑡 , 𝜌 𝑐 𝑖𝑛𝑖𝑡 , 𝑐 𝑓𝑖𝑛 1 = 𝑐 𝑓𝑖𝑛 roadmap
(Farber) Global roadmap exists ⇔ 𝒞 is contractible.
Consider partial roadmaps, defined on subsets of 𝒞×𝒞.
𝑻𝑪 𝓒 := minimal number of partial roadmaps, needed to satisfy all queries. 2

3. Partial roadmaps are partial sections of 𝜋.
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
𝒞×𝒞 𝑃𝒞 𝜋 𝛼
(𝛼 0 ,𝛼 1 ) 𝜌
Partial roadmaps are partial sections of 𝜋. 𝒬
𝜋 is a fibration
𝑇𝐶 𝒞 is the Schwarz genus of 𝜋
In particular, 𝑇𝐶 𝒞 is a homotopy invariant of 𝒞.
Extensive computations of 𝑇𝐶 𝒞 based on explicit roadmaps (upper bounds)
and cohomology obstructions (lower bounds). 3

4. Actual motion planning is more complex:
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
Actual motion planning is more complex:
configuration space 𝒞 (position of joints)
robot device
𝑓 – forward kinematic map
working space 𝒲 (spatial position of the device)
universal joint
planar joints 
(x,y,z)   
𝒞= 𝑇 2
𝒞= 𝑇 2 = ( 𝑒 𝑖𝜑 , 𝑒 𝑖𝜓 )∈ 𝑆 1 × 𝑆 1
𝒲= 𝑆 2
𝒲= (𝑥,𝑦)∈ ℝ 2 𝑟 2 ≤ 𝑥 2 + 𝑦 2 ≤ 𝑅 2
𝑥=𝑟 cos 𝜓 cos 𝜑
𝑦=𝑟 cos 𝜓 sin 𝜑
𝑧=𝑟 sin 𝜓
𝑓 𝑒 𝑖𝜑 , 𝑒 𝑖𝜓 =(𝑅 cos 𝜑+𝑟 cos 𝜑+𝜓 ,
𝑅 sin 𝜑+𝑟 sin 𝜑+𝜓 ) 4

5. 𝒞= 𝑇 3 robot arm wrist 𝒲=𝑆𝑂(3)
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
robot arm wrist
𝒞= 𝑇 3
𝒲=𝑆𝑂(3)
Forward kinematic map 𝑓:𝑇 3 ⟶𝑆𝑂 3 is given In terms of Euler angles as
𝑓 𝑒 𝑖𝜑 , 𝑒 𝑖𝜓 , 𝑒 𝑖𝜃 = cos 𝜑 cos 𝜓 cos 𝜃 − sin 𝜑 sin 𝜃 − cos 𝜃 sin 𝜑 − cos 𝜑 cos 𝜓 sin 𝜃 cos 𝜑 sin 𝜃 + cos 𝜓 cos 𝜃 sin 𝜑 cos 𝜑 cos 𝜃 − cos 𝜓 sin 𝜑 sin 𝜃 cos 𝜑 sin 𝜓 sin 𝜑 sin 𝜓 − cos 𝜓 sin 𝜓 sin 𝜓 sin 𝜃 cos 𝜓 5

6. (Dranishnikov) Global roadmap exists ⇒ 𝑓 admits a section.
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
MOTION PLANNING
𝑓:𝒞⟶𝒲 forward kinematic map from a configuration space 𝒞 to a working space 𝒲
Find robust rules to move the device from a given joint configuration to a desired position in the working space.
𝑐 𝑖𝑛𝑖𝑡 , 𝑤 𝑓𝑖𝑛 ∈𝒞×𝒲 query
𝜌:𝒞×𝒲→𝑃𝒞, 𝜌 𝑐 𝑖𝑛𝑖𝑡 , 𝑤 𝑓𝑖𝑛 0 = 𝑐 𝑖𝑛𝑖𝑡 , 𝑓(𝜌 𝑐 𝑖𝑛𝑖𝑡 , 𝑤 𝑓𝑖𝑛 1 )= 𝑤 𝑓𝑖𝑛 roadmap
(Dranishnikov) Global roadmap exists ⇒ 𝑓 admits a section.
Proof: given 𝜌:𝒞×𝒲→𝑃𝒞, 𝑤⟼𝜌( 𝑐 0 ,𝑤)(1) is a section of 𝑓.
If 𝑓 does not admit a section consider partial roadmaps, defined on subsets of 𝒞×𝒲.
𝑻𝑪 𝒇 := minimal number of partial roadmaps, needed to satisfy all queries. 6

7. partial sections of (1×𝑓)𝜋.
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
INTRODUCTION
Assume 𝒞 ,𝒲,𝒬 are ENRs:
𝒞×𝒲 𝑃𝒞
(1×𝑓)𝜋 𝛼
(𝛼 0 ,𝑓(𝛼 1 )) 𝜌
Partial roadmaps are
partial sections of (1×𝑓)𝜋. 𝒬
If 𝑓 is a fibration, then 𝑇𝐶 𝑓 is the Schwarz genus of(1×𝑓)𝜋.
However, forward kinematic maps that arise in practice:
(1) do not admit inverse kinematic maps (i.e. sections), and
(2) are smooth but always have singular points (where Jacobian does not have
maximal rank). Therefore are not fibrations.
(D. Gottlieb, 1986, 1988)
In general 𝑇𝐶 𝑓 ≥ Schwarz genus of(1×𝑓)𝜋 (and the difference can be arbitrarily big).
Problem: compute 𝑇𝐶 𝑓 for basic robot arm configurations. 7

8. (1) Assume 𝑓:𝒞⟶𝒲 admits section 𝑠:𝒲⟶𝒞:
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
SPECIAL CASES
Observe: 𝑇𝐶 1 𝑋 =𝑇𝐶 𝑋 .
Two special cases:
(1) Assume 𝑓:𝒞⟶𝒲 admits section 𝑠:𝒲⟶𝒞:
Then, to move from 𝒸∈𝒞 to 𝑤∈𝒲 it is sufficient to move from 𝒸 to 𝑠 𝑤 in 𝒞, and to move from 𝑤 to 𝑤′ in 𝒲 it is sufficient to more from 𝑠(𝑤) to 𝑤′.
𝑓 sectioned ⟹ 𝑇𝐶 𝒲 ≤𝑇𝐶 𝑓 ≤𝑇𝐶 𝒞
(2) Assume 𝑓:𝒞⟶𝒲 fibration:
Then, to move from 𝒸∈𝒞 to 𝑤∈𝒲 it is sufficient to find a path in 𝒲 from 𝑓(𝒸) to 𝑤, and lift it to 𝒞 starting at 𝒸.
𝑓 fibration ⟹ 𝑇𝐶 𝑓 ≤𝑇𝐶 𝒲 8

9. (3) 𝑇𝐶 𝑓 𝒬∪𝒬′ ≤𝑇𝐶 𝑓 𝒬 +𝑇𝐶 𝑓 𝒬′
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
RELATIVE COMPLEXITY
In general, avoid singularities and split into subsets that admit local sections.
RELATIVE COMPLEXITY
𝑓:𝒞⟶𝒲, 𝒬⊆𝒞×𝒲
𝑻𝑪 𝒇 𝓠 = minimal number of partial roadmaps, needed to satisfy all queries in 𝒬.
Properties:
(1) 𝑇𝐶 𝑓 Γ(𝑓) =1
(2) 𝒬⊆ 𝒬 ′ ⟹ 𝑇𝐶 𝑓 𝒬 ≤𝑇𝐶 𝑓 𝒬′
(3) 𝑇𝐶 𝑓 𝒬∪𝒬′ ≤𝑇𝐶 𝑓 𝒬 +𝑇𝐶 𝑓 𝒬′
(4) If 𝒬 ′ can be horizontally deformed into 𝒬, then 𝑇𝐶 𝑓 𝒬 ≥𝑇𝐶 𝑓 𝒬′
In particular, if 𝒬 ′ can be horizontally deformed to a subset 𝒬, then 𝑇𝐶 𝑓 𝒬 =𝑇𝐶 𝑓 𝒬′ . 9

10. 𝒲 𝑟 ≔ regular values of 𝑓, 𝒞 𝑟 := 𝑓 −1 ( 𝒲 𝑟 )
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
RELATIVE COMPLEXITY
𝑓:𝒞⟶𝒲 smooth
𝒲 𝑟 ≔ regular values of 𝑓, 𝒞 𝑟 := 𝑓 −1 ( 𝒲 𝑟 )
𝑇𝐶 𝑓 𝒞 𝑟 × 𝒲 𝑟 ≤𝑇𝐶 𝑓 𝒲 𝑟
𝑇𝐶 𝑓 𝒞× 𝒲 𝑟 ≤𝑐𝑎𝑡(𝒞× 𝒲 𝑟 )
𝑓:𝒞⟶𝒲 admits a partial section 𝑠 over 𝒲 ′ , 𝒞 ′ := 𝑓 −1 ( 𝒲 ′ )
𝑇𝐶 𝒲 𝒲× 𝒲 ′ ≤ 𝑇𝐶 𝑓 𝒞× 𝒲 ′ ≤ 𝑇𝐶 𝒞 𝒞× 𝑠(𝒲 ′ )
Moreover,
if 𝒞 ′ can be deformed to 𝑠(𝒲 ′ ), then 𝑇𝐶 𝑓 𝒞× 𝒲 ′ =𝑇𝐶 𝒞 𝒞× 𝑠(𝒲 ′ )
and
if 𝒲 ′ ⊆ 𝒲 𝑟 , then 𝑇𝐶 𝑓 𝒞′× 𝒲 ′ =𝑇𝐶( 𝒲 ′ )
If 𝒞 is a topological group, then 𝑇𝐶 𝒞 𝒞× 𝑠(𝒲 ′ ) =𝑇𝐶 𝒞 =𝑐𝑎𝑡 𝒞 . 10

11. Kinematic map is sectioned and is regular except for 𝜓=0,𝜋.
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
COMPUTATIONS
planar joints
Kinematic map is sectioned and is regular except for 𝜓=0,𝜋.  
⟹ 2=𝑇𝐶 𝑆 1 ×𝐼 ≤𝑇𝐶 𝑓 ≤𝑇𝐶 𝑇 2 =3
Furthermore, for 𝒞 ′ =𝑆 1 ×0 we have
𝑇𝐶 𝑓 ≥𝑇𝐶 𝑓 𝑇 2 ×𝑓( 𝒞 ′ )=𝑇𝐶 𝑇 2 𝑇 2 × 𝒞 ′ =3.
Therefore 𝑇𝐶 𝑓 =3.
universal (Cardan) joint
Kinematic map is regular except for 𝜓=± 𝜋 2 ,
and is sectioned over 𝑆 2 − poles .  
(x,y,z)
Previous results yield 3≤𝑇𝐶 𝑓 ≤5,…
… which may be improved to 4≤𝑇𝐶 𝑓 ≤5, by a careful
analysis of extensions of roadmaps around singular points. 11

12. These are matrices of the form
TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
COMPUTATIONS
robot arm wrist
Singular values of 𝑓 are rotations whose Z-axis coincides with the Z-axis of the reference frame.
These are matrices of the form
corresponding to points on a circle in 𝑆𝑂(3).
cos 𝜃 − sin 𝜃 sin 𝜃 cos 𝜃
Singular points are of the form 𝑒 𝑖𝜑 ,1, 𝑒 𝑖𝜃 and 𝑒 𝑖𝜑 ,−1, 𝑒 𝑖𝜃 , corresponding to points in a disjoint union of two two-dimensional tori in 𝑇 3 .
𝑓 admits a section on the complement of singular points, and the previous estimates yield 4≤𝑇𝐶 𝑓 ≤6. 12

13. TOPOLOGICAL COMPLEXITY OF KINEMATIC MAPS
Further results
We implicitly used cohomological lower bounds for the topological complexity of various configuration and working spaces.
One can also consider the diagram for partial liftings
and use the standard approach to derive the estimate 𝒬
𝒞×𝒲 𝑃𝒞
(1×𝑓)𝜋 𝜌
𝑇𝐶 𝑓 ≥ nil ( Ker (1,𝑓) ∗ : 𝐻 ∗ (𝒞×𝒲)⟶ 𝐻 ∗ (𝒞) )
(Also follows from the fact that 𝑇𝐶 𝑓 is bigger than the Schwarz genus of (1×𝑓)𝜋.) 13

14. THANKS! 14