The diagram represents a common subset of 6R industrial arms: a vertical base, parallel
shoulder/elbow axes, and three concurrent wrist axes whose successive directions are
perpendicular. A horizontal shoulder offset a₁ is common but may be zero. The drawing uses
the representative a₃ = 0 case, so O₂ and O₃ coincide; the table below deliberately
keeps a₃ variable for robots with an elbow offset. Under this frame placement, the family has
the displayed shape of the DH table and α values in {0, ±π/2}. The ABB GoFa is
compared later as a deliberate
near-neighbour, not a member of this subset: its nonzero a₅ separates two wrist axes, so it
has no spherical wrist.
z-axis
x-axis
common normal
displacement along z
origin
A conventional spherical-wrist table shape
With the frame placement used below, the spherical-wrist subset fills these slots. This is a useful template, not a unique table and not a template for every 6R arm.
i
αi
ai
di
θi
1
π/2
a₁
d₁
q₁
2
0
a₂
0
q₂
3
π/2
a₃
0
q₃
4
−π/2
0
d₄
q₄
5
π/2
0
0
q₅
6
0
0
d₆
q₆
Be precise about geometry versus frame choice. The α magnitudes are 0 or π/2 here
because consecutive axes are parallel or perpendicular. In the displayed assignments,
d₂ = d₃ = d₅ = 0: parallel-axis origin freedom permits d₂ = 0, and wrist concurrency permits
the z₄–z₅ intersection to be used for both wrist origins, giving d₅ = 0. The value d₃ = 0
comes from the regional elbow/forearm geometry and chosen origins; it is not forced by
spherical-wrist concurrency.
The α signs depend on the complete frame assignment, and a₃ is not reliably zero:
it carries an elbow offset on several robots and can have either sign. The displayed
spherical-wrist assignments use a₄ = a₅ = a₆ = 0. GoFa's a₅ = −80 mm is exactly why that
robot falls outside the spherical-wrist subset.
The α signs are not invariant. Reorienting a z-axis can affect both adjacent twists,
the positive joint-coordinate sense, and constant offsets, so the whole assignment must stay
consistent. De Luca's own exams show the resulting sign variation directly: the 2019
Kawasaki S030 paper required all twists to be 0 or +π/2, and its published solution
has the α column π/2, 0, π/2, π/2, π/2, 0 — the fourth entry is +π/2 rather than the
RS010N assignment's −π/2. Read every frame and sign constraint before assuming a sign pattern.
Published numerical comparison — four members plus GoFa
Four spherical-wrist members and the GoFa counterexample have full numerical DH tables in
De Luca's published exam solutions. Values are in millimetres and transcribed from those
solutions; the S030 source's apparent row-6 “d₄ = 165” label typo is recorded here as
d₆ = 165. Read a₃ to see physical variation within the pattern, then a₅ to see the family boundary.
Parameter
Kawasaki RS010N 2024-09-19
Kawasaki S030 2019-06-17
Yaskawa GP7 2024-01-24
ABB GoFa 2025-06-11 · counterexample
Fanuc CR-15iA 2022-09-09
a₁ (shoulder offset)
100
150
40
0
75
a₂ (upper arm)
650
870
445
444
640
a₃ (elbow offset)
0
0
40
−110
≈192.55
a₅
0
0
0
−80
0
d₁ (base column)
430
680
330
265
648
d₄ (forearm)
700
1080
440
470
700
d₆ (tool flange)
88
165
80
101
75
α column
π/2, 0, π/2, −π/2, π/2, 0
π/2, 0, π/2, +π/2, π/2, 0
π/2, 0, π/2, −π/2, π/2, 0
π/2, 0, π/2, +π/2, π/2, 0
−π/2, 0, −π/2, π/2, −π/2, 0
Constraint imposed
none on twist signs
all αi must be 0 or +π/2
none on twist signs
all αi nonnegative
none on twist signs
Two lessons sit in this table. The α signs follow the complete frame assignment and the
problem's constraints, not the robot alone. The S030 and GoFa papers require nonnegative
twists, whereas the RS010N paper does not impose that sign rule. The Fanuc goes further: its
published α column differs from the RS010N in four entries — rows 1, 3, 4, and 5 —
while retaining the same spherical-wrist architecture as the RS010N.
And a₃ is where these robots physically differ. The two Kawasakis have zero elbow
offset; the GP7, GoFa and Fanuc do not. On GoFa, a₃ and a₅ are both nonzero and negative.
That a₅ entry is not a small exception to a spherical-wrist template — it records the axis
separation that makes GoFa non-spherical.
The Yaskawa Motoman UP500 exam (2025-02-07) has no published DH solution in the course
archive. Its data sheet is not enough reason to copy this template blindly: establish each
joint-axis relation, common normal, origin, and sign constraint before filling any slot.
Recognizing the spherical-wrist subset
How to decide whether the conventional template is an appropriate starting point.
1. Six revolute joints
All joints are R (no prismatic)
3 + 3 split: regional + wrist
2. Vertical base rotation
Joint 1 axis is vertical
z₀ points up along it
3. Parallel shoulder+elbow
Joints 2 and 3 both horizontal
Same direction → α₂ = 0
4. Orthogonal spherical wrist
Joints 4, 5, 6 intersect at one point
Successive wrist axes are perpendicular; O₄ = O₅ here
5. Regional offsets may vary
a₁ may be zero or nonzero
a₃ may also be zero or nonzero
6. Tool offset
d₆ from wrist center to flange
Usually small (~80-200 mm)
If a 6R robot has the parallel shoulder/elbow pair and three concurrent, successively
perpendicular wrist axes, this table is a useful hypothesis. Verify every axis relation and
common normal before using it. Then apply the exam's frame, sign, origin, and zero-reference
constraints and read all nonzero dimensions from the data sheet — commonly a₁, a₂, a₃,
d₁, d₄, and d₆, with a₁ or a₃ allowed to vanish.
Related robots and counterexamples
Most of these robots fall outside the subset; Unitree is included as a symbolic family member.
ABB GoFa CRB 15000
6R near-neighbour · no spherical wrist
a₅ = −80 mm separates two wrist axes
Official problem explicitly states no spherical wrist
All αi constrained to be nonnegative
UR5 (Universal Robots)
No spherical wrist
Joints 2, 3, 4 all parallel (three-in-a-row)
a₁ = 0 (no shoulder offset)
Different α pattern: π/2, 0, 0, −π/2, π/2, 0
Efort ECR5
Non-spherical wrist
Wrist joints have offsets between them
Negative d₄, d₅ (exam-specific convention)
Panda assignment / FR3 comparison
7R · distinct source assignments
2018 Panda solution gives symbolic signs
2024 FR3 solution uses a separate nonnegative-constant frame assignment
PAL TIAGo
Prismatic joint 1
Torso lift is prismatic (d₁ = q₁, θ₁ = 0)
1P + 7R = 8 dof ⇒ redundancy 2
z₀ ∥ z₁ (α₁ = 0)
Siemens Artis Zeego
α = π on row 6, 7 rows total
Medical C-arm geometry
6R + 1P = 7 dof
Row 6 is prismatic with α = π
Unitree Z1 Pro
exam 2026-06-10 · symbolic spherical-wrist member
Spherical wrist, so it is in the family
a₃ > 0 (elbow offset); d₃ = 0
α column −π/2, 0, −π/2, π/2, −π/2, 0 — every nonzero twist sign flipped vs the Kawasaki