Mathematics on Mulham Fetna

Finger joint angles from three hand landmarks: the dot-product geometry

contact@mulhamfetna.com (Mulham Fetna) — Wed, 16 Sep 2026 00:00:00 +0000

Every knuckle angle in this project comes from three landmarks and one dot product. No learning, no lookup table — just the definition of the angle between two vectors, plus two guards that keep a single glitchy frame from sending NaN into a motor command.

Three points make an angle
#

An angle needs a vertex and two rays. A knuckle is the vertex; the two bones meeting there are the rays:

a base point, where the previous bone starts,
the vertex — the knuckle being measured,
an end point, where the next bone ends.

flowchart LR
    P1(("p1
base")) -- "v1 = p1 − p2" --- P2(("p2
vertex
knuckle"))
    P2 -- "v2 = p3 − p2" --- P3(("p3
end"))

The triplet table
#

self.finger_triplets = {
    "thumb":  [(0, 1, 2), (1, 2, 3), (2, 3, 4)],
    "index":  [(0, 5, 6), (5, 6, 7), (6, 7, 8)],
    "middle": [(0, 9, 10), (9, 10, 11), (10, 11, 12)],
    "ring":   [(0, 13, 14), (13, 14, 15), (14, 15, 16)],
    "pinky":  [(0, 17, 18), (17, 18, 19), (18, 19, 20)]
}

Landmark 0 — the wrist — starts every finger’s first triplet.

Finger	Triplet	Vertex	Joint measured
Index	(0, 5, 6)	5	MCP — joins finger to palm
Index	(5, 6, 7)	6	PIP — middle knuckle
Index	(6, 7, 8)	7	DIP — fingertip knuckle
Thumb	(0, 1, 2)	1	CMC — saddle joint at the wrist
Thumb	(1, 2, 3)	2	MCP
Thumb	(2, 3, 4)	3	IP

Why every first triplet starts at 0. The palm has no landmark of its own, so the wrist → knuckle line stands in for the metacarpal bone. It isn’t exactly collinear with a straight finger — ring and pinky metacarpals fan outward — so a relaxed straight finger rarely measures a full $\pi$. That’s part of why the calibrated “straight” threshold is 3.10 rad rather than 3.14.

From finger flexion to tendon force: linear interpolation onto a MuJoCo motor

contact@mulhamfetna.com (Mulham Fetna) — Wed, 16 Sep 2026 00:00:00 +0000

On the far side of the ROS 2 topic, five flexions arrive and five tendon motors wait. One line of linear interpolation connects them — plus a name lookup that can fail silently, and a string that is allowed to push.

The formula
#

$$F(t) = F_\text{open} + t\,(F_\text{closed} - F_\text{open}) = 50 + t\,(-50 - 50) = 50 - 100\,t$$

FORCE_OPEN = 50.0
FORCE_CLOSED = -50.0

def _lerp(self, start_val, end_val, t):
    return start_val + t * (end_val - start_val)

def apply_flexions(self, flexions):
    for finger, flexion_amount in flexions.items():
        target_force = self._lerp(FORCE_OPEN, FORCE_CLOSED, flexion_amount)
        self.data.ctrl[self.motors[finger]] = target_force

Three parts of one line
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With $t = 0.75$, a finger 75% closed:

Normalizing finger curl: from radians to a 0–1 flexion

contact@mulhamfetna.com (Mulham Fetna) — Wed, 16 Sep 2026 00:00:00 +0000

Radians belong to the camera. Newtons belong to the robot. The number that crosses between them is a plain percentage — and producing it takes one inverted formula and one clip.

The formula
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RAW_STRAIGHT_ANGLE = 3.10  # ~177.6° — open finger
RAW_CURLED_ANGLE   = 1.60  # ~91.7° — fully curled finger

flexion = (straight_limit - avg_angle) / (straight_limit - curled_limit)
flexions[finger_name] = float(np.clip(flexion, 0.0, 1.0))

$$\text{flexion} = \operatorname{clip}\!\left(\frac{\theta_\text{straight} - \bar\theta}{\theta_\text{straight} - \theta_\text{curled}},\; 0,\; 1\right)$$

The three pieces
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Denominator — the range. $3.10 - 1.60 = 1.50$ rad of travel. It sets the scale.

From camera coordinates to mechanical radians

contact@mulhamfetna.com (Mulham Fetna) — Tue, 08 Sep 2026 00:00:00 +0000

Twenty-one points in a camera’s coordinate system on one side. A CAD assembly with hard mechanical stops on the other. This is the arithmetic that makes them agree — and the one row where it currently does not.

The vision layer gives you 21 points floating in a normalized coordinate space that has no physical units and a faked depth axis. The mechanical layer gives you fifteen revolute joints, each with a lower and upper bound in radians that came out of a CAD mate.

Nothing connects them. Building that connection is the actual work of this project, and it happens in about forty lines of Python.

flowchart LR
    A["21 landmarks
(x, y, z) normalized"] --> B["Triplet selection
15 × (p₁, p₂, p₃)"]
    B --> C["Two vectors per joint
v₁ = p₁ − p₂ · v₂ = p₃ − p₂"]
    C --> D["Dot product → arccos
θ in radians"]
    D --> E["Normalize
flexion ∈ [0, 1]"]
    E --> F["Lerp onto URDF limits
θ_urdf"]
    F --> G["JointState
names + positions"]

Step 1 — pick three points
#

To measure a joint you need the joint itself and the two bones meeting at it. In landmark terms: the vertex, plus its two neighbours.

Mathematics on Mulham Fetna

Finger joint angles from three hand landmarks: the dot-product geometry

Three points make an angle #

The triplet table #

From finger flexion to tendon force: linear interpolation onto a MuJoCo motor

The formula #

Three parts of one line #

Normalizing finger curl: from radians to a 0–1 flexion

The formula #

The three pieces #

From camera coordinates to mechanical radians

Step 1 — pick three points #

Three points make an angle
#

The triplet table
#

The formula
#

Three parts of one line
#

The formula
#

The three pieces
#

Step 1 — pick three points
#