Mathematics

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 # 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 # Denominator — the range. \(3.10 - 1.60 = 1.50\) rad of travel. It sets the scale.

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 # With \(t = 0.75\), a finger 75% closed:

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.

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.