robotics-kinematics

name	robotics-kinematics
description	Robotics kinematics — forward/inverse kinematics, DH parameters, transformation matrices, workspace analysis, singularities, Jacobian, velocity kinematics, robot types (serial/parallel/SCARA).
metadata	{"priority":7,"promptSignals":{"phrases":["robotics","robot kinematics","forward kinematics","inverse kinematics","DH parameters","Denavit-Hartenberg","Jacobian","robot arm","workspace","end effector"],"minScore":3}}

Robotics Kinematics — Complete Skill

Coordinate Frames and Transformations

Homogeneous Transformation Matrix

4×4 matrix combining rotation and translation: T = [R | p; 0 0 0 | 1] R = 3×3 rotation matrix [orthogonal: Rᵀ = R⁻¹] p = 3×1 position vector [m]

Composition: T_A→C = T_A→B × T_B→C (multiply left to right)

Inverse transformation: T⁻¹ = [Rᵀ | -Rᵀp; 0 0 0 | 1]

Elementary Rotations

R_x(α) = [1, 0, 0; 0, cα, -sα; 0, sα, cα] R_y(β) = [cβ, 0, sβ; 0, 1, 0; -sβ, 0, cβ] R_z(γ) = [cγ, -sγ, 0; sγ, cγ, 0; 0, 0, 1] c = cosine, s = sine

Denavit-Hartenberg (DH) Parameters

Modified DH Convention (Craig, common in robotics)

Each link described by 4 parameters: [a_i, α_i, d_i, θ_i] a_i = link length (distance along x_i) α_i = link twist (rotation about x_i, from z_{i-1} to z_i) d_i = link offset (distance along z_{i-1}) θ_i = joint angle (rotation about z_{i-1})

For revolute joint: θ_i is variable; d_i is fixed For prismatic joint: d_i is variable; θ_i is fixed

DH Transformation Matrix

ᵢ₋₁Tᵢ = R_x(α_{i-1}) × D_x(a_{i-1}) × R_z(θ_i) × D_z(d_i)

Expanded: ᵢ₋₁Tᵢ = [cθᵢ, -sθᵢ, 0, aᵢ₋₁; sθᵢ cαᵢ₋₁, cθᵢ cαᵢ₋₁, -sαᵢ₋₁, -sαᵢ₋₁ dᵢ; sθᵢ sαᵢ₋₁, cθᵢ sαᵢ₋₁, cαᵢ₋₁, cαᵢ₋₁ dᵢ; 0, 0, 0, 1]

Base to end-effector: ⁰Tₙ = ⁰T₁ × ¹T₂ × ... × ⁿ⁻¹Tₙ

Forward Kinematics

Definition

Given joint variables q = [q₁, q₂, ..., qₙ], find end-effector pose {R, p} ⁰Tₙ(q) = product of DH matrices → position p and orientation R

Example — 2-DOF Planar Robot

Links: a₁, a₂; joint angles θ₁, θ₂ p_x = a₁ cos(θ₁) + a₂ cos(θ₁+θ₂) p_y = a₁ sin(θ₁) + a₂ sin(θ₁+θ₂) φ = θ₁ + θ₂ (end-effector angle)

Example — Standard 6-DOF Robot (Puma-like)

Build DH table: n=6 rows with parameters [aᵢ, αᵢ, dᵢ, θᵢ] Wrist singularity: when axes 4, 5, 6 intersect at same point → ⁰Tₙ = ⁰T₃ × ³T₆ Standard: joints 1-3 position wrist; joints 4-6 orient end-effector

Inverse Kinematics (IK)

Overview

Given desired end-effector pose {R_d, p_d}, find joint variables q Challenges: non-unique solutions, singularities, no-solution regions, nonlinear equations

Number of solutions: 6-DOF robot typically has 8 solutions (elbow up/down, wrist flip, shoulder L/R)

Analytical IK (Preferred When Possible)

Step 1 — Decouple (wrist-partitioned robots): Wrist center: p_wc = p_d - d₆ × R_d[:,2] (d₆ = wrist-to-tool offset along z₆) Joints 1-3 from p_wc using geometric/algebraic relations Joints 4-6 from R₃₋₆ = ⁰R₃ᵀ × R_d (use ZYZ or ZYX Euler angles for R₃₋₆)

2-DOF planar solution: c₂ = (p_x² + p_y² - a₁² - a₂²) / (2a₁a₂) s₂ = ±√(1-c₂²) → two solutions (elbow up/down) θ₂ = atan2(s₂, c₂) θ₁ = atan2(p_y, p_x) - atan2(a₂ s₂, a₁ + a₂ c₂)

Numerical IK (General)

Newton-Raphson: q_{k+1} = q_k + J⁺(q_k) × Δx J⁺ = Jᵀ(JJᵀ)⁻¹ (right pseudo-inverse, for overdetermined) Δx = x_d - f(q_k) (task space error)

Damped least squares (DLS): q̇ = Jᵀ(JJᵀ + λ²I)⁻¹ ẋ (avoids singularity: λ = 0.1-1.0)

Jacobian and Velocity Kinematics

Geometric Jacobian

ẋ = J(q) q̇ ẋ = [v; ω] (6×1 velocity vector, linear + angular) J(q) = 6×n matrix (n = DOF)

For column i (revolute joint): Jv_i = z_{i-1} × (p_n - p_{i-1}) (linear velocity contribution) Jω_i = z_{i-1} (angular velocity contribution) z_{i-1} = rotation axis of joint i (third column of ⁰R_{i-1})

For prismatic joint column i: Jv_i = z_{i-1} (linear only) Jω_i = 0

Force Transformation

τ = Jᵀ × F (joint torques from task-space forces/moments) F = wrench [f; m] applied at end-effector

Singularities

Definition

Singularity: rank(J) < min(m, n) → det(J) = 0 (for square J) At singularity: lost DOF in task space; infinite joint velocities for finite task velocity

Types

Workspace boundary singularity: arm fully extended or fully folded — expected Interior singularity (wrist): when wrist axes are aligned → infinite solutions in null space Puma-type at θ₅ = 0°: all solutions with any θ₄ + θ₆ = const

Singularity avoidance: Monitor manipulability: w = √(det(J Jᵀ)) — decreases near singularity Damped least squares: automatically degrades gracefully near singularity

Workspace

Reachable vs. Dexterous Workspace

Reachable workspace: all positions the end-effector can reach with at least one orientation Dexterous workspace: positions where all orientations are achievable Typical 6-DOF: dexterous workspace ⊂ reachable workspace

Joint limits and workspace: With joint limits: non-spherical reachable workspace Collision constraints: further reduce practical workspace

Workspace Calculation

Monte Carlo sampling: random q in joint limits → compute FK → scatter positions Discretized joint space: grid of joint angles → FK → boundary detection

Robot Types Summary

Type	DOF	Configuration	Workspace Shape	Accuracy	Application
Cartesian (gantry)	3	PPP	Rectangular box	±0.01 mm	Large parts, CNC
Cylindrical	3	RPP	Cylinder	±0.1 mm	Palletizing
Spherical (polar)	3	RRP	Partial sphere	±0.1 mm	Older design
SCARA	4	RRPR	Cylinder, vertical z	±0.01 mm	Assembly, pick&place
Serial 6-DOF	6	RRRRRR	Irregular sphere	±0.05 mm	General industrial
Parallel (Delta)	3-6	Multiple chains	Limited volume	±0.05 mm	High speed pick, food
7-DOF redundant	7	RRRRRRR	Enhanced sphere	±0.05 mm	Human-collaborative

Payload to weight ratio (serial): typically 1:10 to 1:5 (100 kg robot carries 10-20 kg) Repeatability vs. accuracy: robots are highly repeatable (±0.02-0.1 mm) but may have poor absolute accuracy without calibration

Output

Provide: DH table with parameters, ⁰Tₙ FK result (position [m] + rotation matrix), IK solutions (all configurations), Jacobian J at given q, manipulability w, singularity detection (det(J) = 0 check), joint torques τ = Jᵀ F for given wrench.