robot-dynamics

name	robot-dynamics
description	Robot dynamics — Euler-Lagrange equations of motion, Newton-Euler recursive algorithm, inertia matrix, Coriolis and centrifugal matrix, gravity vector, joint torque/force calculation, dynamic model identification (IDIM, least-squares), friction models (Coulomb, viscous, Stribeck), payload estimation, joint torque limits, computed torque control, inverse dynamics feed-forward, trajectory dynamics (joint-space and Cartesian), and KUKA/ABB dynamic model validation.
metadata	{"priority":7,"promptSignals":{"phrases":["robot dynamics","robot equations of motion","Euler-Lagrange robot","Newton-Euler robot","robot torque","inverse dynamics robot"],"minScore":3}}

Robot Dynamics — Complete Skill

Equations of Motion

Lagrangian Formulation

Generalized equations of motion: M(q) × q̈ + C(q,q̇) × q̇ + G(q) = τ [standard robot dynamics; vector equation; n joints]

Variables: q = [q₁, q₂, ..., qₙ]ᵀ: joint positions (rad or m) q̇ = dq/dt: joint velocities; q̈ = d²q/dt²: joint accelerations M(q): n×n inertia matrix (positive definite symmetric) C(q,q̇): n×n Coriolis and centrifugal matrix (Christoffel symbols) G(q): n×1 gravity vector (gradient of potential energy) τ: n×1 joint torque/force vector (control input)

Kinetic energy: T = (1/2) × q̇ᵀ × M(q) × q̇ [scalar; sum over all links] T = (1/2) × Σᵢ [mᵢ × vᵢ_cᵀ × vᵢ_c + ωᵢᵀ × Iᵢ × ωᵢ] [link i: mass mᵢ; CoM velocity vᵢ_c; angular velocity ωᵢ; inertia Iᵢ]

Potential energy: V = Σᵢ mᵢ × g × zᵢ_c [zᵢ_c = height of CoM of link i; g = 9.81 m/s²]

Euler-Lagrange equations: d/dt(∂T/∂q̇) - ∂T/∂q + ∂V/∂q = τ → M(q) × q̈ + C(q,q̇) × q̇ + G(q) = τ

Properties of the Dynamic Model

M(q) properties: Symmetric positive definite; M = M^T; eigenvalues > 0 M(q) varies with configuration; diagonal for some simple structures; off-diagonal = coupling between joints

Ṁ - 2C skew-symmetry: ẋᵀ × (Ṁ - 2C) × x = 0 for all x [key property; used in passivity-based control proof]

Coriolis matrix (Christoffel symbols): C_{ij} = Σₖ c_{kij} × q̇ₖ [c_{kij} = 1/2 × (∂M_{ij}/∂qₖ + ∂M_{kj}/∂qᵢ - ∂M_{ik}/∂qⱼ)]

Gravity vector: G_j = ∂V/∂q_j = Σᵢ mᵢ × g^T × ∂r_ic/∂q_j [r_ic = position of CoM of link i]

Newton-Euler Recursive Algorithm

Forward-Backward Recursion

Outward recursion (base to tool; propagate kinematics): Angular velocity: ω_i = R_{i-1}^i × ω_{i-1} + q̇_i × ẑ_i [ẑ_i = unit vector of joint i axis] Angular acceleration: ω̇_i = R_{i-1}^i × ω̇_{i-1} + q̈_i × ẑ_i + q̇_i × ω_i × ẑ_i Linear acceleration of origin: v̇_i = R_{i-1}^i × v̇_{i-1} + ω̇_i × r_i + ωᵢ × (ωᵢ × r_i) [r_i = position of frame i origin] Linear acceleration of CoM: v̇_ci = v̇_i + ω̇_i × d_ci + ωᵢ × (ωᵢ × d_ci) [d_ci = CoM position from frame i origin]

Inward recursion (tool to base; compute forces/torques): Forces: fᵢ = mᵢ × v̇_ci - mᵢ × g + Rᵢ^{i+1} × f_{i+1} [Newton's 2nd law for link i] Torques: nᵢ = Iᵢ × ω̇ᵢ + ωᵢ × (Iᵢ × ωᵢ) + d_ci × (mᵢ × v̇_ci) + r_{i+1} × (Rᵢ^{i+1} × f_{i+1}) + Rᵢ^{i+1} × n_{i+1}

Joint torque (revolute): τᵢ = nᵢ · ẑᵢ + d_i × fᵢ · ẑᵢ [projection onto joint axis; d_i = motor damping]

Computational complexity: O(n) for n joints; Newton-Euler is more efficient than Lagrange for numerical simulation

Inertia Matrix Calculation

For 2-DOF Planar Robot (Example)

Links: L₁ = 0.5 m, m₁ = 5 kg; L₂ = 0.4 m, m₂ = 3 kg (uniform rods; Ic = mL²/12)

M₁₁(q) = m₁(L₁/2)² + I_c1 + m₂[L₁² + 2L₁(L₂/2)cos(q₂) + (L₂/2)²] + I_c2 M₁₂ = M₂₁(q) = m₂[L₁(L₂/2)cos(q₂) + (L₂/2)²] + I_c2 M₂₂ = m₂(L₂/2)² + I_c2

Coriolis: C₁₁ = 0; C₁₂ = -m₂ × L₁ × (L₂/2) × sin(q₂) × q̇₂; C₂₁ = m₂ × L₁ × (L₂/2) × sin(q₂) × q̇₁; C₂₂ = 0

Gravity: G₁ = (m₁×(L₁/2) + m₂×L₁) × g × cos(q₁) + m₂ × (L₂/2) × g × cos(q₁+q₂) G₂ = m₂ × (L₂/2) × g × cos(q₁+q₂)

Friction Models

Coulomb + Viscous + Stribeck

Coulomb friction: τ_friction = F_c × sgn(q̇) [constant magnitude; reverses with velocity sign] F_c = static friction torque (at breakaway); values 1–20 Nm for industrial robot joints

Viscous friction: τ_viscous = B × q̇ [B = viscous damping coefficient Nm·s/rad; linear with velocity] B: 0.5–5 Nm·s/rad for industrial joints with lubricated gearboxes

Stribeck effect (combined): τ_friction = [F_c + (F_s - F_c) × exp(-(q̇/v_s)²)] × sgn(q̇) + B × q̇ v_s = Stribeck velocity (velocity at friction minimum); typically 0.01–0.05 rad/s At low velocities: stick-slip; friction drops from F_s (static) to F_c (kinetic) above v_s

Friction identification: Run constant-velocity experiments at multiple velocities; measure steady-state τ Fit F_c, F_s, B, v_s from τ = f(q̇) curve using least-squares

Dynamic Model Identification

IDIM (Inverse Dynamics Identification Method)

Regressor form: τ = Y(q, q̇, q̈) × π [linear in parameters; Y = regressor matrix n×p; π = parameter vector p×1] π = [m₁, m₁ × d_c1_x, m₁ × I_c1_xx, ..., F_c1, B₁, ...] [inertial + friction parameters per link]

Standard form: Measure: τ_measured (from motor current × torque constant) Compute: Y from measured q, q̇, q̈ (filtered numerically) Estimate: π̂ = (Y^T Y)^{-1} Y^T × τ_measured [ordinary least squares; 30–60 excitation trajectories needed]

DIDIM (Dynamic IDentification through Inverse Model): More numerically robust; integrates measured velocities to reduce differentiation noise Validation: simulate dynamics with identified π → compare τ_simulated vs. τ_measured; target: RMSE < 10% rated torque

Exciting trajectories: Periodic trajectories designed to excite all joint frequencies Optimal design: maximize condition number of Y^T Y (minimize correlation between parameters) Fourier series: q_j(t) = q₀_j + Σₖ (a_jk sin(kω_f t) + b_jk cos(kω_f t))

Computed Torque Control (CTC)

Feed-Forward + PD Controller

Inverse dynamics control law: τ = M(q) × (q̈_d + K_d × ė + K_p × e) + C(q,q̇) × q̇ + G(q) + τ_friction [q̈_d = desired acceleration; e = q_d - q = position error; ė = velocity error; K_p, K_d = PD gains]

Linearization: with perfect model, closed-loop error dynamics become: ë + K_d × ė + K_p × e = 0 [linear decoupled double integrator per joint] Critically damped: K_d = 2ωₙ, K_p = ωₙ² → ωₙ = desired bandwidth [rad/s]

Robustness: model errors, friction, payload → disturbance force δτ Integral term added: K_i × ∫e dt → steady-state error eliminated

Practical implementation: Sample at ≥ 1 kHz; compute M, C, G online at each sample; ~10⁶ flops per sample for 6-DOF robot Required: fast CPU (FPGA or multi-core real-time controller); pre-compiled Matlab/Simulink or C++ dynamics

Payload Estimation

Static payload identification: With known robot configuration q → measure joint torques τ_static → solve for mass and CoM of payload G(q, payload) = G(q, no_payload) + G_payload(q) [linear superposition] G_payload_j = m_p × g × ∂z_t / ∂q_j [z_t = height of tool CoM; m_p = payload mass] Estimate m_p and d_c_payload from τ_static at multiple q configurations

Standards and References

Standard	Scope
ISO 9283	Robot performance testing (includes dynamic performance)
Siciliano et al. "Robotics" (Springer 2009)	Comprehensive robot dynamics textbook
Khalil & Dombre "Modeling Identification and Control of Robots"	IDIM identification method
Slotine & Li "Applied Nonlinear Control"	Computed torque, adaptive control theory
Craig "Introduction to Robotics"	Newton-Euler algorithm derivation

Output

Provide: robot configuration (n joints; DH parameters; link masses mᵢ [kg]; inertia tensors Iᵢ [kg·m²]), equations of motion (M, C, G expressions or numerical values at a specific q), inverse dynamics at a given trajectory (q, q̇, q̈) → joint torques τ [Nm] computed by Newton-Euler, peak torque per joint [Nm] vs. motor rated torque [Nm] (utilization %), friction model (F_c [Nm]; B [Nm·s/rad]; Stribeck v_s [rad/s]; total friction at operating speed), computed torque controller gains (K_p, K_d per joint; expected bandwidth ωₙ [rad/s]; tracking error e_max [mm]), dynamic model identification (regressor Y; parameter vector π̂; RMSE [Nm]; validation RMSE vs. 10% rated), payload (m_p [kg]; CoM offset d_c [mm]; additional gravity torque per joint [Nm]), and applicable reference (Siciliano Robotics, Khalil IDIM, ISO 9283).