Skip to main content

1.5 The Physics of Motion

Stage I: The Substrate

"Calculus is the study of continuous change. Computers are the study of discrete steps. The bridge between them is where simulation lives."

1. The Derivative (Velocity)​

Position: x(t) Velocity: v(t) = dx/dt Acceleration: a(t) = dv/dt

In a computer, we don't have $dt \to 0$. We have $\Delta t$ (Time Step). v[k] ≈ (x[k] - x[k-1]) / Δt This is the Finite Difference. It creates noise. A bumpy position sensor creates massive spikes in derivative-calculated velocity.

2. The Integral (Dead Reckoning)​

x(t) = ∫ v(t) dt In simulation: x[k] = x[k-1] + v[k] · Δt

Drift (The Enemy)​

Every integration adds a tiny error. Δt is never perfect. v[k] has sensor noise. Error Accumulation: A robot navigating purely by odometry (integrating wheel encoders) will eventually think it is in the next country. This is why we need Loop Closure (Stage III).

3. Numerical Integration​

How do we simulate physics? We predict the future state based on the current state.

Euler Method (First Order)​

x_new = x_old + v · Δt Simple. Fast. Wrong. It assumes velocity is constant during the time step. It is not. It adds energy to the system (instability).

Runge-Kutta 4 (RK4)​

The gold standard. It samples the slope at 4 different points within the time step to get a weighted average. It is computationally expensive but physically stable. Gazebo uses this.

4. Lagrangian Mechanics​

Newton (F = ma) is hard for multi-joint arms. You have to calculate constraint forces at every joint pin. Lagrange says: Forget forces. Look at Energy.

L = T - V

  • T: Kinetic Energy (Motion)
  • V: Potential Energy (Gravity/Springs)

Euler-Lagrange Equation: d/dt(∂L/∂q̇) - ∂L/∂q = 0

This magic formula gives you the Equations of Motion for any robot, no matter how complex, purely by tracking energy. It is the heart of modern physics engines (Dart, MuJoCo).

Deep FAQ​

Q: Why does my simulation explode (robot flies to infinity)? A: Likely the Integration Step ($\Delta t$) is too large. If the robot moves too far in one step, it might penetrate a wall. The physics engine applies a corrective "Spring Force" to push it out. If $\Delta t$ is large, this force is huge. Next step, it shoots past origin. It oscillates and gains infinite energy.

Q: What is a Jacobian? A: It is a matrix of partial derivatives relating Joint Velocities (q̇) to End-Effector Velocities (ẋ). It tells you: "If I move joint 1 by a tiny bit, how much does the hand move?" Singularities occur when the Jacobian loses rank (Hand cannot move in a certain direction).