The equations of motion are the physics that link a robot's joint torques to how it actually accelerates — the model behind realistic simulation, model-based control, and understanding why heavy arms are hard to move precisely.
The equations of motion are the formula that says: given the forces or torques applied, this is how the robot will accelerate. They capture gravity, inertia, and the way joints tug on each other.
🎯 Quick challenge
In the manipulator equation M(θ)θ̈ + C(θ,θ̇)θ̇ + g(θ) = τ, the g(θ) term is…
Kinematics tells you where a robot's parts are; dynamics tells you how it moves when you push it. The equations of motion are the bridge between torque and acceleration — and the foundation of realistic control and simulation.
The manipulator equation
For a robot arm, the whole of its dynamics condenses into one elegant equation:
M(θ) — the inertia matrix: how hard each joint is to accelerate, built from every link's mass and inertia tensor. It changes with the arm's pose.
C(θ,θ̇) — Coriolis and centrifugal terms: the forces that arise because the joints are moving and flinging each other around.
g(θ) — gravity: the torque needed just to hold the arm up in a given configuration.
τ — the joint torques the motors provide.
Torque in, acceleration out
Solve the equation one way (forward dynamics) to simulate motion from torques; solve it the other way (inverse dynamics) to find the torques a desired motion needs.
Why it matters two ways
Forward dynamics (torques → motion) is how a physics simulator predicts what the robot will do — essential for training in sim and testing controllers safely.
Inverse dynamics (motion → torques) is how model-based controllers work: computed-torque control plugs a desired motion into the equation to get exactly the torque that produces it, cancelling gravity and coupling.
The intuition it gives
The equation explains everyday robot behavior: an arm sags without gravity compensation (the g(θ) term), a fast-swinging joint disturbs its neighbors (the C term), and the same joint feels "heavier" to accelerate when the arm is stretched out (the M(θ) term changing with pose). Deriving it comes from Newton-Euler or Lagrangian mechanics.
Why it matters
The equations of motion are where a robot stops being geometry and becomes physics. Every dynamic simulation, every model-based controller, and every serious understanding of why a robot moves the way it does starts here.
