Geometry Without Forces — and Where It Stops
Imagine a robot arm frozen in a photograph. You can measure every joint angle and, with forward kinematics, compute exactly where the gripper sits in space. Kinematics is the geometry of motion: it answers "where?" and "how the pieces line up" using only angles, lengths, and shapes. It never asks whether a real motor could actually hold that pose, or how hard you'd have to push to get there.
Now unfreeze the photo. Switch off the motors and the arm sags — it droops toward the floor under its own weight. Nothing about the geometry changed; the joint angles the kinematics described are still perfectly valid. What changed is that gravity, a force, was finally allowed to act. That sag is the moment kinematics runs out of answers and dynamics begins.
Dynamics is the study of why and how things move — the relationship between forces and the motion they cause. For a robot we usually model each link as a rigid body, a chunk of matter that does not bend or squish, and study its rigid-body dynamics: how its mass responds to pushes and pulls. Kinematics drew the map; dynamics tells you the cost of every trip across it.
The Cast of Characters: Mass, Force, Torque, Momentum
Dynamics has a small cast, and meeting them once makes the rest of the track far easier. The first is mass: how much matter an object contains, and therefore how stubbornly it resists changing speed. A loaded shopping cart is harder to get rolling — and harder to stop — than an empty one. That stubbornness is inertia, and mass is the number that measures it.
Next is force: a push or pull, like gravity tugging the arm down or your hand shoving a door. A force applied along a straight line changes how fast something moves along that line. But robots mostly rotate at their joints, and the rotational cousin of force is torque — a twisting effort. Push a door near its hinge and almost nothing happens; push at the handle and it swings easily. Same force, more torque, because torque depends on how far from the axis you push.
Torque is the star of robot dynamics because it is what a joint motor actually delivers. The twisting effort each motor must supply to move its joint is the joint torque, and almost every dynamics calculation ultimately answers one question: how much joint torque is needed? Too little, and the arm sags or stalls; the motor's limit is its actuation force budget.
The last character is momentum: roughly, mass times velocity — how much "motion" a moving thing is carrying. It is why a fast, heavy object is dangerous and slow to stop. A robot tracks both linear and angular momentum: the straight-line kind that a moving base carries, and the spinning kind that a swinging arm stores. Momentum is what a robot must build up to start moving and shed to stop — and shedding it abruptly is what makes fast robots jerk.
Where Dynamics Earns Its Keep
If a robot moved infinitely slowly and carried nothing, you could almost ignore dynamics and just plan geometry. Real robots are not so polite. Three situations force you to model forces, and they cover most of what hard robotics is about.
- Fast moves. When a joint accelerates quickly, a swinging link flings the other links around — a whip-like effect captured by the Coriolis and centrifugal terms. The faster you go, the larger these velocity-driven forces grow, and a controller that ignored them would overshoot or wobble.
- Heavy payloads. Pick up a full paint can and your whole arm strains differently. The robot must constantly supply torque just to hold position against gravity — a job called gravity compensation — and the heavier the load, the more of every motor's effort is spent fighting weight rather than moving.
- Gentle contact. Sliding a peg into a hole or shaking a hand means the robot touches the world and must regulate how hard it presses. The forces exchanged at the touch point are contact dynamics, and getting them wrong scratches a surface or crushes an egg. Here you cannot plan position alone — you must reason about force.
There is also a quieter villain working against every motor: friction. The rubbing inside gears and bearings — joint friction — quietly eats some of the torque you send before it ever reaches the link. It is small, messy, and hard to model exactly, but ignoring it leaves a robot sluggish and imprecise at low speeds.
The Equation This Track Builds Toward
Everything above can be folded into a single tidy statement that the rest of this track unpacks piece by piece: the robot equation of motion. It is the bookkeeping ledger that says, for every joint at every instant, the torque you supply must exactly account for accelerating the arm, whipping it around at speed, and holding it up against gravity.
tau = M(q) * q_ddot + C(q, q_dot) * q_dot + g(q)
| | | |
torque inertia x velocity x gravity
you send acceleration velocity holding
(push the mass) (whip / Coriolis) termThe first piece, M(q), is the mass (inertia) matrix: it captures how heavy and how spread-out the arm is in its current pose, and therefore how much torque a given acceleration costs. Its building block for a single body is the inertia tensor, which describes how that body's mass is distributed around its axes. The middle term gathers the Coriolis and centrifugal effects, and the last term, g(q), is exactly the gravity-holding effort we met earlier.
Read this equation in two directions and you get the two great questions of robot dynamics. Read left-to-right — given the torques, what motion results? — and you have forward dynamics, the heart of every physics simulator. Read right-to-left — given the motion I want, what torques must I command? — and you have inverse dynamics, the heart of every high-performance controller. The rest of this track teaches you to fill in, compute, and use each term.