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Kinematics describes a robot’s motion. Dynamics relates that motion to its causes. In a tennis match, a camera can show you the ball’s path. To explain how gravity bends that path, how a racket changes it, or how the ball bounces off the court, you also need the ball’s mass and the forces acting on it. A robot’s equations of motion make the same connection. They relate actuation and contact forces to the resulting accelerations and motion. You will express the equations of motion for one rigid body using twists, wrenches, and the small adjoint from calculus on Lie groups. You will also see how the adjoint changes the frame of a spatial inertia. The presentation follows sections 8.1 and 8.2 of Lynch and Park (book pages 270 and 281) (Kevin M. Lynch, 2017) and builds on the wrenches introduced below. The following sections extend these ideas to an entire robot arm through inverse dynamics, forward dynamics, and constrained dynamics.

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Inertial and non-inertial frames

Newton’s laws hold in an inertial frame, one that does not accelerate or rotate. In a frame that does, you have to add forces that no object exerts. The example below shows where these forces come from.
Thomas the train.Picture a coffee mug on a table in a train. Two people watch it: a passenger sitting in the train, and a person standing on the platform. The passenger’s frame moves with the train. The platform is (close enough to) an inertial frame. Whether the train’s frame is inertial depends on what the train is doing.
  1. The train moves at a steady speed on a straight track. The mug sits still on the table. The passenger sees nothing happening. The person on the platform sees the mug gliding along with the train. Both descriptions are simple, and neither person needs any force beyond gravity and the table holding the mug up. A frame that moves at constant velocity is also inertial. You cannot tell you are moving without looking out the window.
  2. The train brakes. The mug slides forward across the table.
    • Platform view: nothing pushed the mug. It simply kept going at the speed it had, while the train slowed down underneath it. The only real horizontal force is friction from the table, which tries to drag the mug along with the slowing train. If friction is too weak, the mug keeps moving forward relative to the table.
    • Passenger view: the mug was at rest, and now it moves forward. Nobody touched it. To explain that with ordinary physics, the passenger has to invent a force pushing the mug forward. This is the inertial force. It is not caused by any object; it appears only because the passenger’s frame is slowing down.
  3. The train speeds up. The same thing happens in reverse. The mug slides toward the back. The platform observer says the train pulled away from the mug. The passenger says a force pushed the mug backward.
  4. The train goes around a curve. The mug slides toward the outside of the curve.
    • Platform view: the mug tried to keep going straight while the train turned away from it. The only thing that can make the mug follow the curve is friction pulling it inward.
    • Passenger view: a force pushed the mug outward. This is the centrifugal force. As with braking, it exists only in the turning frame.
  5. You roll a marble across the aisle while the train is turning. The marble’s path curves as it rolls.
    • Platform view: the marble goes in a straight line. The train rotates underneath it.
    • Passenger view: the marble seems to be steered sideways while it moves. This is the Coriolis force. It acts only on things that are moving relative to the turning frame. That is why the mug sitting still does not feel it, but the rolling marble does.
In the inertial frame there are only real forces, and the mug does what it was already doing. In the non-inertial frame, you must add inertial, centrifugal and Coriolis forces to make Newton’s laws work again. These forces are bookkeeping for the frame’s own motion. No object is pushing, but they correctly predict what the passenger sees. You accept that bookkeeping because the passenger’s frame is the natural one for the passenger. The table, the seats and the mug’s position “on the table” are all simplest to describe from inside the train.Lynch and Park sidestep these apparent forces by expressing body quantities in a stationary frame that coincides with the body at each instant, so every frame in the book is inertial. Rotation representations describes this convention.

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Wrenches

A wrench is a six-dimensional vector that combines a moment and a force, just as a twist combines an angular and a linear velocity: F=[mf]∈R6.\mathcal{F} = \begin{bmatrix} m \\ f \end{bmatrix} \in \mathbb{R}^6 . Here ff is the force and m=r×fm = r \times f is the moment it produces about the frame’s origin, where rr is a point on the force’s line of action. Lynch and Park also call the moment a torque, τ\tau. A wrench changes coordinate frame through the transpose of the adjoint from motion representations: Fa=[AdTba]⊤Fb.\mathcal{F}_a = [\mathrm{Ad}_{T_{ba}}]^\top \mathcal{F}_b .

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References

  • Kevin M. Lynch, F. (2017). Modern Robotics.
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