Agenda

Kinematics v.s. Dynamics

• Kinematics describes the motion of objects. We have been talking about rigid transformation and derivatives w.r.t. time.
• Dynamics describes the cause of motion. We will talk about mass, energy, momentum, and force.
• The basic law of dynamics, Newton's Law, describes the motion of a point mass: \[ \mv{f}=m\mv{a} \]
• But there are caveats that you may not be aware of.

Kinematics v.s. Dynamics

• We start from point mass dynamics and will move on to rigid body dynamics.
• We will provide certain proofs but not all (many are very tricky and lengthy).

Concepts

• Observer's Frame:
  • When we record any motion, we choose the observer's frame \(\cal{F}_{o}\), so that every point would have a coordinate and every vector will have a direction and length.
  • For our symbols, this is on the superscript.
  • If the frame is moving (e.g., taken to be the body frame), when recording motions, we first clone a version of this frame and keep it static for recording.
• Body Frame:
  • An rigid object moves in the space, and we bind a frame \(\cal{F}_{b(t)}\) tightly to it.
• Reference Frame:
  • When recording the movement of objects, we introduce a reference frame so that the notion of movement is relative to this frame.

Some Notes on Reference Frame

• Reference Frame:
  • When recording the movement of objects, we introduce a reference frame so that the notion of movement is relative to this frame.
• We have not discussed this frame much in developing robot kinematics theories.
In dynamics, the choice of reference frame is not arbitrary!

Recording a Relative Velocity

• We introduce \(s(t)\) to denote a reference frame which may be moving.
• Then we denote the relative velocity as below:
• Relative velocity for a point mass
• \(\mv{v}^o_{s(t)\to b(t)}=\mv{v}^o_{o\to b(t)}-\mv{v}^o_{o\to s(t)}\)
• Relative velocity for rigid body
• \(\mv{\xi}^o_{s(t)\to b(t)}=\mv{\xi}^o_{b(t)}-\mv{\xi}^o_{s(t)}\)
• Consistency
\[ \mv{v}^o_{s(t)\to b(t)}=\mv{\xi}^o_{s(t)\to b(t)}p^o \] where \(p^o\) is a point observed in \(\cal{F}_o\)

Inertia Frame

• Inertia frame refers to the choice of the reference frame.
• Only in an inertia frame can Newton's law be written as \(\mv{f}=m\mv{a}\).
• Definition of Inertia frame:
  • Where the law of inertia (Newton's First Law) is satisfied.
  • Any free motion has a constant magnitude and direction.
• A clear notion of Newton's Second Law: \[ \mv{f}^o=m\mv{a}^o_{s(t)\to b(t)} \] where \(s(t)\) is an inertia frame (\(o\) is static).

Fictitious Force

• What if the reference frame is not an inertia frame?
• Assume we have two moving frames, \(\cal{F}_{s(t)} \) and \(\cal{F}_{b(t)}\)
• e.g., the earth and an object sitting on the earth
• We are interested in how the force \(\mv{f}^o\) affects the relative acceleration \(\mv{a}^o_{s(t)\to b(t)}\)

• For simplicity and illustration purpose, assume that \(\cal{F}_{s(t)}\) is moving with an angular velocity without linear acceleration.
• Some intuition that \(\mv{f}^o\neq m \mv{a}^o_{s(t)\to b(t)}\)
  • Since \(\cal{F}_{s(t)}\) is moving with an angular velocity, any object \(b(t)\) moving along with it must also have an acceleration to gain the same angular velocity.
  • Computation shows that some additional force will be consumed to maintain the relative velocity of \(b(t)\) against \(s(t)\).

Fictitious Force

• Computing \(\mv{f}^o=\d{(m\mv{v}^o_{s'\to b(t)})}/\d{t}\) (note: \(s'\) is chosen to be an inertia frame), and we have \[ \mv{f}^o-m\frac{\d{\mv{\omega}^o}}{\d{t}}\times r^o-2m\mv{\omega}^o\times \mv{v}^o-m\mv{\omega}^o\times (\mv{\omega}^o\times r^o)=m\mv{a}^o \]
  where
  • \(\mv{f}^o\): the physical forces acting on the object
  • \(\mv{\omega}^o:=\mv{\omega}^s_{s'\to s(t)}\)
  • \(\mv{v}^o:=\mv{v}^o_{s(t)\to b(t)}\)
  • \(\mv{r}^o:=\mv{r}^o_{s(t)\to b(t)}\)
  • \(\mv{a}^o:=\mv{a}^o_{s(t)\to b(t)}\)

Fictitious Force

\[ \mv{f}^o-m\frac{\d{\mv{\omega}}}{\d{t}}\times r^o-2m\mv{\omega}^o\times \mv{v}^o-m\mv{\omega}^o\times (\mv{\omega}^o\times r^o)=m\mv{a}^o \]

• Euler force: \(-m\frac{\d{\mv{\omega}^o}}{\d{t}}\times r^o\)
• Centrifugal force: \(-m\mv{\omega}^o\times (\mv{\omega}^o\times r^o)\)
• Coriolis force: \(-2m\mv{\omega}^o\times \mv{v}^o\)