# Kinematics

**Kinematics** is the mathematical description of *motion*. It allows us to study how an object will move with relation to time.

On the other hand, dynamics study why objects move in a certain way and is concerned with forces.

Types of Kinematics in Robotics

- Forward Kinematics: joint positions $β$ pose ($qβX_{B}$)
- Inverse Kinematics: pose $β$ joint positions ($X_{B}βq$)
- Differential Kinematics: joint positions, velocities $β$ spatial velocity ($q,vβV_{B}$)
- Differential Inverse Kinematics: spatial velocity, joint positions $β$ joint velocities ($V_{B},qβv$)

### Simple Concepts

In a Cartesian coordinate system, the position of a particle can be written as

$r=xi^+yj^β+zk^$

**Displacement = $Ξr=r_{f}ββr_{i}β$** is a *vector* physical quantity. Note: If you are working in 1D, just use $Ξx=x_{f}βx_{i}$

**Distance = $d$** is a positive *scalar* representing the actual distance traveled.

**Average speed** = $v_{avg}=Ξtdβ=tβt_{0}dβ$ is a *scalar quantity*.

**Average velocity** = $v_{avg}=ΞtΞrβ$ is a vector.

**Average acceleration** = $a_{avg}=ΞtΞvβ$ is a vector.

**instantaneous speed** is the magnitude of the instantaneous velocity.

**instantaneous velocity** is the velocity of an object at a given moment, $v(t)=dtdx(t)β=x_{β²}(t)$

**instantaneous acceleration** is the acceleration of an object at a given moment, $a(t)=dtdv(t)β=v_{β²}(t)=x_{β²β²}(t)$

### Kinematics Equations

### Calculus of Kinematics

$v_{x}(t)=v_{x_{0}}+β«_{0}a_{x}(t)dt$ $x(t)=x_{0}β«_{t_{0}}v_{x}(t)dt=x_{0}+β«_{t_{0}}(a_{x}(t)t+v_{x_{0}})dt$

### Relative Motion

The subscripts may be confusing, so hereβs a reminder If we say $V_{BA}β$, we refer to Frame B having a certain velocity with reference to frame A. $v_{PA}β=v_{PB}β+v_{BA}β$