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 $\to$ pose ($q\to X^B$)
• Inverse Kinematics: pose $\to$ joint positions ($X^B\to q$)
• Differential Kinematics: joint positions, velocities $\to$ spatial velocity ($q,v\to V^B$)
  • Differential Inverse Kinematics: spatial velocity, joint positions $\to$ joint velocities ($V^B,q\to v$)

Simple Concepts

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

$\vec{r}=x\hat{i}+y\hat{j}+z\hat{k}$

Displacement = $\Delta r=\overrightarrow{r_f}-\overrightarrow{r_i}$ is a vector physical quantity. Note: If you are working in 1D, just use $\Delta x=x_f-x_i$

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

Average speed = $v_{avg}=\frac{d}{\Delta t}=\frac{d}{t-t_0}$ is a scalar quantity.

Average velocity = ${\vec{v}}_{avg}=\frac{\Delta \vec{r}}{\Delta t}$ is a vector.

Average acceleration = ${\vec{a}}_{avg}=\frac{\Delta \vec{v}}{\Delta t}$ is a vector.

instantaneous speed is the magnitude of the instantaneous velocity.

instantaneous velocity is the velocity of an object at a given moment, $v(t)=\frac{dx(t)}{dt}=x^{\prime }(t)$

instantaneous acceleration is the acceleration of an object at a given moment, $a(t)=\frac{dv(t)}{dt}=v^{\prime }(t)=x^{\prime \prime }(t)$

Kinematics Equations

Calculus of Kinematics

$v_x(t)=v_{x_0}+{\int }_0^t{a}_x(t)dt$ $x(t)=x_0{\int }_{t_0}^t{v}_x(t)dt=x_0+{\int }_{t_0}^t(a_x(t)t+v_{x_0})dt$

Relative Motion

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