Denavit–Hartenberg Transformation

Serial-link manipulator
- Links are rigid
- Joints connect links
  - can move
  - revolute or prismatic
Every link connects 2 joints
- except the first and last link
Every joint connects 2 links
Each joint is described simply by 4 parameters
- Two constraints
  - $x_j$ axis intersects $z_{j-1}$ axis
  - $x_j$ axis is perpendicular to $z_{j-1}$ axis
Describe the pose of a link frame with respect to the link frame of the previous joint

$N+1$ Links ( $0 \dots n$ )
- First link: (link $0$ ) base
- Last link $n$ : end effector
$N$ Joints ( $1 \dots n$ )

Start at base
Attach a coordinate frame to the end of every link
Enumerate joints and links
- Base link: $0$
- Joint $j$ connects link $j$ with link $j-1$
Base frame $\{0\}$ :
- $z_0$ aligns with axis of first link
Generally: $z_j$ axis aligns with $j+1$ joint axis
$x_j$ axis:
- Along normal between $z_{j-1}$ and $z_j$ axis
- Points (positive direction) from joint $j$ to joint $j+1$
Choose $y_j$ axis to form a right-handed coordinate system

The link frame is not necessarily located on the physical link

Describe pose of a link $j$ with respect to previous link frame $\{j-1\}$
Only 4 parameters: $a$ , $\alpha$ , $d$ and $\theta$
DH notation: constraints
- Axis $x_j$ intersects $z_{j-1}$
- Axis $x_j$ is perpendicular to $z_{j-1}$
Axis of each joint defined by $z$ -axis of previous coordinate frame
- Revolute: rotates about $z$ -axis
- Prismatic: translates about $z$ -axis
Define coordinate system $\{j\}$
- $z_j$ axis: axis of movement of joint ( $j+1$ ) for end of link
- Origin of $\{z_j\}$ frame: intersection point of $z_j$ with common normal of $z_{j-1}$ and $z_j$
- $y$ axis is not considered (defined by $x$ and $z$ axes, through right-handed coordinate system)

What	Symbol	Description	Variable / Constant
Link Length	$a_j$	Distance between $z_j$ and $z_{j+1}$ along $x_{j+1}$ axis. For intersecting axes: parallel to $z_j \times z_{j+1}$	Constant
Link Twist	$\alpha_j$	Angle from $z_j$ axis to $z_{j+1}$ axis about $x_{j+1}$ axis	Constant
Link Offset	$d_j$	Distance from origin of frame $\{j\}$ to $x_{j+1}$ axis along $z_j$ axis	Constant or prismatic joint variable
Joint Angle	$\theta_j$	Angle between $x_j$ and $x_{j+1}$ about $z_{j-1}$ axis	Revolute joint variable or constant
Joint Type	$\sigma_j$	$\sigma_j = R$ for revolute, $\sigma_j = P$ for prismatic	Constant: $R=0$ , $P=1$

The link transform is represented by a homogeneous transformation matrix $A$ .

It comprises four elementary transformations: