Denavit-Hartenberg (D-H) Representation

All explanation will be for this Alpha II robotic arm

We split this up into two-pass procedure:

Phase I (Steps 0 - 7):
we will physically walk along the robot arm from the floor to the gripper, sticking coordinate axis "stickers" ($L_0,L_1,...,L_n$) on each link.

Part II (Steps 8 - 13):
we go back to the beginnning of the robot (base) and measure the exact physical distances and twist the angles between those stickers to fill out our D-H Parameter Table.

Phase I

Step 0: Number the joints
Before placing axis (stickers), we need to count the number of individual Yaw, Pitch and Roll joints seperately. Number them from 1 to $n$, starting from the joint closest to the floor (the base joint) and ending at the final tool position.

Joints are counted from 1 onwards, so joint 1, joint 2, etc.

Step 1: Set up the Base Frame ($L_0$)
Place our first coordinate sticker ($L_0$) at the fixed base of the robot. In D-H, the Z-axes are always joint axes. The joint axes is the physical line in 3-dimensional space about or along which relative motion between 2 connected links occurs. So let's say a particular link rotates about a certain axis, this axis is $z_0$.

Links and Coordinate Frames are counted from 0 onwards.

Therefore,

1. If motion is linear (Prismatic Joint), we point $z_0$ directly along the line of motion
2. If motion is rotational (Revolute Joint), $z_0$ is the axis that the link rotates about

Once $z_0$ is set, we can place $x_0$ and $y_0$ axes in any way we want. Since $L_0$ is the starting point of the entire robot, there are no previous axes or joints to constrain it. However, there are 2 strict rules and 1 convention to follow. Rule 1 is that all 3 axes must be perpendicular to each other, and that $x_0$ and $y_0$ are in the same plane with $z_0$ pointing out of that plane. Rule 2 is that $x_0\times y_0=z_0$. Using right-hand rule, index finger in direction of $x_0$, middle finger in direction of $y_0$, and thumb in $z_0$.

Step 2: Align the next Z-axis ($z_k$)
Now we look at the next joint in the chain, which is joint 2 (Shoulder). Because joint 2 is the shoulder hinge that tilts the arm up/down, its rotational axis is horizontal. Therefore, we must point out $z_1$ horizontally sideways, sticking out of the robot's shoulders.

Step 3: Locate the center (origin) of Frame $L_1$
For this step we need to look at the new $z_1$ (sideway shoulder axis) and the previous $z_0$ (vertical waist axis). We need to check if they intersect. They do intersect, where they cross at a point inside the shoulder's housing. We can therefore place the origin of Frame $L_1$ exactly at this crossing point.

Step 4: Place $x_2$

• ok simple rule to follow: $x_k$ lies along the common normal between $z_{k-1}$ and $z_k$, pointing from $z_{k-1}$ towards $z_k$
• one very important point is that if $z_1$ and $z_2$ are parallel to each other, means we need to pick a point $x_2$ anywhere along this particular parallel plane from $z_1$ towards $z_2$

Step 5: Place $y_2$
Form the right-hand rule.

Phase II

Now we can fill up the physical relationship between our frames and fill out the D-H Table.

$$\begin{array}{ccccc}\mathbf{Axis}& \theta & d& a& \alpha \\ 1& {\theta }_1& 215\text{ mm}& 0& -90^{\circ }\end{array}$$

where:

• Axis refers to the joint or link number $k$ in the robotics chain, joint $k$ connects link $k$ to $k-1$.
• $\theta$ refers to the joint angle, or the angle of rotation about the previous z-axis. The physical meaning of this is "how much we gotta rotate around the previous X-axis $x_{k-1}$ so that it points parallel to the current X-axis $x_k$"
• $d_k$ is the distance of translation along the previous Z-axis. It measures how far we must slide along the joint axis $z_{k-1}$ to get to the previous X-axis ($x_{k-1}$) to physically interact with the current X-axis ($x_k$)
• $a_k$ measures the physical length of the rigid link. It is the distance along the common perpendicular bridge ($x_k$) required to make joint axis $z_{k-1}$ physically intersect with joint axis $z_k$
• ${\alpha }_k$ measures the structural twist of the link. It is the angle we must rotate around $x_k$ to tilt the previous joint axis $z_{k-1}$ so that it aligns parallel with the current joint axis $z_k$

$$\begin{array}{cccccc}\mathbf{Parameter}& \mathbf{Symbol}& \mathbf{Movement}& \mathbf{Measured}\mathbf{Along}/\mathbf{About}& \mathbf{Revolute}\mathbf{Joint}(\mathbf{R})& \mathbf{Prismatic}\mathbf{Joint}(\mathbf{P})\\ \text{Joint Angle}& {\theta }_k& \text{Rotation}& Z\text{-axis}(z_{k-1})& \mathbf{VARIABLE}& \text{Fixed}\\ \text{Joint Distance}& d_k& \text{Translation}& Z\text{-axis}(z_{k-1})& \text{Fixed}& \mathbf{VARIABLE}\\ \text{Link Length}& a_k& \text{Translation}& X\text{-axis}(x_k)& \text{Fixed}& \text{Fixed}\\ \text{Link Twist}& {\alpha }_k& \text{Rotation}& X\text{-axis}(x_k)& \text{Fixed}& \text{Fixed}\end{array}$$

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1 PAGE SUMMARY OF DH

• place $z$ in the same line as joint axes
• start with $z_0$ at first joint, $J_1$
• $x_k$ always lie along the common normal between $z_{k-1}$ and $z_k$, pointing from $z_{k-1}$ towards $z_k$
• one very important point is that if $z_1$ and $z_2$ are parallel to each other, means we need to pick a point $x_2$ anywhere along this particular parallel plane from $z_1$ towards $z_2$

$$\boxed{\begin{array}{ccccc}\mathbf{Axis},k& \theta & d& a& \alpha \\ \text{xk,yk,zk}& \text{angle from xk−1 to xk about }{z}_{k-1}& \text{distance between xk−1 and xk}& \text{distance between b and L}& \text{angle from zk−1 to zk about xk}\end{array}}$$

where:

• $b_k=x_k\cap z_{k-1}$, and origin at $L_k=x_k\cap z_k$
• we view $x_k$ from top view, such that $x_k$ points out of the page, in this view anti-clockwise is positive

$$T_{k-1}^k(q_k)=\left(\begin{array}{cccc}C{\theta }_k& -C{\alpha }_{k}S{\theta }_k& S{\alpha }_{k}S{\theta }_k& a_{k}C{\theta }_k\\ S{\theta }_k& C{\alpha }_{k}C{\theta }_k& -S{\alpha }_{k}C{\theta }_k& a_{k}S{\theta }_k\\ 0& S{\alpha }_k& C{\alpha }_k& d_k\\ 0& 0& 0& 1\end{array}\right)$$ $$T_{base}^{tool}=T_{base}^{wrist}{T}_{wrist}^{tool}$$ $$T_{base}^{wrist}=T_0^1{T}_1^2$$ $$T_{wrist}^{tool}=T_2^3{T}_3^4$$

Intuitive way of thinking about which $T$ comes first:

Looking at $T_{base}^{wrist}$, it is the transformation matrix to transform coordinate frame base to wrist. In otherwords, the wrist coordinate frame with respect to base coordinate frame. We add $T_0^1$ first and then post-multiply $T_1^2$ after, because once we add $T_0^1$, we are already in the coordinate frame $1$, and we are performing the transform with respect to its current coordinate frame $1$, no longer the base coordinate frame $0$.