A Kinematic Chain in ${\mathbb R}^3$

As for a single rigid body, the 3D case is significantly more complicated than 2D due to 3D rotations. Also, several more types of joints are possible, as shown in Figure 2.10. Nevertheless, it is naturally extend the ideas from transformations of 2D kinematic chains to the 3D case. The following steps from Section 2.3.1 will be recycled here: 1) the coordinate frame must be carefully placed to define the model for each ${\cal A}_i$; 2) based on joint relationships, several parameters will be defined; 3) the parameters will be used to define a homogeneous transformation matrix, T_i; 4) the transformation of points on link ${\cal A}_m$ will be given by $T_1 T_2 \cdots T_m$.

 

Figure 2.10:   Types of 3D Joints

Consider a kinematic chain of m links in ${\cal W}= {\mathbb R}^3$, in which each ${\cal A}_i$ for $1 \leq i < m$ is attached to ${\cal A}_{i+1}$ by a revolute joint. Each link can be a complicated, rigid body as shown in Figure 2.11.a. For the 2D problem, the coordinate frames were based on the points of attachment. For the 3D problem, it is convenient to use the axis of rotation of each revolute joint (this is equivalent to the point of attachment for the 2D case). The axes of rotation will generally be skew lines in ${\mathbb R}^3$, as shown in Figure 2.11.b. Let Z_i refer to the axis of rotation for the revolute joint that holds ${\cal A}_i$ to ${\cal A}_{i-1}$. Between each pair of axes in succession, draw a vector, X_i, that joins the two closest pair of points, one from Z_i and the other from Z_i-1 (this choice is unique if the axes are not parallel). The recommended coordinate frame for defining the geometric model for each ${\cal A}_i$ will be given with respect to Z_i and X_i, which are given in Figure 2.11.b. Assuming a right-handed coordinate system, the Y_i axis points away from us in Figure 2.11.b. In the transformations that appear shortly, the coordinate frame given by X_i, Y_i, and Z_i, will be most convenient for defining the model for ${\cal A}_i$, even if the origin of the frame lies outside of ${\cal A}_i$.

 

Figure 2.11:   a) The diagram of a generic link; b) The rotation axes are skew lines in ${\mathbb R}^3$.

 

Figure 2.12:   Definitions of the four DH parameters: d_i, $\theta_i$, a_i-1, $\alpha_{i-1}$

In Section 2.3.1, each T_i was defined in terms of two parameters, a_i-1 and $\theta_i$. For the 3D case, four parameters will be defined: d_i, $\theta_i$, a_i-1, and $\alpha_{i-1}$. These are referred to as Denavit-Hartenberg parameters, or DH parameters for short [2]. The definition of each parameter is indicated in Figure 2.12. Figure 2.12.a shows the definition of d_i. Note that X_i-1 and X_i contact Z_i at two different places. Let d_i denote signed distance between these points of contact. If X_i is above X_i-1 along Z_i, then d_i is positive; otherwise, d_i is negative. The parameter $\theta_i$ is the angle between X_i and X_i-1, which corresponds to the rotation about Z_i that moves X_i-1 to coincide X_i. In Figure 2.12.b, Z_i is pointing outward. The parameter a_i is the distance between Z_i and Z_i-1; recall these are generally skew lines in ${\mathbb R}^3$. The parameter $\alpha_{i-1}$ is the angle between Z_i and Z_i-1. In Figure 2.12.d, X_i-1 is pointing outward.

The homogeneous transformation matrix T_i will be constructed by combining two simpler transformations. The transformation

$$R_i = \pmatrix{ \cos\theta_i & -\sin\theta_i & 0 & 0 \cr \s... ...& \cos\theta_i & 0 & 0 \cr 0 & 0 & 1 & d_i \cr 0 & 0 & 0 & 1 }$$

causes a rotation of $\theta_i$ about the Z_i axis, and a translation of d_i along the Z_i axis. Notice that the effect of R_i is independent of the ordering of the rotation by $\theta_i$ and the translation by d_i, because both operations occur with respect to the same axis, Z_i. The combined operation of a translation and rotation with respect to the same axis is referred to as a screw (as in the motion of a screw through a nut). . The effect of R_i can thus be considered as a screw about Z_i. The second transformation is

$$Q_{i-1} = \pmatrix{ 1 & 0 & 0 & a_{i-1} \cr 0 & \cos\alpha_... ...& \sin\alpha_{i-1} & \cos\alpha_{i-1} & 0 \cr 0 & 0 & 0 & 1 } ,$$

which can be considered as a screw about the X_i-1 axis. A rotation of $\alpha_{i-1}$ about X_i-1 is followed by a translation of a_i-1.

The homogeneous transformation matrix, T_i, for $1 < i \leq m$, is  

$$T_i = Q_{i-1} R_i = \pmatrix{ \cos\theta_i & -\sin\theta_i ... ...} & \cos\alpha_{i-1} & \cos\alpha_{i-1}d_i \cr 0 & 0 & 0 & 1} .$$ (14)

This can be considered as the 3D counterpart to the 2D transformation matrix, (2.10). The following four operations are performed in succession:

1.

Translate by d_i along the Z-axis

2.

Rotate counterclockwise by $\theta_i$ about the Z-axis

3.

Translate by a_i-1 along the X-axis

4.

Rotate counterclockwise by $\alpha_{i-1}$ about the X-axis

The transformation T_i gives the relationship of the frame for ${\cal A}_i$ to the frame for ${\cal A}_{i-1}$. The position of a point (x,y,z) on ${\cal A}_m$ is given by

$$T_1 T_2 \cdots T_m \pmatrix{x \cr y \cr z \cr 1 \cr} .$$

As in the 2D case, the matrix T₁ is the standard rigid-body homogeneous transformation matrix, which is given by (2.11) for the 3D case.

For each revolute joint, $\theta_i$ is treated as the only variable in T_i. A prismatic joints can be simulated by allowing a_i to vary. More complicated joints can be simulated as a sequence of degenerate joints. For example, a spherical joint can be considered as a sequence of three zero-length revolute joints; the joints perform a roll, a pitch, and a yaw. Another option for more complicated joints is to derive a special homogeneous transformation matrix. This might be needed to preserve some of the topological properties that will be discussed in Chapter .