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An articulated six DOF robotic arm uses forward kinematics to position the gripper.

The forward kinematics equations define the trajectory of the end-effector of a PUMA robot reaching for parts.

Forward kinematics refers to the use of the kinematic equations of a robot to compute the position of the end-effector from specified values for the joint parameters. The kinematics equations of the robot are used in robotics, computer games, and animation. The reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics.

Kinematics equations

The kinematics equations for the series chain of a robot are obtained using a rigid transformation [Z] to characterize the relative movement allowed at each joint and separate rigid transformation [X] to define the dimensions of each link. The result is a sequence of rigid transformations alternating joint and link transformations from the base of the chain to its end link, which is equated to the specified position for the end link,

$[T]=[Z_1][X_1][Z_2][X_2]\dots [X_{n-1}][Z_n],$

where [T] is the transformation locating the end-link. These equations are called the kinematics equations of the serial chain.

Link transformations

In 1955, Jacques Denavit and Richard Hartenberg introduced a convention for the definition of the joint matrices [Z] and link matrices [X] to standardize the coordinate frame for spatial linkages. This convention positions the joint frame so that it consists of a screw displacement along the Z-axis

$[Z_i]={\mathrm{Trans}}_{Z_i}(d_i){\mathrm{Rot}}_{Z_i}({\theta }_i),$

and it positions the link frame so it consists of a screw displacement along the X-axis,

$[X_i]={\mathrm{Trans}}_{X_i}(a_{i,i+1}){\mathrm{Rot}}_{X_i}({\alpha }_{i,i+1}).$

Using this notation, each transformation-link goes along a serial chain robot, and can be described by the coordinate transformation,

${}^{i-1}{T}_i=[Z_i][X_i]={\mathrm{Trans}}_{Z_i}(d_i){\mathrm{Rot}}_{Z_i}({\theta }_i){\mathrm{Trans}}_{X_i}(a_{i,i+1}){\mathrm{Rot}}_{X_i}({\alpha }_{i,i+1}),$

where θ_i, d_i, α_i,i+1 and a_i,i+1 are known as the Denavit-Hartenberg parameters.

Kinematics equations revisited

The kinematics equations of a serial chain of n links, with joint parameters θ_i are given by

$[T]={}^0{T}_n=\prod _{i=1}^n{}^{i-1}{T}_i({\theta }_i),$

where ${}^{i-1}{T}_i({\theta }_i)$ is the transformation matrix from the frame of link $i$ to link $i-1$. In robotics, these are conventionally described by Denavit–Hartenberg parameters.

Denavit-Hartenberg matrix

The matrices associated with these operations are:

${\mathrm{Trans}}_{Z_i}(d_i)=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& d_i\\ 0& 0& 0& 1\end{array}\right],{\mathrm{Rot}}_{Z_i}({\theta }_i)=\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i& 0& 0\\ \sin {\theta }_i& \cos {\theta }_i& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right].$

Similarly,

${\mathrm{Trans}}_{X_i}(a_{i,i+1})=\left[\begin{array}{cccc}1& 0& 0& a_{i,i+1}\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right],{\mathrm{Rot}}_{X_i}({\alpha }_{i,i+1})=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \cos {\alpha }_{i,i+1}& -\sin {\alpha }_{i,i+1}& 0\\ 0& \sin {\alpha }_{i,i+1}& \cos {\alpha }_{i,i+1}& 0\\ 0& 0& 0& 1\end{array}\right].$

The use of the Denavit-Hartenberg convention yields the link transformation matrix, [^i-1T_i] as

${\mathrm{}}^{i-1}{T}_i=\left[\begin{array}{cccc}\cos {\theta }_i& -\sin {\theta }_i\cos {\alpha }_{i,i+1}& \sin {\theta }_i\sin {\alpha }_{i,i+1}& a_{i,i+1}\cos {\theta }_i\\ \sin {\theta }_i& \cos {\theta }_i\cos {\alpha }_{i,i+1}& -\cos {\theta }_i\sin {\alpha }_{i,i+1}& a_{i,i+1}\sin {\theta }_i\\ 0& \sin {\alpha }_{i,i+1}& \cos {\alpha }_{i,i+1}& d_i\\ 0& 0& 0& 1\end{array}\right],$

known as the Denavit-Hartenberg matrix.

See also

• Kinematic chain
• Forward kinematic animation
• Robot control
• Mechanical systems
• Robot kinematics

References

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