Forward kinematics

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.

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,

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

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

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

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

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

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

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

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