Forward kinematics

Inrobot kinematics,forward kinematicsrefers to the use of thekinematicequations of arobotto compute the position of theend-effectorfrom specified values for thejointparameters.

An articulated sixDOFrobotic armuses forward kinematics to position the gripper.

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

The kinematics equations of the robot are used inrobotics,computer games,andanimation.The reverse process, that computes the joint parameters that achieve a specified position of the end-effector, is known asinverse kinematics.

Forward vs Backwards Kinematics

Kinematics equations

The kinematics equations for the series chain of a robot are obtained using arigid transformation[Z] to characterize therelative movementallowed at eachjointand 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,

$[T]=[Z_{1}][X_{1}][Z_{2}][X_{2}]\ldots [X_{n-1}][Z_{n}],\!$

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

$[Z_{i}]=\operatorname {Trans} _{Z_{i}}(d_{i})\operatorname {Rot} _{Z_{i}}(\theta _{i}),$

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

$[X_{i}]=\operatorname {Trans} _{X_{i}}(a_{i,i+1})\operatorname {Rot} _{X_{i}}(\ Alpha _{i,i+1}).$

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

${}^{i-1}T_{i}=[Z_{i}][X_{i}]=\operatorname {Trans} _{Z_{i}}(d_{i})\operatorname {Rot} _{Z_{i}}(\theta _{i})\operatorname {Trans} _{X_{i}}(a_{i,i+1})\operatorname {Rot} _{X_{i}}(\ Alpha _{i,i+1}),$

whereθ_i,d_i,α_i,i+1anda_i,i+1are known as theDenavit-Hartenberg parameters.

Kinematics equations revisited

The kinematics equations of a serial chain ofnlinks, with joint parametersθ_iare 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 byDenavit–Hartenberg parameters.

Denavit-Hartenberg matrix

The matrices associated with these operations are:

$\operatorname {Trans} _{Z_{i}}(d_{i})={\begin{bmatrix}1&0&0&0\\0&1&0&0\\0&0&1&d_{i}\\0&0&0&1\end{bmatrix}},\quad \operatorname {Rot} _{Z_{i}}(\theta _{i})={\begin{bmatrix}\cos \theta _{i}&-\sin \theta _{i}&0&0\\\sin \theta _{i}&\cos \theta _{i}&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}}.$

Similarly,

$\operatorname {Trans} _{X_{i}}(a_{i,i+1})={\begin{bmatrix}1&0&0&a_{i,i+1}\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}},\quad \operatorname {Rot} _{X_{i}}(\ Alpha _{i,i+1})={\begin{bmatrix}1&0&0&0\\0&\cos \ Alpha _{i,i+1}&-\sin \ Alpha _{i,i+1}&0\\0&\sin \ Alpha _{i,i+1}&\cos \ Alpha _{i,i+1}&0\\0&0&0&1\end{bmatrix}}.$

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

$\operatorname {} ^{i-1}T_{i}={\begin{bmatrix}\cos \theta _{i}&-\sin \theta _{i}\cos \ Alpha _{i,i+1}&\sin \theta _{i}\sin \ Alpha _{i,i+1}&a_{i,i+1}\cos \theta _{i}\\\sin \theta _{i}&\cos \theta _{i}\cos \ Alpha _{i,i+1}&-\cos \theta _{i}\sin \ Alpha _{i,i+1}&a_{i,i+1}\sin \theta _{i}\\0&\sin \ Alpha _{i,i+1}&\cos \ Alpha _{i,i+1}&d_{i}\\0&0&0&1\end{bmatrix}},$

known as theDenavit-Hartenbergmatrix.

Computer animation

The forward kinematic equations can be used as a method in3D computer graphicsfor animating models.

The essential concept of forward kinematic animation is that the positions of particular parts of the model at a specified time are calculated from the position and orientation of the object, together with any information on the joints of an articulated model. So for example if the object to be animated is an arm with the shoulder remaining at a fixed location, the location of the tip of the thumb would be calculated from the angles of theshoulder,elbow,wrist,thumbandknucklejoints. Three of these joints (the shoulder, wrist and the base of the thumb) have more than onedegree of freedom,all of which must be taken into account. If the model were an entire human figure, then the location of the shoulder would also have to be calculated from other properties of the model.

Forward kinematic animation can be distinguished frominverse kinematic animationby this means of calculation - in inverse kinematics the orientation of articulated parts is calculated from the desired position of certain points on the model. It is also distinguished from other animation systems by the fact that the motion of the model is defined directly by the animator - no account is taken of anyphysical lawsthat might be in effect on the model, such as gravity or collision with other models.

See also

• Inverse kinematics
• Kinematic chain
• Robot control
• Mechanical systems
• Robot kinematics
• Kinematic synthesis