Contents
What are forward kinematics and inverse 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 reverse process that computes the joint parameters that achieve a specified position of the end-effector is known as inverse kinematics.
How is forward kinematics different from inverse kinematics?
Forward kinematics (for a robot arm) takes as input joint angles, and calculates the Cartesian position and orientation of the end effector. Inverse kinematics takes as input the Cartesian end effector position and orientation, and calculates joint angles. Inverse kinematics is used for trajectory planning.
How are transformation matrices used in forward kinematics?
The forward kinematics mapping is therefore, in the general case, FK : q ↦ T. As discussed above, transformation matrices can represent the position/orientation of a rigid body with respect to the (absolute) laboratory frame. They can also represent relative position/orientation between two rigid bodies (or two frames).
What are the joint angles in forward kinematics?
The joint angles are denoted by q: = (θ1, θ2). The end-effector is a parallel gripper (in blue). The position and the orientation of the end-effector are denoted by p: = (x, y, θ). The forward kinematics problem is then to compute the mapping FK : q ↦ p.
Which is the Jacobian matrix of the forward kinematics mapping?
Let us now introduce a fundamental object, the Jacobian matrix of the forward kinematics mapping. The Jacobian matrix of the forward kinematics mapping at a given configuration q0 is defined by: J(θ1, θ2) = ( ∂x ∂θ1 ∂x ∂θ2 ∂y ∂θ1 ∂y ∂θ2 ∂θ ∂θ1 ∂θ ∂θ2) = (− d1sin(θ1) − d2sin(θ1 + θ2) − d2sin(θ1 + θ2) d1cos(θ1) + d2cos(θ1 + θ2) d2cos(θ1 + θ2) 1 1).
Which is the formula for the forward kinematics function fk?
One has thus obtained the explicit formulae for the forward kinematics function FK. Let us now introduce a fundamental object, the Jacobian matrix of the forward kinematics mapping. The Jacobian matrix of the forward kinematics mapping at a given configuration q0 is defined by: