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The forward kinematic equations can be used as a method in 3D computer graphics for 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 ...
The first of these equations express the Newton's law and is the equivalent of the vector equation = (force equal mass times acceleration) plus = ˙ + (angular acceleration in function of inertia and angular velocity); the second equation permits the evaluation of the linear and angular momentum when velocity and inertia are known.
From this point of view the kinematics equations can be used in two different ways. The first called forward kinematics uses specified values for the joint parameters to compute the end-effector position and orientation. The second called inverse kinematics uses the position and orientation of the end-effector to compute the joint parameters ...
The combination of a rotation and translation in the plane R 2 can be represented by a certain type of 3×3 matrix known as a homogeneous transform. The 3×3 homogeneous transform is constructed from a 2×2 rotation matrix A(φ) and the 2×1 translation vector d = (d x, d y), as: [(,)] = [()] = [ ].
For each joint of the kinematic chain, an origin point q and an axis of action are selected for the zero configuration, using the coordinate frame of the base. In the case of a prismatic joint, the axis of action v is the vector along which the joint extends; in the case of a revolute joint, the axis of action ω the vector normal to the rotation.
The matrix [S] is an element of the Lie algebra se(3) of the Lie group SE(3) of homogeneous transforms. The components of [S] are the components of the twist screw, and for this reason [S] is also often called a twist. From the definition of the matrix [S], we can formulate the ordinary differential equation,
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Informally, the Kolmogorov forward equation addresses the following problem. We have information about the state x of the system at time t (namely a probability distribution p t ( x ) {\displaystyle p_{t}(x)} ); we want to know the probability distribution of the state at a later time s > t {\displaystyle s>t} .