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Screw theory is the algebraic calculation of pairs of vectors, also known as dual vectors [1] – such as angular and linear velocity, or forces and moments – that arise in the kinematics and dynamics of rigid bodies.
Vectors are defined in cylindrical coordinates by (ρ, φ, z), where ρ is the length of the vector projected onto the xy-plane, φ is the angle between the projection of the vector onto the xy-plane (i.e. ρ) and the positive x-axis (0 ≤ φ < 2π), z is the regular z-coordinate. (ρ, φ, z) is given in Cartesian coordinates by:
In engineering, for instance, kinematic analysis may be used to find the range of movement for a given mechanism and, working in reverse, using kinematic synthesis to design a mechanism for a desired range of motion. [8] In addition, kinematics applies algebraic geometry to the study of the mechanical advantage of a mechanical system or mechanism.
The first step in the process is to show that this condition implies that the infinitesimal rotation tensor is uniquely defined. To do that we integrate ∇ w {\displaystyle {\boldsymbol {\nabla }}\mathbf {w} } along the path X A {\displaystyle \mathbf {X} _{A}} to X B {\displaystyle \mathbf {X} _{B}} , i.e.,
Rotation matrices can either pre-multiply column vectors (Rv), or post-multiply row vectors (wR). However, R v produces a rotation in the opposite direction with respect to w R . Throughout this article, rotations produced on column vectors are described by means of a pre-multiplication.
Traditionally the Newton–Euler equations is the grouping together of Euler's two laws of motion for a rigid body into a single equation with 6 components, using column vectors and matrices. These laws relate the motion of the center of gravity of a rigid body with the sum of forces and torques (or synonymously moments) acting on the rigid body.
The first step is to select a "zero configuration" where all the joint angles are defined as being zero. The 4x4 matrix () describes the transformation from the base frame to the tool frame in this configuration. It is an affine transform consisting of the 3x3 rotation matrix R and the 1x3 translation vector p. The matrix is augmented to create ...
The second called inverse kinematics uses the position and orientation of the end-effector to compute the joint parameters values. Remarkably, while the forward kinematics of a serial chain is a direct calculation of a single matrix equation, the forward kinematics of a parallel chain requires the simultaneous solution of multiple matrix ...