![]() ![]() When used to represent orientations, rotation quaternions are typically called orientation quaternions or attitude quaternions. With respect to rotation vectors, they can be more easily converted to and from matrices. They are equivalent to rotation matrices and rotation vectors. Main article: Quaternions and spatial rotationĪnother way to describe rotations is using rotation quaternions, also called versors. Determine the resultant force and torque at a reference point R, to obtain In this case, Newton's laws (kinetics) for a rigid system of N particles, P i, i=1., N, simplify because there is no movement in the k direction. If a system of particles moves parallel to a fixed plane, the system is said to be constrained to planar movement. ![]() The formulation and solution of rigid body dynamics is an important tool in the computer simulation of mechanical systems. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the system itself, as a function of time. The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law ( kinetics) or their derivative form, Lagrangian mechanics. This excludes bodies that display fluid, highly elastic, and plastic behavior. they do not deform under the action of applied forces) simplifies analysis, by reducing the parameters that describe the configuration of the system to the translation and rotation of reference frames attached to each body. The assumption that the bodies are rigid (i.e. In the physical science of dynamics, rigid-body dynamics studies the movement of systems of interconnected bodies under the action of external forces. Note that p^0 and p^1 point to the same location although p^0 \neq p^1 due to the choice of different coordinate frames.Movement of each of the components of the Boulton & Watt Steam Engine (1784) can be described by a set of equations of kinematics and kinetics The following script specifies the coordinates of a point p with respect to two different coordinate frames o_0 x_0 y_0 and o_1x_1y_1 on the two-dimensional space. Mathematically, the position of a point p can be represented as n-tuple such that p \in \mathbb^n where n = 2 or 3. A coordinate frame consists of an origin (a single point in space), and two or three orthogonal coordinate axes, for two- and three-dimensional spaces, respectively.Ī point is a geometric entity which corresponds to a specific location in the space. The mathematical representation of those entities requires the choice of a reference coordinate frame. Furthermore, representing the forces and the torques is essential to analyze and design robots. ![]() To accomplish certain tasks, engineers need to represent the positions and the velocities of those special points. Representing Positions and VectorsĪ robot has several special points on its body such as the center of mass, the tip of its end effector, etc. Finally, it presents homogeneous transformations. This chapter introduces mathematical representations of positions and vectors, rotations and displacements. ![]() Understanding rigid motions allow robot programmers to describe the position and orientation of the end effector of a robot. A homogeneous transformation matrix combines a rotation matrix with a displacement vector to represent those two properties simultaneously. Therefore, there are two essential kinematic properties of a rigid body: (i) orientation, (ii) position. Owing to this assumption, a rigid body motion is a combination of rotation and translation. Consequently, the distance between any two points on a rigid body is assumed to remain constant in time during any motion. A rigid body is a solid body in which deformation is negligibly small. ![]()
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