SimMechanics

Representing Body Positions and Orientations

Machines are composed of bodies, which have relative degrees of freedom (DoFs). Bodies have positions, orientations, mass properties, and sets of Body coordinate systems. Joints represent the motions of the bodies.

• A machine's geometry consists of its static body features before starting a simulation: positions, orientations, and Body coordinate systems.
• A machine's kinematics consists of the positions/orientations and their derivatives of all degrees of freedom (DoFs) of all bodies at any instant during the machine's motion.

The full description of a machine's motion includes not only its kinematics, but specification of its observers, who define reference frames (RFs) and coordinate systems (CSs) for measuring the machine motion.

This section summarizes observer coordinate systems, measuring body motion, forms of body rotation, and how SimMechanics represents the state of a machine's motion.

• Reference Frames and Coordinate Systems
• Relating Moving Coordinate Systems
• Observing Motion in Different Coordinate Systems
• Observing a Translating, Rotating Rigid Body
• How SimMechanics Represents Body Rotations
• Kinematics and the Machine State

All vectors and tensors (matrices), unless otherwise noted, are Cartesian with three and nine, respectively, spatial components measured by orthogonal or rectangular coordinate axes. This section assumes basic knowledge of vector algebra and analysis. The books of Goldstein [2] and Murray, Li, and Sastry [11] present coordinate transformations, rotations, and rigid body kinematics in detail.

Reference Frames and Coordinate Systems

The reference frame of an observer is an observer's state of motion, which has to be measured by other observers. The Newtonian dynamics of a mechanical system take their simplest form in the special set of inertial RFs, the set of all frames moving at uniform velocities with respect to one another. Within an RF, you can pick any point as a coordinate system origin, then set up Cartesian (orthogonal) axes there.

SimMechanics uses a master inertial RF called World. A CS origin and axis triad are also defined in World. World can mean either the RF or the CS, although in most contexts, the coordinate system is indicated. For SimMechanics, World defines absolute rest and a universal coordinate origin and axes independent of any bodies and grounds in a machine.

Relating Moving Coordinate Systems

Add a second CS, called O, whose origin is translating with respect to the World origin and whose axes are rotating with respect to the World axes.

A vector C represents the origin of O. Its head is at the O origin and its tail at the World origin. The O origin translates as some arbitrary function of time C(t).

The orthogonal unit vectors {u(x), u(y), u(z)} define the coordinate axes of O.

• This set is tilted with respect to the World coordinate axes X, Y, Z, with unit vectors {e(x), e(y), e(z)}. The tilt changes with time.
• You can express the set {u(x), u(y), u(z)} as a linear combination of the basis {e(x), e(y), e(z)} in terms of nine coefficients:

• u(x) = R_xx e(x) + R_yx e(y) + R_zx e(z)

  u(y) = R_xy e(x) + R_yy e(y) + R_zy e(z)

  u(z) = R_xz e(x) + R_yz e(y) + R_zz e(z)

  These are relationships between vectors (not vector components) and are independent of reference frame and coordinate system.

• You obtain the components of the u's in World by taking the scalar products of the u's with the e's:

• u_x(x) = R_xx , u_y(x) = R_yx , u_z(x) = R_zx

  u_x(y) = R_xy , u_y(y) = R_yy , u_z(y) = R_zy

  u_x(z) = R_xz , u_y(z) = R_yz , u_z(z) = R_zz

  The time-dependent R coefficients represent the orientation of the u's with respect to the e's. You can replace the labels (x,y,z) by (1,2,3).

• The components of any vector v measured in World are e(i)*v. Represent them by a column vector v_World. The components of v in O are u(i)*v. Represent them by a column vector v_O. The two sets of components are related by the matrix transformation v_World = R* v_O. The coefficients R form a matrix whose columns are the components of the u's in World:
  
  

• The orthogonality and unit length of the u's guarantee that R is an orthogonal rotation matrix satisfying RR^T = R^TR = I, the identity matrix. R^T is the transpose of R (switch rows and columns). Thus R^-1 = R^T.

• Rotations always follow the right-hand rule.

Observing Motion in Different Coordinate Systems

To the two observer CSs, World and O, now add a third point p(t) in arbitrary motion. p could represent a point mass, the center of gravity (CG) of an extended body, or a point fixed in a moving rigid body, for example. The two observers describe the motion of this point in different ways, related to one another by time-dependent World-to-O coordinate transformations.

The motion of p is given by its column vector components in some CS. The components of p as measured in World are a column vector p_World and, measured in O, are a column vector p_O. The two descriptions are related by

• p_World(t) = C_World(t) + R(t) * p_O(t)

Thus the motion as measured by p_World, when transformed and observed by O as p_O, has additional time dependence arising from C(t) and R(t).

Relating Velocities Observed in Different Coordinate Systems

Differentiate once with respect to time the relationship between p_World and p_O. The result relates the velocity of p as measured by O to the velocity as measured in World.

• dp_World/dt = dC_World/dt + R * dp_O/dt + dR/dt * p_O

The section The Angular Velocity of a Body from Its Rotation Matrix following explains how to express the third term in a simpler form.

Understanding Mechanical Concepts

Observing a Translating, Rotating Rigid Body