X3D-Based Robot Kinematics Simulation

Posture Mathematical Model

Denavit and Hartenberg in “ASME Journal of Applied Mechanics” published a paper. They use the paper to representation and modeling for the robot, and deduced the equation of motion, this has become a current of robots and robot motion modeling standard. Denavi Hartenberg (DH) model of the robot modeling connecting rods and joints of a very simple method, the robot can be used in any configuration, regardless of the structure of the robot how to order and complexity in any coordinate transformation, such as rectangular, cylindrical coordinate, spherical coordinates, the ruler Angle coordinate and twisted coordinates, etc., DH model can be represent. In addition, it can also be used for the rotating chain robot, SCARA robot or any possible joints and connecting rod combination.

DH model assumes that the robot is composed of a series of joints and connecting rod. These joints may be sliding (linear) or rotation (rotation), they can be placed in any order and in any plane. The length of the connecting rod can be arbitrary (including zero), it can attack distorted or curved, may also at any plane. So any set of joints and connecting rod can constitute a said we want to modeling of the robot (Xu, Hong Zhong, &Xu, 2004).

Using DH modeling, the first specify a reference coordinate system for each joint, and then determined from a joint to the next step, to conduct a joint transform namely from a joint coordinate system to the next joint coordinate system transformation. Joints from base to top, and then from the first joint until the last of all joints transformation, from the total transformation matrix of the robot (Koren, 1985).

Specifies the coordinate system for each joint of the steps are as follows (Yi, 2007):

• 1.

  For each joint use z axis for represent. For rotary joints, z positive direction according to the rules of right hand rotation direction of the axis. For smooth joints, z axis is the direction of motion along a straight line. In each case, the z axis of the joints (as well as the local reference coordinate system) of the joint of the subscript for n-1. For example the number n+1 of joint z axis is z_n. This is the simple rule allows us to quickly define all the joints of the shaft. For rotary joints. For the rotation of the z axis (Angle) is the joint variables. For sliding joints, along the z axis of the connecting rod length is joint variables represent as d.

• 2.

  Most of the joint does not necessarily parallel or intersecting in reality (Jing Jun, 2008). As a result, the z axis is usually a slash, but there is always a shortest distance and vertical line; It is orthogonal to the arbitrary two diagonal lines. Usually in vertical direction is defined on the x axis of local reference frame. Adjacent joints between the common normal don't intersect or collinear, therefore, the position of two adjacent coordinate system origin may also not in the same place. In order for all the joints specified x axis.

• 3.

  If two joint z axis parallel, between them there are many common normal lines. At this time can be selected from the previous joint the common normal of collinear any a common normal, can simplify the model to do so.

• 4.

  If it is the intersection of two adjacent joint z axis, so between them, there is no common normal (or perpendicular distance is zero). At this moment can be perpendicular to the axis of the two constitutes a straight line is defined as the x axis plane. In other words, the common normal is perpendicular to the contains two axis plane straight line, it is equivalent to choose the direction of the cross product of two z axis as the x axis. This method can simplify the model.

Transform a coordinate system to the next one, can use the steps in turn: