## 2 Point-to-Point Joint Motion via Multiple Points with Constrained Velocities
### Point-to-Point Joint Motion  
PTP (point to point) motion of the industrial robot is suitable for spot welding, handling, loading and unloading, circuit board insertion and other operations. In PTP motion mode, the end-effector moves from a start point to an end point. PTP motion only needs to specify positions and orientations of end-effector at the start point and end point, but the trajectory and orientations between the start point and end point are not specified or required.  
When the end-effector moves in PTP mode, joints rotates normally in PTP mode as well. As shown in Figure 1, take joint 1 as an example. When the end-effector wants to move from start point \(P_{a}\) to end point \(P_{b}\), joint 1, which is one of the six joints, needs to rotate from the start point \(A\) to end point \(B\). In general, other joints rotate similarly. Finally, rotations of six joints work together to form the movement of the end-effector.  
There is a corresponding functional relationship between the position and orientation of the end-effector and the angular displacement of each joint, as shown in formula (1). There is also a corresponding functional relationship between the end-effector velocity and the angular velocities of each joint, as shown in formula (2).  
\[f(\theta_{i})=[x\ y\ z\ \alpha\ \beta\ \gamma]^{T} \tag{1}\]  
\[\mathbf{f}\cdot\mathbf{\omega}_{i}^{T}=[v_{x}\ v_{y}\ v_{z}\ \omega_{x}\ \omega_{y}\ \omega_{z}]^{T} \tag{2}\]  
In formula (1), \([x\ y\ z]\) and \([\alpha\ \beta\ \gamma]\) are position and orientation of end-effector, \(\mathbf{\theta}_{i}\) are angular displacements of each joint. In formula (2), \([v_{x}\ v_{y}\ v_{z}]\) and \([\omega_{x}\ \omega_{y}\ \omega_{z}]\) are linear velocity and angular velocity of end-effector, \(\omega_{i}\) are angular velocities of each joint, and \(\mathbf{J}\) is the velocity Jacobian matrix.\(f(\theta_{i})\) and \(J\) vary in different robots. Formulas (1) and (2) are called forward motion functions, by which the end-effector motion can be obtained from joint motion.  
According to the position, orientation and speed of the end-effector, angular displacement and velocity of each joint can be obtained, which is called inverse kinematics, as shown in formulas (3) and (4) below. The speeds of the end-effector at the start and end points are zero in PTP movement so that angular velocities at the start and end points of  
Figure 1: Point-to-Point movement.  
each joint are also zero. As seen in Figure 1, the angular velocities of joint 1 at the start and end points are all equal to zero, \(\omega_{a}=\omega_{b}=0\).  
\[\boldsymbol{\theta}_{i}^{T}=f^{-1}(\boldsymbol{\theta}_{i})\cdot[x\ y\ z\ \alpha\ \beta\ \gamma]^{T} \tag{3}\]  
\[\boldsymbol{\omega}_{i}^{T}=J^{-1}\cdot[v_{x}\ v_{y}\ v_{z}\ \omega_{x}\ \omega_{y}\ \omega_{z}]^{T} \tag{4}\]  
On the basis of angular displacement and velocity constraints at the start point and end point of each joint, motion plannings of each joint can be carried out and these joint motion plannings are finally synthesized into the end-effector motion, as shown in Figure 2.