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4.1 Joint Interpolated Trajectories
In joint interpolated trajectory for a robot arm showed the following considerations:
1. When picking up an object, motion of the hand must be directed away form an object otherwise the hand may crash into the supporting surface of the object.
2. If a lift of point along the normal vector to the surface out form the initial position and if further specify the time required to reach this position then the the sped can be control at which the object to be lifted.
3. The same set at lift of requirement for the arm motion set down point of the final position motion, so that the correct approach direction can be obtained and controlled.
4. From the above, there are four positions for each motion: Initial, lift off, set down and final. Refer figure 4.2 below.
5. Position constraints,
a. Initial position: velocity and acceleration are give (normally zero)
b. Lift off position: continuous motion for intermediate points.
c. Set down position: same as lift off position.
d. Final position: velocity and acceleration are given (normally zero)
6. Joint trajectories must be within the physical and geometric limits of each joint.
7. Time consideration:
a. Initial and final trajectory segments: time is based of approach of the hand to and from the surface and is some fixed constant based on the characteristic of the joints motors.
b. Intermediate and trajectory segments: time is based on maximum velocity and acceleration of the joints and the maximum of these times is used.
Figure 4.1 Position conditions for a joint trajectory.
Table 4.1 Constraints for planning joint interpolated trajectory
Initial position:
1. Position (given)
2. Velocity (given, normally zero)
3. Acceleration (given, normally zero)
Intermediate position:
4. Lift off position (given)
5. Lift off position (continuous with previous trajectory segment)
6. Velocity (continuous with previous trajectory segment)
7. Acceleration (continuous with previous trajectory segment)
8. Set down position (given)
9. Set down position (continuous with trajectory segment)
10. Velocity (continuous with next trajectory segment)
11. Acceleration (continuous with next trajectory segment)
Final position:
12. Position (given)
13. Velocity (given, normally zero)
14. Acceleration (given, normally zero)
The constraints of a typical joint trajectory are listed in table 4.1 above. Based on these constraints, with selecting a class of polynomial function of degree n or less such that the required joint position, velocity and acceleration at these knot points. Below are the seventh polynomials for each joint,
it is difficult to find its extreme and it tends have extraneous motion. An alternative approach is to split the entire joint trajectory into several trajectory segments so that different interpolating polynomials of a lower degree can be used in each trajectory segment.
4.1.1 Cubic Spline Trajectory (Five Cubics)
Cubic spline after several advantages. First, it is the lowest degree polynomial function that allow continuity in velocity and acceleration.
Second, low degree polynomial reduce the efforts of competition and the possibility of numerical instabilities.
The general equator of five cubic polynomial for each joint trajectory segment is,
The boundary conditions of trajectory segment polynomial must satisfy:
1. Position constraint at the initial, lift off, set down and final position.
2. Continuity of velocity and acceleration at all interpolation points.
The boundary conditions for a five cubic joint trajectory are shown in figure 4.2 below.
Figure 4.2 Boundary conditions for a cubic joint trajectory
The first and second derivatives of the polynomial with respect to real time are:
and
Given the position, velocities and acceleration at the initial and final positions, the polynomial equators for the initial and final trajectory segments [h1(t) and hn(t)] are completely determined using the position constraints and continuity conditions.
For the first trajectory segment, the governing polynomial equation is,
At t=0 satisfying the boundary conditions at this point,
At t=1 satisfying the position constraint at this position,
From which a13 is found to be
Then the first trajectory segment polynomial is determined:
For the last trajectory segment, the polynomial equation is,
At t=0 and t=1 satisfying the boundary conditions,
By this can obtain,
For the second trajectory segment, equation is
At t=0 satisfying the position constraint,
The polynomial equation becomes,
For third trajectory segment, the equation is
At t=0 satisfying the continuity of velocity and acceleration
Then the polynomial equation become
At t=1 the velocity and acceleration can be obtain
For the fourth trajectory segment, the equation is
At t=0 satisfying the position constraint and continuity of velocity and acceleration
The polynomial equation become
