Task-space Separation Principle by Paolo Tommasino

Author:Paolo Tommasino
Language: eng
Format: epub, pdf
Publisher: Springer Singapore, Singapore

(3.33)

subject tothe task-space dynamic:

(3.34)

where has been expressed as in (3.15);

the boundary conditions:

(3.35)

The advantage of this formulation, in presence of kinematic redundancy, is that the optimal control problem is solved on a lower dimensional space (the task space) rather than on the full configuration space of the manipulator.

We parametrized the above problem with B-splines polynomials as proposed in [31, 32]. In particular, task-space velocities were parametrized with fourth-order B-Splines and task-space forces with second-order B-splines. Spline parameters were optimized by using sequential quadratic programming (fmincon Matlab function).

Results

In this section, we compare the -PMP with the optimal control approach presented in the previous section and that for brevity we will indicate as . As for the velocity-resolution control, both models solve the centre-out task in the four damping and stiffness conditions reported previously and with the same parameter settings. For the -PMP, the task-space spring was set as in Eq. 3.32. The final time () for the model was set equal to the target time s.

The task-space paths predicted by the two models, in the low stiffness condition, are shown in Fig. 3.14. The high stiffness condition presents similar results, and therefore, it is not reported. When the damping matrix W is anisotropic, the -PMP generates curved task trajectories with different paths for outbound and inbound movements (task-space hysteresis). This is because the -PMP uses only an elastic (see Table 3.1(3)) and therefore, differently from the OLVRC, does not compensate for the task-space damping B with the result that the two task-space velocity components, and , have different magnitudes along the trajectory (see Eq. (3.36)). The instead, optimizes the task-space force according to the task-space dynamic and therefore produces straight-line paths independently of the damping matrix W.

Fig. 3.14Task-space paths simulated with the isotropic (a)–(b) and the anisotropic (c)–(d) joint damping W

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