Invariant Hybrid Force/Position Control of a Velocity Controlled Robot with Compliant End Effector Using Modal Decoupling

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Abstract
A new method based on modal decoupling is proposed for the hybrid force/position control of a robot during constrained mo tion. The robot is equipped with an internal velocity controller, i.e., the control commands issued by the hybrid force/position controller are desired (joint) velocities. This velocity controller is supposed to be very stiff or of high bandwidth, as is the case for most commercial industrial robots. As a result, the robot dynamics are negligible compared to the contact dynam ics, which have low frequency because passive compliance is concentrated in the robot end effector. The new method incor porates results from the control of flexible robots, to damp out undesired vibrations of the end effector through active control. The contact dynamics during constrained motion are de scribed by a MIMO system: the inputs are the joint velocities; the outputs are the forces and torques measured by the wrist sensor. Using modal decoupling, this MIMO system is decom posed into its vibrational modes, i.e., a set of decoupled SISO systems. These modes are either unconstrained real vibrational modes, or constrained, and hence degenerated modes. Both are easy to control to satisfy the overall requirements of desired contact forces and vibration free motion of the manipulated object. Attention is paid to the invariance of the approach; i.e., the resulting robot behavior is independent of the choice of reference frame and the choice of units. The article contains experimental results of free space and partially constrained motion. Both confirm the theoretical re sults. 2. Usually, forces and torques are measured using strain gauges or dis placement transducers. 3. In practice, the effect of w g can easily be compensated for, by using a gravity model of the MO. 4. Proportional damping is an invariant concept; i.e., α and β are indepen dent of the representation of C, M and K, but few real systems exhibit this characteristic. Nevertheless, the assumption of proportional damp ing is frequently used for the ease of mathematical treatment. Indeed, Caughey (1960) showed that proportional damping is a sufficient—but not a necessary—condition for the existence of real modes. 5. Mathematically, this is easily verified using the positive definiteness of M and K, and the positive semidefiniteness of projection operator JJ †M. 6. In practice, such discontinuities are avoided by proper trajectory genera tion, but here they are deliberately introduced.

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