# Stiffness based trajectory planning and feedforward based vibration suppression control of parallel robot machines

##### Li, Ming (2014-12-08)

Väitöskirja

Li, Ming

08.12.2014

Lappeenranta University of Technology

Acta Universitatis Lappeenrantaensis

**Julkaisun pysyvä osoite on**

http://urn.fi/URN:ISBN:978-952-265-703-9

#### Tiivistelmä

The dissertation proposes two control strategies, which include the trajectory planning

and vibration suppression, for a kinematic redundant serial-parallel robot machine, with

the aim of attaining the satisfactory machining performance.

For a given prescribed trajectory of the robot's end-effector in the Cartesian space, a set

of trajectories in the robot's joint space are generated based on the best stiffness

performance of the robot along the prescribed trajectory.

To construct the required system-wide analytical stiffness model for the serial-parallel

robot machine, a variant of the virtual joint method (VJM) is proposed in the dissertation.

The modified method is an evolution of Gosselin's lumped model that can account for the

deformations of a flexible link in more directions. The effectiveness of this VJM variant

is validated by comparing the computed stiffness results of a flexible link with the those

of a matrix structural analysis (MSA) method. The comparison shows that the numerical

results from both methods on an individual flexible beam are almost identical, which, in

some sense, provides mutual validation. The most prominent advantage of the presented

VJM variant compared with the MSA method is that it can be applied in a flexible

structure system with complicated kinematics formed in terms of flexible serial links and

joints. Moreover, by combining the VJM variant and the virtual work principle, a systemwide

analytical stiffness model can be easily obtained for mechanisms with both serial

kinematics and parallel kinematics. In the dissertation, a system-wide stiffness model of a

kinematic redundant serial-parallel robot machine is constructed based on integration of

the VJM variant and the virtual work principle. Numerical results of its stiffness

performance are reported.

For a kinematic redundant robot, to generate a set of feasible joints' trajectories for a

prescribed trajectory of its end-effector, its system-wide stiffness performance is taken as the constraint in the joints trajectory planning in the dissertation. For a prescribed

location of the end-effector, the robot permits an infinite number of inverse solutions,

which consequently yields infinite kinds of stiffness performance. Therefore, a

differential evolution (DE) algorithm in which the positions of redundant joints in the

kinematics are taken as input variables was employed to search for the best stiffness

performance of the robot. Numerical results of the generated joint trajectories are given

for a kinematic redundant serial-parallel robot machine, IWR (Intersector

Welding/Cutting Robot), when a particular trajectory of its end-effector has been

prescribed. The numerical results show that the joint trajectories generated based on the

stiffness optimization are feasible for realization in the control system since they are

acceptably smooth. The results imply that the stiffness performance of the robot machine

deviates smoothly with respect to the kinematic configuration in the adjacent domain of

its best stiffness performance.

To suppress the vibration of the robot machine due to varying cutting force during the

machining process, this dissertation proposed a feedforward control strategy, which is

constructed based on the derived inverse dynamics model of target system. The

effectiveness of applying such a feedforward control in the vibration suppression has

been validated in a parallel manipulator in the software environment. The experimental

study of such a feedforward control has also been included in the dissertation. The

difficulties of modelling the actual system due to the unknown components in its

dynamics is noticed. As a solution, a back propagation (BP) neural network is proposed

for identification of the unknown components of the dynamics model of the target system.

To train such a BP neural network, a modified Levenberg-Marquardt algorithm that can

utilize an experimental input-output data set of the entire dynamic system is introduced in

the dissertation. Validation of the BP neural network and the modified Levenberg-

Marquardt algorithm is done, respectively, by a sinusoidal output approximation, a

second order system parameters estimation, and a friction model estimation of a parallel

manipulator, which represent three different application aspects of this method.

and vibration suppression, for a kinematic redundant serial-parallel robot machine, with

the aim of attaining the satisfactory machining performance.

For a given prescribed trajectory of the robot's end-effector in the Cartesian space, a set

of trajectories in the robot's joint space are generated based on the best stiffness

performance of the robot along the prescribed trajectory.

To construct the required system-wide analytical stiffness model for the serial-parallel

robot machine, a variant of the virtual joint method (VJM) is proposed in the dissertation.

The modified method is an evolution of Gosselin's lumped model that can account for the

deformations of a flexible link in more directions. The effectiveness of this VJM variant

is validated by comparing the computed stiffness results of a flexible link with the those

of a matrix structural analysis (MSA) method. The comparison shows that the numerical

results from both methods on an individual flexible beam are almost identical, which, in

some sense, provides mutual validation. The most prominent advantage of the presented

VJM variant compared with the MSA method is that it can be applied in a flexible

structure system with complicated kinematics formed in terms of flexible serial links and

joints. Moreover, by combining the VJM variant and the virtual work principle, a systemwide

analytical stiffness model can be easily obtained for mechanisms with both serial

kinematics and parallel kinematics. In the dissertation, a system-wide stiffness model of a

kinematic redundant serial-parallel robot machine is constructed based on integration of

the VJM variant and the virtual work principle. Numerical results of its stiffness

performance are reported.

For a kinematic redundant robot, to generate a set of feasible joints' trajectories for a

prescribed trajectory of its end-effector, its system-wide stiffness performance is taken as the constraint in the joints trajectory planning in the dissertation. For a prescribed

location of the end-effector, the robot permits an infinite number of inverse solutions,

which consequently yields infinite kinds of stiffness performance. Therefore, a

differential evolution (DE) algorithm in which the positions of redundant joints in the

kinematics are taken as input variables was employed to search for the best stiffness

performance of the robot. Numerical results of the generated joint trajectories are given

for a kinematic redundant serial-parallel robot machine, IWR (Intersector

Welding/Cutting Robot), when a particular trajectory of its end-effector has been

prescribed. The numerical results show that the joint trajectories generated based on the

stiffness optimization are feasible for realization in the control system since they are

acceptably smooth. The results imply that the stiffness performance of the robot machine

deviates smoothly with respect to the kinematic configuration in the adjacent domain of

its best stiffness performance.

To suppress the vibration of the robot machine due to varying cutting force during the

machining process, this dissertation proposed a feedforward control strategy, which is

constructed based on the derived inverse dynamics model of target system. The

effectiveness of applying such a feedforward control in the vibration suppression has

been validated in a parallel manipulator in the software environment. The experimental

study of such a feedforward control has also been included in the dissertation. The

difficulties of modelling the actual system due to the unknown components in its

dynamics is noticed. As a solution, a back propagation (BP) neural network is proposed

for identification of the unknown components of the dynamics model of the target system.

To train such a BP neural network, a modified Levenberg-Marquardt algorithm that can

utilize an experimental input-output data set of the entire dynamic system is introduced in

the dissertation. Validation of the BP neural network and the modified Levenberg-

Marquardt algorithm is done, respectively, by a sinusoidal output approximation, a

second order system parameters estimation, and a friction model estimation of a parallel

manipulator, which represent three different application aspects of this method.

##### Kokoelmat

- Väitöskirjat [851]