VIBRATION CONTROL OF FLEXIBLE
BASE MANIPULATORS
Combined Position and Base Vibration Control Scheme
Interaction Force/Torque Predictions and Comparisons with Measured Lab Data
Nonlinear Interaction Force and Torque Effects
Description of Proposed Research and Expected Contributions
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A rigid (micro) robot mounted serially to the tip of a long, flexible (macro) manipulator is often used to increase reach capability, but flexibility in the macro manipulator can interfere with positioning accuracy. Macro/micro manipulators are desirable for certain uses, because the macro manipulator can provide long reach capability by moving the robot to the general area of interest where it can be used for fine-tuned positioning. For example, in the nuclear waste industry macro-micromanipulators have been used by the Dept of Energy to remove nuclear waste from underground storage tanks and in space applications, where long reach capability is needed but weight is crucial.
Since the macro links are long and lightweight, they are susceptible to vibrations, either induced by movement of the robot arms itself or by external disturbances (Figure 1). This research considers a rigid robot attached to a flexible but unactuated base (Figure 2) to study a control scheme to achieve accurate positioning of the micro-manipulator combined with enhanced vibration damping of the macro manipulators. The base motion is similar to that due to vibration at the tip of a flexible macro manipulator with locked joints. The assumption is that six degree-of-freedom base vibration is possible, making this work applicable to a wide variety of problems.
The objective of this research is to develop a combined position and active vibration control scheme for a flexible base manipulator. If the inertial interaction forces and torques acting between the robot and its base can be determined, they can be used to damp the vibration.
Why use the rigid robot to damp the vibrations? The macro manipulator actuators are not the best option due to bandwidth limitations – they typically are designed to move very slowly due to the large masses/inertias involved. In addition, non-collocation of the actuators, which would usually be at the joints, and the macro end point vibration, can be a problem. The use of the micro manipulator’s actuators has proven to be a promising area – they are close to the vibration, the links are rigid so only rigid body dynamics are involved, and they can respond very quickly to create large inertial forces and torques. It is also advantageous because the forces/torques can be applied directly to the tip of the macro-manipulator, where the flexible mode displacements are at a maximum (at least for the first few modes).
Why not use end point positioning? I suppose you could if you have an accurate end-point sensor, but a more typical (and cheaper) scenario uses optical encoders at the joints to measure joint rotations. If the links are rigid, the end point position can easily be determined from the robot’s geometry. If the links are flexible, as in the case of the macro links, the end point position is a combination of the flexible and rigid coordinates, thus it is very difficult to accurately predict and control. However, if the tip of the macro-manipulator is maintained at its equilibrium position, only the rigid coordinates are needed to describe the end point position.
In this work, the rigid robot control scheme must perform the dual task of damping unwanted base vibration (macro-manipulator vibration) while providing position control of the end effector. Initial work has been done on modeling a flexible base manipulator and modal testing of a flexible manipulator.
Position and Base Vibration Control Scheme
The control variables considered here are the interaction forces and torques. The goal is to command the rigid robot motion to provide the interaction forces and torques to damp the vibration. This scheme takes the form shown below in Figure 3:
or, it may be more effective to specify the motion in the form of desired joint position.
For this research, the assumption is that the requirement will be for a specified end point position so the focus is maintaining a steady position while damping vibrations. The vibrations may be induced by initial conditions on the macro-manipulator (moving it into place) or due to a disturbance. The inverse kinematics/damping performance index check will choose the best joint space configuration for damping performance given a desired end point position.
A general flex compensator of the form:
is proposed for this research where ID is an inverse dynamics algorithm that
estimates the significant rigid robot control and coupled rigid/flex dynamic
parameters. For more detailed
information on the control scheme, including simplified simulations and 3 DOF
simulations results, see control summary. It was determined through further research
that the inertia effects are most significant and are the only terms that need
to be included provided 1) the robot is not operating near or about an inertial
singularity (described in Performance Index
section) and the appropriates control gain limits are used to limit the
prescribed joint motion (described in Nonlinear Effects
section). For a demonstration of this
scheme in action, see the Experimental Results
section.
Interaction
Force/Torque Predictions and Comparisons with Measured Lab Data
The goal of this work is to verify predicted interaction forces and torques, determine if we are modeling them adequately, and analyze these effects in order to determine how to more effectively control these interactions. The equations of motion take the form:
q
represents rigid joint angles and q represents flexible base motion. We are particularly interested in the
inertia effects BF and Bt0, but the force/torque sensor picks up all of the
effects. Thus, in order to better isolate
the interactions due to the rigid robot (which are the effects that are
directly controllable), I braced the flexible
beam as shown in Figure 4. An ATI force/torque sensor (Delta) was mounted
between the robot and the flexible base (beam in this case) and used to compare
predicted interaction forces/torques to measured interaction forces/torques due
to the rigid robot’s movement. Input
signals were sent to the 3 DOF rigid robot, SAMII
(small articulated manipulator, Jr) similar to the motion expected during
active inertial damping.
The first part of this work involved actuating each
joint independently, with sinusoidal inputs at 1 Hz, slightly lower than the
first natural frequency of the flexible link.
The model for the inertia forces is:
An example of predicted and measured forces due to joint 1's motion, along with predicted nonlinear rigid effects (NF), is shown in figure 5. These tests were performed for each term in the inertia matrix and gave good results, which verifies that we can get good inertial forces in limited degrees of freedom. The next activity involved extending this to multiple degrees of freedom. Figure 6 shows predicted and measured interaction forces due to joints 1, 2, and 3 actuating simultaneous (at 1, 1.5, 2 Hz, respectively). In configurations where we have predicted good inertial forces, the results are clean (results for torques are similar). Unfortunately, in other configurations the relationship becomes more complicated and the ability of the scheme to accurately predict the interaction forces by considering the inertia effects alone does not work well with unconstrained multi-degree of freedom actuation and we begin to see more higher frequency phenomena (Figure 7). In certain configurations, nonlinear effects can become significant compared to the inertia effects. For more detail on this subject, along with more test results, see force/torque testing. A method of addressing this is discussed below in the Nonlinear Iteraction Force/Torque Effects Section.
Performance Index
One inherent problem with inertial damping schemes is that there are locations in the workspace at which the micromanipulator’s ability to create inertial forces and torques is diminished or nonexistent in one or more degrees of freedom. This research includes developing a performance index to evaluate the robot’s ability to effectively damp vibrations in the base. One possibility is to compare the inertia terms associated with the flexible robot with those associated with the rigid robot.
For example, the ability of the anthropomorphic type robot to generate forces in three degrees of freedom can be evaluated by examining the determinate of the Bf matrix from the coupled equations of motion:
This can be visualized by plotting the determinant of (BfT*Bf) for varying joint angles as shown in Figure 8:
The regions with very low values represent locations in the workspace at which
the robot loses inertial damping capability in one or more degrees of
freedom. These will be termed “inertial
singularities”. The inertial
singularity regions are important for two reasons. First, the inertia matrices are inverted in the control scheme,
which becomes a numerical problem when the matrices are singular. The more important consideration, however,
is that these represent physical limitations in that an inertial force
or torque cannot be created in one of more degrees of freedom. These singularities are coincident with some
of the kinematic singularities plus additional dynamically singular
configurations. For this particular
example, the singularities at theta 3 = 0 degrees indicates a region where
joints 2 and 3 are aligned, indicating the inertia forces created by the two
joints are parallel. These also
coincide with the robot’s kinematic singularities, where the velocities
generated by the two links are parallel.
These are not a major concern since they would be normal operating
regions. The other regions of low
inertial damping performance correspond to a column of zeros in the matrix,
which indicates a location in the workspace where the motion of a joint cannot
create any inertial forces. In general,
these occur when the system center of mass is aligned along an axis of
rotation, rendering that joint ineffective.
These are, in general, different from the kinematic singularities
and are functions of the robot's joint space configuration.
This led to the realization that this measure of performance would be a good check of predicted inertial damping performance, which was the inspiration of the first block in the proposed scheme. The robot can then be commanded to the operational joint space position that provides the best predicted inertial damping performance. As an example, consider the anthropomorphic robot position with a specific desired end point position of [x y z]. This end point position may be reached by four different end point configurations, given by the inverse kinematics solutions. The two different configurations of joint 2 and 3 provide very different inertial damping performance as demonstrated in the figures below (the other two cases are the same with joint 1 rotated 180°).
The first joint configuration provides better inertial damping capability. The second figure above is a near-singularity position because the robot’s CG is near the z-axis, rendering joint 1 ineffective and thus unable to control vibration in the out-of-plane direction.
Nonlinear Interaction Force and Torque Effects
There are configurations where nonlinear effects can become large in a particular direction compared with inertial forces, and this was verified via lab testing. For forces, centrifugal effects are of concern; for torques, coriolis effects are the concern in certain configurations. Inertia forces vary throughout the workspace as shown in figure 6. Centrifugal and coriolis effects vary throughout the workspace also. As an extreme example, consider simulations of the interaction forces due to inertia, coriolis, and centrifugal effects in a workspace location where predictions indicated the nonlinear effects would be largest, shown in worst. An extreme example is shown in figure 10.
It is easy to see why this occurs if we consider the configuration (Figure 11). Most of the inertia force in the x direction is generated only from link 3, whereas the all three links contribute to centrifugal forces and hence become a problem in multi-DOF.
The nonlinear forces occur at a frequency twice that of inertia effects (since functions of cos2, or cos(2*wn)). If there happens to be a macromanipulator mode at 2 times the mode of interest, the nonlinear effects can be of great concern. Even if not, when the nonlinear forces dominate the interaction forces the scheme will clearly not work based on controlling inertia effects only.
The solution is two-fold. First, it is important to use the performance index to ensure the robot operates in better regions for inertial damping. Second, it is also important to establish limits on control gains to limit the magnitude of the joint motion. The recommended limits on control gains for the ith degree of freedom are:
Rationale for these limits is only briefly introduced and more detail will be added at a later date. The lower limit, z and w represent damping ratios and natural frequencies of the maximum mode of concern in the ith degree of vibrational freedom. This lower limit was based on ensuring total energy will be removed from the system in the worst-case scenario that a higher mode of the flexible system becomes directly excited by the vibration controller. The upper limit is based on ensuring the magnitude of the rigid joint accelerations will be larger than the square of the joint velocities, thereby further ensuring limited significance of the nonlinear effects. w and X are the natural frequency and maximum expected amplitude of vibration of the lowest mode of interest. B and A are user established minimum allowable average inertia and maximum allowable arm amplitudes.
For more information on this topic, see nonlinear effects.
Ga Tech's current micro/macro manipulator test bed consists of a 20' aluminum beam suspended from an I-beam in the laboratory with a 5' extension for a second macromanipulator link. SAMII (Small Articulated Manipulator II)/wrist, a 6 degree of freedom hydraulically operated micromanipulator is attached to the tip of the second link. The complete setup allows investigation into micro/macro robot interaction for multiple degrees of freedom of vibration. Base acceleration is measured and used in an additional feedback control loop to command SAM II's joints to cancel the beam vibration.
This is a diagram
showing the range of motion of the first three joints of the rigid robot:
Another flexible link robot our lab is designated RALF (Robotic Arm, Large and Flexible). It consists of two flexible links, each approximately ten feet long. It has been used many times in the past on various research projects involving the dynamics and control of flexible robot arms.
To see the damping capability of this control scheme, see the clip below. The robot is initially under joint PID control when a disturbance initiates vibration in the flexible beam. The robot is then vibration freely – although it is able to keep joint position control, its base is vibration too much. The damping controller described above is turned on about 3 seconds into the clip, reducing the vibration significantly.
On the other hand, another joint configuration that gives the same end point position is unable to damp the vibration as well. This is an extreme case, but demonstrates that a check of predicted damping capability can make a tremendous difference if this sort of control scheme is used. Notice this location is not at a kinematic singularity.
This is a plot of the free vibration response of the system – very low damping:
Compare this with the damped response (the damping controller is turned on at approximately two seconds in this example).
Combined joint position control and vibration control also works well, as shown in the plots below. In this case, the robot is commanded from one position to another in the workspace and the measured base vibration with and without the vibration controller are compared below.
Compare the same movement with the vibration controller:
Active Vibration Control of a Flexible Base Manipulator
Ph.D. Thesis Proposal
Presented to the Faculty
18 Sept 2001
By
Lynnane E. George
A rigid (micro) robot mounted serially to the tip of a long, flexible (macro) manipulator is often used to increase reach capability, but flexibility in the macro manipulator can interfere with positioning accuracy. A rigid manipulator attached to a flexible but unactuated base will be used to study a scheme to achieve accurate positioning of the micro- manipulator combined with enhanced vibration damping of the macro manipulator. Inertial interaction forces and torques acting between the robot and its base will be determined and used to damp the vibration.
First, a complete derivation of the coupled equations of motion will be developed and analyzed for typical six degree-of-freedom robot manipulators mounted on flexible bases. A performance index will be developed to predict the ability of the robot to damp vibrations throughout its usable workspace and will be implemented in parallel with a position control scheme. The ability of the method to provide combined position and base vibration control will be demonstrated via Matlab simulations. Finally, the control scheme will be demonstrated experimentally.
For a complete report on this proposed area of research, see Proposal
Expected
Contributions
This proposed research is to develop a control scheme to provide combined position and vibration damping of a macro micromanipulator. The configuration of a rigid manipulator attached to a flexible base was presented as a similar configuration that will be used to develop the control scheme. The goal is to achieve accurate positioning of the micromanipulator combined with enhanced vibration damping of the macro manipulator. Inertial interaction forces and torques acting between the robot and its base will be investigated and used to damp the vibration.
Expected contributions of this work are:
1. Extension of the macro/micromanipulator control problem to multiple degrees of freedom by considering the analogous problem of a flexible manipulator mounted on a fully flexible base
2. Development of a performance index that will predict the ability of the micromanipulator to effectively minimize base vibrations using an inertial damping scheme
3. Developments of a control scheme that will provide active base vibration damping in parallel with position control throughout the usable workspace. This will include the use of the performance index to maximize the available vibration damping capability given a desired end point position
4. Verification of the above control scheme via simulation
5. Verification
of the above control scheme via experimental verification
Committee
The reading committee for this
work is:
Dr. Wayne Book (my advisor) – Mechanical
Engineering
Dr. Anthony Calise –
Aerospace Engineering
Dr.
Aldo Ferri – Mechanical Engineering
Dr. William Singhose – Mechanical
Engineering
Dr. David Taylor
– Electrical Engineering
Additional
Information
Background on active damping control schemes
Modal testing and model verification
Presentations
The following are some presentations I have given on macro/micromanipulator vibration control. The presentation will be rotated initially so click once on “rotate view clockwise.”
Active Vibration Control of a Flexible Base Manipulator – Ph.D. Defense, 10 July 2002
Current Progress on Research Activities – IMDL Group Presentation, 17 April 2002
Active Vibration Control of a Flexible Base Manipulator – Ph.D. Proposal presentation, 18 Sept 2001
Using
Incremental Optical Encoders to Estimate Joint Velocities, 19 April 2001
This is a side project I've been working on in conjunction with a class I took Spring 2001. I use US Digital E3 incremental encoders to get position measurements for each joint of SAMII. These are very accurate and, in conjunction with a quadrature pulse counter, give precise position measurements (8192 quadrature pulses/rev, or a resolution of .000767 rads). Velocity measurements can be obtained simply by counting the number of pulses that occur over a fixed time interval. Since the control loop runs at 200 Hz, velocity estimates are based on the change in position over .005s. This "pulse counting" method works well at relatively high speeds (> 1 rad/s) but at lower speeds it does not because few pulses occur during the time period of interest. One way around this is to sample fast enough to measure the period of a single square wave output (note this is more accurate than measuring the period between your quadrature pulses because they occur four times as fast). Although this "period counting" method works well at low velocities, it doesn't work as well at high velocities unless you can sample extremely fast. So, is there a way to best estimate velocities over a wide range of operating speeds?
One way around this is to compare the uncertainty associated with each method and determine a transition speed; "pulse counting" is better above this point and "period counting" better below this point. Here is a presentation summarizing the concept and some experimental results. If anyone else has run into the problem and has discovered a good solution, please let me know.
Macro/Micromanipulator Vibration Control: Limitations and Issues – IMDL Group Presentation, 12 Feb 2001
Macro/Micromanipulator Vibration Control: Introduction and Overview – IMDL Group Presentation, 1 Nov 2000
Reports and Papers
Active Vibration Control of a Flexible Base Manipulator, Ph.D. Dissertation, July 2002
Active Vibration Control of a Flexible Base Manipulator, Ph.D. Proposal Report, Sept 2001
Accepted
Papers:
"Inertial Vibration Damping of a Flexible Base Manipulator." Lynnane E. George and Wayne J. Book. Sixth International Conference on Motion and Vibration (MOVIC 2002), The Dynamics, Measurement and Control Division of the JSME, August 20-23, 2002, Saitama, Japan.
"Practical Implementation of A Dead Zone Inverse on a Hydraulic Wrist." Joel D. Fortgang, Lynnane E. George, and Wayne J. Book. Fluid Power Systems and Technology Division, 2002 ASME International Mechanical Engineering Congress and Exposition, November 17-22, 2002, New Orleans, Louisiana.
" Inertial Vibration Damping Control for a Flexible Base Manipulator." Lynnane E. George and Wayne J. Book. Symposium on Active Control of Vibration and Noise, 2002 ASME International Mechanical Engineering Congress and Exposition, November 17-22, 2002, New Orleans, Louisiana.
Hardware and Software Links
Links to some of the hardware used for this research:
Moog – hydralic servovalves
Micro Precision Textron – vane type rotary hydraulic actuators
US Digital – optical encoders
PCB Piezotronics – accelerometers and support equipment
Motorola – microprocessors
Acromag – VME I/O Boards
ATI Industrial Automation – six axis force/torque sensor
Software in support of this research was provided under following companies' University Programs:
Wind River Systems - VX Works real-time operating system environment
Real-Time Innovations, Inc. - Control Shell version 7.0, a software framework for real-time systems software development