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| Learn More | Kinematics and Dynamics: Nonlinear Dynamics |
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| Should we be concerned with Nonlinear Dynamic Behaviors? |
| When do we have to be concerned? |
| What does it all mean? |
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| Should we be concerned with Nonlinear Dynamic Behaviors? |
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Even apparently simple systems can result in non-linear equations. Most undergraduate courses introduce the simple pendulum (left) as a linear system that is easy to deal with.
In reality, it is a slightly non-linear system, that is very closely linear only when a small angle approximation to that system is valid. Robotic Systems can exhibit dynamic behaviors that are far more complex. |
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From a control point of view, these nonlinearities are a computational problem, and in serial robots this translates into difficulties in creating an accurate dynamic model of the robotic system.
While several efforts have been made in this area, much work remains to be done, particularly with the advent of more and more complex robotic systems. As a result, most robotic operations are restricted to low speed movements that allow the control system to "keep up" with the nonlinearities and correct for them, but as higher and higher speeds are desired, the nonlinear dynamics of robotic systems will become increasingly important.
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| When do we have to be concerned? |
Nonlinear behaviors are readily apparent in simulation, although the effects of friction and actuator resistance in joints tend to make these behaviors more difficult to observe in laboratory settings.
Using a 6-DOF Puma robot as a model, a dynamic model can demonstrate dramatic nonlinear behaviors. For convenience, the puma joints were defined in the following manner: (This convention is adhered to throughout the remaining discussions).
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Joint 1 is vertical in the shoulder, joint 2 is horizontal through the shoulder, and joint 3 is in the elbow. |
Joints 4, 5 & 6 are in the wrist with 4 and 6 initially concurrent along the forearm and 5 perpendicular to 4 and 6.
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| The scenario is that the joints have been released with the arm at an extended position, allowing the arm to fall naturally. From this simulation, several features can be noted: |
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- First, the plot in the upper left hand corner is of joint position (in radians). Note that the purple joint (joint 4 located in the wrist), rapidly rotates enough so that it leaves the plot window. The second plot in the top center is of the joint velocities. Several sharp spikes occur in the velocities, indicative of sudden and dramatic velocity changes.
Finally, in the top right is a plot of the joint accelerations. It too shows spikes due to rapid and dramatic changes in the torque applied to the joints by the links outbound from them. All three plots are synchronized with the motion of the puma.
- The pattern is clearly nonperiodic, although like most chaotic systems, there are periods of quasi-periodicity in certain joints. If you observe the movie a few times, you will notice that the spikes in the plot windows tend to correspond with distinct changes in the motion of the arm, or the passage of the arm through singular configurations.
- Get a Dynamic Simulation Movie
- The motion of the arm begins as if it were a single pendulum, until the should link passes horizontal. At this time, the elbow begins to move away from the camera, applying a torque to the elbow joint.
This torque causes the first spike in the velocity and acceleration windows (corresponding roughly with the dramatic motion of joint four in the angular position window), and represents the onset of compound pendulum motion.
This movement leads to the passage through a workspace and mathematical singularity that occurs when the elbow reaches full extension.
- Passage through that singularity changes the torque transmitted to the forearm link again changes, altering the arm's motion again. The end-effector begins moving upwards (the forearm link rotates about its center of mass) while the robot begins turning about joint 1.
- The system then appears to stabilize into a period of quasi-stability, but during this period the arm remains very close to the elbow singularity point.
This period ends when the arm again begins to fall (on the other end of the pendulum swing now) first as a simple pendulum, and then as a compound pendulum leading to the second set of spikes in the velocity and acceleration windows.
- At this point the motion continues as a compound pendulum throughout the remainder of the movie, although the spikes are not as dramatic as they were previously.
- Finally, this model does not include the very real joint resistance effects present in real robotic systems, but the nonlinear dynamics are present within the system.
A more detailed study of this motion can be made with the movies of each individual component provided below.
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| That's neat, but what does it all mean? |
Every time there is a spike to the robotic system in terms of a dramatic change in position, velocity, or acceleration, there is a corresponding loss of precision in the system's task due to the shock to the system.
Furthermore, as operational speeds are increased, these shocks become more powerful and the stability of control systems are challenged by the increasing importance of nonlinear dynamic effects.
While seeing the wild motions of the puma pictured in the simulation is unlikely, even momentary errors in the control of robotic systems can have catastrophic results. To improve the speed and the performance of robotic systems a greater understand of nonlinear dynamics and nonlinear controls will be crucial.
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