Review: Neural Network Control of Robot Manipulators; Frank L. Lewis; 1996

Sub-topics Control Theory/ System Theory.  Closed loop controllers.  Adaptive Control.  PID Neural Networks. Dynamics in Robotics.

Control Theory Systems that use feed-back loops to regulate a control parameter about a set value. For example: The core temperature of the human body must be regulated about a set value, and there are biological sensors and actuators that allow this. Also consider how you adjust the tap in the shower to regulate the temperature of the water to a comfortable temperature. Also: Notice how your most comfortable temperature normally rises during your shower and you adjust the tap accordingly.

Below: Adaptive control is important where tuning parameters are uncertain.

Proportion + Integration + Differentiation. Closed loop system to maintain a small error e between the set value and the actual value for the control parameter. The parameters K v, K i, K d are tuned for the particular task as they represent the strength of the effectors. PID

Neural Networks in Control Theory Rejected. Why? - The author claims previous attempts to introduce Neural Networks into Control Theory have lacked theoretical proofs and repeatable design algorithms. The challenges: providing repeatable design algorithms; online learning algorithms (no offline tuning); demonstrating closed loop trajectory following, computing various weight tuning gradients and demonstrate the weights remained bounded given unmodelled dynamics. Interested in NN because of function approximation property – which fails to hold for adaptive control.

Neural Network Architecture 3 layered architecture. Model free. Continuous Differentiable activation function. Equation 1 shows f(x) the true function and Equation 2 shows approximation, ^f(x).

Dynamics in Robotics Once we know the target position of the end effectors (having used inverse kinematics), dynamics deals with what forces are required to perform that action, ie; of moving the joints along a trajectory. The matrices : M is the inertia, V is the centripetal/coriolis, G is the gravity and F is the friction, whilst is the input torque and d represents bounded unknown disturbances.

Calculating the trajectory. The joint force vector is q R n for n joints. Quantities are imperfectly known and difficult to determine

Robotic Controller Structure How about try to estimate unknown information using adaptive control? We have ˜f is the tracking error which is found using adaptive control and v(t) is the robustifying signal to compensate for unstructured/ un-modelled dynamics.

Use NN instead of Adaptive Control

Parameter Tuning Not just neural network weights need to be “tuned”. Tuned by backpropagation as in Table and using 12.

NN controller proven to track trajectory. Trajectory error bounded. Outer loop is a PD controller and K v represents the PD gains. Neural Network weights are initialised to zero. The larger the NN, the smaller the PD gains. Removal of the NN causes the system to become a PD Controller. But errors due to parameter-uncertainty will be high too.

The more nodes in the network, the more difficult to implement because each node needs an 'integrator'. Option to incorporate a partitioned Neural Networks to enhance controller structure and to increase the speed of the weight tuning algorithm.

Comparison of Adaptive Control and NN, given un-modelled dynamics Neural Network Controller. Left: Ideal Conditions; Right Adaptive Controller

Hebbian Tuning Hebbian tuning provides very good closed loop performance. Works well because of ^V update and e modification second terms.

Application: Force and position controllers Grinding, Milling and Surface Finishing.