Constructing the Optimization Problem

Any optimization problem has three main components:

• Objective function to be minimized or maximized.
• Optimization variables.
• Set of constraints.

A set of objective functions that can be used in the optimization problem are specified. This set will form the database for the formation of the final objective functions for some of the parameters using the task specification and the performance requirements.

Some of the criteria that can be used to form objective functions are:

• Work space.
• Manipulability.
• Speed.
• Accuracy.
• Power consumption of motors.

To form the objective functions, we need to find quantitative measures for the manipulator specification and the performance requirements. In some cases, a closed form expression is not available. In such cases, the simulation programs can be used to determine the required quantitative measure. For example, the maximum velocity is a function of most of the parameters (link lengths, masses, friction, motor parameters), but it is not easy to get a closed form expression for the velocity as a function of all of these parameters; therefore, the simulation program can be used to measure the maximum velocity for different values of these parameters.

In addition to these quantitative measures, there are some rules and assumptions that will be used to solve for some of the parameters, and to give guidance during the design cycle. Some of the assumptions we made to simplify the problem are:

• The robot type and the degrees of freedom are given.
• Only revolute and prismatic joints are considered.
• The links are uniform with rectangular or cylindrical cross section.
• There is a finite set of materials used to build the robot with known densities.
• There is a finite number of actuators and sensors with known specifications that can be used in the design.

Some of the rules that can serve as additional constraints are:

• Select the solution with equal link lengths or masses because this will simplify the manufacturing process (i.e., minimize the cost).
• Choose the feedback controls that give critically damped behavior ( ).
• Consider a minimum length for each link to satisfy some assembly and manufacturing constraints, such as actuator and sensors sizes.

Our strategy for solving this optimization problem will be to divide it into stages, at each stage solve for some of the parameters, then use the values obtained for these parameters in the following stage. The reason for choosing this strategy is that, some of the robot parameters must be determined before we can start solving for other parameters. For example, the robot type must be determined first. The other parameters are largely affected by the choice of the robot type. The selection of the robot type depends on the tasks and performance requirements. For the time being, we assume that the robot type is given, and later the selection of the robot type can be added to the system.

There are many algorithms for solving the optimization problem. In our case most of the objective functions will have more than one variable. In this case multidimensional optimization techniques are recommended. One of the simplest methods is pattern search which alternates sequences of local exploratory moves with extrapolations (or pattern moves). Another method is simple random search which selects random search points and evaluates the function at each of those points. More details about these methods and other optimization techniques can be found in [16, 53].

The following are some quantitative measures that can be used as objective functions for some of the design parameters with some examples of forming the optimization problem from the robot specification.

• Structural Length Index
• Manipulability
• Force Transmissibility
• Accuracy