Core Control Algorithms

Energid Technologies' Actin approach is multifaceted, with algorithmic, language, and software-implementation components. This section gives an overview of the algorithms.

The core velocity framework is based on the manipulator Jacobian equation:

Here V is an m -length vector representation of the motion of the hand or hands (usually some combination of linear and angular velocity referenced to points rigidly attached to parts of the manipulator); q is the n -length vector of joint positions; and J(q) is the mxn manipulator Jacobian, a function of q. (For spatial arms with a single end effector, V is often the frame velocity with three linear and three angular components. In Actin, it takes on a larger meaning that includes the concatenation of point, frame, or other motion of multiple end-effectors.) This is illustrated in the figure below.

Figure 1 An illustration, based on the RRC K-1207i manipulator, of the parameters for velocity control. The column vector V represents hand motion (for positioning and orienting, it would be 6x1), and q represents the concatenated joint values (for the RRC K-1207i shown, it would be 7x1). The Jacobian J is the matrix that makes equation (1) true for all possible values ofthe derivative of q.

For any physical manipulator that is not self-connecting, a manipulator Jacobian can be defined to make equation (1) true. When the manipulator is kinematically redundant, the dimension of V is less than the dimension of q (m less than n ), and (1) is underconstrained when V is specified. By using V to represent relative motion, (1) can support self-connecting mechanisms by setting the relative motion to zero.

The velocity control question is the following: given a desired hand motion V, what are the joint rates that best achieve this motion? To answer this, Actin's framework uses a scalar a, a matrix function W, and a scalar function f to solve for the joint rates through the following formula:

where nabla-f is the gradient of f and N_J is an n x (n-m) set of vectors that spans the null space of J. That is, JN_{J= 0}, and N_J has rank (n-m ). By changing the values of a, W, and f, many new and most established velocity-control techniques can be implemented.

Actin, however, goes beyond the formulation in equation (2) to create a more general framework. Instead of insisting on the use of the gradient of a function, it uses a general column vector F(q). Not all vector functions are gradients. This minor, but important, modification yields the following formula:

Equation (3) is the core velocity-control algorithm used in the Actin Toolkit. Mathematically, it achieves the desired V while minimizing the following metric:

Through Actin's patented approach, the parameters a, W, and F can be defined using XML to give many different types of behaviors.

Would you like to apply these algorithms to your problem? Please contact us.