By no doubt, the problem of inverse kinematics is the most important part of studying the motion of mechanical systems. In this article, a mathematical solution to the problem of inverse kinematics in a mechanical manipulator is introduced, developed and generalized for various conventions of representing errors between actual and desired poses of the endeffector. This method is based on constrained optimization using lagrangian multipliers and is a general solution to the problem of inverse kinematics for any class of chained link moving mechanisms which can deal with kinematic or even dynamic-based performance criteria.
In a pure kinematic optimization, in a redundant manipulator, the cost function is by default, defined as the norm of joint deviations or joit velocities which is independent of any dynamic terms and only leads to kinematic inversion. However, it will be shown that one can use a different cost function dealing with dynamic terms like jerk or energy consumption and exploit the redundancy to generate a dynamically optimized trajectory in the joint space.
The optimization is subject to multiple specified constraints changing over the time. In each time step of motion, the constraints are set so that the position and orientation of the endeffector (which can be any set of manipulator points and/or frames in the operational task space) be taken as close as possible to its desired pose in the next time step.
The main concept of this method is that error functions evaluating the distance between actual and desired endeffector poses, are considered as constraints of optimization and do not appear in the cost function, therefore it is possible to increase or decrease the physical degree of redundancy in the mathematical system of equations. This feature, provides freedom of choice for the designer to give arbitrary weights to three important factors: (cost function minimization, convergence rate and complexity).
This algorithm also has the capability of combining multiple constraints in the operational task space, (like multiple endeffector position and orientation tracking, posture and stability control and obstacle avoidance), fulfilling additional constraints like limits of joint parameters together with a trajectory optimization based on any desired kinematic or dynamic cost function.
These features, guarantee that the best use of redundancy in the control of any moving mechanism is ensured by an appropriate selection of taskspace parametrization, vector of error functions and relevant coefficients.