# A Novel Approach for a Inverse Kinematics Solution of a Redundant Manipulator
## 3 Inverse Kinematic Model and Computing Algorithm
### Final Inverse Kinematic Model
The following figures compare generalized variables \(\mathrm{q}=[\mathrm{q}_{1},\,\mathrm{q}_{2},\ldots,\mathrm{q}_{n}]^{\mathrm{ T}}\) from the simulation model to those of the experimental model. Figure 10 shows that the real manipulator, except for small deviations, almost exactly copies the simulated values of the generalized variables. The small deviations are caused by the control system

Figure 9: Graphical representations of the simulation of the five-link manipulator. of the servomechanisms not being properly tuned [34]. Figure 11 shows the end-effector positioning error. The error of simulation is roughly 5 mm. This error was caused by the predetermined tolerance of positioning, which is 5 mm. Figure 12 shows images of the motion of the experiment using the five-link manipulator according to the simulation from \[\dot{\mathrm{q}}=\left(\mathrm{J}^{-1}\mathrm{WJ}+\mathrm{J}_{\mathrm{c}}^{ \mathrm{T}}\mathrm{WJ}_{\mathrm{c}}+\mathrm{J}_{\mathrm{l}}^{-1}\mathrm{WJ}_{ \mathrm{l}}+\mathrm{W}_{\mathrm{s}}\right)^{-1}\left(\mathrm{J}^{\mathrm{T}} \mathrm{W}\dot{\mathrm{x}}\right) \tag{22}\]0 The video sequence was recorded using MATLAB image equipment [35]. Figure 12 corresponds to the time behavior of the experimentally-given generalized variables shown in \[\dot{\mathrm{q}}=\left(\mathrm{J}^{-1}\mathrm{WJ}+\mathrm{J}_{\mathrm{c}}^{ \mathrm{T}}\mathrm{WJ}_{\mathrm{c}}+\mathrm{J}_{\mathrm{l}}^{-1}\mathrm{WJ}_{ \mathrm{l}}+\mathrm{W}_{\mathrm{s}}\right)^{-1}\left(\mathrm{J}^{\mathrm{T}} \mathrm{W}\dot{\mathrm{x}}\right) \tag{22}\]1

\[\dot{\mathrm{q}}=\left(\mathrm{J}^{-1}\mathrm{WJ}+\mathrm{J}_{\mathrm{c}}^{ \mathrm{T}}\mathrm{WJ}_{\mathrm{c}}+\mathrm{J}_{\mathrm{l}}^{-1}\mathrm{WJ}_{ \mathrm{l}}+\mathrm{W}_{\mathrm{s}}\right)^{-1}\left(\mathrm{J}^{\mathrm{T}} \mathrm{W}\dot{\mathrm{x}}\right) \tag{22}\]2 The time behavior of generalized variables. ### Comparison with Other Methods

In the previous section, the WLN original approach working with task priorities was compared with our developed FPS approach. In this section, analyses considering the next methods are presented. Different simulation conditions will also be used. The first method analyzed is from the gradient projection methods (GPM) class. GPM are probably the most frequently discussed and used method for inverse kinematics of kinematically-redundant manipulators. GPM was firstly introduced by [PERSON] [8] to utilize kinematic redundancy to avoid joint limits. By extension of the basic inverse kinematic model, a new model was derived [8]:

\[\dot{\mathrm{q}}=\left(\mathrm{J}^{-1}\mathrm{WJ}+\mathrm{J}_{\mathrm{c}}^{ \mathrm{T}}\mathrm{WJ}_{\mathrm{c}}+\mathrm{J}_{\mathrm{l}}^{-1}\mathrm{WJ}_{ \mathrm{l}}+\mathrm{W}_{\mathrm{s}}\right)^{-1}\left(\mathrm{J}^{\mathrm{T}} \mathrm{W}\dot{\mathrm{x}}\right) \tag{22}\]3

where \(\mathrm{J}^{\dagger}\) is the pseudoinverse matrix of \(\mathrm{J}\), \(\mathrm{I}_{\mathrm{n}}\) is the identity matrix \(\mathrm{I}_{\mathrm{n}}\in\mathrm{R}^{\mathrm{n}\times\mathrm{n}}\), \(\
abla\mathrm{z}\) is the vector of objective function, and \(\alpha\) is a weighting parameter larger than zero. The term \(\mathrm{J}^{\dagger}\dot{\mathrm{x}}\) is a minimum-norm solution and the term \(\left(\mathrm{J}^{\dagger}\mathrm{J}-\mathrm{I}_{\mathrm{n}}\right)\) is a null-space projection matrix. The homogeneous term \(\left(\mathrm{J}^{\dagger}\mathrm{J}-\mathrm{I}_{\mathrm{n}}\right)\
abla \mathrm{z}\) is orthogonal to \(\mathrm{J}^{\dagger}\dot{\mathrm{x}}\) and is referred to as the self-motion of the mechanism (manipulator) in joint space with any influence on end-effector motion in task space. An arbitrary secondary task can be applied for the objective function z according to requirements. For following case study, the same secondary tasks that were introduced in previous sections were used. Equation (23) is presented in the following tables as generalized GPM (GGPM).